Zip shift Space

TL;DR

This paper introduces zip shift spaces in symbolic dynamics, establishing their properties, conjugacies with hyperbolic horseshoe maps, and detailed orbit structure analysis, advancing understanding of complex dynamical systems.

Contribution

It defines zip shift spaces, explores their conjugacy with certain endomorphisms, and analyzes orbit structures and pre-image properties in hyperbolic dynamics.

Findings

Zip shift spaces are conjugate to N-to-1 hyperbolic horseshoe maps.

Pre-image structures and stable/unstable sets are characterized.

The paper provides a detailed description of homoclinic orbits in this context.

Abstract

We introduce a new extension in symbolic dynamics on two sets of alphabets, called the zip shift space. In finite case, it represents a finite-to-1 local homeomorphism called zip shift map. Such extension, offers a conjugacy between some endomorphisms and some zip shift map over two-sided space with finite sets of alphabets. As an application, the topological conjugacy of an N-to-1 uniformly hyperbolic horseshoe map with a zip shift map and its orbit structure is investigated. Moreover, the pre-image studies over zip shift space and the concepts of stable and unstable sets and homoclinic orbits, with a precise description for N-to-1 horseshoe are illustrated.