[Paper Review] Symbolic dynamics for surface diffeomorphisms with positive topological entropy

Omri Sarig|arXiv (Cornell University)|May 9, 2011

Mathematical Dynamics and Fractals22 references3 citations

TL;DR

This paper constructs a countable Markov partition for surface diffeomorphisms with positive topological entropy, enabling symbolic dynamics on a full-measure invariant set for any entropy threshold below the topological entropy. The key result proves J. Buzzi's conjecture that such systems have at most countably many ergodic measures of maximal entropy and confirms A. Katok's conjecture on the exponential growth of periodic points by showing limsup of e^{-nh_top}P_n(f) > 0 for C^1+β diffeomorphisms.

ABSTRACT

Suppose f is a C^{1+ε} surface diffeomorphism with positive topological entropy. For every positive δstrictly smaller than the topological entropy of f we construct an invariant Borel set E such that (a) f|E has a countable Markov partition; and (b) E has full measure with respect to any ergodic invariant probability measure with entropy larger than δ. This allows us to prove the following conjecture of A. Katok: if f is C^\infty with topological entropy h>0, and if P_n(f)=#{x:f^n(x)=x}, then limsup P_n(f)/exp(nh)>0.

Motivation & Objective

• To establish symbolic dynamics for C^1+β surface diffeomorphisms with positive topological entropy via a countable Markov partition.
• To resolve J. Buzzi's conjecture on the countability of ergodic measures of maximal entropy.
• To prove A. Katok's conjecture that limsup e^{-nh_top}P_n(f) > 0 for C^∞ diffeomorphisms on compact surfaces.
• To develop a Markov extension with finite pre-images and Hölder continuous coding map.

Proposed method

• Constructs a locally finite countable Markov cover using Pesin theory and non-uniform hyperbolicity.
• Employs Pesin charts and Lyapunov change of coordinates to linearize dynamics near hyperbolic points.
• Defines overlapping charts and ε-chains to model orbits via admissible manifolds under the graph transform.
• Applies a Bowen–Sinai refinement to construct a countable Markov partition with the symbolic Markov property.
• Uses scaling and window parameters to control the size and overlap of dynamical windows in the chart system.
• Establishes a finite-to-one Markov extension via a directed graph and Hölder continuous coding map π_χ.

Experimental results

Research questions

• RQ1 Does every C^1+β surface diffeomorphism with positive topological entropy admit at most countably many ergodic measures of maximal entropy?
• RQ2 Is the limsup of e^{-nh_top(f)} times the number of n-periodic points bounded below by a positive constant for C^∞ surface diffeomorphisms?
• RQ3 Can a countable Markov partition be constructed for the set of points with entropy greater than χ < h_top(f)?
• RQ4 Can the dynamics be semi-conjugated to a topological Markov shift with finite pre-images on a full-measure set for each χ < h_top(f)?
• RQ5 Do regular chains shadowing the same orbit remain exponentially close under iteration?

Key findings

• For every 0 < χ < h_top(f), there exists a locally compact topological Markov shift Σ_χ and a Hölder continuous map π_χ: Σ_χ → M such that π_χ ∘ σ = f ∘ π_χ and π_χ[Σ_χ^#] is χ-large.
• The system has at most countably many ergodic invariant probability measures with maximal entropy, confirming Buzzi’s conjecture.
• For C^1+β diffeomorphisms with positive entropy, if a measure of maximal entropy exists, then liminf_{n→∞, p|n} e^{-nh_top(f)} P_n(f) > 0.
• The number of periodic points satisfies limsup_{n→∞} e^{-nh_top(f)} P_n(f) > 0 for C^∞ surface diffeomorphisms, confirming Katok’s conjecture.
• Regular chains shadowing the same orbit are exponentially close under iteration, with distance decaying as e^{-nχ/2} times a size ratio.
• The Markov extension is finite-to-one, with every point in the image of π_χ having finitely many pre-images.