2014 Houston Summer School on Dynamical Systems

Probability

Andrew Ferguson and Mark Pollicott, Escape rates for Gibbs measures (2011).

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

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

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

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.

In the third lecture in this series we will discuss equilibrium states for topological pressure.  This paper gives conditions on the system under which there is a unique equilibrium state, 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.

In the second lecture in this series we will describe Markov partitions and state the result that every Axiom A diffeomorphism has a Markov partition.  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.


Cones

Marcelo Viana, Stochastic dynamics of deterministic systems, Chapter 2.

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

A further reference for this project is: Carlangelo Liverani, Decay of correlations, Annals of Mathematics 142 (1995), p. 239-301.

Carlangelo Liverani, Saussol, Vaienti: Conformal measure and decay of correlations for covering weighted systems, Ergodic Theory Dynam. Systems 18 (1998), p. 1399-1420

The above deal with iterates of a single map.  For nonequilibrium dynamical systems:

William Ott, Mikko Stenlund, and Lai-Sang Young, Memory loss for time-dependent dynamical systems, Math. Res. Lett. 16 (2009), p. 463-475.

A further reference for this project is: Chinmaya Gupta, William Ott, and Andrew Torok, Memory loss for time-dependent piecewise expanding systems in higher dimension, Math. Res. Lett. 20 (2013), p. 141-161.

Anushaya Mohapatra and William Ott, Memory loss for nonequilibrium open dynamical systems, Disc. and Cont. Dyn. Sys. (2014), to appear.

Mark Demers, Lai-Sang Young, Escape rates and conditionally invariant measures, Nonlinearity, 2006


Spectral theory

G. Keller and C. Liverani, A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps, Lect. Notes Phys. 671 (2005), p. 115-151


C* algebras

Efren Ruiz and Mark Tomforde, Classification of unital simple Leavitt path algebras of infinite graphs (2013).  Section 3 is the focus of this project.

John Franks, Flow equivalence of subshifts of finite type, ETDS 4 (1984), 53-66.

Will Ott, Mark Tomforde, Paulette Willis, One-sided shift spaces over infinite alphabets, New York Journal of Mathematics Monographs, 2014