COHOMOLOGY OF DYNAMICAL SYSTEMS

AND RIGIDITY OF PARTIALLY HYPERBOLIC

ACTIONS OF HIGHER RANK LATTICES

Viorel Niţică and Andrei Török

§1. Introduction

Let SL(n, ℤ) be the group of n × n integer matrices of determinant one, and let π be the standard action by linear automorphisms of SL(n, ℤ) on the torus 𝕋^{n}. Let d be a positive integer and consider the action ρ_{0} of SL(n, ℤ) on 𝕋^{n+d} = 𝕋^{n} × 𝕋^{d} given by ρ_{0}(A)(x, y) = (π(A)x, y) (where x ∈ 𝕋^{n}, y ∈ 𝕋^{d} and A ∈ SL(n, ℤ)). Then ρ_{0} is a (non-irreducible and non-transitive) partially hyperbolic action.

This paper emerges from an attempt to understand the differentiable actions close to ρ_{0}. After Zimmer initiated the program to classify volume-preserving actions of higher rank lattices on compact manifolds (see [Z1], [Z2]), Hurder, Katok, Lewis, Qian, Spatzier, Zimmer and others tried to understand local, global and infinitesimal rigidity of these actions (see [H1], [H2], [K-L1], [K-L2], [K-L-H-Z], [K-L-Z], [Q1], [Q2], [Z3]). In the great majority of these papers, an essential use was made of the presence of an Anosov element. Here we show for the first time the deformation rigidity of non-irreducible and non-transitive partially hyperbolic actions (for terminology, see §7). One example is ρ_{0}, for n ≥ 3 and any integer d; others are given in §7. Precise statements are Theorems 7.1 and 7.8.

As turns out from this study, there is a close connection between the rigidity of the natural actions of lattices on tori and the cohomology of hyperbolic dynamical systems. The first nontrivial results about this cohomology were obtained by Livsic and Sinai in [L1], [L2] and [L-S]. We recall some definitions and Livsic’s main results.

Let G be a discrete group acting on a closed Riemannian manifold M, and let Γ be some topological group with unit 1_{Γ}. A cocycle β is a continuous function β : G × M → Γ such that

β(g_{1}g_{2}, x) = β(g_{1}, g_{2}x)β(g_{2}, x),

for all g_{1}, g_{2} ∈ G, x ∈ M. Two cocycles β_{1} and β_{2} are called cohomologous if there exists a continuous map P : M → Γ such that

β_{1}(g, x) = P (gx)β_{2}(g, x)P (x)^{−1},

for all g ∈ G, x ∈ M.

Typeset by AMS-TEX

We say that a cocycle β satisfies the closing conditions if the following holds:

gx = x for some x ∈ M and g ∈ G implies that β(g, x) = 1_{Γ}. In the case G = ℤ the cocycle is determined by the function \overline{β}(x) := β(1, x). We recall also the definition of an Anosov diffeomorphism. Note that this def-

inition is equivalent to the standard one (see [K]). The Riemannian metric which appears is called the Lyapunov metric.

Definition. Let M be a compact manifold and T : M → M a C^{1}-diffeomorphism. The diffeomorphism T is called Anosov if there is a continuous splitting of the tangent bundle into DT -invariant subbundles T M = E^{s} ⊕ E^{u} and a Riemannian metric such that:

∥DT v^{s}∥ ≤ λ∥v^{s}∥, v^{s} ∈ E^{s},
∥DT^{−1}v^{u}∥ ≤ λ∥v^{u}∥, v^{u} ∈ E^{u},

for some 0 < λ < 1.

We denote by W^{s}(x) (resp. W^{u}(x)) the stable (resp. unstable) manifold of the point x ∈ M.

Theorem 1.1. (Livsic [L2]) If the group G is ℤ and its action on the Riemannian manifold M is given by a topologically transitive C^{1} Anosov diffeomorphism, then a Hölder function \overline{β} : M → ℝ determines a null-cohomological cocycle β(g, x) if and only if β satisfies the closing conditions. The corresponding function P is also Hölder.

A topological group Γ is called “good” if it admits right-invariant and leftinvariant metrics ρ_{R} and ρ_{L}, which are equivalent in a neighborhood of the identity, and such that Γ metrized by these metrics is a complete metric space. Moreover, for any points (x_{1}, x_{2}) ∈ Γ × Γ

\frac{ρ_{L}(x_{1}, x_{2})}{ρ_{L}(gx_{1}g^{−1}, gx_{2}g^{−1})} → 1 and \frac{ρ_{R}(x_{1}, x_{2})}{ρ_{R}(gx_{1}g^{−1}, gx_{2}g^{−1})} → 1

as g → 1_{Γ}, this convergence being uniform for (x_{1}, x_{2}) in a bounded set*. The main example of a good group is a finite dimensional Lie group.

Theorem 1.2. (Livsic [L2]) If the group G is ℤ and its action on the Riemannian manifold M is given by a topologically transitive C^{1} Anosov diffeomorphism, then for each Hölder class α and in each good group Γ there exists a neighborhood of the identity U such that an α-Hölder function \overline{β} : M → U defines a null-cohomological cocycles β(g, x) if and only if β satisfies the closing conditions. The corresponding function P is also α-Hölder.

Livsic proved similar results for Anosov flows and was able to extend Theorem 1.1 to cocycles with values in finite dimensional nilpotent Lie groups. After Livsic’s

* This is the wording in the translation of [L2]. However, it seems that one uses uniform convergence with respect to all (x_{1}, x_{2}) ∈ Γ × Γ. work, the main results obtained were about the regularity of P produced by Theorem 1.1, assuming some regularity for β (see [L1], [C-E-G], [G-K], [L-M-M], [Lla]). De la Llave-Marco-Moriyon showed in [L-M-M] that P is C^{∞} if β is C^{∞}, and de la Llave showed in [Lla] that P is analytic if β is analytic. A somehow different direction is taken by A. Katok, S. Katok and Spatzier in [K-S] and [K-K], where real-valued cocycles over certain ℤ^{k} and ℝ^{k} Anosov actions (k ≥ 2) are shown to be cohomologous to constant cocycles (in these cases, the periodic conditions are automatically satisfied!).

Our first theorem, Theorem 3.1, which is maybe the most important result of the paper, extends Livsic’s Theorem 1.2 to cocycles with values in the group of C^{K}-diffeomorphisms of a compact closed manifold with stably trivial tangent bundle (4 ≤ K < ∞). The statement and the proof are in §3. The strategy of the proof is similar to the one used by Livsic. The main difference is that on the group of C^{K}- diffeomorphisms the natural metric is neither left nor right invariant. Nevertheless, in a small neighborhood of the identity one can show that this metric is “quasiinvariant”, and this is enough for our purposes. These properties of the metric are presented in §2.

In order to show deformation rigidity for ρ_{0}, one needs a Livsic type result for a cocycle over an SL(n, ℤ)-action, instead of a ℤ-action. We obtain much more. Using a generalization of the Anosov Closing Lemma to a generic class of Anosov group actions (see Theorem 4.1), we show that if a cocycle over an Anosov group action satisfies the closing conditions and if the restriction of the cocycle to the action of a (special) Anosov element is null-cohomological, then the whole cocycle is null-cohomological. The Anosov group action is not necessarily the action of a large group. Results like this seems to be the missing link between the theory of ℤ-actions and the theory of actions of “big groups”. This result and various combinations of it with Livsic’s theorems and with Theorem 3.1 appear in §5. In particular, we are able to generalize the Livsic-Sinai criterion about the existence of a smooth invariant measure for an Anosov ℤ-action, to more general Anosov group actions.

In §6 we prove a version of the Hirsch-Pugh-Shub structural stability theorem (see [H-P-S], Theorem 7.1), namely we show that for some partially hyperbolic automorphism of the torus, the leaf-wise conjugation between the automorphism and a small C^{K+Lip}-perturbation is Hölder, differentiable of order K along the leaves and these derivatives are Hölder. This result is necessary in order to produce from a perturbation of ρ_{0} a cocycle which is Hölder in the d_{K}-metric. Note that our approach differs from that in [H-P-S].

The rigidity of ρ_{0} and of other partially hyperbolic higher rank lattice actions is proved in §7, Theorems 7.1 and 7.8. The proofs involve the Livsic type theorem for cocycles with values in diffeomorphisms, our Noncommutative Closing Lemma, and the version of the Hirsch-Pugh-Shub theorem.

In the paper [N-T] we prove the regularity of the solution of Livsic’s equation with values in diffeomorphism groups and with values in a matrix Lie group.

We are indebted to A. Katok for introducing us to dynamical systems, for encouragement and for helpful conversations during the development of this work. We also would like to thank N. Qian and E. Wayne for helpful conversations.

§2. Metrics on Diffeomorphism Groups

The purpose of this section is to define some metrics on the space of C^{K} or C^{K+Lip} functions between “flat spaces” and then to study their invariance properties with respect to composition. These metrics are almost invariant in a small neighborhood of the identity. We use this “quasi-invariance” property in two places. First, it is used in §3 to generalize Livsic’s Theorem 1.2 to cocycles with values in diffeomorphism groups. Second, it is used in §6 to prove a version of the Hirsch-Pugh-Shub Theorem (Theorem 7.1 in [H-P-S]).

Definition 2.1. A flat space is either a finite dimensional linear space ℝ^{d} (with the Euclidean metric), or a flat torus 𝕋^{d} = ℝ^{d}/ℤ^{d} with the metric induced from ℝ^{d}.

Note that if X is a flat space and x ∈ X, then we have a canonical identification of the tangent space at x, T_{x}X, with ℝ^{dim X}.

Definition 2.2. Let U be an open connected set in a flat space. For any path γ in U, denote by l(γ) the length of γ, and for any x, y in U, define their distance in U by

d_{U}(x, y) : = inf_{γ} l(γ),

where the infimum is taken over all paths γ in U connecting x and y. Let X and Y be flat spaces, U ⊂ X and V ⊂ Y open connected sets. For any integer K ≥ 0 denote by C^{K}(U, V) the set of C^{K}-functions from U into V. Take a ∈ C^{K}(U, V) and for 1 ≤ k ≤ K denote by D_{x}^{k}a the k-th derivative of a at x ∈ U, which is a symmetric k-linear map D_{x}^{k}a : (ℝ^{dim U})^{k} → ℝ^{dim V}. Denote by ∥D_{x}^{k}a∥ the operatorial norm of D_{x}^{k}a. Denote by C^{K+Lip}(U, V) the set of K-differentiable functions from U into V with the K-th derivative Lipschitz, i.e. ϕ ∈ C^{K}(U, V) for which there exists C > 0 such that

∥D_{x′}^{K}ϕ − D_{x′′}^{K}ϕ∥ ≤ C d_{U}(x′, x′′), for all x′, x′′ ∈ U.

The smallest constant C with this property is called the Lipschitz norm of D^{K}ϕ and is denoted by ∥D^{K}ϕ∥_{Lip}.

For a, b ∈ C^{K}(U, V) and any 1 ≤ k ≤ K, define:

d_{0}(a) : = sup_{x∈U} d_{V} (a(x), 0), if 0 ∈ V,
d_{0}(a, b) : = sup_{x∈U} d_{V} (a(x), b(x)),
d_{(k)}(a) : = sup_{x∈U} ∥D_{x}^{k}a∥,

d′_{K}(a) : = ∑_{k=1}^{K} \frac{d_{(k)}(a)}{k!},

d′′_{K}(a) : = ∑_{k=2}^{K} \frac{d_{(k)}(a)}{k!},

d′_{K+Lip}(a) : = d′_{K}(a) + \frac{∥D^{K}a∥_{Lip}}{(K + 1)!} ,
d′′_{K+Lip}(a) : = d′′_{K}(a) + \frac{∥D^{K}a∥_{Lip}}{(K + 1)!} ,
d_{K}(a) : = d_{0}(a) + d′_{K}(a), if 0 ∈ V,
d_{K}(a, b) : = d_{0}(a, b) + d′_{K}(a − b),
d_{K+Lip}(a, b) : = d_{0}(a, b) + d′_{K+Lip}(a − b).

Lemma 2.3. If a ∈ C^{K+1}(U, V), then for any 0 ≤ k ≤ K

∥D^{k}a∥_{Lip} ≤ d_{(k+1)}(a).

In particular, d′_{K+Lip}(a) ≤ d′_{K+1}(a).

Proof. Let η_{1}, η_{2}, ⋯, η_{k} be vectors in ℝ^{dim U}. Let x′, x′′ be two points in U, and let γ : [0, 1] → U be a path joining x′ and x′′. Then:

∥D_{x′}^{k}a(η_{1}, η_{2}, ⋯, η_{k}) − D_{x′′}^{k}a(η_{1}, η_{2}, ⋯, η_{k})∥ =

= ∥ ∫_{0}^{1} \frac{d}{dt}[D_{γ(t)}^{k}a(η_{1}, η_{2}, ⋯, η_{k})]dt∥ =

= ∥ ∫_{0}^{1} [D_{γ(t)}^{k+1}a(γ′(t), η_{1}, η_{2}, ⋯, η_{k})]dt∥ ≤

≤ sup_{t∈[0,1]}∥D_{γ(t)}^{k+1}a∥( ∫_{0}^{1} ∥γ′(t)∥dt)∥η_{1}∥∥η_{2}∥ ⋯ ∥η_{k}∥ ≤

≤ l(γ)d_{(k+1)}(a)∥η_{1}∥∥η_{2}∥ ⋯ ∥η_{k}∥. □

Notation. If r_{1}, r_{2}, ⋯, r_{n} is a set of real numbers, denote

r_{1} ∨ r_{2} ∨ ⋯ ∨ r_{n} := max_{1≤i≤n}r_{i}.

This notation will be used throughout the rest of the paper.

Proposition 2.4. Let K ≥ 1 be an integer. Let X, Y and Z be flat spaces and let U ⊂ X, V ⊂ Y and W ⊂ Z be open sets.

(i) If a ∈ C^{K}(U, U), then:

d′_{K}(a) ≤ 1 + d_{K}(a, Id),  (D1)
d′′_{K}(a) = d′′_{K}(a, Id).  (D2)

(ii) If a, b ∈ C^{K}(V, W) and u ∈ C^{K}(U, V), then:

d′_{K}(au) ≤ d′_{K}(a) · [max{1, d′_{K}(u)}]^{K},  (R1)
d′_{K}(au, bu) ≤ d′_{K}(a, b) · [max{1, d′_{K}(u)}]^{K},  (R2)
d_{K}(au, bu) ≤ d_{K}(a, b) · [max{1, d′_{K}(u)}]^{K}.  (R3)

(iii) If a, b ∈ C^{K}(U, V), u ∈ C^{K+Lip}(V, W) and v ∈ C^{K+1}(V, W), then:

d_{K}(ua, ub) ≤ d_{K}(a, b)(K + 1)d′_{K+Lip}(u)·
· [1 ∨ (max{d′_{K}(a), d′_{K}(b)} + d′′_{K}(a) + d′′_{K}(b))]^{K},  (L1)
d_{K}(va, vb) ≤ d_{K}(a, b)(K + 1)d′_{K+1}(v)·
· [1 ∨ (max{d′_{K}(a), d′_{K}(b)} + d′′_{K}(a) + d′′_{K}(b))]^{K}.  (L2)

In particular, if a, b ∈ C^{K}(V, V), then:

d_{K}(ua, ub) ≤ d_{K}(a, b)(K + 1)d′_{K+Lip}(u)[1 + 2d′_{K}(a, Id) + 2d′_{K}(b, Id)]^{K}, (L3)
d_{K}(va, vb) ≤ d_{K}(a, b)(K + 1)d′_{K+1}(v)[1 + 2d′_{K}(a, Id) + 2d′_{K}(b, Id)]^{K}. (L4)

(iv) If a ∈ C^{K+Lip}(V, W) and b ∈ C^{K+Lip}(U, V), then

d′_{K+Lip}(ab) ≤ d′_{K+Lip}(a)[max{1, d′_{K}(b)}]^{K}·
· [d′_{K+Lip}(b) + K^{2}d′′_{K}(b)],  (Lip1)

therefore, if 0 ∈ W,

d_{K+Lip}(ab) ≤ d_{K+Lip}(a)[max{1, d′_{K}(b)}]^{K}·
· max{1, d′_{K+Lip}(b) + K^{2}d′′_{K}(b)}.  (Lip2)

In particular, if b ∈ C^{K+Lip}(V, V) and 0 ∈ W, then

d_{K+Lip}(ab) ≤ d_{K+Lip}(a)[1 + d_{K+Lip}(b, Id)]^{K}·
[1 + (K^{2} + 1)d_{K+Lip}(b, Id)].  (Lip3)

Remark. One can obtain in (iii) an estimate which is sharp for u = Id. Since we do not need it, we do not describe it here.

Proof. Part (i) follows from the definitions. We prove (ii), (iii) and (iv). We introduce first a formula for the derivative of a composite function. Assume a ∈ C^{K}(V, W), u ∈ C^{K}(U, V), ℓ ≤ K and ξ, ξ_{1}, …, ξ_{ℓ} ∈ ℝ^{dim U}. Using

Taylor’s formula, one has

\frac{1}{ℓ!}D_{x}^{ℓ}(au)(ξ^{(ℓ)}) =
= ∑_{k=1}^{ℓ}_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} \frac{1}{k!}D_{u(x)}^{k}a (\frac{D_{x}^{i_{1}}u(ξ^{(i_{1})})}{i_{1}!}, \frac{D_{x}^{i_{2}}u(ξ^{(i_{2})})}{i_{2}!}, …, \frac{D_{x}^{i_{k}}u(ξ^{(i_{k})})}{i_{k}!}) ,

where ξ^{(k)} stands for the k-tuple (ξ, ξ, …, ξ).

Therefore, D_{x}^{ℓ}(au) is given by the symmetrization of the ℓ-linear map

(ξ_{1}, …, ξ_{ℓ}) ↦ ℓ! · ∑_{k=1}^{ℓ} _{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} \frac{1}{k!}D_{u(x)}^{k}a (\frac{1}{i_{1}!}D_{x}^{i_{1}}u(ξ_{1}, …, ξ_{i_{1}}) ,

\frac{1}{i_{2}!}D_{x}^{i_{2}}u(ξ_{i_{1}+1}, …, ξ_{i_{1}+i_{2}}), …, \frac{1}{i_{k}!}D_{x}^{i_{k}}u(ξ_{i_{1}+···+i_{k−1}+1}, …, ξ_{ℓ})) . (2.1)

Note that as far as the norms computed below are concerned, this symmetrization has no effect.

(ii) Take a ∈ C^{K}(V, W) and u ∈ C^{K}(U, V). From (2.1) it follows that

\begin{matrix}d_{(ℓ)}(au) = sup_{x∈U} ∥D_{x}^{ℓ}(au)∥ ≤ \\ ≤ ℓ! ∑_{k=1}^{ℓ} d_{(k)}(a) · \frac{1}{k!} (_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} k ; ∏_{j=1} \frac{1}{i_{j}!} · ∥D^{i_{j}}u∥) . \end{matrix}

Hence  \frac{1}{ℓ!} · d_{(ℓ)}(au) ≤ ∑_{k=1}^{ℓ} \frac{1}{k!}d_{(k)}(a) ·_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} k ; ∏_{j=1} \frac{∥D^{i_{j}}u∥}{i_{j}!}.  (2.2)

In order to simplify the above expressions, we introduce the following notations:

X_{i} := ∥D^{i}u∥ · \frac{1}{i!} , deg X_{i} = i, X := X_{1} + X_{2} + ⋯ + X_{K}

and for a polynomial F in some “graded” variables, by F |_{deg≤k} we mean the sum of the monomials of total degree less or equal to k.

Using (2.2) one has

d′_{K}(au) ≤ ∑_{k=1}^{K} \frac{1}{k!}d_{(k)}(a) · (X^{k}|_{deg≤K}) ≤

≤ d′_{K}(a) · [max{1, d′_{K}(u)}]^{K},

hence (R1). The formulas (R2) and (R3) follow immediately from (R1).

(iii) Take a, b ∈ C^{K}(U, V) and u ∈ C^{K+Lip}(V, W). We use now (2.1) for ua and ub:

d_{(ℓ)}(ua, ub) = sup_{x∈U} ∥D_{x}^{ℓ}(ua) − D_{x}^{ℓ}(ub)∥ ≤

≤ ℓ! · ∑_{k=1}^{ℓ} _{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} \frac{1}{k!} · \frac{1}{i_{1}!} … \frac{1}{i_{k}!}·

·∥D_{a(x)}^{k}u(D_{x}^{i_{1}}a, D_{x}^{i_{2}}a, …, D_{x}^{i_{k}}a) − D_{b(x)}^{k}u(D_{x}^{i_{1}}b, D_{x}^{i_{2}}b, …, D_{x}^{i_{k}}b)∥.

