[Paper Review] Bowen factors, their degree, and codings of surface diffeomorphisms

Jérôme Buzzi|arXiv (Cornell University)|Jul 11, 2018

Mathematical Dynamics and Fractals8 references3 citations

TL;DR

This paper establishes that symbolic finite-to-one extensions of surface diffeomorphisms—constructed by O. Sarig—induce H{"o}lder-continuous conjugacies on large invariant sets, sometimes preserving transitivity, via the Bowen property and degree theory. It improves Sarig's lower bound on periodic points and characterizes surface diffeomorphisms admitting H{"o}lder-continuous coding of all aperiodic hyperbolic measures.

ABSTRACT

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\older-continuous conjugacies on large sets, sometimes preserving transitivity. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We use the notion of degree from finite equivalence theory and magic word isomorphisms. As an application, we improve Sarig's lower bound on the number of periodic points for surface diffeomorphisms. Finally we characterize surface diffeomorphisms admitting a H\older-continuous coding of all their aperiodic hyperbolic measures.

Motivation & Objective

• To understand the dynamical implications of symbolic finite-to-one extensions in surface diffeomorphisms, particularly their regularity and structural preservation.
• To generalize R. Bowen's classical result on Markov partitions using the notion of the Bowen property in a broader symbolic extension framework.
• To improve the lower bound on the number of periodic points for surface diffeomorphisms using finite equivalence theory and magic word isomorphisms.
• To characterize surface diffeomorphisms that admit H{"o}lder-continuous coding for all aperiodic hyperbolic measures, using degree and extension theory.

Proposed method

• Utilize the Bowen property—originally for Markov partitions and extended here to symbolic finite-to-one extensions—to establish H{"o}lder-continuous conjugacies on large invariant sets.
• Apply the notion of degree from finite equivalence theory to analyze the structure and multiplicity of symbolic extensions.
• Employ magic word isomorphisms to relate symbolic extensions to the underlying dynamics and facilitate conjugacy construction.
• Use the interplay between the Bowen property and degree to derive quantitative bounds on periodic point growth.
• Characterize the existence of H{"o}lder-continuous coding for aperiodic hyperbolic measures via structural and regularity conditions on the extension.

Experimental results

Research questions

• RQ1 Under what conditions do symbolic finite-to-one extensions of surface diffeomorphisms induce H{"o}lder-continuous conjugacies on large invariant sets?
• RQ2 How does the Bowen property in symbolic extensions generalize classical results for Markov partitions in hyperbolic dynamics?
• RQ3 What is the precise improvement to Sarig's lower bound on the number of periodic points using degree and magic word isomorphisms?
• RQ4 Which surface diffeomorphisms admit a H{"o}lder-continuous coding for all aperiodic hyperbolic measures, and what characterizes them?

Key findings

• Symbolic finite-to-one extensions of surface diffeomorphisms induce H{"o}lder-continuous conjugacies on large invariant sets, preserving transitivity in certain cases.
• The Bowen property for such extensions generalizes classical results of R. Bowen on Markov partitions, extending their applicability to broader symbolic systems.
• The paper improves Sarig's lower bound on the number of periodic points in surface diffeomorphisms using degree theory and magic word isomorphisms.
• A complete characterization is obtained for surface diffeomorphisms admitting a H{"o}lder-continuous coding of all aperiodic hyperbolic measures, based on extension structure and regularity.