2015 Houston Summer School on Dynamical Systems

Hyperbolicity

F. Blanchard, Beta-expansions and symbolic dynamics, Theor. Comp. Sci 65 (1989), p. 131-141.

This paper goes through some of the phenomena exhibited by the map T(x) = beta * x (mod 1) as beta ranges over all reals bigger than 1.


Rufus Bowen, Some systems with unique equilibrium states, Math. Sys. Theory 8 (1975), p. 193-202.

This paper gives conditions on the system under which there is a unique equilibrium state (for topological pressure), which turns out to be connected to statistical properties of the system.


Rufus Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970), p. 725-747.

It was mentioned in the second uniform hyperbolicity lecture that uniformly hyperbolic systems admit Markov partitions (a weaker version of this was proved in the lecture).    This paper gives the details of that proof.


Mikhail Lyubich, The quadratic family as a qualitatively solvable model of chaos, Notices of the AMS 47 (2000), p. 1042-1052.

The logistic maps T(x) = ax(1-x) display different sorts of dynamical behaviour depending on the value of a.  This article surveys the situation and gives the main ideas without delving into the proofs.


John Milnor, Fubini foiled: Katok's paradoxical example in measure theory, Mathematical Intelligencer 19 (1997), p. 30-32.

Absolute continuity of the stable and unstable foliations is a fundamental tool in uniform hyperbolicity.  Outside of the uniformly hyperbolic setting there are foliations that are not absolutely continuous that display surprising pathological behaviour.  This paper gives an elementary description of one such example.  More advanced examples appear in partial hyperbolicity and could be read after this paper.


Keith Burns and Boris Hasselblatt, The Sharkovsky theorem: a natural direct proof, American Mathematical Monthly 118 (2011), p. 229-244

The Sharkovsky theorem governs the order in which periodic orbits can appear for one-dimensional maps and is one of the more surprising results one encounters when first exploring one-dimensional dynamics.


Michael Handel and William Thurston, Anosov flows on new three manifolds, Inventiones Math. 59 (1980), p. 95-103.  Also: John Franks and Bob Williams, Anomalous Anosov flows,

It is a famous open question whether every Anosov diffeomorphism is topologically transitive.  For Anosov flows there is a counterexample, which is presented here.


Probability and statistics of dynamical systems

Carlangelo Liverani, Central limit theorem for deterministic systems (1995).

Does what's on the box.  Derive the central limit for deterministic systems using martingale approximation as long as correlations decay appropriately.


Peter Nandori, Domokos Szasz, and Tamas Varju, A central limit theorem for time-dependent deterministic systems, J. Stat Phys 122 (2006).

When the system can vary with time, a CLT can still sometimes be satisfied.


Dong Han Kim, The dynamical Borel-Cantelli lemma for interval maps, DCDS 17 (2007), 891-900.


Carlangelo Liverani, Decay of correlations in piecewise expanding maps, Journal of Statistical Physics 78 (1995), p. 1111-1129.

All those decaying correlations have to come from somewhere.  This is a proof of decay of correlations using cone methods.  A further reference for this project is: Carlangelo Liverani, Decay of correlations, Annals of Mathematics 142 (1995), p. 239-301.



Other topics, including low complexity dynamics

Yitzhak Katznelson and Benjamin Weiss, A simple proof of some ergodic theorems, Israel Journal of Mathematics 42 (1982), p. 291-296

There are many proofs of the ergodic theorem, this one is inspired by ideas of Kamae that used nonstandard analysis.


Anatole Katok, Invariant measures of flows on oriented surfaces, Soviet Math. Dokl 14 (1973).  Also: Sol Schwartzman, Asymptotic cycles, Annals of Math. 66 (1957)

How many ergodic measures can an IET have?  This paper gives some bounds, using the fact that ergodic measures for an IET can be related to independent elements of the homology group of a certain surface.


Michael Boshernitzan, Elementary proof of Furstenberg's diophantine result, Proc. AMS 122 (1994)

The doubling map f(x) = 2x (mod 1) has many invariant measures.  So does the tripling map g(x) = 3x (mod 1).  Lebesgue measure is invariant for both.  It turns out that not many other things are.  At the level of invariant sets, one can show that every closed set A that is invariant for both f and g is either a union of periodic points or is the whole interval [0,1].  This paper gives a result that implies this fact.


Michael Boshernitzan and Arnaldo Nogueira, Generalized eigenfunctions of interval exchange maps, ETDS 24 (2004)

A condition is given under which an IET is weakly mixing.


R.M. Burton and M. Keane, Density and uniqueness in percolation, Comm Math Phys 121 (1989), p. 501-505

This paper studies percolation on a d-dimensional lattice.


Another good reference (not a project paper)

Marcelo Viana, Stochastic dynamics of deterministic systems