In order to estimate the norm of the differences, first add and subtract for each term D_{a(x)}^{k}u(D_{x}^{i_{1}}a, ⋯) an intermediate term D_{b(x)}^{k}u(D_{x}^{i_{1}}a, ⋯). Then apply the triangle inequality. It remains to estimate terms of the form

∥D_{a(x)}^{k}u(D_{x}^{i_{1}}a, D_{x}^{i_{2}}a, …, D_{x}^{i_{k}}a) − D_{b(x)}^{k}u(D_{x}^{i_{1}}a, D_{x}^{i_{2}}a, …, D_{x}^{i_{k}}a)∥,  (2.3)

and

∥D_{b(x)}^{k}u(D_{x}^{i_{1}}a, D_{x}^{i_{2}}a, ⋯, D_{x}^{i_{k}}a) − D_{b(x)}^{k}u(D_{x}^{i_{1}}b, D_{x}^{i_{2}}b, ⋯, D_{x}^{i_{k}}b)∥.  (2.4)

The quantity in (2.3) can be estimated using

∥D_{a(x)}^{k}u − D_{b(x)}^{k}u∥ ≤ ∥D^{k}u∥_{Lip}d_{0}(a, b).  (2.5)

For (2.4), introduce again intermediate terms, use the linearity of the derivatives and the triangle inequality to obtain

∥D_{b(x)}^{k}u(D_{x}^{i_{1}}a, D_{x}^{i_{2}}a, …, D_{x}^{i_{k}}a) − D_{b(x)}^{k}u(D_{x}^{i_{1}}b, D_{x}^{i_{2}}b, …, D_{x}^{i_{k}}b)∥ ≤

≤ ∥D^{k}u∥ [d_{(i_{1})}(a, b) · ∥D^{i_{1}}a∥ ⋯ ∥D^{i_{k}}a∥ + ∥D^{i_{1}}b∥ · d_{(i_{2})}(a, b) · ∥D^{i_{3}}b∥ ⋯ ∥D^{i_{k}}b∥+

+ ⋯ + ∥D^{i_{1}}b∥∥D^{i_{2}}b∥ ⋯ ∥D^{i_{k−3}}b∥ · d_{(i_{k})}(a, b)] .

Therefore

\begin{matrix}d_{(ℓ)}(ua, ub) ≤ ℓ! ∑_{k=1}^{ℓ} \{_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=ℓ} \frac{1}{k!}[∥D^{k}u∥_{Lip} · d_{0}(a, b)· \\ · k ; ∏_{j=1} \frac{1}{i_{j}!}∥D^{i_{j}}a∥ + ∥D^{k}u∥ · ∑_{j=1}^{k} (\frac{d_{(i_{j})}(a, b)}{i_{j}!} · j−1 ; ∏_{α=1} \frac{∥D^{i_{α}}b∥}{i_{α}!}· \\ · k ; ∏_{β=j+1} \frac{∥D^{i_{β}}a∥}{i_{β}!})]\} . \end{matrix}  (2.6)

In order to simplify the above formula, we introduce the following notations:

δ_{i} : = \frac{d_{(i)}(a, b)}{i!},
α_{i} : = max \{\frac{∥D^{i}a∥}{i!}, \frac{∥D^{i}b∥}{i!}\} ,
X_{i} : = \frac{∥D^{i}u∥}{i!},
X_{K+1} : = \frac{∥D^{K}u∥_{Lip}}{(K + 1)!} ,

α : = ∑_{i=1}^{K} α_{i},  δ := ∑_{i=1}^{K} δ_{i},
deg α_{i} = deg δ_{i} = i, 1 ≤ i ≤ K.

It is clear that:

\begin{aligned}α_{1} & ≤ max\{d′_{K}(a), d′_{K}(b)\}, \\ α_{i} & ≤ \frac{d_{(i)}(a)}{i!} + \frac{d_{(i)}(b)}{i!}, 2 ≤ i ≤ K. \end{aligned}

Therefore

α ≤ max{d′_{K}(a), d′_{K}(b)} + d′′_{K}(a) + d′′_{K}(b).  (2.7)

By (2.6) one has

\begin{matrix}\frac{1}{ℓ!} d_{(ℓ)}(ua, ub) ≤ δ_{0} · ∑_{k=1}^{ℓ} (k + 1)X_{k+1} · (α_{1} + ⋯ + α_{ℓ})^{k}|_{deg= ℓ}+ \\ + ∑_{k=1}^{ℓ} X_{k}(δ_{1} + ⋯ + δ_{ℓ} + α_{1} + ⋯ + α_{ℓ})^{k}|_{deg=ℓ,#(δ)=1} , \end{matrix}  (2.8)

where the subscript “deg = ℓ, #(δ) = 1” means that we take the monomials of total degree ℓ which have exactly one δ_{i} factor.

Using (2.8) and the inequality d_{0}(ua, ub) ≤ ∥Du∥d_{0}(a, b) = X_{1} · δ_{0}, one has

\begin{matrix}d_{K}(ua, ub) = d_{0}(ua, ub) + d′_{K}(ua, ub) ≤ \\ ≤ X_{1} · δ_{0} + δ_{0} · ∑_{ℓ=1}^{K} ∑_{k=1}^{ℓ} (k + 1) · X_{k+1} · α^{k}|_{deg=ℓ}+ \\ + ∑_{ℓ=1}^{K} ∑_{k=1}^{ℓ} X_{k} · (δ + α)^{k}|_{deg=ℓ,#(δ)=1}. \end{matrix}  (2.9)

But

(δ + α)^{k}|_{deg=ℓ,#(δ)=1} = ∑_{i≥1} kδ_{i}(α^{k−1}|_{deg= (ℓ−i)}),

hence, if we denote by B the second double sum which appears in the RHS of (2.9), one has:

B = ∑_{ℓ=1}^{K} ∑_{k=1}^{ℓ} ∑_{i≥1} kδ_{i}X_{k}(α^{k−1}|_{deg=(ℓ−i)}) =

= ∑_{i≥1} δ_{i} ∑_{k=1}^{K} X_{k} ∑_{ℓ=k}^{K} k(α^{k−1}|_{deg= (ℓ−i)}) =

= ∑_{i≥1} δ_{i} ∑_{k=1}^{K} kX_{k}(α^{k−1}|_{k−i≤deg≤K−i}) =

= ∑_{i≥1} δ_{i} ∑_{k=1}^{K} kX_{k}(α^{k−1}|_{deg≤K−i}).

Then inequality (2.9) becomes

d_{K}(ua, ub) ≤ δ_{0} · ∑_{k=2}^{K+1} kX_{k}α^{k−1}|_{deg≤K}+

+ ∑_{i≥1} δ_{i} ∑_{k=1}^{K} kX_{k}α^{k−1}|_{deg≤K−i} ≤

≤ d_{K}(a, b)(K + 1)d′_{K+Lip}(u)[max{1, α}]^{K}

which together with (2.7) yields (L1). (L2) follows from (L1) using Lemma 2.3. In order to obtain (L3) and (L4), notice that for a, b : V → V one has:

max{d′_{K}(a), d′_{K}(b)} + d′′_{K}(a) + d′′_{K}(b) ≤
≤ 1 + max{d′_{K}(a, Id), d′_{K}(b, Id)} + d′_{K}(a, Id) + d′_{K}(b, Id) ≤
≤ 1 + 2d′_{K}(a, Id) + 2d′_{K}(b, Id).

(iv) Take a ∈ C^{K+Lip}(V, W) and b ∈ C^{K+Lip}(U, V). We estimate ∥D_{x′}^{K}(ab) − D_{x′′}^{K}(ab)∥. Proceeding as in (iii) and making the appropriate changes in the formula (2.6) above, we have

\frac{1}{K!}∥D^{K}(ab)∥_{Lip} ≤ ∑_{k=1}^{K} \{_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=K} [\frac{∥D^{k}a∥_{Lip}}{k!} · ∥b∥_{Lip}·

· k ; ∏_{j=1} \frac{1}{i_{j}!}∥D^{i_{j}}b∥ + \frac{∥D^{k}a∥}{k!}(∑_{j=1}^{k} \frac{∥D^{i_{j}}b∥_{Lip}}{i_{j}!} ·_{k ; ∏_{α=1} ; α≠j} \frac{∥D^{i_{α}}b∥}{i_{α}!})]\} .

We introduce now the following notations:

X_{i} : = \frac{d_{(i)}(b)}{i!}, 1 ≤ i ≤ K,
X_{K+1} : = \frac{∥D^{K}b∥_{Lip}}{(K + 1)!} ,
X : = X_{1} + X_{2} + ⋯ + X_{K},
X′′ : = X_{2} + X_{3} + ⋯ + X_{K},
α_{i} : = \frac{d_{(i)}a}{i!}, 1 ≤ i ≤ K,
α_{K+1} : = \frac{∥D^{K}a∥_{Lip}}{(K + 1)!} ,
degX_{i} = i, 1 ≤ i ≤ K + 1.

COHOMOLOGY OF DYNAMICAL SYSTEMS AND RIGIDITY 11

From Lemma 2.3 it follows that

\frac{∥D^{i}a∥_{Lip}}{i!} ≤ \frac{d_{i+1}(a)}{i!} = (i + 1)α_{i+1},

and similarly for b. Hence:

\begin{matrix}\frac{1}{K!}∥D^{K}(ab)∥_{Lip} & ≤ ∑_{k=1}^{K}(k + 1)α_{k+1}X_{1}(_{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=K} k ; ∏_{j=1}X_{i_{j}})+ \\ + ∑_{k=1}^{K} α_{k} & _{∑_{i_{1},...,i_{k}≥1} ; i_{1}+···+i_{k}=K} ∑_{j=1}^{k}(i_{j} + 1)X_{i_{j}+1}_{k ; ∏_{α=1} ; α≠j} X_{i_{α}} = \end{matrix}

= ∑_{k=1}^{K}(k + 1)α_{k+1}X_{1}(X^{k}|_{deg=K}) + ∑_{k=1}^{K} α_{k}[k( ∑_{i=2}^{K+1} iX_{i})X^{k−1}|_{deg=K+1}]=
= ∑_{k=2}^{K+1} kα_{k}X_{1}(X^{k−1}|_{deg=K}) + ∑_{k=1}^{K} kα_{k}[( ∑_{i=2}^{K+1} iX_{i})X^{k−1}|_{deg=K+1}]=
= ∑_{k=2}^{K} kα_{k}[( ∑_{k=1}^{K+1} iX_{i})X^{k−1}|_{deg=K+1}]+(K + 1)α_{K+1}X_{1}(X^{K}|_{deg=K})+

+ α_{1}( ∑_{k=1}^{K+1} iX_{i}|_{deg=K+1}) =

= ∑_{k=2}^{K} kα_{k}[(∑_{i=1}^{K} iX_{i})X^{k−1}|_{deg=K+1}]+(K + 1)α_{K+1}X_{1}(X^{K}|_{deg=K})+

+ α_{1}(K + 1)X_{K+1}.

But, for 2 ≤ k ≤ K, one has:

(∑_{i=1}^{K} iX_{i})X^{k−1}|_{deg=K+1} ≤ KX^{k}|_{deg=K+1} = K(X^{k} − X_{1}^{k})|_{deg=K+1} =

= K(X − X_{1})(X^{k−1} + X^{k−2}X_{1} + ⋯ + X_{1}^{k−1})|_{deg=K+1} ≤ ≤ Kk(X′′X^{k−1}|_{deg=K+1}).

Also, X^{K}|_{deg=K} = X_{1}^{K}. Therefore

\frac{1}{K!}∥D^{K}(ab)∥_{Lip} ≤ ∑_{k=2}^{K} α_{k}k^{2}K(X′′X^{k−1}|_{deg=K+1})+
+(K + 1)(α_{K+1}X_{1}^{K+1} + α_{1}X_{K+1}).

From the proof of (ii), we know that

d′_{K}(ab) ≤ ∑_{k=1}^{K} α_{k}(X^{k}|_{deg≤K}).

Then

d′_{K+Lip}(ab) = d′_{K}(ab) + \frac{∥D^{K}(ab)∥_{Lip}}{(K + 1)!} ≤

\begin{aligned}≤ ∑_{k=1}^{K} α_{k}(X^{k}|_{deg≤K})+ & ∑_{k=2}^{K} α_{k} \frac{k^{2}K}{K + 1}(X′′X^{k−1}|_{deg=K+1})+ \\ & + α_{K+1}X_{1}^{K+1} + α_{1}X_{K+1} ≤ \end{aligned}

\begin{matrix}≤ ∑_{k=2}^{K} α_{k}[(X^{k}|_{deg≤K}+ & K^{2}(X′′X^{k−1}|_{deg=K+1})]+ \\ + α_{1}(X + X_{K+1}) + α_{K+1}X_{1}^{K+1} ≤ \end{matrix}

≤ ∑_{k=2}^{K} α_{k}X^{k−1}(X + K^{2}X′′) + α_{1}(X + X_{K+1}) + α_{K+1}X_{1}^{K+1} ≤

≤ ( ∑_{k=1}^{K+1} α_{k})[max{1, X}]^{K}(X + X_{K+1} + K^{2}X′′), q.e.d. □

Proposition 2.5. Let U be an open connected subset of a flat space. Then Diff_{bdd}^{K}(U) := {a ∈ Diff^{K}(U) | d_{K}(a^{±1}, Id) < ∞} is a group.

Proof. We only have to show that Diff_{bdd}^{K}(U) is closed under composition. Since d_{K} = d_{0} + d′_{K}, a diffeomorphism a ∈ Diff^{K}(U) is in Diff_{bdd}^{K}(U) if and only if

d′_{K}(a^{±1}) < ∞ and d_{0}(a^{±1}, Id) < ∞.

The first condition is clearly composition invariant, while for the second one, by the right-invariance of d_{0}, one has:

d_{0}(ab, Id) ≤ d_{0}(ab, b) + d_{0}(b, Id) = d_{0}(a, Id) + d_{0}(b, Id). □

Proposition 2.6. The group Diff_{bdd}^{K}(U) introduced in Proposition 2.5 is complete with respect to the metric

\overline{d}_{K}(a, b) := max \{d_{K}(a, b), d_{(1)}(a^{−1}, b^{−1})\}.

Proof. Assume \{a_{n}\}_{n≥0} ⊂ Diff_{bdd}^{K}(U) is a Cauchy sequence in the metric \overline{d}_{K}. In particular, it is Cauchy in d_{K}, so it has a limit

a = lim_{n→∞} a_{n} ∈ C^{K}(U)

with d_{K}(a, Id) < ∞.

It is enough to show that a ∈ Diff^{1}(U), because d′_{K}(a) < ∞ and d_{(1)}(a^{−1}) < ∞ implies d′_{K}(a^{−1}) < ∞ (see the proof of Proposition 2.9 (b)).

In order to obtain this, note that the family {a_{n}^{−1}}_{n≥0} is uniformly equicontinuous (because the first derivatives are uniformly bounded) and pointwise bounded (because d_{0}(a_{n}^{−1}, Id) = d_{0}(Id, a_{n}), hence sup_{n} d_{0}(a_{n}^{−1}, Id) < ∞). Therefore, by Ascoli-Arzela’s theorem (see [R], which works for our “vector valued” functions too), there is a subsequence {a_{n_{k}}^{−1}}_{k≥0} which converges uniformly (i.e., in d_{0}) to some b ∈ C^{0}(U). Clearly d_{0}(b, Id) < ∞ and since {a_{n_{k}}^{−1}}_{k≥0} converges actually in d_{1}, we see that b ∈ C^{1}(U) and d_{1}(b, Id) < ∞.

We have to show that b = a^{−1}.

But this follows from the continuity of the composition with respect to the the convergence we have here: if f_{n} → f, g_{n} → g in d_{0} and f is uniformly continuous, then f_{n}g_{n} → fg in d_{0}.

(Indeed, for x ∈ U

d_{U}(f_{n}g_{n}(x), fg(x)) ≤

\begin{matrix}≤ d_{U}(f_{n}(g_{n}(x)), f(g_{n}(x))) + d_{U}(f(g_{n}(x)), f(g(x))) ≤ \\ ≤ d_{0}(f_{n}, f) + d_{U}(f(g_{n}(x)), f(g(x))), \end{matrix}

and now use the uniform convergence of {g_{n}} and the uniform continuity of f.) Therefore:

ab = lim_{k→∞} a_{n_{k}}a_{n_{k}}^{−1} = Id_{U},
ba = lim_{k→∞} a_{n_{k}}^{−1}a_{n_{k}} = Id_{U},

and we are done. □

The following definition describes manifolds on whose diffeomorphism groups one can still define a family of metrics with the desired properties.

Definition 2.7. Let N be a connected C^{∞} manifold without boundary. N is called a good manifold if there is an embedding of N into an Euclidean space ℝ^{d} such that N has a tubular neighborhood U which is an open set in ℝ^{d} and which is C^{∞}-diffeomorphic to N × D, where D is an open bounded disk in ℝ^{d−dim N}.

Remark. Any good manifold has stably trivial tangent bundle and for closed manifolds the converse is also true. Examples of such manifolds are the n–torus, the n–sphere, or any closed compact hypersurface in ℝ^{d}.

Take now an integer K ≥ 1 and let N be a good manifold. Then any diffeomorphism a in Diff^{K}(N) can be extended to a diffeomorphism ã in Diff^{K}(U) by ã(x, y) := (a(x), y), for x ∈ N and y ∈ D (here U ≅ N × D is the open set given by the Definition 2.7).

By Propositions 2.4 and 2.6 we obtain the following:

Proposition 2.8. For any diffeomorphisms a, b in Diff^{K}(N) define:

\begin{matrix}d_{0}(a, b) := & d_{0}(ã, b̃), \\ d_{K}(a, b) := & d_{K}(ã, b̃), \\ d′_{K}(a, b) := & d′_{K}(ã, b̃), \\ d′_{K}(a) := & d′_{K}(ã), \\ \overline{d}_{K}(a, b) := & \overline{d}_{K}(ã, b̃). \end{matrix}

Denote by Diff_{U,bdd}^{K}(N) the set of C^{K}-diffeomorphisms a of N such that d_{K}(a^{±1}, Id_{U}) < ∞.

Then, for any diffeomorphisms a, b, u ∈ Diff_{U,bdd}^{K}(N) and v ∈ Diff_{U,bdd}^{K+1}(N), the properties (D1), (D2), (R1), (R2), (R3) and (L4) hold.

Also, Diff_{U,bdd}^{K}(N) is a complete group with respect to the metric \overline{d}_{K}. □

Remark. We use the same notation for the metrics on Diff^{K}(N) and Diff^{K}(U), for N a good manifold and U an open set in a flat space, but this will produce no confusion.

The next proposition will be used in §6.

Proposition 2.9. (a) Consider a ∈ C^{1}(𝕋^{d}, 𝕋^{d}). If d_{1}(a, Id) < \frac{1}{2}, then a ∈ Diff^{1}(𝕋^{d}).

(b) Let K ≥ 1 be an integer. Then there exists a constant L_{K} > 0 such that for any a ∈ C^{K+Lip}(𝕋^{d}) with d_{K+Lip}(a, Id) < \frac{1}{2}, one has

d_{K+Lip}(a^{−1}, Id) ≤ L_{K}.

(c) Define

\begin{matrix}M′_{K} := (K^{2} + 1)(1 + L_{K})^{K+1}, \\ M_{K} := & (1 + K)(1 + L_{K})^{K+1}[1 + 3(1 + L_{K})^{K}]^{K}. \end{matrix}

Then, for any a, b ∈ C^{K+Lip}(𝕋^{d}) with d_{K+Lip}(a, Id) < \frac{1}{2} and d_{K+Lip}(b, Id) < \frac{1}{2}, one has:

\begin{matrix}(i) & d_{K+Lip}(a^{−1}, Id) ≤ M′_{K}d_{K+Lip}(a, Id), \\ (ii) & d_{K}(a^{−1}, b^{−1}) ≤ M_{K}d_{K}(a, b). \end{matrix}

Proof. (a) If d_{0}(a, Id) < \frac{1}{2}, then a is homotopic to the identity map. Hence a is surjective (use the topological degree). If d_{(1)}(a, Id) < \frac{1}{2}, then for any x, y ∈ 𝕋^{d}:

\begin{matrix}∥a(x) − a(y)∥ ≥ ∥x − y∥ − ∥(a(x) − x) − (a(y) − y∥ ≥ \\ ≥ & ∥x − y∥ − ∥x − y∥ sup_{z∈𝕋^{d}}∥D_{z}(a − Id)∥ ≥ \frac{1}{2}∥x − y∥, \end{matrix}

so a is injective. Since a is also a local diffeomorphism, a ∈ Diff^{1}(𝕋^{d}).

(b) By differentiating the equality aa^{−1} = Id, one can express D_{x}^{l}a^{−1} as a “polynomial” in D_{a^{−1}(x)}^{s}a, 1 ≤ s ≤ l, and (D_{a^{−1}(x)}^{1}a)^{−1}. For example:

\begin{matrix}D_{x}^{1}a^{−1} & = (D_{a^{−1}(x)}^{1}a)^{−1}, \\ D_{x}^{2}a^{−1}(ξ, η) = −(D_{a^{−1}(x)}^{1}a)^{−1} & D_{a^{−1}(x)}^{2}a((D_{a^{−1}(x)}^{1}a)^{−1}ξ, (D_{a^{−1}(x)}^{1}a)^{−1}η). \end{matrix}

(See formula (2.1)).

Use now that if X ∈ End(V) (V some normed vector space) is such that ∥I − X∥ < 1, then ∥I − X^{−1}∥ ≤ \frac{∥I−X∥}{1−∥I−X∥} (by the power series expansion).

Hence, if d_{K}(a, Id) < \frac{1}{2}, then

∥D_{x}a^{−1} − Id∥ = ∥(D_{a^{−1}(x)}a)^{−1} − Id∥ < 2∥D_{a^{−1}(x)}a − Id∥ < 1

and d_{(1)}(a^{−1}) < 2.

Therefore, for 1 ≤ l ≤ K, d_{(l)}(a^{−1}) is bounded by some constant which does not depend on a.

In order to estimate ∥D^{K}a^{−1}∥_{Lip}, notice that it will be bounded by some polynomial expression of ∥a^{−1}∥_{Lip}, d_{(1)}(a^{−1}), ∥(D^{1}a)^{−1}∥_{Lip}, d_{(s)}(a) and ∥D^{s}a∥_{Lip} (1 ≤ s ≤ K). This proves our assertion.

(c) (i)

d_{K+Lip}(a^{−1}, Id) = d_{K+Lip}((Id − a)a^{−1}) ≤

≤^{(Lip3)} d_{K+Lip}(Id − a)[1 + d_{K+Lip}(a^{−1}, Id)]^{K}·

·[1 + (K^{2} + 1)d_{K+Lip}(a^{−1}, Id)] ≤

≤ (1 + L_{K})^{K}[1 + (K^{2} + 1)L_{K}]d_{K+Lip}(a, Id) ≤

≤ (1 + L_{K})^{K+1}(K^{2} + 1)d_{K+Lip}(a, Id).

