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Quasi-shadowing property for nonuniformly partially hyperbolic systems

Gang Liao, Xuetong Zu

TL;DR

The paper develops a quasi-shadowing theory for nonuniformly partially hyperbolic diffeomorphisms by constructing fake center foliations and proving Hölder continuity of invariant bundles on Pesin blocks, enabling shadowing that adapts to pseudo-orbit movement. It derives quasi-specification and quasi-closing results as corollaries, illustrating robust orbit concatenation and near-periodic behavior under center-direction motion. The main application shows that the exponential growth rate of quasi-periodic points matches the metric entropy for ergodic measures, extending Katok’s entropy-periodic point connection to a broader nonuniform setting. Together, these results provide a framework for analyzing stability, orbit approximation, and complexity in nonuniformly partially hyperbolic dynamics with potential implications for entropy and periodic-point theory.

Abstract

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a $C^{1+α}$ diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property and quasi-closing property are also investigated. As an application of quasi-closing property, we extend Katok's reslut on the growth of periodoc orbits for hyperbolic ergodic measure to any ergodic measure: the number of quasi-periodic points grows exponentially at least the metric entropy.

Quasi-shadowing property for nonuniformly partially hyperbolic systems

TL;DR

The paper develops a quasi-shadowing theory for nonuniformly partially hyperbolic diffeomorphisms by constructing fake center foliations and proving Hölder continuity of invariant bundles on Pesin blocks, enabling shadowing that adapts to pseudo-orbit movement. It derives quasi-specification and quasi-closing results as corollaries, illustrating robust orbit concatenation and near-periodic behavior under center-direction motion. The main application shows that the exponential growth rate of quasi-periodic points matches the metric entropy for ergodic measures, extending Katok’s entropy-periodic point connection to a broader nonuniform setting. Together, these results provide a framework for analyzing stability, orbit approximation, and complexity in nonuniformly partially hyperbolic dynamics with potential implications for entropy and periodic-point theory.

Abstract

In this paper, we establish a new quasi-shadowing property for any nonuiformly partially hyperbolic set of a diffeomorphism, which is adaptive to the movement of the pseudo-orbit. Moreover, the quasi-specification property and quasi-closing property are also investigated. As an application of quasi-closing property, we extend Katok's reslut on the growth of periodoc orbits for hyperbolic ergodic measure to any ergodic measure: the number of quasi-periodic points grows exponentially at least the metric entropy.
Paper Structure (6 sections, 9 theorems, 102 equations, 4 figures)

This paper contains 6 sections, 9 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Uniform partial hyperbolicity
  3. Hölder continuity of Oseledets splitting
  4. Invariant foliations
  5. Quasi-shadowing property
  6. the growth of quasi-periodic orbits

Key Result

Theorem A

Let $f$ be a $C^{1+\alpha}$ diffeomorphism on $M$ and $\Lambda=\bigcup\limits_{k\ge 1}\Lambda_k(\lambda,\mu,\lambda',\mu',\varepsilon)$ be a nonuniformly partially hyperbolic set. Then $f$ has the quasi-shadowing property for $\Lambda$ in the following sense: for any $\eta>0$ there exists a sequence

Figures (4)

  • Figure 1: Quasi-shadowing
  • Figure 2: Quasi-specification
  • Figure 3: Quasi-closing
  • Figure 4:

Theorems & Definitions (16)

  • Remark
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • ...and 6 more