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Limit Bi-Shadowing for Semi-Hyperbolic Systems

Linying Cheng, Haiye Guo

Abstract

This paper investigates the shadowing properties in semi-hyperbolic systems. We introduce three classes of shadowing properties defined on families of manifolds, and prove that a semi-hyperbolic family possesses the $L^p$ bi-shadowing property, the limit bi-shadowing property, and the asymptotic bi-shadowing property under certain conditions. The proof strategy is to transform the shadowing problem into a fixed-point problem, and then apply the Brouwer fixed-point theorem to complete the verification.

Limit Bi-Shadowing for Semi-Hyperbolic Systems

Abstract

This paper investigates the shadowing properties in semi-hyperbolic systems. We introduce three classes of shadowing properties defined on families of manifolds, and prove that a semi-hyperbolic family possesses the bi-shadowing property, the limit bi-shadowing property, and the asymptotic bi-shadowing property under certain conditions. The proof strategy is to transform the shadowing problem into a fixed-point problem, and then apply the Brouwer fixed-point theorem to complete the verification.

Paper Structure

This paper contains 9 sections, 4 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notations
  4. Statements of main results
  5. --bi-shadowing property
  6. Finite --bi-shadowing property
  7. Proof of the --bi-shadowing property
  8. Limit bi-shadowing property
  9. Asymptotic bi-shadowing property

Key Result

Lemma 3.1

Let $\lambda_u=\inf\limits_{i\in\mathbb{Z}}\{\lambda_{ui}\}>1$. Then for sufficiently small $\delta>0$, there exists a constant $\tilde{\lambda}_u\in (1, \lambda_u)$ such that for any $w_i, w_i'\in E_{x_i}^u(L\delta)$, we have

Theorems & Definitions (11)

  • Definition 2.1
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 1 more