(ii)

\begin{matrix}d_{K}(a^{−1}, b^{−1}) & = d_{K}(a^{−1}Id, a^{−1}(ab^{−1})) ≤ \\ ≤^{(L1)} d_{K}(Id, ab^{−1})(K + 1) & d′_{K+Lip}(a^{−1})[1 + 2d′_{K}(ab^{−1}))]^{K} ≤ \\ ≤^{(R1)} d_{K}((bb^{−1}, ab^{−1})(K + 1) & (1 + L_{K}) · (1 + 2d′_{K}(a)[1 + d_{K}(b^{−1}, Id)]^{K})^{K} ≤ \\ ≤^{(R3)} d_{K}(b, a)[1 + d_{K}(b^{−1}, & Id)]^{K}(K + 1)(L_{K} + 1)· \\ ·\{1 + 2[1 + d_{K}(a, Id) & ][1 + d_{K}(b^{−1}, Id)]^{K}\}^{K} ≤ \\ ≤ (1 + L_{K})^{K+1}(1 + K) & [1 + 3(1 + L_{K})^{K}]^{K}d_{K}(a, b). □ \end{matrix}

§3. A Livsic’s Type Theorem for Cocycles with Values in Diffeomorphism Groups

The goal of this section is to prove the following theorem:

Theorem 3.1. Let M be a compact closed manifold, let ℤ × M → M, (g, x) ↦ gx be an action given by a C^{1} transitive Anosov diffeomorphism T : M → M. Endow M with a Lyapunov metric d_{M} and for k ≥ 0 let G^{k} be one of the groups:

(a) Diff_{bdd}^{k}(U) endowed with the metric d_{k} described in Definition 2.2, where U is an open connected subset in ℝ^{d};

(b) Diff_{U,bdd}^{k}(N) endowed with the metric d_{k} described in Proposition 2.8, where N is a good manifold.

Fix 0 < α ≤ 1, K ∈ ℕ, K ≥ 4.

Then there exists δ > 0 with the following property: if \overline{β} is any ℤ-cocycle over the considered action

\overline{β} : ℤ × M → G^{K}

such that:

(1) d_{K}(\overline{β}(1, x)^{±1}, Id) < δ, for any x ∈ M;

(2) \overline{β} fulfills the closing conditions:

x ∈ M, g ∈ ℤ and gx = x implies that \overline{β}(g, x) = Id;

(3) \overline{β}(1, ·) : M → G^{K} is α-Hölder,

then there is an α-Hölder map P : M → G^{K−2} which trivializes \overline{β}, i.e.

\overline{β}(g, x) = P (gx)P (x)^{−1}.  (T)

Remarks. 1. The proof uses the idea of Livsic (compare the proof of Theorem 1.2 in [L2] with Claim 1 in the proof of Theorem 3.1). Nevertheless, our situation is more complicated due to the absence of invariant metrics. We replace these by the “quasi-invariance” properties exhibited in Propositions 2.4 and 2.8. The proof of the theorem will be the same in the cases (a) and (b).

2. One can view a connected matrix Lie group G ⊂ GL(n, ℝ) as a subgroup of Diff_{bdd}^{k}(GL(n, ℝ)^{0}), via its action by left translations. [GL(n, ℝ)^{0} is the connected component of the identity in the group of n×n invertible matrices.] Since GL(n, ℝ)^{0} is an open subset of ℝ^{n^{2}}, one can try to find a proof of Theorem 1.2 using directly Theorem 3.1 (a). The problem is that the natural metric on the Lie group might not be equivalent to the one we introduced in §2.

3. There is also a similar statement for cocycles over actions given by diffeomorphisms with an invariant locally maximal hyperbolic set.

Before beginning the proof, we record the following ingredients, also used by Livsic in the proof of Theorem 1.2.

Anosov’s Closing Lemma. ( [K-H]) Given T : M → M an Anosov diffeomorphism on a compact manifold, there are constants ϵ_{CL} > 0, K_{CL} > 0 with the following property:

for 0 < δ < ϵ_{CL}, given a periodic δ-pseudo-orbit (x_{0}, x_{2}, …, x_{n−1}), there is a periodic orbit of the same length (x_{∗}, T x_{∗}, …, T^{n−1}x_{∗}), T^{n}x_{∗} = x_{∗} such that

d_{M}(x_{k}, T^{k}x_{∗}) ≤ K_{CL}δ, 0 ≤ k ≤ n − 1.

Local Product Structure. ( [K-H]) Given T : M → M an Anosov diffeomorphism on a compact manifold, there are ϵ_{P S} > 0, K_{P S} > 0 such that:

for any x, y ∈ M with d_{M}(x, y) < ϵ_{P S}, the intersection W^{s}(x) ∩ W^{u}(y) contains exactly one element, whose distance to both x and y is at most K_{P S} · d_{M}(x, y).

Since in our case the metrics d_{k} are not as well behaved as those of Livsic, we isolate the main computational part in the following lemma:

Lemma 3.2. Assume 0 < μ < 1 and δ > 0 are such that μ(1 + 4δ)^{2K} < 1. Let N ≥ 1 be an integer and, for any 1 ≤ k ≤ N, let a_{k}, b_{k}, c_{k} ∈ G^{K} be such that

(i) d_{K−1}(a_{k}, b_{k}) ≤ C_{0} · μ^{k−1} · ω_{0}, (ii) d_{K−2}(b_{k}, c_{k}) ≤ C_{0} · μ^{N−k} · ω_{0}, (iii) d_{K−1}(a_{k}^{±1}, Id) < δ, d_{K}(b_{k}^{±1}, Id) < δ, d_{K−2}(c_{k}^{±1}, Id) < δ. Denote a = a_{1}a_{2} ⋯ a_{N}, b = b_{1}b_{2} ⋯ b_{N}, c = c_{1}c_{2} ⋯ c_{N}. Then:

(a) d_{K−1}(a, b) ≤ ω_{0} · \frac{C_{0}K}{1−μ(1+4δ)^{2K}} max\{1, d′_{K−1}(a)\}^{K−1};

(b) d_{K−2}(b, c) ≤ ω_{0} · \frac{C_{0}(K−1)}{1−μ(1+4δ)^{2K}} · d′_{K−1}(b).

(Note that d′_{K−1}(b) ≤ 1 + d_{K−1}(b, Id)).

Proof. (a) d_{K−1}(a, b) = d_{K−1}(a_{1}a_{2} ⋯ a_{N}, b_{1}b_{2} ⋯ b_{N}) ≤ Δ_{1} + Δ_{2} + ⋯ + Δ_{N}, where

Δ_{k} = d_{K−1}(b_{1} ⋯ b_{k−1}a_{k}a_{k+1} … a_{N}, b_{1} … b_{k−1}b_{k}a_{k+1} … a_{N}).

Replace in Δ_{k} a_{k+1} ⋯ a_{N} by a_{k}^{−1}a_{k−1}^{−1} ⋯ a_{1}^{−1} a. Then apply (R3) successively for the final a-factors, and then (L4) once for the product of the leading b-factors to get:

Δ_{k} ≤ d_{K−1}(b_{1} … b_{k−1}a_{k}, b_{1} … b_{k})·

· [ k ; ∏_{j=1} (1 ∨ d′_{K−1}(a_{j}^{−1}))^{K−1}] (1 ∨ d′_{K−1}(a))^{K−1} ≤

≤ d_{K−1}(a_{k}, b_{k}) · K · [ k ; ∏_{j=1} (1 ∨ d′_{K−1}(a_{j}^{−1}))^{K−1}] · (1 ∨ d′_{K−1}(a))^{K−1}·

·d′_{K}(b_{1} ⋯ b_{K−1}) · [1 + 2d′_{K−1}(a_{k}, Id) + 2d′_{K−1}(b_{k}, Id)]^{K−1}.

18 VIOREL NIŢICĂ AND ANDREI TÖRÖK

Applying successively (R1) for each factor, we get (for k ≥ 2) that

d′_{K}(b_{1} ⋯ b_{k−1}) ≤ d′_{K}(b_{1}) · k−1 ; ∏_{ℓ=2} [1 ∨ d′_{K}(b_{ℓ})]^{K}.

By (D1) and (iii):

Δ_{k} ≤ d_{K−1}(a_{k}, b_{k}) · K · [(1 + δ)^{K−1}]^{k} · (1 ∨ d′_{K−1}(a))^{K−1}·

·(1 + δ)^{1+K(k−2)} · (1 + 4δ)^{K−1} ≤
≤ d_{K−1}(a_{k}, b_{k}) · K · (1 + 4δ)^{2kK}(1 ∨ d′_{K−1}(a))^{K−1}.

Together with (i), this yields (a).

(b) Write d_{K−2}(b, c) = d_{K−2}(b_{1}b_{2} ⋯ b_{N}, c_{1}c_{2} ⋯ c_{N}) ≤ Δ_{1} + Δ_{2} + ⋯ + Δ_{N},

with

Δ_{k} = d_{K−2} (b_{1} ⋯ b_{N−k}b_{N−k+1}c_{N−k+2} ⋯ c_{N}, b_{1} ⋯ b_{N−k}c_{N−k+1}c_{N−k+2} ⋯ c_{N}) . (Here we start interchanging factors at the right side of the products.)

Replace in Δ_{k} b_{1} ⋯ b_{N−k} by b b_{N}^{−1} ⋯ b_{N−k+1}^{−1}, apply (R3) successively for the last c-factors and (L4) once for the product of the leading b-factors to get

\begin{matrix}Δ_{k} ≤ d_{K−2}(b_{N−k−1}, c_{N−k+1}) · (K − 1) · [ N ; ∏_{j=N−k+2} (1 ∨ d_{K−2}(c_{j}))^{K−2}] · \\ ·d′_{K−1}(b b_{N}^{−1} ⋯ b_{N−k+1}^{−1}) · [1 + d′_{K−2}(b_{N−k+1}, Id) + d′_{K−2}(c_{N−k+1}, Id)]^{K−2}. \end{matrix}

By successive use of (R1) we obtain that

d′_{K−1}(b b_{N}^{−1} ⋯ b_{N−k+1}^{−1}) ≤ d′_{K−1}(b) · N ; ∏_{j=N−k+1}(1 ∨ d′_{K−1}(b_{N−k+1}))^{K−1},

hence, by (D1) and (iii), it follows that:

Δ_{k} ≤ d_{K−2}(b_{N−k+1}, c_{N−k+1}) · (K − 1) · (1 + δ)^{(K−2)(k−1)} · d′_{K−1}(b)·

·(1 + δ)^{(K−1)k} · (1 + 4δ)^{K−2} ≤

\begin{matrix}≤ d_{K−2}(b_{N−k+1}, c_{N−k+1})(K − 1)(1 + 4δ)^{(2kK−3k)} · d′_{K−1}(b) ≤ \\ ≤ d_{K−2}(b_{N−k+1}, c_{N−k+1})(K − 1)(1 + 4δ)^{2kK} · d′_{K−1}(b) \end{matrix}

which together with (ii) implies (b). Proof of Theorem 3.1.

Let T M = E^{u} ⊕ E^{s} be the Anosov splitting of T M and 0 < λ < 1 such that ∥DT |_{E^{s}}∥ < λ and ∥DT^{−1}|_{E^{u}}∥ < λ. Take δ > 0 such that λ^{α}(1 + 4δ)^{2K} < 1.

Let x_{0} ∈ M be a transitive point for T, i.e. one has \overline{O}_{+} = M, where x_{n} := T^{n}x, and O_{+} = {x_{n}}_{n≥0}.

Denote β(·) := \overline{β}(1, ·) : M → G^{K}, where \overline{β} : ℤ × M → G^{K} fulfills the conditions (1), (2) and (3) in the statement of Theorem 3.1.

The equation (T) has a unique solution on O_{+} up to a choice of P (x_{0}). For P (x_{0}) = Id we obtain the solution P_{0} : O_{+} → G^{K}, P_{0}(x_{n}) =^{not.} φ_{n} = β(T^{n−1}x_{0}) β(T^{n−2}x_{0}) ⋯ β(x_{0}).

We will show that P_{0} can be extended to M to give the desired P.

Claim 1. There exist 1 ≥ ϵ > 0 and A ≥ 1 such that whenever

d_{M}(x_{n}, x_{m}) < ϵ for some n > m, then
d_{K−2}(φ_{n}φ_{m}^{−1}, Id) ≤ A (d_{M}(x_{n}, x_{m}))^{α}.

Proof. Set ϵ = min \{1, ϵ_{CL}, \frac{ϵ_{P S}}{K_{CL}} \}; A will be specified later.

Assume n > m are such that d_{M}(x_{n}, x_{m}) =^{not.} ω < ϵ. Denote x = x_{m}, N = n−m. Therefore (x, T x, ⋯, T^{N−1}x) is a periodic ω-pseudo-orbit, so there is an N-periodic point x_{∗} such that

d_{M}(T^{k}x, T^{k}x_{∗}) < K_{CL} ω < ϵ_{P S}, for 0 ≤ k ≤ N.

Let \overline{x} be the unique element of W^{u}(x_{∗}) ∩ W^{s}(x). Then one has the inequalities:

d_{M}(T^{k}x, T^{k}\overline{x}) ≤ C · λ^{k} · ω,
d_{M}(T^{k}\overline{x}, T^{k}x_{∗}) ≤ C · λ^{N−k} · ω , 0 ≤ k ≤ N,

where C depends only on T.

Denote:

a_{k} = β(T^{N−k}x_{∗}),
b_{k} = β(T^{N−k}\overline{x}),
c_{k} = β(T^{N−k}x), 1 ≤ k ≤ N,
a = a_{1}a_{2} ⋯ a_{N},
b = b_{1}b_{2} ⋯ b_{N},
c = c_{1}c_{2} ⋯ c_{N}.

Then φ_{n}φ_{m}^{−1} = c, while a = Id due to the closing conditions. We have to show that c is close to a.

Since β is α-Hölder,

\begin{aligned}d_{K}(a_{k}, b_{k}) & ≤ C′ μ^{k} · ω^{α}, \\ d_{K}(b_{k}, c_{k}) & ≤ C′ μ^{N−k} · ω^{α}, \end{aligned}

where C′ = C^{α} · ∥β∥ _{Hölder} and μ = λ^{α}.

Now we can apply Lemma 3.2 to obtain

d_{K−2}(φ_{n}φ_{m}^{−1}, Id) = d_{K−2}(c, a) ≤
≤ d_{K−1}(b, a) + d_{K−2}(c, b) ≤ A ω^{α} = A · d_{M}(x_{m}, x_{n})^{α},

where

A = \frac{C′K}{1 − μ(1 + 4δ)^{2K}} +

+ \frac{C′(K − 1)}{1 − μ(1 + 4δ)^{2K}} · (1 + \frac{C′K}{1 − μ(1 + 4δ)^{2K}} · ϵ^{α}) + 1

(Note: the last 1 appears only because we want A ≥ 1.)

Claim 2. If a ∈ G^{2} and d_{2}(a, Id) < \frac{1}{2}, then

d′_{2}(a^{−1}, Id) ≤ 8 d_{2}(a, Id) < 4.

Proof. From the proof of Proposition 2.9 (b), it follows that ∥D_{x}a^{−1} − I∥ < 2 and

D_{x}^{2}a^{−1} = (D_{a^{−1}(x)}a)^{−1}D_{a^{−1}(x)}^{2}a((D_{a^{−1}(x)}a)^{−1}, (D_{a^{−1}(x)}a)^{−1}).

Claim 3. One has:

(a) M_{1} := sup_{n} d_{K−2}(φ_{n}, Id) < ∞,

(b) M_{2} := sup_{n} d′_{2}(φ_{n}^{−1}) < ∞.

Proof. Define ϵ′ = max\{ϵ, \frac{1}{(2A)^{1/α}} \}. Since M is compact, there is a finite set I ⊂ ℕ such that the d_{M}-balls of radius \frac{1}{2}ϵ′ centered in \{x_{i}\}_{i∈I} cover M. Call these balls U_{i} (i ∈ I) respectively.

(a) Let n ∈ ℕ be such that x_{n} ∈ U_{i}, n ≥ i. Then:

d_{K−2}(φ_{n}, φ_{i}) = d_{K−2}((φ_{n}φ_{i}^{−1})φ_{i}, φ_{i}) ≤^{(R3)}

≤ d_{K−2}(φ_{n}φ_{i}^{−1}, Id) · [1 + d_{K−2}(φ_{i}, Id)]^{K−2} ≤

≤^{Claim 1} A(\frac{ϵ′}{2} )^{α}[1 + d_{K−2}(φ_{i}, Id)]^{K−2}.

Note that the values of n which are not covered by the above argument are in J = {n ∈ ℕ; x_{n} ∈ U_{i}, n < i}, which is a finite set. Therefore:

sup_{n∈ℕ} d_{K−2}(φ_{n}, Id) ≤ max_{i∈I} {d_{K−2}(φ_{i}, Id)+

+ A(\frac{ϵ′}{2} )^{α}[1 + d_{K−2}(φ_{i}, Id)]^{K−2}\} ∨ max_{n∈J} \{d_{K−2}(φ_{n}, Id)\} < ∞.

(b) Consider again n ∈ ℕ such that x_{n} ∈ U_{i}, n ≥ i. Then:

d′_{2}(φ_{n}^{−1}, φ_{i}^{−1}) ≤ d_{2}(φ_{i}^{−1}(φ_{n}φ_{i}^{−1})^{−1}, φ_{i}^{−1}) ≤^{(L4)}

≤ 3d_{2}((φ_{n}φ_{i}^{−1})^{−1}, Id)d′_{3}(φ_{i}^{−1})[1 + 2d′_{2}((φ_{n}φ_{i}^{−1})^{−1}, Id) + 2d′_{2}(Id, Id)]^{2} ≤

≤ d′_{3}(φ_{i}^{−1}))[1 + 3d_{2}((φ_{n}φ_{i}^{−1})^{−1}, Id)]^{3}.

Using now Claim 1 and Claim 2, it follows that

d_{2}(φ_{n}φ_{i}^{−1}, Id) ≤ A(\frac{ϵ′}{2} )^{α} < \frac{1}{2}, for 2 ≤ K − 2,

and

d_{2}((φ_{n}φ_{i}^{−1})^{−1}, Id) ≤ 4,

and from here we proceed as at (a).

Claim 4. P_{0} : O_{+} → G^{K} is α-Hölder with respect to \overline{d}_{K−2}.

Proof. Let n, m be such that d_{M}(x_{n}, x_{m}) < ϵ′. Assume n ≥ m. We consider separately the two metrics which define \overline{d}_{K−2}.

d_{K−2}(φ_{n}, φ_{m}) = d_{K−2}((φ_{n}φ_{m}^{−1})φ_{m}, φ_{m}) ≤

≤^{(R3)} d_{K−2}(φ_{n}φ_{m}^{−1}, Id)[1 + d_{K−2}(φ_{m}, Id)]^{K−2} ≤ (M_{1})^{K−2}A (d_{M}(x_{m}, x_{n}))^{α};

d_{(1)}(φ_{n}^{−1}, φ_{m}^{−1}) ≤ d_{1}(φ_{n}^{−1}, φ_{n}^{−1}(φ_{n}, φ_{m}^{−1})) ≤

≤^{(L4)} d_{1}(φ_{n}φ_{m}^{−1}, Id) d′_{2}(φ_{n}^{−1})[1 + 2d_{1}(φ_{n}φ_{m}^{−1}, Id)] ≤

≤ M_{2}(1 + 2Aϵ)A(d_{M}(x_{m}, x_{n}))^{α}.

Conclusion. Due to the α-Hölder continuity in the complete metric space (G^{K−2}, \overline{d}_{K−2}), P_{0} can be extended by continuity to an α-Hölder function P : M → G^{K−2}. Since P fulfills equation (T) on O_{+} and both sides of (T) are continuous functions into G^{K−2}, P will trivialize \overline{β} over M. □

Remark. The correspondence “cocycle \overline{β} ↦ trivialization P ” constructed in the Theorem 3.1 is continuous in the sense described below. (The notations are the same as in the Theorem 3.1.)

Notice first of all that a cocycle \overline{β} : ℤ×M → G^{K} is completely determined by the function β := \overline{β}(1, ·) : M → G^{K}. From now on we will speak about such functions when we mean a ℤ-cocycle. On the space of ℤ-cocycles β : M → G^{K} consider the topology given by the uniform convergence (note that for functions β : M → G^{K} in a given Hölder class with uniformly bounded Hölder norm, this topology is the same as that of pointwise convergence).

Fix a degree of smoothness K ∈ ℕ, K ≥ 4, and a Hölder class 0 < α ≤ 1. Let δ > 0 be the constant provided by Theorem 3.1.

Corollary (of the proof) 3.3.

Assume that β, β_{ι} are cocycles with values in G^{K}, in the same Hölder class α, and ∥β∥_{Hölder}, ∥β_{ι}∥_{Hölder} ≤ C_{0}, where C_{0} > 0 is a constant. If β_{ι} converges to β, and all these cocycles satisfy the conditions (1) and (2) of the theorem, then P_{ι} converges uniformly to P, as functions from M into G^{K−2} (where P, P_{ι} are the trivializations constructed for β, respectively β_{ι}).

