Statistical properties in hyperbolic
Gerhard Keller and Carlangelo Liverani:
A spectral gap for a one-dimensional lattice of
coupled piecewise expanding interval maps, Section 2,
Lect. Notes Phys. 671 (2005), p. 115-151
Self-contained proof - with background on BV - of exponential decay of
correlations for piecewise-expanding maps on the interval, including
the Lasota-Yorke inequality.
Ian Melbourne and Matthew Nicol:
Almost sure invariance principle for nonuniformly hyperbolic
systems, Sections 2 (a)-(b),
Commun. Math. Phys. 260 (2005) 131-146.
A clean discussion of exponential decay of correlations for
Gibbs-Markov maps (includes full-branch piecewise expanding interval
Ott, William; Stenlund, Mikko; Young, Lai-Sang.
Memory loss for time-dependent dynamical
Math. Res. Lett. 16 (2009), no. 3, 463–475.
This paper uses coupling to prove decay of correlations for
non-stationary (piecewise) expanding systems.
Bressaud, Xavier; Liverani, Carlangelo.
Anosov diffeomorphisms and coupling,
Theory Dynam. Systems 22 (2002), no. 1, 129–152.
This is an elegant coupling approach to Anosov diffeomorphisms, to
prove exponential decay of correlations.
Central limit theorem for deterministic
in International Conference on Dynamical Systems (Montevideo, 1995),
56-75, Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996.
on Liverani's page
Once it has been proved that correlations decay quickly enough, the
CLT can be derived using martingale approximations.
A gentle introduction to anisotropic Banach
Chaos, Solitons and Fractals 116 (2018), 29-42.
"Further reading", which explains how to deal with
hyperbolic systems without reducing first to the expanding direction.
To obtain a spectral gap in this setting, different Banach spaces are
needed. The paper starts with a very illuminating example of a
What are SRB measures, and which dynamical
systems have them?
J. Stat Phys 108, No 5/6 (2002), 733-754.
The use of mathematical methods from
statistical mechanics to construct a physically relevant SRB measure
for smooth hyperbolic dynamical systems is one of the crowning
achievements of thermodynamic formalism. This survey paper gives an
Uniform and Non-uniform Hyperbolicity
Equilibrium states and the ergodic theory of
Anosov diffeomorphisms, Chapter 3.
Second revised edition. With a preface by David Ruelle. Edited by
Jean-René Chazottes. Lecture Notes in Mathematics, 470.
Springer-Verlag, Berlin, 2008. viii+75 pp.
A proof that uniformly hyperbolic systems can be modeled with subshifts
of finite type.
This is the approach that Sarig generalizes to surface diffeos.
D.V. Anosov, A.V. Klimenko, G. Kolutsky.
On the hyperbolic automorphisms of the 2-torus
and their Markov partitions,
Starts with a discussion of how to
treat deterministic systems as random processes, then constructs Markov
partitions on the 2-torus for hyperbolic automorphisms, which lets
these systems be viewed as Markov shifts. This was the first example of
Markov partitions historically and is simpler than the general
construction by Bowen in Hyp1. A.
The paper also includes some results on classification of Markov
Differentiable dynamical systems,
Bull. Amer. Math. Soc. 73 1967, 747–817,
Emphasis on Sections I.6-7 which have the spectral decomposition.
This is one of the fundamental first papers on uniform hyperbolicity.
Rodriguez Hertz, F.; Rodriguez Hertz, M. A.; Tahzibi, A.; Ures, R.
New criteria for ergodicity and nonuniform hyperbolicity,
Duke Math. J. 160 (2011), no. 3, 599–629.
This paper has some results in non-uniform hyperbolicity related to
the spectral decomposition, extending the spectral decomposition of
uniformly hyperbolic systems.
Fubini foiled: Katok's paradoxical example in
Math. Intelligencer 19 (1997), no. 2, 30–32.
Shub, Michael; Wilkinson, Amie.
Pathological foliations and removable zero
Invent. Math. 139 (2000), no. 3, 495–508.
These explain how absolute continuity of foliations can fail beyond
Barreira, L.; Pesin, Ya.
Lectures on Lyapunov exponents and smooth
ergodic theory. Appendix A by M. Brin and Appendix B by D. Dolgopyat,
H. Hu and Pesin,
Proc. Sympos. Pure Math., 69, Smooth ergodic
theory and its applications (Seattle, WA, 1999), 3–106, Amer. Math. Soc.,
Providence, RI, 2001.
This is a survey of the Pesin theory and its application, see
Barreira, Luis; Pesin, Yakov B.
