- Statistical properties in hyperbolic dynamics
- Uniform and Non-uniform Hyperbolicity
- Introduction to Quantum walks
- Bratteli diagrams, flat surfaces and the hierarchical structure of minimal systems

**Statistical properties in hyperbolic
dynamics**

S1.

Gerhard Keller and Carlangelo Liverani: A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Section 2, Lect. Notes Phys.

Self-contained proof - with background on BV - of exponential decay of correlations for piecewise-expanding maps on the interval, including the Lasota-Yorke inequality.

S2.

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 maps)

S3.

Ott, William; Stenlund, Mikko; Young, Lai-Sang. Memory loss for time-dependent dynamical systems, Math. Res. Lett. 16 (2009), no. 3, 463–475.

preprint at https://www.math.uh.edu/~ott/Publications/docs/ott_5.pdf

This paper uses coupling to prove decay of correlations for non-stationary (piecewise) expanding systems.

S4.

Bressaud, Xavier; Liverani, Carlangelo. Anosov diffeomorphisms and coupling, Ergodic Theory Dynam. Systems 22 (2002), no. 1, 129–152.

This is an elegant coupling approach to Anosov diffeomorphisms, to prove exponential decay of correlations.

S5.

C. Liverani, Central limit theorem for deterministic systems, in International Conference on Dynamical Systems (Montevideo, 1995), 56-75, Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996.

preprint on Liverani's page

Once it has been proved that correlations decay quickly enough, the CLT can be derived using martingale approximations.

S6.

Mark Demers A gentle introduction to anisotropic Banach spaces, Chaos, Solitons and Fractals 116 (2018), 29-42.

preprint at http://faculty.fairfield.edu/mdemers/research/2018.09.12.normsurvey.proofed.pdf

"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 contracting system.

S7.

Lai-Sang Young. 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 overview.

**Uniform and Non-uniform Hyperbolicity**

Hyp1.

A: Bowen, Rufus. 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.

B: D.V. Anosov, A.V. Klimenko, G. Kolutsky. On the hyperbolic automorphisms of the 2-torus and their Markov partitions, (2008), arXiv:0810.5269

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 partitions.

Hyp2.

Steve Smale 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.

Hyp3.

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.

Hyp4.

A: Milnor, John. Fubini foiled: Katok's paradoxical example in measure theory, Math. Intelligencer 19 (1997), no. 2, 30–32.

B: Shub, Michael; Wilkinson, Amie. Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508.

These explain how absolute continuity of foliations can fail beyond uniform hyperbolicity

Hyp5.

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 the TOC.

Hyp6.

Barreira, Luis; Pesin, Yakov B. Lyapunov exponents and smooth ergodic theory, University Lecture Series, 23. American Mathematical Society, Providence, RI, 2002. xii+151 pp. ISBN: 0-8218-2921-1

"Further reading", a more complete version of Hyp5.

**Introduction to Quantum walks**

QW1.

Barry Simon. 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.

QW2.

Cantero, María-José; Grünbaum, F. Alberto; Moral, Leandro; Velázquez, Luis. Matrix-valued Szegő polynomials and quantum random walks, 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.

QW3.

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.

QW4.

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 almost-Mathieu operator, 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.

QW5.

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
systems
**

BR1.

Two excellent references for introduction to the classical theory of Teichmuller dynamics and flat surfaces:

Flat Surfaces by Anton Zorich: https://arxiv.org/pdf/math/0609392.pdf

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards by Giovanni Forni and Carlos Matheus: https://arxiv.org/abs/1311.2758

BR2.

Construction of flat surfaces from Bratteli diagrams:

Infinite type flat surface models of ergodic systems by Kathryn Lindsey and Rodrigo Treviño: https://www.aimsciences.org/article/doi/10.3934/dcds.2016043

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.

BR3.

Perfect Orderings on Finite Rank Bratteli Diagrams by S. Bezuglyi, J. Kwiatkowski and R. Yassawi: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/perfect-orderings-on-finite-rank-bratteli-diagrams/B1D9D7B30AF354808BE5E4BAE6632722

Contains conditions for when randomly-ordered Bratteli diagrams contain perfect orderings (i.e. for which the Vershik map is a homeomorphism).

BR4.

The invariant measures of some infinite interval exchange maps by Pat Hooper: https://msp.org/gt/2015/19-4/p03.xhtml

Contains a general construction of flat surfaces of infinite type and classifies their invariant measures.

BR5. Veech groups

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, http://www.orst.edu/%7Eschmidtt/ourPapers/Hubert/intVchGpsX04.pdf

It contains a proof for the discreteness of the Veech group for finite genus surfaces.

Veech Groups of Loch Ness Monsters by Piotr Przytycki; Gabriela Schmithüsen; Ferrán Valdez: https://eudml.org/doc/219716

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 Randecker: https://arxiv.org/abs/1612.06877

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

Funding for this event is provided by the NSF grant DMS-1900964.