Proof. Let P_{0}, P_{0}^{(ι)} : O_{+} → G^{K} be the trivializations constructed for β, respectively β_{ι}, on the orbit of the transitive point x_{0}, such that their value at x_{0} is Id. Since all these are finite products of the values of the cocycles, P_{0} → P_{0}^{(ι)} pointwise on O_{+}. But the functions P_{0}, P_{0}^{(ι)} : O_{+} → G^{K−2} are all α-Hölder, with uniformly bounded norms, (the estimate obtained in Claim 4, (a) depends “continuously on the cocycle”). Therefore the functions P, P_{ι} : M → G^{K−2} are uniformly equicontinuous, and then pointwise convergence on the dense set O_{+} implies uniform convergence on the compact set M. □

§4. Noncommutative Closing Lemma

In this section we generalize the Anosov Closing Lemma. This will be used later to extend Livsic’s theorems to Anosov group actions.

Theorem 4.1. (“Noncommutative Closing lemma”) Let M be a smooth closed manifold (i.e. compact and without boundary). Let A, B ∈ Diff^{1}(M), A Anosov. Let T M = E^{u} ⊕ E^{s} be the splitting induced by DA, the differential of A. Endow M with an A-adapted metric d_{M}. Assume that

P_{+}(D B)|_{E^{u}} : E^{u} → E^{u} and P_{−}(D B^{−1})|_{E^{s}} : E^{s} → E^{s}

can be inverted fiberwise (hence the norm of the inverses is bounded uniformly on M), where P_{±} : T M → E^{u,s} is the projection along E^{s,u}. (Equivalently, that there is a constant α_{B} > 0 such that

∡ (E^{s}, D B(E^{u})) > α_{B}

and

∡ (E^{u}, D B^{−1}(E^{s})) > α_{B}.)

Then there are N_{0} ∈ ℕ, δ_{0} > 0, K > 0 such that for any N ⩾ N_{0}, 0 < δ < δ_{0}, and for any sequence (\overline{x}_{0}, \overline{x}_{1}, ⋯, \overline{x}_{N}) such that d_{M}(A\overline{x}_{k}, \overline{x}_{k+1}) < δ, for 0 ≤ k ≤ N −1, and d_{M}(B\overline{x}_{N}, \overline{x}_{0}) < δ, there is a point x ∈ M with the following properties:

(a) B A^{N}x = x;

(b) the pseudo-orbit (\overline{x}_{0}, \overline{x}_{1}, ⋯, \overline{x}_{N}) and the orbit of x are Kδ-close, i.e. d_{M}(\overline{x}_{k}, A^{k}x) < Kδ, 0 ≤ k ≤ N.

Moreover, in a neighborhood of radius δ_{0} of \overline{x}_{0}, there is only one point x that satisfies (a) and (b).

Notations. 1. A diffeomorphism B as above will be said to be in generic position with respect to A.

2. By abuse of notation, a sequence (\overline{x}_{0}, \overline{x}_{1}, ⋯, \overline{x}_{N}) as in the theorem will be called a δ-pseudo-orbit.

Remarks. 1. The restriction on B is very weak (the required property is generic).

2. The theorem holds for a hyperbolic set too, provided the appropriate changes are done. Also, if the considered fiber bundle maps are not globally invertible, one can obtain a local result.

3. The proof of the theorem is an adaptation of the classic proof of Anosov’s Closing Lemma as presented in [K-H], Theorem 6.4.15. Our contribution is to show that the analog of the operator denoted (1 − L) there, is also invertible in our case, and the norm of the inverse is uniformly bounded with respect to N. For the reader’s convenience, we present the complete proof.

4. Note that the proof is much simpler if A and B are linear automorphisms of the torus. This is the case we need to prove rigidity of ρ_{0}. The general case is used to generalize the Livsic-Sinai theorem to Anosov group actions (see §5).

Proof. We first introduce local coordinates on M.

Given δ > 0, denote by (T M)_{δ} the bundle of balls of radius δ centered at the origin. By compactness, there are δ′_{1}, δ_{1} > 0 such that the map

exp : (T M)_{δ′_{1}} → M

given in each fiber by

ϕ_{x} = exp_{x} : (T_{x}M)_{δ′_{1}} → M

has the following properties:

– for any x in M, the image of ϕ_{x} contains a ball of radius δ_{1} centered at x;

– the dependence x ↦ ϕ_{x} is continuous from M to the C^{1}-topology on the maps {ϕ_{x}}_{x∈M};

– ∥ϕ_{x}^{±1}∥_{Lip} ≤ 2 for each x ∈ M. Recall that D_{0}ϕ_{x} = Id_{T_{x}M}. Since ∡(E^{u}, E^{s}) > α_{A} > 0, there is a constant C_{1} > 0 with the property that

for each x ∈ M one can choose a linear isomorphism Λ_{x} : T_{x}M → ℝ^{d} such that:

– Λ_{x} takes the decomposition T_{x}M = E_{x}^{u} ⊕ E_{x}^{s} into the decomposition ℝ^{d} = ℝ^{d_{+}} ⊕ ℝ^{d_{−}};

– Λ_{x}|_{E_{x}^{u,s}} : E_{x}^{u,s} → ℝ^{d_{±}} are isometries, where ℝ^{d} is endowed with the standard Euclidean structure;

– ∥Λ_{x}^{±1}∥ ≤ C_{1}. Having done all these, we describe the procedure by which we reduce the maps

along a pseudo-orbit to ones from ℝ^{d} to ℝ^{d}.

Consider any \frac{δ_{1}}{2} -pseudo-orbit (\overline{x}_{0}, ⋯, \overline{x}_{N}). Describe (locally) the maps A and B by:

A_{k}^{0} = Λ_{\overline{x}_{k+1}}ϕ_{\overline{x}_{k+1}}^{−1}Aϕ_{\overline{x}_{k}}Λ_{\overline{x}_{k}}^{−1}, 0 ≤ k ≤ N − 1;
B_{N}^{0} = Λ_{\overline{x}_{0}}ϕ_{\overline{x}_{0}}^{−1}Bϕ_{\overline{x}_{N}} Λ_{\overline{x}_{N}}^{−1}.

Using the decomposition ℝ^{d} = ℝ^{d_{+}} ⊕ ℝ^{d_{−}}, write

D_{0}A_{k}^{0} = ( \begin{matrix}A_{k}^{+} & ∗ \\ ∗ & A_{k}^{−} \end{matrix} ) ,
D_{0}B_{N}^{0} = ( \begin{matrix}B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix} ) =^{not.} B_{N},
(D_{0}B_{N}^{0})^{−1} = ( \begin{matrix}\overline{B}_{11} & \overline{B}_{12} \\ \overline{B}_{21} & \overline{B}_{22} \end{matrix} ) =^{not.} \overline{B}_{N},

where we denoted by ∗ the entries which are not used in the sequel.

By the properties of the chosen coordinates and of the diffeomorphisms A and B, there are δ_{2} > 0, 0 < λ_{1} < 1 and C_{2}, C_{3} ≥ 1 which satisfy the following:

for any 0 < ϵ, there is 0 < δ_{3}(ϵ) < \frac{δ_{1}}{2} such that if (\overline{x}_{0}, ⋯, \overline{x}_{N}) is a δ_{3}(ϵ)-pseudo-orbit then:

– the maps A_{k}^{0}, 0 ≤ k ≤ N −1, B_{N}^{0} are defined on a ball of radius δ_{2} centered at the origin; – on these balls

\begin{matrix}∥A_{k}^{0} − ( \begin{matrix}A_{k}^{+} & 0 \\ 0 & A_{k}^{−} \end{matrix} ) ∥_{C^{1}} < ϵ, \\ ∥B_{N}^{0} − ( \begin{matrix}B_{11} & B_{12} \\ B_{21} & B_{22} \end{matrix} ) ∥_{C^{1}} < ϵ; \end{matrix}  (4.1)

– the linear parts satisfy

∥A_{k}^{−}∥ ≤ λ_{1}, ∥(A_{k}^{+})^{−1}∥ ≤ λ_{1},
∥(B_{11})^{−1}∥ ≤ C_{2}, ∥(\overline{B}_{22})^{−1}∥ ≤ C_{2},
∥B_{N}∥ ≤ C_{3}, ∥(B_{N})^{−1}∥ ≤ C_{3}.

In order to make the following steps simpler, we will extend the maps A_{k}^{0} and B_{N}^{0} to the whole space ℝ^{d}. We use the following fact (see [K-H]):

Let U, V be open bounded sets in ℝ^{d} with \overline{U} ⊂ V. For any ϵ > 0, there is δ_{4}(ϵ) such that if f ∈ C^{1}(V, V) and Λ is a linear map on ℝ^{d} with ∥f − Λ∥_{C^{1},V} < δ_{4}(ϵ), then there is a function f̃ ∈ C^{1}(ℝ^{d}, ℝ^{d}) such that

∥f̃− Λ∥_{C^{1},ℝ^{d}} < ϵ and f̃|_{U} = f|_{U}.

(Indeed, let χ : ℝ^{d} → [0, 1] be a C^{1} function such that χ|_{U} ≡ 1 and χ|_{ℝ^{d}∖V} ≡ 0, and set f̃ = χ · f + (1 − χ)Λ.)

Take U, V to be the balls of radii \frac{δ_{2}}{2} , respectively δ_{2}, centered at the origin. Denote the extended maps by A_{k} and B_{N}.

Using the extended maps, the problem becomes one about small perturbations of linear maps between Euclidean spaces of dimension d: if the ϵ used in (4.1) is less than δ_{4}(ω), then

A_{k}(x^{+}, x^{−}) = (A_{k}^{+}x^{+} + α_{k}^{+}(x^{−}, x^{+}), A_{k}^{−}x^{−} + α_{k}^{−}(x^{−}, x^{+})),  0 ≤ k ≤ N − 1,
B_{N}(x^{+}, x^{−}) = (B_{11}x^{+} + B_{12}x^{−} + β^{+}(x^{−}, x^{+}), B_{21}x^{+} + B_{22}x^{−} + β_{N}^{−}(x^{−}, x^{+})),

where A_{k}^{±}, B_{ij} are linear maps and α_{k}^{±}, β^{±} are such that ∥α_{k}^{±}∥_{C^{1}} ≤ ω for all k and ∥β_{N}^{±}∥_{C^{1}} ≤ ω.

Let

C_{4} := \frac{1}{1 − λ_{1}} (2C_{3} + 1)^{3} max\{λ_{1}^{−1}, C_{3}\}.

After these preliminaries, we are ready to start the proof itself.

The constants mentioned in the theorem are determined by:

– δ_{0} = min\{\frac{δ_{2}}{16C_{1}C_{4}} , δ_{3}(δ_{4}(\frac{1}{2}C_{4}^{−1}))\};

– K = 8C_{1}^{2};

– N_{0} satisfies

\begin{matrix}λ_{1}^{N_{0}}C_{2} < \frac{1}{2}, \\ λ_{1}^{N_{0}}C_{3}(1 + 2C_{2}C_{3}) < \frac{1}{2}. \end{matrix}  (4.2)

Consider a δ-pseudo-orbit (\overline{x}_{0}, \overline{x}_{1}, ⋯, \overline{x}_{N}), with δ < δ_{0} and N ≥ N_{0}. The equation we want to solve (BA^{N}x = x) becomes

F (x̃) = x̃,  (4.3)

where x̃ = column(x_{0}, x_{1}, ⋯, x_{N}) ∈ (ℝ^{d})^{N+1}, and

F (x_{0}, x_{1}, ⋯, x_{N}) := (A_{0}x_{1}, A_{1}x_{2}, ⋯, A_{N−1}x_{N}, B_{N}x_{0}).

(Via our change of coordinates, x_{k+1} corresponds to A^{k}x and x_{0} to A^{N}x.) Write F = L_{N} + S_{N}, where L_{N}, S_{N} : (ℝ^{d})^{N+1} → (ℝ^{d})^{N+1} are given by:

\begin{matrix}L_{N}(x_{0}, x_{1}, ⋯, x_{N}) = (A_{0}x_{1}, ⋯, A_{N−1}x_{N}, B_{N}x_{0}), \\ S_{N}(x_{0}, x_{1}, ⋯, x_{N}) = (α_{0}(x_{1}), ⋯, α_{N−1}(x_{N}), β_{N}(x_{0})). \end{matrix}

Consider on (ℝ^{d})^{N+1} the norm

∥(x_{0}, x_{1}, ⋯, x_{N})∥ = max_{0≤i≤N}∥x_{i}∥.

Note that ∥S_{N}∥_{C_{1}} ≤ \frac{1}{2}C_{4}^{−1} and

∥S(˜0)∥ = ∥F (˜0)∥ = ∥(Λ_{\overline{x}_{1}}ϕ_{\overline{x}_{1}}^{−1}A\overline{x}_{0}, Λ_{\overline{x}_{2}}ϕ_{\overline{x}_{2}}^{−1}A\overline{x}_{1}, ⋯, Λ_{\overline{x}_{0}}ϕ_{\overline{x}_{0}}^{−1}B\overline{x}_{N})∥ ≤ 2C_{1}δ.

We will show that if N ≥ N_{0}, then 1 − L_{N} is invertible and

∥(1 − L_{N})^{−1}∥ ≤ C_{4}.  (4.4)

Then the equation (4.3) is equivalent to

x̃ = F(x̃),

where F = (I −L_{N})^{−1}S_{N}. Since F is a contraction, the equation above has a unique solution x̃_{∗}, and

∥x̃_{∗}∥ ≤ \frac{∥F(˜0)∥}{1 − ∥F∥_{Lip}} ≤ \frac{C_{4}∥S(˜0)∥}{1 − ∥F∥_{Lip}} ≤ 4C_{1}C_{4}δ ≤ \frac{δ_{2}}{4} .

Therefore x̃_{∗} corresponds to a solution of BA^{N}x = x on the manifold, which will satisfy claim (b) as well.

The last claim of the theorem follows from the fact that the M-balls of radius δ_{0} centered at the \overline{x}_{k}’s are mapped inside balls of radius 2C_{1}δ_{0} ≤ 2C_{1} \frac{δ_{2}}{16C_{1}C_{4}} < \frac{δ_{2}}{2} centered at the origin in the corresponding ℝ^{d}’s. Therefore, on this set, the action on the manifold is conjugated to the F -action on (ℝ^{d})^{N+1} and F has a unique fixed point.

It remains to prove (4.4).

COHOMOLOGY OF DYNAMICAL SYSTEMS AND RIGIDITY 27

Denoting η̃=column(η_{0}, η_{1}, ⋯, η_{N}) and ỹ=column(y_{0}, y_{1}, ⋯, y_{N}) we have to solve in ỹ the equation

(I − L_{N})ỹ = η̃

⟺  \{ \begin{matrix}y_{0} − A_{0}y_{1} = η_{0} \\ y_{1} − A_{1}y_{2} = η_{1} \\ ⋯ \\ y_{N−1} − A_{N−1}y_{N} = η_{N−1} \\ y_{N} − B_{N}y_{0} = η_{N}. \end{matrix}  (4.5)

Denote

Ã:= A_{0}A_{1} ⋯ A_{N−1}

and

E :=B_{N}A_{0} ⋯ A_{N−2}η_{N−1} + B_{N}A_{0} ⋯ A_{N−3}η_{N−2} + ⋯
⋯ + B_{N}A_{0}η_{1} + B_{N}η_{0} + η_{N}.  (4.6)

Decompose both variables and maps with respect to ℝ^{d} = ℝ^{d_{+}} ⊕ ℝ^{d_{−}}:

Ã = ( \begin{matrix}Ã_{+} & 0 \\ 0 & Ã_{−} \end{matrix} ) ,

y_{k} = ( \begin{matrix}y_{k}^{+} \\ y_{k}^{−} \end{matrix} ) , η_{k} = ( \begin{matrix}η_{k}^{+} \\ η_{k}^{−} \end{matrix} ) , 0 ≤ k ≤ N,

E = ( \begin{matrix}E^{+} \\ E^{−} \end{matrix} ) .

Note that ∥Ã_{−}∥ ≤ λ_{1}^{N}, ∥(Ã_{+})^{−1}∥ ≤ λ_{1}^{N}.

Consider the equations (4.5). One obtains, by eliminating the other variables,

y_{N} = B_{N} Ãy_{N} + E.  (4.7)

One can separate (4.7) into two parts, according to the decomposition of ℝ^{d}:

\{ \begin{matrix}y_{N}^{+} & = B_{11}Ã_{+}y_{N}^{+} + B_{12}Ã_{−}y_{N}^{−} + E^{+} \\ y_{N}^{−} & = B_{21}Ã_{+}y_{N}^{+} + B_{22}Ã_{−}y_{N}^{−} + E^{−} \end{matrix}  \begin{matrix}(4.8) \\ (4.9) \end{matrix}

Rewrite (4.8) as

\begin{matrix}(I − B_{11}Ã_{+})y_{N}^{+} = B_{12}Ã_{−}y_{N}^{−} + E^{+} \\ ⟺ F_{0}Ã_{+}y_{N}^{+} = B_{12}Ã_{−}y_{N}^{−} + E^{+}, \end{matrix}

where F_{0} = (Ã_{+}^{−1}(B_{11})^{−1} − I)B_{11}.

Now N ≥ N_{0} implies that ∥Ã_{+}^{−1}(B_{11})^{−1}∥ ≤ \frac{1}{2} (see (4.2)). Therefore F_{0} is invertible,

∥F_{0}^{−1}∥ ≤ 2∥(B_{11})^{−1}∥ ≤ 2C_{2}  (4.10)

and

y_{N}^{+} = Ã_{+}^{−1}(F_{0}^{−1}B_{12}Ã_{−}y_{N}^{−} + F_{0}^{−1}E^{+}).  (4.11)

Substituting into (4.9), one has

y_{N}^{−} = F_{1}y_{N}^{−} + B_{21}^{−1}F_{0}^{−1}E^{+} + E^{−},  (4.12)

where

F_{1} = B_{22}Ã_{−} + B_{21}F_{0}^{−1}B_{12}Ã_{−}.

Compute

\begin{matrix}∥F_{1}∥ ≤ ∥B_{22}∥ · ∥Ã_{−}∥ + ∥B_{21}∥ · ∥F_{0}^{−1}∥ · ∥B_{12}∥ · ∥Ã_{−}∥ ≤ \\ ≤^{(4.10)} ∥B_{N}∥λ_{1}^{N} + ∥B_{N}∥ · 2C_{2} · ∥B_{N}∥ · λ_{1}^{N}. \end{matrix}

By (4.2), if N ≥ N_{0} then ∥F_{1}∥ ≤ \frac{1}{2}. So I − F_{1} is invertible, ∥(I − F_{1})^{−1}∥ ≤ 2, and the equation (4.12) has the (unique) solution

y_{N}^{−} = (I − F_{1})^{−1}(B_{21}^{−1}F_{0}^{−1}E^{+} + E^{−})

with

∥y_{N}^{−}∥ ≤ 2(∥B_{N}∥ · 2 · ∥E^{+}∥ + ∥E^{−}∥) ≤ (4C_{3} + 2)∥E∥  (4.13)

In conclusion, for N ≥ N_{0} the system (4.5) has a unique solution, hence 1 − L_{N} is invertible.

It remains to estimate the norm of (1 − L_{N})^{−1}. Denote V^{+} := (ℝ^{d_{+}})^{N+1} and V^{−} := (ℝ^{d_{−}})^{N+1}. In order to make the computation easier, we show separately that ∥(1 − L_{N})^{−1}|_{V^{±}}∥ are bounded by C_{4}, and hence so is ∥(1 − L_{N})^{−1}∥.

Consider η̃ ∈ V^{−}. For 0 ≤ k ≤ N − 1, by (4.5) and (4.11):

y_{k}^{−} = A_{k}^{−} ⋯ A_{N−1}^{−}y_{N}^{−} + A_{k}^{−} ⋯ A_{N−2}^{−}η_{N−1}^{−} + ⋯ + A_{k}^{−}η_{k+1}^{−} + η_{k}^{−},
y_{k}^{+} = A_{k}^{+} ⋯ A_{N−1}^{+}Ã_{+}^{−1}(F_{0}^{−1}B_{12}Ã_{−}y_{N}^{−} + F_{0}^{−1}E^{+})
= (A_{k−1}^{+})^{−1} ⋯ (A_{0}^{+})^{−1}(F_{0}^{−1}B_{12}Ã_{−}y_{N}^{−} + F_{0}^{−1}E^{+}).

Hence

∥y_{k}^{−}∥ ≤ λ_{1}^{N−k}∥y_{N}^{−}∥ + (λ_{1}^{N−k−1}∥η̃∥ + λ_{1}^{N−k−2}∥η̃∥ + ⋯ + ∥η̃∥) ≤

≤ λ_{1}^{N−k}∥y_{N}^{−}∥ + \frac{1}{1 − λ_{1}} ∥η̃∥,

∥y_{k}^{+}∥ ≤ λ_{1}^{k}(2C_{3}λ_{1}^{N}∥y_{N}^{−}∥ + 2∥E∥).

By (4.13), it is enough to estimate ∥E∥. By (4.6), we see that for η̃ ∈ V^{−}

∥E∥ ≤ C_{3} \frac{1}{1 − λ_{1}} ∥η̃∥.

Hence

∥(1 − L_{N})^{−1}|_{V^{−}}∥ ≤ \frac{1}{1 − λ_{1}} (2C_{3} + 1)^{3}.

In order to estimate ∥(1 − L_{N})^{−1}|_{V^{+}}∥, note that this case can be reduced to the one studied above. Indeed:

– (1 − L_{N})^{−1}|_{V^{+}} = −L_{N}^{−1}(1 − L_{N}^{−1})^{−1}|_{V^{+}} and ∥L_{N}^{−1}∥ ≤ max{λ_{1}^{−1}, C_{3}};

– the matrix of J(L_{N}^{−1})J^{−1} is of same form as that of L_{N}, where J = (j_{kl})_{0≤k,l≤N} is the isometry of (ℝ^{d})^{N+1} given by j_{kl} = δ_{k+l,N+1}Id_{ℝ^{d}};

– the “contracting space” of L_{N}^{−1} is V^{+} (which is invariated by J^{−1}), and the corresponding entry of \overline{B} is \overline{B}_{22}.

