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S-Expansiveness and Zip Shift Maps in Symbolic Dynamics

S. Lamei, P. Mehdipour, W. Vargas

TL;DR

The paper addresses how to characterize S-expansiveness for local homeomorphisms and to represent such dynamics symbolically via zip shift maps. It develops the inverse limit framework $X^f$ and the bilateral extension $\tilde{f}$, introduces $m$-to-1 local homeomorphisms, and uses generators and topological partitions to link expansivity to symbolic coding. The main contributions show that zip shift maps are S-expansive and possess the shadowing property, and that any S-expansive local homeomorphism $f$ is a factor of a zip shift map through a surjection $\pi:\Sigma\to X$ with $\pi\circ\sigma_{\tau}=f\circ\pi$. This establishes a universal symbolic model for a broad class of non-invertible systems and clarifies how expansive behavior can be captured by zip shifts.

Abstract

In this paper, we introduce the concept of S-expansiveness for local homeomorphisms and demonstrate that a class of extended symbolic dynamics, known as zip shift maps, are S-expansive and possess the shadowing property. Furthermore, we prove that any S-expansive local homeomorphism is a factor of a zip shift map.

S-Expansiveness and Zip Shift Maps in Symbolic Dynamics

TL;DR

The paper addresses how to characterize S-expansiveness for local homeomorphisms and to represent such dynamics symbolically via zip shift maps. It develops the inverse limit framework and the bilateral extension , introduces -to-1 local homeomorphisms, and uses generators and topological partitions to link expansivity to symbolic coding. The main contributions show that zip shift maps are S-expansive and possess the shadowing property, and that any S-expansive local homeomorphism is a factor of a zip shift map through a surjection with . This establishes a universal symbolic model for a broad class of non-invertible systems and clarifies how expansive behavior can be captured by zip shifts.

Abstract

In this paper, we introduce the concept of S-expansiveness for local homeomorphisms and demonstrate that a class of extended symbolic dynamics, known as zip shift maps, are S-expansive and possess the shadowing property. Furthermore, we prove that any S-expansive local homeomorphism is a factor of a zip shift map.
Paper Structure (6 sections, 13 theorems, 16 equations)

This paper contains 6 sections, 13 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Preliminary and Main results
  3. Zip shift spaces
  4. Dataavailability
  5. Conflicts of interest
  6. Founding

Key Result

Theorem 2.2

Let $X$ be compact and $f: X\to X$ a continuous surjection. If $f$ is a local homeomorphism, then there exist two positive numbers $\lambda$ and $\mu$ such that each $D\subset X$ with diameter less than $\lambda$ determines a decomposition of the set $f^{-1}(D)$ with the following properties: If, in addition, $X$ is connected, then there is a constant $m > 0$ such that $f : X\to X$ is an m-to-1 m

Theorems & Definitions (37)

  • Definition 2.1: m-to-1 local homeomorphism
  • Theorem 2.2
  • Lemma 2.3: Lebesgue number
  • Definition 2.4: Principal Domain
  • Definition 2.5
  • Definition 2.6: Domain topological partition ML
  • Definition 2.7: Image topological partition ML
  • Definition 2.8: Extended topological partition ML
  • Remark 2.9
  • Definition 2.10: Generator/Weak generator
  • ...and 27 more