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Symbolic dynamics for certain non-invertible $C^{1+β}$ maps

Jing Xun, Yifan Zhang, Yujun Zhu

TL;DR

This work extends symbolic dynamics to non-invertible $C^{1+\beta}$ maps with zero Lyapunov exponents and singularities by combining ideas from Sarig, Ovadia, and Lima–Poletti. It constructs a countable Markov partition for the 0-summ subset of the natural extension and proves the existence of a finite-to-one symbolic extension that semi-conjugates the natural extension to a topological Markov shift, while lifting the geometric potential with summable variations. The approach hinges on refined Pesin theory in the presence of singularities, a ladder-function temperability framework, and a graph-transform analysis of admissible manifolds to control distortions and overlaps. The results yield entropy and periodic-point counts on the 0-summ set and provide a concrete example illustrating the applicability to singular non-invertible dynamics on compact manifolds.

Abstract

Let $f$ be a non-invertible $C^{1+β}(β>0)$ map with zero Lyapunov exponents and singularities on a closed Riemannian manifold $M$. We consider the symbolic dynamics of $f$. Combining the techniques in recent works of Sarig, Ovadia and Araujo-Lima-Poletti, we construct a countable Markov partition for the invariant set consisting of summable points of the inverse limit space of $(M, f)$ and show that there exists a finite-to-one symbolic extension for $f$ on the corresponding subset of $M$.

Symbolic dynamics for certain non-invertible $C^{1+β}$ maps

TL;DR

This work extends symbolic dynamics to non-invertible maps with zero Lyapunov exponents and singularities by combining ideas from Sarig, Ovadia, and Lima–Poletti. It constructs a countable Markov partition for the 0-summ subset of the natural extension and proves the existence of a finite-to-one symbolic extension that semi-conjugates the natural extension to a topological Markov shift, while lifting the geometric potential with summable variations. The approach hinges on refined Pesin theory in the presence of singularities, a ladder-function temperability framework, and a graph-transform analysis of admissible manifolds to control distortions and overlaps. The results yield entropy and periodic-point counts on the 0-summ set and provide a concrete example illustrating the applicability to singular non-invertible dynamics on compact manifolds.

Abstract

Let be a non-invertible map with zero Lyapunov exponents and singularities on a closed Riemannian manifold . We consider the symbolic dynamics of . Combining the techniques in recent works of Sarig, Ovadia and Araujo-Lima-Poletti, we construct a countable Markov partition for the invariant set consisting of summable points of the inverse limit space of and show that there exists a finite-to-one symbolic extension for on the corresponding subset of .
Paper Structure (13 sections, 30 theorems, 166 equations)

This paper contains 13 sections, 30 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Statement of the Main results
  3. Preliminaries
  4. Natural extension
  5. The subset with Markovian symbolic dynamics
  6. Pesin Theory
  7. Coarse Graining
  8. Admissible Manifold
  9. Inverse theorem
  10. Locally finite Markov Partition
  11. Applications and Example
  12. The growth rate of Pn(f) and the number of M.M.E.
  13. An Example

Key Result

Corollary 1.1

Assume the system $(M,f)$ satisfies the above assumptions, there exists a finite-to-one symbolic extension for $(M,f)$, which lifts the geometric potential with summable variations.

Theorems & Definitions (79)

  • Corollary 1.1
  • Definition 3.1
  • Proposition 3.1
  • Remark 1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.2
  • Proof 1
  • Remark 2
  • Definition 3.4
  • ...and 69 more