This completes the proof of Theorem 4.1. □

§5. Livsic Type Results for Anosov Smooth Group Actions

In this section we extend Livsic’s results about the cohomology of Anosov ℤ-actions to more general Anosov smooth group actions.

Definition 5.1. Let G be a discrete group and let α : G × M → M be a C^{k}- action (k ≥ 1) of G by diffeomorphisms on a compact manifold M. We say that α is Anosov if there is an element g ∈ G such that α(g) is Anosov. Let Γ be a topological group and β : G × M → Γ a continuous cocycle. We say that β satisfies the isolated closing conditions if for any g ∈ G and any x ∈ M isolated fixed point of g, one has β(g, x) = 1_{Γ}.

Remark. The theorem below shows that the notion of isolated closing conditions is natural. If we have an Anosov ℤ-action, then the closing conditions are equivalent with the isolated closing conditions. As one will see in §7, the isolated closing conditions can be easily obtained for some natural examples.

The main result is the following theorem which is based essentially on the Noncommutative Closing Lemma, Theorem 4.1:

Theorem 5.2. Let M be a smooth closed manifold. Let A, B ∈ Diff^{1}(M), A a topologically mixing Anosov and B in generic position with respect to A. Let G = ⟨A, B⟩ be the group generated by A and B in Diff^{1}(M). Let Γ be a topological group and β : G × M → Γ a continuous cocycle which satisfies the isolated closing conditions. Denote by ⟨A⟩ the group generated by A in Diff^{1}(M) and assume that β|_{⟨A⟩×M} is null cohomological, i.e. there is P : M → Γ continuous such that β(A, x) = P (Ax)P (x)^{−1} for x ∈ M. Then β is null cohomological, being trivialized by P.

Proof. For any g ∈ G, define \overline{β}(g, x) = P (gx)^{−1}β(g, x)P (x). Then \overline{β}(A, x) = 1 and we have to show that \overline{β}(B, x) = 1 for any x ∈ M.

Pick x_{0} close to Bx such that A^{n}x_{0} is close to x for some large n. Then, by the continuity of B, BA^{n}x_{0} is close to x_{0}. Apply now Theorem 4.1 (Noncommutative Closing Lemma) and find x_{∗} such that BA^{n}x_{∗} = x_{∗}, x_{∗} is close to x_{0} and A^{n}x_{∗} is close to A^{n}x_{0}. Hence A^{n}x_{∗} is also close to x_{0}. But

\overline{β}(B, A^{n}x_{∗}) = \overline{β}(BA^{n}, x_{∗}) · \overline{β}(A^{n}, x_{∗})^{−1} = \overline{β}(BA^{n}, x_{∗}) = 1_{Γ},

because BA^{n}x_{∗} = x_{∗}. Using now the fact that A^{n}x_{∗} approaches x when A^{n}x_{0} approaches x, and the continuity of \overline{β}, we have \overline{β}(B, x) = 1_{Γ}. □

It is clear that for many Anosov actions of “big groups” we can apply Theorem 5.2 several times for generators and obtain Livsic type results. This motivates the following definition:

Definition 5.3. We say that a smooth action G × M → M is generically Anosov, (GA), if G contains a set of generators {A, B_{1}, …, B_{n}} such that A is transitive Anosov and each B_{k} is in generic position with respect to A (1 ≤ k ≤ n).

Note that Theorem 5.2 has a straightforward analogue for cocycles over (GA)- actions.

The following theorems extend to (GA)-actions the theorems of Livsic and our Livsic type theorem for cocycles with values in diffeomorphisms. The element A in G which appears in Theorems 5.4, 5.5, 5.6 and 5.9 is the Anosov element specified in the definition of a (GA)-action.

Theorem 5.4. (a) Let ρ : G × M → M be a C^{1} (GA)-action. Let β : G × M → ℝ be a continuous cocycle over ρ which satisfies the isolated closing condition. Assume that β(A, ·) is Hölder. Then there exists a Hölder function P : M → ℝ such that β(g, x) = P (gx) − P (x).

(b) Let K ≥ 1. If ρ(B) and β(B, ·) are C^{K} for some Anosov ρ(B), then P is C^{K−1}. In particular, if ρ(B) and β(B, ·) are C^{∞}, then P is C^{∞}. The same holds in the class C^{ω}.

Proof. The first part follows from Livsic’s Theorem 1.1, and Theorem 5.2. The regularity is a result in [L-M-M] (Theorem 2.1), respectively [Lla]. □

Theorem 5.5. (a) Let ρ : G × M → M be a C^{1} (GA)-action. Let Γ be a good group (see §1) and α ∈ (0, 1]. Then there is a neighborhood U of 1_{Γ} with the property that for any continuous cocycle β : G × M → Γ such that:

(1) β(A, ·) is α-Hölder;

(2) β(A, M) ⊂ U;

(3) β satisfies the isolated closing conditions, there exists an α-Hölder function P : M → Γ with β(g, x) = P (gx)P (x)^{−1}.

(b) Let K ≥ 1 and Γ be a matrix Lie group. If ρ(B) and β(B, ·) are C^{K} for some Anosov ρ(B), then P is C^{K−1}. In particular, if ρ(B) and β(B, ·) are C^{∞}, then P is C^{∞}.

Proof. The first part follows from Livsic’s Theorem 1.2, and Theorems 5.2. The regularity follows from [N-T]. □

Theorem 5.6. (a) Let ρ : G × M → M be a C^{1} (GA)-action. Fix K ∈ ℕ, K ≥ 4 and let α ∈ (0, 1], G^{K}, d_{K} be as in Theorem 3.1. Then there is δ > 0 with the property that for any continuous cocycle β : G × M → G^{K} such that:

(1) β(A, ·) is α-Hölder;

(2) d_{K}(β(A, x)^{±1}, Id) < δ, for any x ∈ M;

(3) β satisfies the isolated closing conditions, there is an α-Hölder map P : M → G^{K−2} which trivializes β.

(b) If, in addition, ρ(B) and β(B, ·)(·) are C^{K−2} (the latter seen as a function M × U → U, respectively M × N → N) for some Anosov ρ(B), then P (·)(·) is C^{K−3}.

Proof. The first part follows from Theorems 3.1 and 5.2. The regularity follows from [N-T]. □

We give now an application of Theorem 5.4, which is a natural generalization of the Livsic-Sinai theorem (see [L-S]).

Theorem 5.7. Let ρ : G × M → M be a C^{2} (GA)-action. Then there is a Ginvariant measure on M which is equivalent to the Lebesgue measure if and only if for any x in M and g in G with gx = x, one has |J_{g}(x)| = 1. Here J_{g} is the Jacobian of g.

Proof. Choose a Riemannian metric on M. Let m be the measure induced by it. Using the Riemannian metric we can introduce the C^{1}-cocycle

(g, x) ∈ G × M → β(g, x) = log |J_{g}(x)| ∈ ℝ.

Assume that μ is a measure on M which is equivalent to m, dμ = ϕdm, where 0 < ϕ < ∞. Note that μ is g-invariant for some g ∈ G if and only if

ϕ(x)dm(x) = ϕ(gx)|J_{g}(x)|dm(x), a.e. with respect to m.  (5.1)

Let A ∈ G be the Anosov element given by the (GA)-property.

If μ is G-invariant, it is in particular A-invariant, hence by [L-S], ϕ is continuous. Then (5.1) implies the desired conditions for β.

Conversely, if the conditions are satisfied, β admits a continuous trivialization ψ, by Theorem 5.4. Then e^{ψ}dm is a G-invariant measure. □

Remark. Using a de la Llave, Marco, Moriyon result (see [L-M-M]) it follows that if ρ is C^{∞}, then so is the invariant measure.

Take now M = 𝕋^{n} and let the group G be a subgroup of SL(n, ℤ). Then G acts on 𝕋^{n} by diffeomorphisms. Many actions G × 𝕋^{n} → 𝕋^{n} are (GA)-actions. Here we prove only a typical example, namely for G = SL(n, ℤ).

Theorem 5.8. The standard linear action SL(n, ℤ) × 𝕋^{n} → 𝕋^{n} is (GA).

Proof. We define first an Anosov element A in SL(n, ℤ).

The following matrices are diagonalizable Anosov elements:

A_{1} = ( \begin{matrix}2 & 3 \\ 1 & 2 \end{matrix} ) , A_{2} = ( \begin{matrix}2 & 3 & 0 \\ 1 & 2 & −1 \\ −1 & −2 & 2 \end{matrix} ) .

If n is even, take A = diag(A_{1}, …, A_{1}), and if n is odd take A = diag(A_{1}, …, A_{1}, A_{2}). Then A is diagonalizable and there exists B, which has the same block-diagonal form as A, such that A = B · diag(λ_{1}, …, λ_{n}) · B^{−1}. Let E^{+} ⊕ E^{−} be the splitting of ℝ^{n} into expanding and contracting parts for diag(λ_{1}, …, λ_{n}). Denote by P_{±} the projection onto E^{±}. It is known that SL(n, ℤ) is generated by {E_{ij}|1 ≤ i, j ≤ n, i ≠ j}, E_{ij} being the matrix with 1 on the diagonal and in the position (i, j) and 0 everywhere else. Therefore it is enough to prove that P_{+} B^{−1}E_{ij}B|_{E^{+}} and P_{−} B^{−1}E_{ij}B|_{E^{−}} are invertible for any i ≠ j.

But

B^{−1}E_{ij}B = I + n ; ∑_{k,ℓ=1}\overline{b}_{ki}b_{jℓ}e_{kℓ} ,

where B = (b_{ij}), B^{−1} = (\overline{b}_{ij}) and \{e_{kℓ}\}_{k,ℓ} are the matrix units of Mat(n, ℝ). Because B and B^{−1} are block-diagonal, in the last sum nonzero terms can appear only in the block to which the position (i, j) belongs (see the n × n matrices as made of blocks, according to the pattern of A). Hence, unless (i, j) lies in one of the diagonal blocks, the matrix B^{−1}E_{ij}B is triangular, and therefore P_{±}B^{−1}E_{ij}B|_{E^{±}} are invertible. The few remaining cases can be checked numerically. □

Putting together Theorems 5.6 and 5.8, we obtain:

Theorem 5.9. (a) Let π : SL(n, ℤ) × 𝕋^{n} → 𝕋^{n} be the standard linear action, K ≥ 4 an integer and let β : SL(n, ℤ) × 𝕋^{n} → Diff^{K}(𝕋^{d}) be a continuous cocycle over π, close to the identity, which satisfies the isolated closing conditions. Assume that β(A, ·) is Hölder. Then there is a Hölder function P : 𝕋^{n} → Diff^{K−2}(𝕋^{d}) which trivializes β.

(b) If, in addition, there is B ∈ SL(n, ℤ) hyperbolic such that β(B, ·)(·) is C^{K−2} as a function 𝕋^{n} × 𝕋^{d} → 𝕋^{d} , then P is C^{K−3}. □

§6. A Version of the Hirsch-Pugh-Shub Structural Stability Theorem

In this section we prove a version of the Hirsch-Pugh-Shub structural stability theorem (Theorem 7.1 in [H-P-S]) about the leaf-wise conjugation between a partially hyperbolic diffeomorphism with a C^{1} neutral foliation and a perturbation of it. This result is necessary in order to obtain rigidity for ρ_{0}. As one will see in §7, we use the conjugation to reduce the problem to one about cocycles with values in the diffeomorphism groups. The fact that such a cocycle is Hölder in the d_{K}-metric can be easily obtained once it is known that the conjugation in the Hirsch-Pugh-Shub Theorem is Hölder, differentiable of certain degree along the neutral leaves and with all these derivatives Hölder. We were not able to derive the fact that the derivatives are Hölder from [H-P-S]. We restrict our attention to the case of a partially hyperbolic automorphism of the torus and prove the following theorem:

Theorem 6.1. Let A ∈ SL(n, ℤ), n ≥ 2, be a hyperbolic linear automorphism of 𝕋^{n}, let d ≥ 1 be an integer and let f be the linear automorphism of 𝕋^{n} × 𝕋^{d} given by f(x, y) = (Ax, y) for x ∈ 𝕋^{n} and y ∈ 𝕋^{d}. Let K ≥ 1 be an integer and let f̃ be a C^{K+Lip} diffeomorphism of 𝕋^{n} × 𝕋^{d}, which is C^{K+Lip} close to f. Then there are functions \overline{h} : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} and \overline{q} : 𝕋^{n} × 𝕋^{d} → 𝕋^{d} such that:

— \overline{h}(·, ·), \overline{q}(·, ·) are uniformly C^{K+Lip} in the second variable and both functions, together with their first K derivatives in the second variable, are Hölder;

— the function H : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} × 𝕋^{d} given by H(x, y) = (x + \overline{h}(x, y), y) is a homeomorphism;

— the function Q : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} × 𝕋^{d} given by Q(x, y) = (x, y + \overline{q}(x, y)) is a homeomorphism;

— the following equation holds: HQf = f̃H. (6.1)

Remark. H is a leaf-wise conjugacy between the neutral foliation of f and the neutral foliation of f̃.

Preliminaries. Write the perturbation f̃ as

f̃(x, y) = (Ax + \overline{α}(x, y), y + \overline{β}(x, y)), (x, y) ∈ 𝕋^{n} × 𝕋^{d},

where \overline{α} : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} and \overline{β} : 𝕋^{n} × 𝕋^{d} → 𝕋^{d} are C^{K+Lip} functions which are C^{K+Lip} close to the constant zero function. Then, decomposing the equation (6.1) in components along 𝕋^{n} and 𝕋^{d}, one obtains the equivalent system

\{ \begin{matrix}\overline{h}(Ax, \overline{q}(Ax, y) + y) = A\overline{h}(x, y) + \overline{α}(x + \overline{h}(x, y), y) \\ \overline{q}(Ax, y) = \overline{β}(x + \overline{h}(x, y), y),  (x, y) ∈ 𝕋^{n} × 𝕋^{d}. \end{matrix}  (6.2)

Substituting \overline{q} in the first equation of (6.2) yields an equation only in \overline{h}:

\overline{h}(Ax, \overline{β}(x + \overline{h}(x, y), y) + y) = A\overline{h}(x, y) + \overline{α}(x + \overline{h}(x, y), y).  (6.3)

We lift \overline{h} to a function h : ℝ^{n} ×𝕋^{d} → ℝ^{n} which is ℤ^{n}-periodic in the first variable (because \overline{h} is homotopic to the constant function zero). Since \overline{α} is close to zero, one can lift it to a function α : 𝕋^{n} × 𝕋^{d} → ℝ^{n}. Note that d_{K+Lip}(α) = d_{K+Lip}(\overline{α}). Let p be the canonical projection p : ℝ^{n} → 𝕋^{n}. Then we obtain the equation

\begin{matrix}Ah(x, y) − h(Ax, \overline{β}(p(x + h(x, y)), y) + y) = −α(p(x + h(x, y)), y), \\ (x, y) ∈ ℝ^{n} × 𝕋^{n} \end{matrix}  (6.4)

We can see equation (6.4) as a fixed point equation for the operator F : χ ↦ h given by:

\begin{matrix}A h(x, y) − h(Ax, \overline{β}(p(x + χ(x, y)), y) + y) = \\ = −α(p(x + χ(x, y)), y). \end{matrix}

One can give an explicit formula for h in terms of χ (see (6.6) below).

We will show that, on a suitable space of functions, F is a contraction. Therefore equation (6.4) has a solution.

Denote by C_{per}^{0}(ℝ^{n} × 𝕋^{d}, X) the space of continuous functions φ : ℝ^{n} × 𝕋^{d} → X which are ℤ^{n}-periodic in the first variable. By C_{per}^{K+Lip} we mean C^{K+Lip} ∩ C_{per}^{0}.

We have to study the operator F : χ ↦ h, which is determined by the functional equation

Ah − h ∘ F_{χ} = g_{χ},  (6.5)

where:

• A is a hyperbolic matrix in SL(n, ℤ);

• h, χ ∈ C_{per}^{0} (ℝ^{n} × 𝕋^{d}, ℝ^{n}) and are C^{K+Lip} in the second variable;

• α ∈ C^{K+Lip} (𝕋^{n} × 𝕋^{d}, ℝ^{n});

• \overline{β} ∈ C^{K+Lip} (𝕋^{n} × 𝕋^{d}, 𝕋^{d});

• F_{χ}(x, y) = (Ax, φ_{x}(y)) (x ∈ ℝ^{n} and y ∈ 𝕋^{d});

• φ_{x}(y) = y + \overline{β} ∘ S_{χ}(x, y) (hence φ_{x} ∈ C^{K+Lip}(𝕋^{d}));

• S_{χ}(x, y) = (p(x + χ(x, y)), y);

• g_{χ} = −α ∘ S_{χ}.

These notations will be valid throughout the rest of this paragraph.

Recall that M′_{K} and M_{K} where introduced in Proposition 2.9.

For t > 0, define the function

Δ(t) = (1 + t)^{K}[1 + (1 + K^{2})t].

Notation. Denote δ_{0} = max\{d_{K+Lip}(\overline{α}), d_{K+Lip}(\overline{β})\},

C_{Lip,ω} = sup_{v∈𝕋^{d}} \frac{∥v∥}{∥v∥^{ω}} .

Lemma 6.2. Let r > 0 and δ_{0} > 0 be constants. Let χ and \overline{β} be such that:

\begin{matrix}d_{K+Lip}(χ(x, ·)) ≤ r , \\ d_{K+Lip}(\overline{β}(x, ·)) ≤ δ_{0} \end{matrix}

for all x ∈ ℝ^{n}. Denote δ_{1} = δ_{0}Δ(r), δ_{2} = M′_{K}δ_{1} > δ_{1}.

Then

d_{K+Lip}(φ_{x}, Id) ≤ δ_{1},

for all x ∈ ℝ^{n}. Hence, by Proposition 2.9, if δ_{1} < \frac{1}{2}, then φ_{x} ∈ Diff^{K+Lip}(𝕋^{d}) and

d_{K+Lip}(φ_{x}^{−1}, Id) ≤ δ_{2}.

Proof. We derive first some estimates for S_{χ}(x, y) = (p(x + χ(x, y)), y).

d′′_{K+Lip}(S_{χ}(x, ·)) ≤ d′′_{K+Lip}(x + χ(x, ·)) + d′′_{K+Lip}(Id) = d′′_{K+Lip}(χ(x, ·)) ≤ r,
d′_{K+Lip}(S_{χ}(x, ·)) ≤ d′_{K+Lip}(x + χ(x, ·)) + d′_{K+Lip}(Id) =
= d′_{K+Lip}(χ(x, ·)) + 1 ≤ r + 1.

Then

d_{K+Lip}(φ_{x}, Id) = d_{K+Lip}(\overline{β} ∘ S_{χ}(x, ·)) ≤^{(Lip2)}

≤ d_{K+Lip}(β)[1 ∨ d′_{K+Lip}(S_{χ}(x, ·))]^{K}·

·[1 ∨ (d′_{K+Lip}(S_{χ}(x, ·)) + K^{2} · d′′_{K+Lip}(S_{χ}(x, ·)))] ≤

≤ δ_{0}(1 + r)^{K} · [1 + r + K^{2}r] = δ_{1}. □

Remark. In what follows, we always choose χ and β such that φ_{x} is invertible for all x ∈ ℝ^{m}. Then F_{χ} is invertible and

F_{χ}^{−1}(x, y) = (A^{−1}x, φ_{A^{−1}x}^{−1}(y)).

We decompose ℝ^{n} = E^{+} ⊕ E^{−} in expansive and contractive subspaces for A. With respect to this decomposition, A can be written as

A = ( \begin{matrix}A_{+} & 0 \\ 0 & A_{−} \end{matrix} )

where ∥(A_{+})^{−1}∥ < 1, ∥A_{−}∥ < 1. Denote λ = max{∥(A_{+})^{−1}∥, ∥A_{−}∥}.

If u is a function with values in ℝ^{n}, denote by either u_{+}, u_{−} or u^{+}, u^{−} the components of u with respect to the decomposition ℝ^{n} = E^{+} ⊕ E^{−}.

We introduce ω ∈ (0, 1) and 0 < λ_{1} < 1 such that

\begin{matrix}∥A∥^{ω}∥(A_{+})^{−1}∥ < λ_{1}, \\ ∥A^{−1}∥^{ω}∥A_{−}∥ < λ_{1}. \end{matrix}

The equation (6.5) can be solved explicitly in C^{0} as follows: h = F(χ) has the components

\{ \begin{matrix}h_{+} & = ∞ ; ∑_{k=0}A_{+}^{−k−1} · g_{χ}^{+} ∘ F_{χ}^{k}, \\ h_{−} & = − ∞ ; ∑_{k=0}A_{−}^{k}g_{χ}^{−} ∘ F_{χ}^{−k−1}. \end{matrix}  (6.6)

Since F_{χ} ∈ C_{per}^{0}(ℝ^{n}×𝕋^{d}, ℝ^{n}×𝕋^{d}) and g_{χ} are bounded, the series converge uniformly. Therefore F : C_{per}^{0}(ℝ^{n} × 𝕋^{d}, ℝ^{n}) → C_{per}^{0}(ℝ^{n} × 𝕋^{d}, ℝ^{n}) is well defined. In order to obtain the desired result, we consider the restriction of F to a subspace, where it is a contraction.

For C_{0} > 0 and r > 0 denote:

B_{K+Lip}(r) = {χ ∈ C_{per}^{0}(ℝ^{n} × 𝕋^{d}, ℝ^{n}) | χ(x, ·) ∈ C^{K+Lip}(𝕋^{d}, ℝ^{n}) and
d_{K+Lip}(χ(x, ·)) ≤ r, for all x ∈ ℝ^{n}},

B_{ω}(C_{0}) = {χ ∈ C_{per}^{0}(ℝ^{n} × 𝕋^{d}, ℝ^{n}) | χ(x, ·) ∈ C^{K}(𝕋^{d}, ℝ^{n}) for all x ∈ ℝ^{n},

and d_{K}(χ(x′, ·), χ(x′′, ·)) ≤ C_{0}∥x′ − x′′∥^{ω}, for all x′, x′′ ∈ ℝ^{n}},
X_{r,C_{0}} = B_{K+Lip}(r) ∩ B_{ω}(C_{0}).