Lyapunov exponents and smooth ergodic
University Lecture Series, 23. American Mathematical Society,
Providence, RI, 2002. xii+151 pp. ISBN: 0-8218-2921-1
"Further reading", a more complete version
Introduction to Quantum walks
OPUC on one foot,
Bull. Amer. Math. Soc. (2005)
Simon 2005 (BAMS) is an expository paper with some background about
CMV matrices and orthogonal polynomials on the unit circle (OPUC).
The most relevant portions for the course are contained in the first
half of the paper.
Cantero, María-José; Grünbaum, F. Alberto; Moral, Leandro; Velázquez,
Matrix-valued Szegő polynomials and quantum
Comm. Pure Appl. Math. 63 (2010), no. 4, 464–507.
CGMV 2010 (CPAM) is the seminal paper that connects 1D quantum walks
with CMV matrices and OPUC.
Damanik, David; Fillman, Jake; Vance, Robert.
Dynamics of unitary operators,
J. Fractal Geom. 1 (2014), no. 4, 391–425.
Damanik, David; Fillman, Jake; Ong, Darren C.
Spreading estimates for quantum walks on the
integer lattice via power-law bounds on transfer matrices,
J. Math. Pures Appl. (9) 105 (2016), no. 3, 293–341.
DFV 2014 (JFG), and DFO 2016 (JMPA) discuss some general methods
connecting spectral theory and dynamics of quantum walks. They both
apply their methods to the Fibonacci QW.
Shikano, Yutaka; Katsura, Hosho.
Localization and fractality in inhomogeneous
quantum walks with self-duality,
Phys. Rev. E (3) 82 (2010), no. 3, 031122, 7 pp.
Fillman, Jake; Ong, Darren C.; Zhang, Zhenghe.
Spectral characteristics of the unitary critical
Comm. Math. Phys. 351 (2017), no. 2, 525–561.
SK and FOZ 2017 (CMP) are about another quasiperiodic quantum walk,
called the Unitary Almost-Mathieu Operator, which we will not discuss
very much in the lectures. These papers will be pretty difficult, but
might provide a nice challenge for a stronger student.
Damanik, David; Erickson, Jon; Fillman, Jake; Hinkle, Gerhardt; Vu, Alan.
Quantum intermittency for sparse CMV matrices
with an application to quantum walks on the half-line,
J. Approx. Theory 208 (2016), 59–84.
DEHFV 2016 (JAT) is about quantum walks with sparse, high barriers.
We won't talk about it in the lectures, but it would make for
pleasant reading for the students. In fact, three of the authors of
that paper were undergraduates! (Erickson, Hinkle, and Vu)
Bratteli diagrams, flat surfaces and the hierarchical structure of minimal
Two excellent references for introduction to the classical theory of
Teichmuller dynamics and flat surfaces:
Flat Surfaces by Anton
Introduction to Teichmüller theory and its
applications to dynamics of interval exchange transformations, flows on
surfaces and billiards by Giovanni Forni and Carlos
Construction of flat surfaces from Bratteli diagrams:
Infinite type flat surface models of ergodic
by Kathryn Lindsey and Rodrigo
Contains construction, examples, and conjectures. There is also a proof
that every ergodic, aperiodic, finite entropy flow can be realized as the
vertical flow on a flat surface of finite area.
Perfect Orderings on Finite Rank Bratteli
by S. Bezuglyi, J. Kwiatkowski and R. Yassawi:
Contains conditions for when randomly-ordered Bratteli diagrams contain
perfect orderings (i.e. for which the Vershik map is a
The invariant measures of some infinite interval
Contains a general construction of flat surfaces of infinite type and
classifies their invariant measures.
For the more algebraically-leaning participants, although I may not
actually mention Veech groups in my lecture (depending on time), here is a
great intro to them:
An introduction to Veech surfaces, by Pascal
Hubert and Thomas Schmidt,
It contains a proof for the discreteness of the Veech group for finite
Veech Groups of Loch Ness Monsters by Piotr
Przytycki; Gabriela Schmithüsen; Ferrán
Proves that your your favorite subgroup of \(SL(2,R)\) is most likely the
Veech group of a flat surface of infinite genus and infinite area.
Notes on the Veech group of the Chamanara
surface by Frank Herrlich, Anja
Computes the Veech group of Chamanara's surface, which is a surface of
infinite genus and finite area.
Open question: Can the Veech group of a flat surface of infinite type
(\(\pi_1\) is infinitely generated) and finite area be a
lattice in \(SL(2,R)\)?
Funding for this event is provided by the NSF grant DMS-1900964.