On X_{r,C_{0}} we introduce the metric

d_{K}^{Y}(χ, χ′) := sup_{x∈ℝ^{n}}d_{K}(χ(x, ·), χ′(x, ·)) (≤ 2r).

We remark that X_{r,C_{0}} is complete with respect to this metric.

Theorem 6.1 follows from Propositions 6.3, 6.4 and 6.5 below, using the contraction principle.

Notation. For r > 0 and δ_{0} > 0 introduced above, let

δ_{1} := δ_{0}Δ(r),
δ_{2} := M′_{K}δ_{1} > δ_{1}.

Proposition 6.3. If r > 0 and δ_{0} > 0 are such that δ_{1} < \frac{1}{2} and λΔ(δ_{2}) < 1, then

F(B_{K+Lip}(r)) ⊂ B_{K+Lip} (δ_{0} \frac{\sqrt{2}Δ(r)Δ(δ_{2})}{1 − λΔ(δ_{2})}) .

Proposition 6.4. If r > 0 and δ_{0} > 0 are such that δ_{1} < \frac{1}{2} and λ_{1}Δ(δ_{2}) < 1, then

F(X_{r,C_{0}}) ⊂ B_{ω} (δ_{0} \frac{\sqrt{2}∥A_{−}∥^{−1}M_{K}C_{1}Δ(r)(1 + 4δ_{2})^{K}(K + 1)}{(1 − λ_{1}Δ(δ_{2}))^{2}}) ,

where C_{1} = (C_{Lip,ω} + C_{0})(K + 1)(1 + 3r)^{K}.

Proposition 6.5. Assume that r > 0 and δ_{0} > 0 are such that δ_{1} < \frac{1}{2} and λΔ(δ_{2}) < 1. Let χ, \overline{χ} ∈ B_{K+Lip}(r), and denote h = F(χ), \overline{h} = F(\overline{χ}). According to Proposition 6.3, h, \overline{h} ∈ B_{K+Lip}(r_{1}) for r_{1} = δ_{0} \frac{\sqrt{2}Δ(r)Δ(δ_{2})}{1−λΔ(δ_{2})}.

If λ(1 + δ_{2})^{K} < 1, then

d_{K}^{Y}(h, \overline{h}) ≤ δ_{0} \frac{C_{K}(δ_{0}, r, r_{1})}{1 − λ(1 + δ_{2})^{K}} · d_{K}^{Y}(χ, \overline{χ}),

where C_{K}(δ_{0}, r, r_{1}) is a polynomial in δ_{0}, r, r_{1}, which has positive coefficients depending on ∥A_{−}∥, ∥A_{+}^{−1}∥, M_{K} and M′_{K}. (The precise expression of C_{K}(δ_{0}, r, r_{1}) can be determined from Claims 2 and 3 in the proof of Proposition 6.5).

Proof of Theorem 6.1.

Let ϵ_{XP} > 0 be the expansiveness constant of the matrix A (i.e., for any x′, x′′ ∈ 𝕋^{n}, x′ ≠ x′′, there is an integer N such that d_{𝕋^{n}}(A^{N}x′, A^{N}x′′) > ϵ_{XP}).

Choose r > 0, C_{0} > 0 and δ_{0} > 0 such that:

r < \frac{ϵ_{XP}}{2} ,

δ_{1} < \frac{1}{2},

λ_{1}Δ(δ_{2}) < 1,

λ(1 + δ_{2})^{K} < 1,

δ_{0} \frac{\sqrt{2}Δ(r)Δ(δ_{2})}{1 − λΔ(δ_{2})} < r,

δ_{0} \frac{\sqrt{2}∥A_{−}∥^{−1}M_{K}C_{1}Δ(r)(1 + 4δ_{2})^{K}(K + 1)}{(1 − λ_{1}Δ(δ_{2}))^{2}} < C_{0},

δ_{0} \frac{C_{K}(δ_{0}, r, r_{1})}{1 − λ(1 + δ_{2})^{K}} < 1.

(Recall that r_{1}, C_{1} and C_{K} were introduced in Propositions 6.3–6.5, δ_{1} = δ_{0}Δ(r) and δ_{2} = M′_{K}δ_{1}.)

Then, by Propositions 6.3 and 6.4, X_{r,C_{0}} = B_{K+Lip}(r) ∩ B_{ω}(C_{0}) is F-invariant, and, by Proposition 6.5, F is a contraction on X_{r,C_{0}}, with respect to the metric d_{K}^{Y}. Therefore, by the contraction principle, F has a unique fixed point h in X_{r,C_{0}}.

This h, being in C_{per}^{0}(ℝ^{n}×𝕋^{d}, ℝ^{n}), gives the desired function \overline{h} ∈ C^{0}(𝕋^{n}×𝕋^{d}, 𝕋^{n}). Then we can construct \overline{q} using the second formula in (6.2):

\overline{q}(x, y) = \overline{β}(A^{−1}x + \overline{h}(A^{−1}x, y), y).

We obtain a solution for the system (6.2), hence a solution of (6.1).

By a similar computation to that done in Claim 1 of Proposition 6.3, we obtain that \overline{q} ∈ B_{K+Lip}(δ_{0}Δ(r)). Adapting the computations of Claim 2 of Proposition 6.4, we obtain that \overline{q} ∈ B_{ω}(δ_{0}C_{1}∥A∥^{−ω}).

It remains to show that H and Q are homeomorphisms. Clearly H and Q are continuous and close to the identity. Then surjectivity follows using the topological degree, and it remains to prove that they are one-to-one. We first prove that Q is one-to-one. Consider two points (x′, y′), (x′′, y′′) ∈ 𝕋^{n} × 𝕋^{d} such that Q(x′, y′) = Q(x′′, y′′). Then x′ = x′′ and y′ + \overline{q}(x′, y′) = y′′ + \overline{q}(x′, y′′). Since the function y ↦ y + \overline{q}(x′, y) is C^{K+Lip} and C^{K+Lip}-close to the identity, it follows from Proposition 2.9 (a) that y′ = y′′. Hence Q is a homeomorphism.

Consider now (x′, y′), (x′′, y′′) ∈ 𝕋^{n} × 𝕋^{d} such that H(x′, y′) = H(x′′, y′′). Then clearly y′ = y′′. Assume by contradiction that x′ ≠ x′′. Then there exists an integer N such that d_{𝕋^{n}}(A^{N}x′, A^{N}x′′) > ϵ_{XP}. If N ≥ 0, one has:

H(Qf)^{N}(x′, y′) =^{(6.1)} f̃^{N}H(x′, y′) = f̃^{N}H(x′′, y′′) =^{(6.1)} H(Qf)^{N}(x′′, y′′).

Hence, introducing y_{1}, y_{2} ∈ 𝕋^{d} such that:

(Qf)^{N}(x′, y′) = (A^{N}x′, y_{1}),
(Qf)^{N}(x′′, y′′) = (A^{N}x′′, y_{2})

we obtain

A^{N}x′ + \overline{h}(a^{N}x′, y_{1}) = A^{N}x′′ + \overline{h}(a^{N}x′′, y_{2})

and therefore

ϵ_{XP} < d_{𝕋^{n}}(A^{N}x′, A^{N}x′′) ≤ 2d_{0}(\overline{h}) < 2r,

which contradicts our initial choice of r.

If N < 0, then

H(f^{−1}Q^{−1})^{−N}(x′, y′) =^{(6.1)} f̃^{−N}H(x′, y′) =

= f̃^{−N}H(x′′, y′′) =^{(6.1)} H(f^{−1}Q^{−1})^{−N}(x′′, y′′),

and now proceed as in the previous case. □

Notation. Denote, for s ∈ ℤ, F_{χ}^{s}(x, y) = (A^{s}x, ϕ_{x}^{s}(y)). Hence

ϕ_{x}^{s} = \{ \begin{matrix}φ_{A^{s−1}x} φ_{A^{s−2}x} ⋯ φ_{x} & if s > 0 \\ Id_{𝕋^{d}} & if s = 0 \\ φ_{A^{s}x}^{−1} φ_{A^{s+1}x}^{−1} ⋯ φ_{A^{−1}x}^{−1} & if s < 0 \end{matrix}

and the following recurrence relation holds:

ϕ_{x}^{s} = \{ \begin{matrix}ϕ_{Ax}^{s−1} φ_{x} & if s > 0 \\ ϕ_{A^{−1}x}^{s+1} φ_{A^{−1}x}^{−1} & if s < 0. \end{matrix}

Proof of Proposition 6.3.

Claim 1. d_{K+Lip}(g_{χ}(x, ·)) ≤ δ_{0}Δ(r), for all x ∈ ℝ^{n}.

Proof. Indeed, d_{K+Lip}(g_{χ}(x, ·)) = d_{K+Lip}(α ∘ S_{χ}(x, ·)) ≤

≤^{(Lip2)} d_{K+Lip}(α)[1 ∨ d′_{K+Lip}(S_{χ}(x, ·)]^{K}·

·[1 ∨ (d′_{K+Lip}(S_{χ}(x, ·)) + K^{2}d′′_{K+Lip}(S_{χ}(x, ·)))] ≤

≤ δ_{0} · (1 + r)^{K}[1 ∨ (1 + r + K^{2}r)] = δ_{0}Δ(r),

where in the last inequality we use estimates from Lemma 6.2.

Claim 2. d_{K+Lip}(g_{χ} ∘ F_{χ}^{s}(x, ·)) ≤ δ_{0}Δ(r)Δ(δ_{∗})^{|s|}, for any x ∈ ℝ^{n} and any s ∈ ℤ, where

δ_{∗} = \{ \begin{matrix}δ_{1}, & if s > 0 \\ δ_{2} = M′_{K}δ_{1}, & if s < 0. \end{matrix}

Proof. Indeed, if s > 0, then

d_{K+Lip}(g_{χ} ∘ F_{χ}^{s}(x, ·)) = d_{K+Lip}(g_{χ}(A^{s}x, ϕ_{x}^{s}(·))) =

= d_{K+Lip}(g_{χ}(A^{s}x, ϕ_{Ax}^{s−1}ϕ_{x}(·))) ≤

≤^{(Lip3)} d_{K+Lip}(g_{χ}(A^{s}x, ϕ_{Ax}^{s−1}(·)) · [1 + d_{K+Lip}(φ_{x}, Id)]^{K}·

·[1 + (K^{2} + 1)d_{K+Lip}(φ_{x}, Id)] ≤

≤ d_{K+Lip}(g_{χ}(A^{s}x, ϕ_{Ax}^{s−1}(·))) · (1 + δ_{1})^{K}[1 + (K^{2} + 1)δ_{1}] =

= Δ(δ_{1})d_{K+Lip}(g_{χ} ∘ F_{χ}^{s−1}(Ax, ·)).

Iterate and then use Claim 1 to obtain the desired estimate.

The case s < 0 is dealt with similarly.

Claim 3. If h = F(χ), then:

d_{K+Lip}^{Y}(h_{+}) ≤ δ_{0} · \frac{Δ(r) · ∥A_{+}^{−1}∥}{1 − ∥A_{+}^{−1}∥Δ(δ_{1})},
d_{K+Lip}^{Y}(h_{−}) ≤ δ_{0} · \frac{Δ(r)Δ(δ_{2})}{1 − ∥A_{−}∥Δ(δ_{2})}.

Proof. Use formula (6.6), and the previous two claims. □

Proof of Proposition 6.4. Since λ < λ_{1}, the hypothesis implies that λΔ(δ_{2}) < 1, hence the conclusion of

Proposition 6.3 holds.

Consider χ ∈ X_{r,C_{0}}. Claim 1. For any x′, x′′ ∈ ℝ^{n}, one has

d_{K}(φ_{x′}, φ_{x′′}) ≤ δ_{0}C_{1} · ∥x′ − x′′∥^{ω},

where C_{1} = (C_{Lip,ω} + C_{0})(K + 1)(1 + 3r)^{K} and (conform Proposition 2.9 (c), (ii))

d_{K}(φ_{x′}^{−1}, φ_{x′′}^{−1}) ≤ δ_{0}C_{1} · M_{K} · ∥x′ − x′′∥^{ω}.

Proof. Using the periodicity of x ↦ φ_{x}, one has:

d_{K}(φ_{x′}, φ_{x′′}) = d_{K}(β ∘ S_{χ}(x′), β ∘ S_{χ}(x′′)) ≤  (6.7)

≤^{(L1)} d_{K}(S_{χ}(x′, ·), S_{χ}(x′′, ·)) · (K + 1) · d′_{K+Lip}(β)·

·{1 ∨ [d′_{K}(S_{χ}(x′, ·)) ∨ d′_{K}(S_{χ}(x′′, ·)) + d′′_{K}(S_{χ}(x′, ·)) + d′′_{K}(S_{χ}(x′′, ·))]}^{K} ≤

≤ ∥p(x′ − x′′)∥ + d_{K}(pχ(x′, ·) , pχ(x′′, ·)) · δ_{0}(K + 1)(1 + 3r)^{K} ≤
≤ ∥x′ − x′′∥^{ω} · (C_{Lip,ω} + C_{0})δ_{0}(K + 1)(1 + 3r)^{K}.

Claim 2. For any x′, x′′ ∈ ℝ^{n}, one has

d_{K}(g_{χ}(x′, ·), g_{χ}(x′′, ·)) ≤ δ_{0}C_{1}∥x′ − x′′∥^{ω}.

Proof. d_{K}(g_{χ}(x′, ·), g_{χ}(x′′, ·)) = d_{K}(α ∘ S_{χ}(x′) , α ∘ S_{χ}(x′′)), which is exactly (6.7) in the proof of Claim 1, written with α instead of β. We can use the same estimates.

Claim 3. For s > 0 and x′, x′′ ∈ ℝ^{n}, one has

\begin{matrix}d_{K}(g_{χ} ∘ F_{χ}^{s}(x′, ·), g_{χ} ∘ F_{χ}^{s}(x′′, ·)) ≤ \\ ≤ δ_{0}C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)(s + 1)∥A∥^{ωs} · Δ(δ_{1})^{s−1}∥x′ − x′′∥^{ω}. \end{matrix}

Proof.

d_{K}(g_{χ}F_{χ}^{s}(x′, ·), g_{χ}F_{χ}^{s}(x′′, ·)) =

= d_{K}(g_{χ}(A^{s}x′, ϕ_{x′}^{s}(·)), g_{χ}(A^{s}x′′, ϕ_{x′′}^{s}(·))) ≤

≤ d_{K}(g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1} ∘ φ_{x′}(·)), g_{χ}(A^{s}x′′, ϕ_{Ax′′}^{s−1} ∘ φ_{x′′}(·))) ≤
≤ d_{K}(g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1} ∘ φ_{x′}(·)), g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1} ∘ φ_{x′′}(·)))+
+d_{K}(g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1} ∘ φ_{x′′}(·)), g_{χ}(A^{s}x′′, ϕ_{Ax′′}^{s−1} ∘ φ_{x′′}(·))) ≤

≤^{(L3),(R3)} d_{K}(φ_{x′}, φ_{x′′})d′_{K+Lip}(g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1}(·)))·

·(K + 1)[1 + 2d′_{K}(φ_{x′}, Id) + 2d′_{K}(φ_{x′′}, Id)]^{K}+

+d_{K}(g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1}(·)), g_{χ}(A^{s}x′′, ϕ_{Ax′′}^{s−1}(·)))[1 + d_{K}(φ_{x′′}, Id)]^{K} ≤

≤^{(∗)} δ_{0}C_{1}∥x′ − x′′∥^{ω} · δ_{0} · Δ(r)Δ(δ_{1})^{s−1}(1 + 4δ_{1})^{K}(K + 1)+

+d_{K}(g_{χ}F_{χ}^{s−1}(Ax′, ·), g_{χ}F_{χ}^{s−1}(Ax′′, ·)) · (1 + δ_{1})^{K}.

For the inequality (∗), note that g_{χ}(A^{s}x′, ϕ_{Ax′}^{s−1}(·)) = g_{χ} ∘ F_{χ}^{s−1}(Ax′, ·), and use also Claim 2 from the proof of Proposition 6.3.

We iterate now the recurrence relation obtained above and obtain

d_{K}(g_{χ}F_{χ}^{s}(x′, ·), g_{χ}F_{χ}^{s}(x′′, ·)) ≤

≤ s−1 ; ∑_{ℓ=0}δ_{0}C_{1}∥A^{ℓ}x′ − A^{ℓ}x′′∥^{ω} · δ_{0}Δ(r) · Δ(δ_{1})^{s−ℓ−1} · (K + 1)(1 + δ_{1})^{Kℓ}(1 + 4δ_{1})^{K}+

+δ_{K}(g_{χ}(A^{s}x′, ·), g_{χ}(A^{s}x′′, ·))(1 + δ_{1})^{Ks} ≤

≤^{Claim 2} s−1 ; ∑_{ℓ=0}∥A^{ℓ}x′ − A^{ℓ}x′′∥^{ω} · δ_{0}^{2} · C_{1} · Δ(r)(K + 1) · (1 + 4δ_{1})^{K}·

·{(1 + δ_{1})^{K}[1 + (1 + K^{2})δ_{1}]}^{s−ℓ−1}(1 + δ_{1})^{Kℓ}+
+∥A^{s}x′ − A^{s}x′′∥^{ω} · δ_{0} · C_{1}(1 + δ_{1})^{Ks} ≤

≤^{Δ(r)>1,δ_{0}≤1} [ s ; ∑_{ℓ=0}∥A^{ℓ}x′ − A^{ℓ}x′′∥^{ω}] δ_{0}C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)·

42 VIOREL NIŢICĂ AND ANDREI TÖRÖK

·(1 + δ_{1})^{K(s−1)}[1 + (1 + K^{2})δ_{1}]^{s−1}∥x′ − x′′∥^{ω} ≤

≤^{∥A∥≥1} δ_{0}C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)(s + 1)∥A∥^{ωs} · {(1 + δ_{1})^{K}[1 + (1 + K^{2})δ_{1}]}^{s−1}·

·∥x′ − x′′∥^{ω} =

= δ_{0}C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)(s + 1)∥A∥^{ωs} · Δ(δ_{1})^{s−1}∥x′ − x′′∥^{ω}.

Claim 4. For s < 0 and x′, x′′ ∈ ℝ^{n}, one has

d_{K}(g_{χ}F_{χ}^{s}(x′, ·)), g_{χ}F_{χ}^{s}(x′′, ·)) ≤

≤ δ_{0} · M_{K} · C_{1} · Δ(r)(1 + 4δ_{2})^{K}(K + 1)(|s| + 1)∥A^{−1}∥^{ω|s|} · Δ(δ_{2})^{|s|} · ∥x′ − x′′∥^{ω}. Proof. As in Claim 3, one has the recurrence

d_{K}(g_{χ}F_{χ}^{s}(x′, ·), g_{χ}F_{χ}^{s}(x′′, ·)) ≤

≤ δ_{0}C_{1}M_{K}∥A^{−1}x′ − A^{−1}x′′∥^{ω}δ_{0}Δ(r)Δ(δ_{2})^{−s−1}(K + 1)(1 + 4δ_{2})^{K}+
+d_{K}(g_{χ}F_{χ}^{s+1}(A^{−1}x′, ·), g_{χ}F_{χ}^{s+1}(A^{−1}x′′, ·)) · (1 + δ_{2})^{K}.

Hence

d_{K}(g_{χ}F_{χ}^{s}(x′, ·), g_{χ}F_{χ}^{s}(x′′, ·)) ≤

≤ −s ; ∑_{ℓ=1}δ_{0}C_{1}M_{K}∥A^{−ℓ}x′ − A^{−ℓ}x′′∥δ_{0}Δ(r)Δ(δ_{2})^{−s−ℓ}(1 + δ_{2})^{Kℓ}·

·(K + 1)(1 + 4δ_{2})^{K} + d_{K}(g_{χ}(A^{−s}x′, ·), g_{χ}(A^{−s}x′′, ·))(1 + δ_{2})^{K(−s)} ≤

≤ −s ; ∑_{ℓ=1}∥A^{−ℓ}x′ − A^{−ℓ}x′′∥^{ω} · δ_{0}^{2} · C_{1} · M_{K}(K + 1)(1 + 4δ_{2})^{K} · (1 + δ_{2})^{K(−s)}·

·[1 + (1 + K^{2})δ_{2}]^{−s−ℓ} + ∥A^{−s}x′ − A^{−s}x′′∥^{ω} · δ_{0} · C_{1} · (1 + δ_{2})^{K(−s)} ≤ ≤ δ_{0}M_{K}C_{1}Δ(r)(1 + 4δ_{2})^{K}(K + 1)(|s| + 1)∥A^{−1}∥^{ω(−s)}Δ(δ_{2})^{−s}∥x′ − x′′∥^{ω}. Claim 5. Let h = F(χ). Then, for any x′, x′′ ∈ ℝ^{n}:

d_{K}(h_{+}(x′, ·), h_{+}(x′′, ·)) ≤

≤ δ_{0} · \frac{∥A_{+}^{−1}∥C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)}{(1 − ∥A_{+}^{−1}∥ · ∥A∥^{ω} · Δ(δ_{1}))^{2}} · ∥x′ − x′′∥^{ω},

d_{K}(h_{−}(x′, ·), h_{−}(x′′, ·)) ≤

≤ δ_{0} · \frac{∥A_{−}∥^{−1} · M_{K} · C_{1} · Δ(r)(1 + 4δ_{2})^{K}(K + 1)}{(1 − ∥A_{−}∥ ∥A^{−1}∥^{ω}Δ(δ_{2}))^{2}} · ∥x′ − x′′∥^{ω}.

Proof. We use relation (6.6):

d_{K}(h_{+}(x′, ·), h_{+}(x′′, ·)) ≤ ∞ ; ∑_{k=0}∥A_{+}^{−k−1}∥d_{K}(g_{χ}F_{χ}^{K}(x′, ·), g_{χ}F_{χ}^{K}(x′′, ·)) ≤^{Claim 2, Claim 3}

≤ ( ∞ ; ∑_{k=0}∥A_{+}^{−}∥^{k+1}δ_{0}C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)(k + 1)∥A∥^{ωk}Δ(δ_{1})^{k}) ∥x′ − x′′∥^{ω} ≤

≤ δ_{0}∥A_{+}^{−1}∥C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)∥x′ − x′′∥^{ω} ∞ ; ∑_{k=0}(k + 1)(∥A_{+}^{−1}∥∥A∥^{ω}Δ(δ_{1}))^{k} =

= δ_{0} · \frac{∥A_{+}^{−1}∥ · C_{1}Δ(r)(1 + 4δ_{1})^{K}(K + 1)}{(1 − ∥A_{+}^{−1}∥ · ∥A∥^{ω}Δ(δ_{1}))^{2}} · ∥x′ − x′′∥,

and

d_{K}(h_{−}(x′, ·), h_{−}(x′′, ·)) ≤ ∞ ; ∑_{k=0}∥A_{−}^{k}∥d_{K}(g_{χ}F_{χ}^{−k−1}(x′, ·), g_{χ}F_{χ}^{−k−1}(x′′, ·)) ≤

≤^{Claim 4} ( ∞ ; ∑_{k=0}∥A_{−}∥^{k} · δ_{0} · M_{K} · C_{1}Δ(r) · (1 + 4δ_{2})^{K}(K + 1)(k + 2)·

·∥A^{−1}∥^{ω(k+1)} · Δ(δ_{2})^{k+1}) ∥x′ − x′′∥^{ω} =

= δ_{0}∥A_{−}∥^{−1} · M_{K} · C_{1} · Δ(r)(1 + 4δ_{2})^{K}(K + 1) · ∥x′ − x′′∥^{ω}·

· ∞ ; ∑_{k=0}(k + 2)(∥A_{−}∥ · ∥A^{−1}∥^{ω} · Δ(δ_{2}))^{k+1} <

< δ_{0} · \frac{∥A_{−}∥^{−1}M_{K}C_{1}Δ(r)(1 + 4δ_{2})^{K}(K + 1)}{(1 − ∥A_{−}∥ · ∥A^{−1}∥^{ω}Δ(δ_{2}))^{2}}∥x′ − x′′∥^{ω}. □

Proof of Proposition 6.5.

Denote the “φ_{x}” corresponding to \overline{χ} by \overline{φ}_{x}.

Claim 1. One has:

d_{K+Lip}(φ_{x}, Id) ≤ δ_{1},
d_{K+Lip}(\overline{φ}_{x}, Id) ≤ δ_{1},
d_{K}(φ_{x}, \overline{φ}_{x}) ≤ δ_{0}(K + 1)(1 + 4r)^{K} d_{K}(χ(x, ·), \overline{χ}(x, ·)),

therefore, applying Proposition 2.9, it follows that

d_{K}(φ_{x}^{−1}, Id) ≤ M′_{K}δ_{1}, and similarly for \overline{φ}_{x}^{−1},
d_{K}(φ_{x}^{−1}, \overline{φ}_{x}^{−1}) ≤ M_{K}δ_{0}(K + 1)(1 + 4r)^{K}d_{K}(χ(x, ·), \overline{χ}(x, ·)).

Proof. The first two inequalities follow from Lemma 6.2. We prove the third

44 VIOREL NIŢICĂ AND ANDREI TÖRÖK

inequality

d_{K}(φ_{x}, \overline{φ}_{x}) = d_{K}(β ∘ S_{χ}(x, ·) , β ∘ S\overline{χ}(x, ·)) ≤^{(L3)}

≤ d_{K}(S_{χ}(x, ·), S\overline{χ}(x, ·))d′_{K+Lip}(β)(K + 1)·

· [1 + 2d′_{K}(S_{χ}(x, ·), Id) + 2d′_{K}(S\overline{χ}(x, ·), Id)]^{K} ≤

≤ d_{K}(χ(x, ·), \overline{χ}(x, ·))δ_{0}(K + 1)(1 + 4r)^{K}.

Claim 2.

d_{K}^{Y}(h_{+}, \overline{h}_{+}) ≤

≤ \frac{δ_{0}∥A_{+}^{−1}∥(1 + 4r)^{K}(K + 1)[(K + 1)r_{1}(1 + 4δ_{1})^{K} + 1]}{1 − ∥A_{+}^{−1}∥(1 + δ_{1})^{K}} · d_{K}^{Y}(χ, \overline{χ}).

Proof. From equation (6.5) one has

h_{+} − \overline{h}_{+} = A_{+}^{−1}(h_{+}F_{χ} − \overline{h}_{+}F\overline{χ}) + A_{+}^{−1}(g_{χ}^{+} − g+ ; \overline{χ}),

therefore

d_{K}^{Y}(h_{+}, \overline{h}_{+}) ≤ ∥A_{+}^{−1}∥[d_{K}^{Y}(h_{+} F_{χ}, \overline{h}_{+} F\overline{χ}) + d_{K}^{Y}(g_{χ}, g\overline{χ})].

The desired estimates follow from the inequalities (6.8) – (6.11) below:

d_{K}^{Y}(h_{+} F_{χ}, \overline{h}_{+} F\overline{χ}) ≤ d_{K}^{Y}(h_{+} F_{χ}, h_{+} F\overline{χ}) + d_{K}^{Y}(h_{+} F\overline{χ}, \overline{h}_{+} F\overline{χ}),  (6.8)
\begin{matrix}d_{K}(g_{χ}(x, ·), \overline{g}_{χ}(x, ·)) = d_{K}(α ∘ S_{χ}(x, ·), α ∘ S\overline{χ}(x, ·)) ≤^{(L3)} \\ ≤ d_{K}(S_{χ}(x, ·), S\overline{χ}(x, ·))(K + 1)d′_{K+Lip}(α)· \\ · [1 + 2d′_{K}(S_{χ}(x, ·), Id) + 2d′_{K}(S\overline{χ}(x, ·), Id)]^{K} ≤ \\ ≤ d_{K}(χ(x, ·), \overline{χ}(x, ·))(K + 1) δ_{0}(1 + 4r)^{K}, \end{matrix}  (6.9)

d_{K}(h_{+} F_{χ}(x, ·), h_{+} F\overline{χ}(x, ·) = d_{K}(h_{+}(Ax, φ_{x}(·)) , h_{+}(Ax, \overline{φ}_{x}(·))) ≤

≤^{(L3)} d_{K}(φ_{x}(·), \overline{φ}_{x}(·)) d′_{K+Lip}(h_{+}(Ax, ·))(K + 1)·

·[1 + 2d′_{K}(φ_{x}, Id) + 2d′_{K}(\overline{φ}_{x}, Id)]^{K} ≤

≤^{Claim 1} d_{K}^{Y}(χ, \overline{χ}) δ_{0} · (K + 1)^{2}(1 + 4r)^{K} · r_{1}(1 + 4δ_{1})^{K},

(6.10)

\begin{matrix}d′_{K}(F\overline{χ}(x, ·)) ≤ 1 + d′_{K}(\overline{φ}_{x}, Id) ≤ 1 + δ_{1}, \\ d_{K}(h_{+}F\overline{χ}(x, ·), \overline{h}_{+}F\overline{χ}(x, ·)) ≤^{(R3)} d_{K}^{Y}(h_{+}, \overline{h}_{+})[max\{1, d′_{K}(F\overline{χ}(x, ·))\}]^{K} ≤ \\ ≤ d_{K}^{Y}(h_{+}, \overline{h}_{+}) · (1 + δ_{1})^{K}. \end{matrix} (6.11)

Claim 3.

d_{K}^{Y}(h_{−}, \overline{h}_{−}) ≤ \frac{δ_{0}d_{K}^{Y}(χ, \overline{χ})}{1 − ∥A_{−}∥(1 + M′_{K}δ_{1})^{K}} ·

· {∥A_{−}∥ · M_{K} δ_{0}(K + 1)^{2}r_{1}(1 + 4r)^{K} (1 + 4M′_{K}δ_{1})^{K} + C_{∞}} , where

C_{∞} = (K + 1) · (1 + 4r)^{K} · [M_{K}δ_{0} · (K + 1) (1 + 4r)^{K}·
· {1 + r(K + 1)[1 + 4M′_{K}δ_{1}]^{K}} · [1 + 4(M′_{K}δ_{1} + r(1 + δ_{1})^{K})]+
+(1 + M′_{K}δ_{1})^{K}]

Proof. From equality (6.5) one has

h_{−} − \overline{h}_{−} = A_{−}(h_{−}F_{χ}^{−1} − \overline{h}_{−}F −1 ; \overline{χ}) + g_{χ}^{−}F_{χ}^{−1} − g− ; \overline{χ}F −1 ; \overline{χ}.

Therefore

d_{K}^{Y}(h_{−}, \overline{h}_{−}) ≤ ∥A_{−}∥ d_{K}^{Y}(h_{−} F_{χ}^{−1}, \overline{h}_{−} F −1 ; \overline{χ}) + d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g\overline{χ}F −1 ; \overline{χ}).

Since

d_{K}^{Y}(h_{−} F_{χ}^{−1}, \overline{h}_{−} F −1 ; \overline{χ}) ≤ d_{K}^{Y}(h_{−} F_{χ}^{−1}, h_{−} F −1 ; \overline{χ}) + d_{K}^{Y}(h_{−} F −1 ; \overline{χ}, \overline{h}_{−} F −1 ; \overline{χ}),

the statement follows if we have estimates in terms of d_{K}^{Y}(χ, \overline{χ}) and d_{K}^{Y}(h_{−}, \overline{h}_{−})

for d_{K}^{Y}(h_{−} F_{χ}^{−1}, h_{−} F −1 ; \overline{χ}), d_{K}^{Y}(h_{−} F −1 ; \overline{χ}, \overline{h}_{−} F −1 ; \overline{χ}) and d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g\overline{χ}F −1 ; \overline{χ}). These are

given below in (6.12), (6.13) and (6.19).

Recall first that

F_{χ}^{−1}(x, ·) = (A^{−1}x, φ_{A^{−1}x}^{−1}(·)).

Then

d_{K}(h_{−} F_{χ}^{−1}(x, ·), h_{−} F −1 ; \overline{χ}(x, ·)) ≤^{(L3)} d_{K}^{Y}(F_{χ}^{−1}, F −1 ; \overline{χ})d′_{K+Lip}(h_{−}(A^{−1}x, ·))·

·(K + 1)[1 + 2d′_{K}(F_{χ}^{−1}(x, ·), Id) + 2d′_{K}(F −1 ; \overline{χ}(x, ·), Id)]^{K}.

Using now Claim 1, one has:

\begin{matrix}d_{K}(h_{−} F_{χ}^{−1}, h_{−} F −1 ; \overline{χ}) ≤ d_{K}^{Y}(χ, \overline{χ})M_{K}δ_{0}(K + 1)^{2}(1 + 4r)^{K}r_{1}· \\ ·(1 + 4M′_{K}δ_{1})^{K}, \end{matrix}  (6.12)

\begin{matrix}d_{K}^{Y}(h_{−} F −1 ; \overline{χ}(x, ·), \overline{h}_{−} F −1 ; \overline{χ}(x, ·)) ≤^{(R3)} d_{K}^{Y}(h_{−}, \overline{h}_{−}) · [max\{1, d′_{K}(F −1 ; \overline{χ}(x, ·))\}]^{K} ≤ \\ ≤ d_{K}^{Y}(h_{−}, \overline{h}_{−})(1 + M′_{K}δ_{1})^{K}. \end{matrix}

(6.13)

To estimate d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g\overline{χ}F −1 ; \overline{χ}), we divide it into two parts:

d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g\overline{χ}F −1 ; \overline{χ}) ≤ d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g_{χ}F −1 ; \overline{χ}) + d_{K}^{Y}(g_{χ}F −1 ; \overline{χ}, g\overline{χ}F −1 ; \overline{χ}).

The two terms of the RHS are estimated by (6.17) and (6.18) below.

To obtain (6.17), we proceed as follows:

\begin{matrix}d_{K}(g_{χ}F_{χ}^{−1}(x, ·), g_{χ}F −1 ; \overline{χ}(x, ·)) = d_{K}(α S_{χ}F_{χ}^{−1}(x, ·), αS_{χ}F −1 ; \overline{χ}(x, ·)) ≤ \\ ≤^{(L3)} d_{K}(S_{χ}F_{χ}^{−1}(x, ·), S_{χ}F −1 ; \overline{χ}(x, ·))(K + 1)d′_{K+Lip}(α)· \\ [1 + 2d′_{K}(S_{χ}F_{χ}^{−1}(x, ·), Id) + 2d′_{K}(S_{χ}F −1 ; \overline{χ}(x, ·), Id)]^{K}. \end{matrix}  (6.14)

Now

\begin{matrix}d′_{K}(S_{χ}F_{χ}^{−1}(x, ·), Id) ≤ d′_{K}(φ_{A^{−1}x}^{−1}(·), Id) + d′_{K}(χ(A^{−1}x, φ_{A^{−1}x}^{−1}(·))) ≤ \\ ≤^{(R1)} M′_{K} · δ_{1} + d′_{K}(χ(A^{−1}x, ·))[max\{1, d′_{K}(φ_{A^{−1}x}^{−1}(·))\}]^{K} ≤ \\ ≤ M′_{K}δ_{1} + r(1 + M′_{K}δ_{1})^{K}, \end{matrix} (6.15)

and a similar estimate holds for d_{K}(S_{χ}F −1 ; \overline{χ}(x, ·), Id). Also,

d_{K}(S_{χ}F_{χ}^{−1}(x, ·), S_{χ}F −1 ; \overline{χ}(x, ·)) ≤

≤ d_{K}(φ_{A^{−1}x}^{−1}(·), \overline{φ}_{A^{−1}x}^{−1}(·)) + d_{K}(χ(A^{−1}x, φ_{A^{−1}x}^{−1}(·)), χ(A^{−1}x, \overline{φ}_{A^{−1}x}^{−1}(·))) ≤

≤^{(L3)} d_{K}(φ_{A^{−1}x}^{−1}(·), \overline{φ}_{A^{−1}x}^{−1}(·))+

+d_{K}(φ_{A^{−1}x}^{−1}(·), \overline{φ}_{A^{−1}x}^{−1}(·))d′_{K+Lip}(χ(A^{−1}x, ·))(K + 1)·

·[1 + 2d′_{K}(φ_{A^{−1}x}^{−1}(·), Id) + 2d′_{K}(\overline{φ}_{A^{−1}x}^{−1}(·), Id)]^{K} ≤

≤^{Claim 1} M_{K} · δ_{0}(K + 1)(1 + 4r)^{K}d_{K}^{Y}(χ, \overline{χ})\{1 + r(K + 1)[1 + 4M′_{K}δ_{1}]^{K}\}.

(6.16)

Therefore, by (6.14), (6.15) and (6.16),

\begin{matrix}d_{K}^{Y}(g_{χ}F_{χ}^{−1}, g_{χ}F −1 ; \overline{χ}) ≤ \\ ≤ M_{K} δ_{0}^{2}(K + 1)^{2}(1 + 4r)^{K}\{1 + r(K + 1)[1 + 4M′_{K}δ_{1}]^{K}\}· \\ ·[1 + 4(M′_{K}δ_{1} + r(1 + M′_{K}δ_{1})^{K})]^{K} · d_{K}^{Y}(χ, \overline{χ}). \end{matrix}  (6.17)

For the second part:

\begin{matrix}d_{K}(g_{χ}F −1 ; \overline{χ}(x, ·), g\overline{χ}F −1 ; \overline{χ}(x, ·)) ≤^{(R3)} d_{K}(g_{χ}(A^{−1}x, ·), g\overline{χ}(A^{−1}x, ·))· \\ ·[max\{1, d′_{K}(\overline{φ}_{A^{−1}x}^{−1}(·))]^{K} ≤^{(6.9)} d_{K}^{Y}(χ, \overline{χ})(K + 1)δ_{0}· \\ ·(1 + 4r)^{K}(1 + M′_{K}δ_{1})^{K}. \end{matrix}  (6.18)

From (6.17) and (6.18) it follows that

\begin{matrix}d_{K}^{Y}(g_{χ}F_{χ}^{−1}g\overline{χ}F −1 ; \overline{χ}) ≤ \\ ≤ d_{K}^{Y}(χ, \overline{χ})(K + 1)δ_{0}(1 + 4r)^{K}[M_{K}δ_{0}(K + 1)(1 + 4r)^{K}· \\ ·\{1 + r(K + 1)[1 + 4M′_{K}δ_{1}]^{K}\}[1 + 4(M′_{K}δ_{1} + r(1 + M′_{K}δ_{1})^{K})]^{K}+ \\ +(1 + M′_{K}δ_{1})^{K}] = d_{K}^{Y}(χ, \overline{χ})δ_{0}C_{∞}. □ \end{matrix}  (6.19)

In §7 we will need a parametric version of Theorem 6.1. This follows from the next two lemmas.

Lemma 6.6. Let (X, d) be a complete metric space, let F_{ι} : X → X, be a family of uniformly contractive operators on X, ∥F_{ι}∥_{Lip} ≤ λ < 1. Then the application F_{ι} ↦ x_{ι}, where x_{ι} is the unique fixed point of F_{ι}, is continuous with respect to the metric d(F, G) := sup_{x∈X} d(F(x), G(x)).

Lemma 6.7. For δ_{0}, r and C_{0} chosen as at the beginning of the proof of Theorem 6.1, the application

f̃ ∈ {g ∈ Diff^{K+Lip}(𝕋^{n} × 𝕋^{d})|d_{K+Lip}(f, g) ≤ δ_{0}} ↦ F ∈ C^{0}(X_{r,C_{0}}; X_{r,C_{0}})

given by (6.5) is continuous (the first space has the metric d_{K+Lip}, and the second the metric described in Lemma 6.6), and the range is a family of uniformly contractive operators.

The first lemma is standard. The second lemma follows using similar estimates as in Proposition 6.5, of which we spare the reader.

§7. Deformation Rigidity of Certain Partially Hyperbolic Actions

In this section, as an application of the Livsic type theorem for cocycles with values in diffeomorphism groups, we show the deformation rigidity of the product action ρ_{0}. First we introduce some definitions.

Let Γ be a finitely-generated discrete group, M a compact manifold and φ : Γ × M → M a C^{∞}-action. Let k ≥ 1, ℓ ≥ 0 be integers. Fix a set of generators {γ_{1}, …, γ_{p}} of Γ. We say that a C^{∞}-action φ̃ is C^{k+Lip}-close to φ if the C^{∞}- diffeomorphisms φ(γ_{i}) and φ̃(γ_{i}) are C^{k+Lip}-close for all i. A C^{k+Lip}-deformation of the action φ is a d_{k+Lip}-continuous path of C^{∞}-actions φ_{t}, (0 ≤ t ≤ 1), with φ_{0} = φ. An action φ is said to be C^{k+Lip,ℓ}-deformation rigid if every C^{k+Lip}- deformation of φ, with φ_{t} contained in a sufficiently small C^{k+Lip}-neighborhood of φ, is C^{ℓ}-conjugate to φ by a continuous path of C^{ℓ}-diffeomorphisms.

The main result is the following:

Theorem 7.1. Let π be the standard linear action of SL(n, ℤ) on 𝕋^{n}, n ≥ 3. Let d ≥ 1 be an integer and consider ρ_{0} to be the action of SL(n, ℤ) on 𝕋^{n+d} = 𝕋^{n} ×𝕋^{d} given by ρ_{0}(A)(x, y) = (Ax, y), for x ∈ 𝕋^{n}, y ∈ 𝕋^{d} and A ∈ SL(n, ℤ). Then, for any integer K ≥ 4, ρ_{0} is C^{K+Lip,K−3}-deformation rigid.

The proof of this theorem will be done in the rest of this paragraph. It relies on our results about cocycles, and uses the techniques of [H1] and [K-L1].

An important fact used in the proof is Stowe’s theorem (see [S1] and [S2]). We recall it here in the form used in [H1] and [K-L1].

Let Γ and M be as above. Let Rep(Γ, Diff^{1}(M)) be the set of all homomorphisms from Γ to Diff^{1}(M) with the compact–open topology. Assume α is in Rep(Γ, Diff^{1}(M)). Denote by F (α) = {x ∈ M|α(γ)x = x for any γ ∈ Γ} the fixedpoint set of the action α and fix x ∈ F (α). Then α induces a finite-dimensional representation of Γ on T_{x}M, the tangent space to M at x. Denote by H^{1}(Γ, T_{x}M) the group cohomology with coefficients in this representation.

Theorem 7.2. (Stowe) If F ⊂ F (α) is a compact connected component of F (α) such that H^{1}(Γ, T_{x}M) = 0 for any x ∈ F, then F is a stable C^{1}-submanifold of M. More precisely, there is a tubular neighborhood Uof F in M, a neighborhood V of α in Rep(Γ, Diff^{1}(M)), and for each β ∈ V a closed embedding q_{β} ∈ C^{1}(F, U), such that:

(i) q_{β}(F) = U ∩ F (β) and

(ii) β ∈ V ↦ q_{β} ∈ C^{1}(F, U) is continuous. □

Using that H^{1}(Γ, V) = 0 for any finite-dimensional representation V of a subgroup Γ of finite index of SL(n, ℤ) (see Margulis [M]) and Theorem 7.2, one has:

Proposition 7.3. Consider {ρ_{t}}_{t}, a small C^{1}-deformation of ρ_{0}. Fix 0 ≤ s ≤ 1 and suppose that G ⊂ SL(n, ℤ) is a subgroup of finite index. Denote by F_{s} ⊂ 𝕋^{n}×𝕋^{d} a connected component of the set of fixed points of ρ_{s}(G) (hence H^{1}(G, T_{y}(𝕋^{n} × 𝕋^{d})) = 0 for all y ∈ F_{s}).

Then there exists ϵ > 0, a tubular neighborhood U of F_{s} in 𝕋^{n}×𝕋^{d} and embeddings q_{t,s} ∈ C^{1}(F_{s}, U), close to the inclusion F_{s} ↪ U and defined for s − ϵ ≤ t ≤ s + ϵ, such that F_{t} := q_{t,s}(F_{s}) is equal to U ∩ F (ρ_{t}) (and is connected). Moreover, the embeddings q_{t,s} depend continuously on t. □

We return now to the proof of Theorem 7.1. First we recall a general situation when a cocycle appears naturally. Let G be a group and α : G → Diff^{k}(M × N) an action of G by bundle morphisms on the trivial bundle M × N → M. Then the action is equivalent to a cocycle over the action in the base M, with values in the diffeomorphism group of the fiber N:

α_{g}(x, y) = (\overline{α}_{g}(x), β(g, x)(y)) , g ∈ G, x ∈ M, y ∈ N,

where β : G × M → Diff^{k}(N) is the cocycle and \overline{α} : G × M → M is the action induced in the base.

Therefore, once we can show that a conjugate of ρ_{t} (or its restriction) preserves the neutral foliation of ρ_{0}, we reduce the problem to one about cocycles.

In order to achieve this, consider the hyperbolic element A ∈ SL(n, ℤ) given in Theorem 5.8. Assume that the perturbation {ρ_{t}}_{0≤t≤1} is small enough in the C^{K+Lip}-metric. Then we can apply the parametric version of the Theorem 6.1 (the version of the Hirsch-Pugh-Shub theorem; we use the Lemmas 6.6 and 6.7). We obtain two continuous paths, {H_{t}}_{t} and {Q_{t}}_{t}, of Hölder homeomorphisms of 𝕋^{n} × 𝕋^{d}, with the following property: H_{t} takes the leaves of the neutral foliation of ρ_{0}(A) to those of ρ_{t}(A), and

H_{t}^{−1} ∘ ρ_{t}(A) ∘ H_{t} = Q_{t} ∘ ρ_{0}(A).  (7.1)

I.e., we obtain a path of ℤ-cocycles β_{t} : ℤ × 𝕋^{n} → Diff^{K+Lip}(𝕋^{d}) over the action of A on 𝕋^{n}, given by

β_{t}(1, x)(y) = y + \overline{q}_{t}(Ax, y).

Because \{\overline{q}_{t}\}_{t} are C^{K+Lip} along the neutral leaves and the first K derivatives are Hölder, we see immediately that β_{t}(1, ·) : 𝕋^{n} → Diff^{K+Lip}(𝕋^{d}) are Hölder, if we consider on Diff^{K+Lip}(𝕋^{d}) the d_{K}-metric.

The next step is to show that H_{t}^{−1} ∘ ρ_{t}(γ) ∘ H_{t} preserves the neutral foliation of ρ_{0}(A), for any γ ∈ SL(n, ℤ). Here we follow a strategy similar to that used by Hurder to prove topological deformation rigidity in [H1]. We recall some of his definitions and results.

Consider the linear action π : SL(n, ℤ) × 𝕋^{n} → 𝕋^{n}. A periodic point for π is a point in 𝕋^{n} which has a finite orbit under π(SL(n, ℤ)). The set Λ of periodic points of π is the set of rational points in 𝕋^{n}. A π-filtration of Λ is an increasing sequence of finite sets Λ_{1} ⊂ Λ_{2} ⊂ ⋯ ⊂ Λ_{p} ⊂ ⋯ whose union is all of Λ, and such that x_{i} ∈ Λ_{i} implies π(γ)(x_{i}) ∈ Λ_{i} for all γ ∈ SL(n, ℤ). From Lemma 3.2 in [H1] we know that Λ admits a natural π-filtration. Denote by G_{p} the π-stabilizer of Λ_{p} in SL(n, ℤ), which is a normal subgroup of finite index. For x ∈ 𝕋^{n}, denote by G_{x} the π-stabilizer of x in SL(n, ℤ).

The following proposition is a generalization of Proposition 3.6 from [H1].

Proposition 7.4. For x ∈ Λ, the set F_{t} := H_{t}({x}×𝕋^{d}) is an isolated submanifold of fixed points of the action ρ_{t}(G_{x}), for all 0 ≤ t ≤ 1.

Moreover, for any subgroup of finite index G ⊂ SL(n, ℤ),

F (ρ_{t}(G)) ⊂ ∪_{x∈Λ′}H_{t}({x} × 𝕋^{d}),

where Λ′ is a finite subset of 𝕋^{d}.

Proof. Since the π-orbit of x is finite, G_{x} has finite index in SL(n, ℤ) and there is m > 0 such that A^{m} ∈ G_{x}. We apply Proposition 7.3 for G = G_{x} and s = 0 to obtain:

• a value t_{1} > 0;

• a continuous path {F̃_{t} | 0 ≤ t ≤ t_{1}} with F̃_{0} = F_{0};

• that F̃_{t} is a connected compact submanifold of fixed points for ρ_{t}(G_{x}), isolated in F (ρ_{t}).

In particular, F̃_{t} is a connected submanifold of the fixed points of ρ_{t}(A^{m}). We claim that the path F_{t} coincides with F̃_{t}, for 0 ≤ t ≤ t_{1}. Denote by F_{t} the neutral foliation of ρ_{t}(A^{m}), 0 ≤ t ≤ 1. Since F_{t} is invariant

under ρ_{t}(A^{m}), one obtains a continuous action on 𝕋^{n} × 𝕋^{d}/∼_{F_{t}}, and, by (7.1), H_{t} induces a continuous conjugacy between the actions ρ_{0}(A^{m})/∼_{F_{0}} and ρ_{t}(A^{m})/∼_{F_{t}}. But ρ_{0}(A^{m})/∼_{F_{0}} is Anosov, hence x is an isolated fixed point in the base. Therefore, due to the conjugacy of the actions on the quotient spaces, H_{t}({x} × 𝕋^{d}) is the unique neutral leaf close to F_{0} which is invariated by ρ_{t}(A^{m}). Hence F̃_{t} is included in H_{t}({x} × 𝕋^{d}). But F̃_{t} is the image of {x} × 𝕋^{d} by a diffeomorphism and H_{t} is a homeomorphism, hence by a topological argument (e.g., the degree) it follows that F̃_{t} = H_{t}({x} × 𝕋^{d}) for 0 ≤ t ≤ t_{1}, which proves our claim.

Let s > 0 be the supremum of the values of ϵ ≤ 1 with the property that F_{t} is an isolated compact connected submanifold of fixed points for ρ_{t}(G_{x}) for all 0 ≤ t < ϵ. Then s < 1 leads to a contradiction by Proposition 7.3.

The last assertion follows from the fact that F (ρ_{t}(G)) ⊂ F (ρ_{t}(A^{m})), where m ∈ ℤ is such that A^{m} ∈ G, and the latter set is included in the union of those neutral leaves of ρ_{t}(A^{m}) which are over the fixed points of the action ρ_{t}(A^{m})/∼_{F_{t}}. This action is conjugated by H_{t}/∼ to the action π(A^{m}) on 𝕋^{n}, which has a finite number of fixed points. □

The previous proposition implies:

Proposition 7.5. For each x ∈ Λ and γ ∈ SL(n, ℤ)

H_{t}^{−1} ∘ ρ_{t}(γ) ∘ H_{t}({x} × 𝕋^{d}) = ρ_{0}(γ)({x} × 𝕋^{d}) for all 0 ≤ t ≤ 1.  (7.2)

Proof. Choose p such that x ∈ Λ_{p}. Denote, for 0 ≤ t ≤ 1, F_{t} := H_{t}({x} × 𝕋^{d}). Then (7.2) is equivalent to

H_{t}ρ_{0}(γ)(F_{0}) = ρ_{t}(γ)(F_{t}), for 0 ≤ t ≤ 1.

We will show that

t ↦ H_{t}ρ_{0}(γ)(F_{0}) and t ↦ ρ_{t}(γ)(F_{t})

are two continuous paths of connected components of F (ρ_{t}(G_{p})), equal to neutral leaves of ρ_{t}(A). Since such components are isolated (by Proposition 7.4) and the two paths are equal at t = 0, the paths have to coincide, thus proving (7.2).

It remains to prove the statement about the two paths.

For the first, we obtain

ρ_{0}(γ)(F_{0}) = ρ_{0}(γ)({x} × 𝕋^{d}) = {γx} × 𝕋^{d}.

But γx ∈ Λ_{p}, hence Proposition 7.4 applied for γx gives the desired conclusion.

For the second path, notice that F_{t} ⊂ F (ρ_{t}(G_{p})) (by Proposition 7.4) and γ^{−1}gγ ∈ G_{p} for g ∈ G_{p} (because G_{p} is normal in SL(n, ℤ)). Therefore

ρ_{t}(g)ρ_{t}(γ)(u, v) = ρ_{t}(γ)ρ_{t}(γ^{−1}gγ)(u, v) = ρ_{t}(γ)(u, v)

for any (u, v) ∈ F_{t}. Hence ρ_{t}(γ)(F_{t}) ⊂ F (ρ_{t}(G_{p})), and, by Proposition 7.4, is a subset of a neutral leaf of ρ_{t}(A). But both the leaf and this subset are homeomorphic to 𝕋^{d}, which can happen only if they are equal (a proper subset of a torus has the top homology group trivial). □

Corollary 7.6. For each γ ∈ SL(n, ℤ) and 0 ≤ t ≤ 1, we have

H_{t}^{−1} ∘ ρ_{t}(γ) ∘ H_{t}({x} × 𝕋^{d}) = ρ_{0}(γ)({x} × 𝕋^{d}) , for all x ∈ 𝕋^{d}.

Proof. Because the set of periodic points Λ is dense in 𝕋^{n}, we can use the continuity of both sides in (7.2). □

It follows that H_{t}^{−1} ∘ ρ_{t}(γ) ∘ H_{t} invariates the foliation {{x} × 𝕋^{d}}_{x∈𝕋^{n}} for any γ ∈ SL(n, ℤ), and the action induced on the base is exactly π. By our previous remark, we get a (continuous!) path of cocycles over π

β_{t} : SL(n, ℤ) × 𝕋^{n} → Diff^{K}(𝕋^{d})

such that

H_{t}^{−1} ∘ ρ_{t}(γ) ∘ H_{t} = (γx, β_{t}(γ, x)(y))

for all γ ∈ SL(n, ℤ), x ∈ 𝕋^{n}, y ∈ 𝕋^{d}. (We use the same notation for the cocycle over the SL(n, ℤ)-action because it is an extension of the previous cocycle over the action of A.) Moreover, from Proposition 7.4, the isolated closing conditions are satisfied:

if g ∈ SL(n, ℤ) and x ∈ 𝕋^{n} is an isolated fixed point of g, then β_{t}(g, x) =

Id_{𝕋^{d}}, for all t ∈ [0, 1].

Here we use the fact that if x ∈ 𝕋^{n} is an isolated fixed point for some g ∈ SL(n, ℤ), then x is a rational point, hence is in Λ. We know that β_{t}(A, ·) is Hölder in the d_{K}-metric on Diff^{K}(𝕋^{d}). Therefore, by Corollary 3.3, Theorem 5.8 and Theorem 5.2, there is a Hölder function P_{t} : 𝕋^{n} → Diff^{K−2}(𝕋^{d}) such that β_{t}(γ, x) = P_{t}(γx)P_{t}(x)^{−1} for any γ ∈ SL(n, ℤ), x ∈ 𝕋^{n}. Define now P_{t} : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} × 𝕋^{d} by P_{t}(x, y) := (x, P_{t}(x, y)). Then

P_{t}^{−1} ∘ H_{t}^{−1} ∘ ρ_{1}(γ) ∘ H_{t} ∘ P_{t}(x, y) = (γx, y),

for all x ∈ 𝕋^{n}, y ∈ 𝕋^{d}, γ ∈ SL(n, ℤ), 0 ≤ t ≤ 1.

Hence there is a continuous path of homeomorphisms ϕ_{t} := P_{t}^{−1} ∘ H_{t}^{−1} conjugating ρ_{0} and ρ_{t}. Recall that H_{t} is C^{K}-along the neutral foliation and P_{t} is C^{K−2} along the neutral foliation, so ϕ_{t} is C^{K−2} along the neutral foliation. The regularity of ϕ_{t} transverse to the neutral foliation can be obtained using the same method Katok and Lewis used to show regularity for conjugation between certain Anosov ℤ^{k}-actions (see [K-L1]). Because for a partially hyperbolic element the distribution E^{ss} ⊕ E^{uu} is not, in general, integrable, we have to make a minor change in proof. The following lemma was also used in [K-L1].

Lemma 7.7. ( [P-R]) Let Γ = SL(n, ℤ), n ≥ 2. Then there is a subgroup A ⊂ Γ such that

(i) the elements of A are simultaneously diagonalizable over ℝ;

(ii) A is isomorphic to a free abelian group of rank n − 1; (iii) if v_{1}, …, v_{n} ∈ ℝ^{n} is a basis of simultaneous eigenvectors for the group A,

and λ_{i} : A → ℝ^{n} is the character of A defined via Av := λ_{i}(A)v_{i}, A ∈ A, then for each i, 1 ≤ i ≤ n, there exists A_{i} ∈ A such that λ_{i}(A_{i}) < 1 and λ_{j}(A_{i}) > 1 for each j ≠ i;

(iv) there exist C_{i} ∈ A, 1 ≤ i ≤ n − 1, such that λ_{j}(C_{i}) < 1, j ≤ i, and λ_{j}(C_{i}) > 1, j > i. □

We finish now the proof of Theorem 7.1.

Let A_{i}, C_{i} ∈ SL(n, ℤ), 1 ≤ i ≤ n, be as given by Lemma 7.7. For a small enough perturbation, we can apply Theorem 6.1 in [H-P-S] for the images of A_{i} and C_{i} through ρ_{t}, 0 ≤ t ≤ 1.

Denote by F_{i,t}^{s}, G_{i,t} the strong stable foliation of ρ_{t}(A_{i}), respectively the weak stable foliation of ρ_{t}(C_{i}). These foliations have uniform C^{∞}-leaves of dimension one, respectively uniform C^{K−2}-leaves of dimension i + n. Note that G_{i,t} and F_{i+1,t}^{s} are Hölder foliations whose tangent spaces span the tangent space of G_{i+1,t}.

Fix 0 ≤ t ≤ 1.

Since ϕ_{t} is a topological conjugacy, it carries F_{i,t}^{s} to F_{i,0}^{s} and G_{i,t} to G_{i,0}. Then Lemma 4.10 from [K-L1] shows that ϕ_{t}, ϕ_{t}^{−1} : 𝕋^{n} × 𝕋^{d} → 𝕋^{n} × 𝕋^{d} are uniformly C^{∞} along the leaves of F_{i,t}^{s} and F_{i,0}^{s}, respectively.

We use now an induction argument. We describe it only for ϕ_{t}, as ϕ_{t}^{−1} is dealt with similarly.

We apply first Journé’s Theorem (see [J]), on each leaf of G_{1,t}, for F_{1,t}^{s} and the neutral foliation of ρ_{t}(A_{1}). We get that ϕ_{t} is C^{K−2−α_{1}} along the leaves of G_{1,t}, for some 0 < α_{1} < 1. Assume now that ϕ_{t} is C^{K−2−α_{i}}, 0 < α_{i} < 1, along the leaves of G_{i,t}. Then we can apply again Journé’s Theorem on each leave of G_{i+1,t}, for the foliations G_{i,t} and F_{i+1,t}^{s}, to obtain that ϕ_{t} is C^{K−2−α_{i+1}}, 0 < α_{i+1} < 1, along the leaves of G_{i+1,t}. At the end, we obtain that ϕ_{t} is C^{K−2−β}, 0 < β < 1, on 𝕋^{n} × 𝕋^{d}. □

Note that the proof depends on only a few properties of the action. We summarize these in Theorem 7.8.

We say that Γ is a higher rank lattice if Γ is a discrete subgroup of a connected semisimple algebraic ℝ-group G, with the ℝ-split rank of each factor of G at least two, G having finite center, G_{ℝ}^{0} having no compact factors and such that G/Γ has finite volume with respect to the Haar measure (see [Z4]). Note that from Margulis [M], H^{1}(Γ, V) = 0 for any finite-dimensional representation V of Γ.

Theorem 7.8. (a) Let Γ ⊂ SL(n, ℤ) be a higher rank lattice. Let π be the standard linear action of Γ on 𝕋^{n} and assume that π is generically Anosov. Let d ≥ 1 be an integer and let ρ_{0} be the action of Γ on 𝕋^{n+d} = 𝕋^{n} × 𝕋^{d} given by ρ_{0}(A)(x, y) = (Ax, y) (where x ∈ 𝕋^{n}, y ∈ 𝕋^{d} and A ∈ Γ). Let K ≥ 4 be any integer. Then ρ_{0} is C^{K+Lip,0}-deformation rigid.

(b) If, in addition, Γ contains an abelian subgroup A which satisfies properties (i)–(iv) of Lemma 7.7, then ρ_{0} is C^{K+Lip,K−3}-deformation rigid. □

References

[C-E-G] P. Collet, H. Epstein, G. Gallavoti, Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties, Com. Math. Phys. 95 (1984), 61–112.

[G-K] V. Guillemin, D. Kazhdan, On the cohomology of certain dynamical systems, Topology

19 (1980), 291–299.

[H1] S. Hurder, Rigidity of Anosov actions of higher rank lattices, Annals of Math. 135 (1992), 361–410.

[H2] S. Hurder, Infinitesimal rigidity of hyperbolic actions, preprint (1993). [H-K-L-Z] S. Hurder, A. Katok, J. Lewis, R. Zimmer, Rigidity of Cartan actions of higher rank

lattices, preprint (1991).

[H-P-S] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Lect. Notes in Math. 583, Springer

Verlag, 1977.

[J] J. -L. Journé, On a regularity problem occurring in connection with Anosov diffeomorphisms, Com. Math. Phys. 106 (1986), 345-352.

[K] A. Katok, Dynamical systems with hyperbolic structure, Three Papers in Dynamical Systems, American Mathematical Society Translations Series 2, vol. 116, Providence, Rhode Island, 1981.

[K-H] A. Katok, B. Hasselblat, Introduction to the modern theory of dynamical systems,

Cambridge University Press, 1995.

[K-K] A. Katok, S. Katok, Higher cohomology for abelian groups of toral automorphisms,

preprint (1993).

[K-L1] A. Katok, J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J.

Math. 75(2-3) (1992), 203-242.

[K-L2] A. Katok, J. Lewis, Global rigidity results for lattice actions on tori and new examples

of volume-preserving actions, preprint (1991).

[K-L-Z] A. Katok, J. Lewis, R. Zimmer, Cocycle superrigidity and rigidity for lattice actions

on tori, preprint (1992).

[K-S] A. Katok, R. Spatzier, Differentiable rigidity of hyperbolic abelian actions, preprint

(1992).

[L1] A. Livsic, Homology properties of U systems, Math. Notes 10 (1971), 758–763.

[L2] A. Livsic, Cohomology of dynamical systems, Math. U.S.S.R., Izvestia 6 (1972), 1278– 1301.

[L-S] A. Livsic, J. Sinai, On invariant measures compatible with the smooth structures for

transitive U systems, Soviet Math. Doklady 13 (1972), no. 6, 1656–1659.

[Lla] R. de la Llave, Analytic regularity of solutions of Livsic’s cohomology equations and some applications to analytic conjugacy of hyperbolic dynamical systems, preprint (1989).

[L-M-M] R. de la Llave, J. M. Marco, R. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for Livsic cohomology equation, Annals of Math. 123 (1986), 537–612.

[M] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag, 1991. [N-T] V. Niţică, A. Török, Regularity results for the solution of Livsic cohomology equation

with values in diffeomorphism groups (to appear in Erg. Th. & Dyn. Syst.).

[P-R] G. Prasad, M. S. Raghunathan, Cartan subgroups and lattices in semisimple groups,

Annals of Math. 96 (1972), 296–317.

[Q1] N. Qian, Local rigidity of Anosov actions on tori, preprint (1993).

[Q2] N. Qian, Topological deformation rigidity of actions of higher rank lattices on tori, preprint (1993).

[R] W. Rudin, Principles of mathematical analysis, 3rd ed., McGraw-Hill, 1976.

[S1] D. Stowe, The stationary set of a group action, Proc. A.M.S. 79 (1980), 139–146.

[S2] D. Stowe, Stable orbits of differentiable group actions, Trans. A.M.S. 277 (1983), 665– [Z1] R. Zimmer, Actions of semisimple groups and discrete groups, Proc. Int. Congress

Math., Berkeley (1986), 1247–1254.

[Z2] R. Zimmer, Discrete groups in geometry and analysis, Progress in Mathematics, Birkhäuser,, Boston, 1987.

[Z3] R. Zimmer, Infinitesimal rigidity for smooth actions of discrete subgroups, J. Diff. Geom. 31 (1990), 301–322.

[Z4] R. Zimmer, Ergodic theory and semisimple groups, Birkhauser, Boston, 1984.

Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania

   E-mail address: nitica@math.psu.edu, torok@math.psu.edu