Zip shift Space

TL;DR

Zip shift spaces $\Sigma_{\mathcal{A},\mathcal{A}'}$ provide finite-alphabet, two-sided symbolic models for non-invertible dynamics by pairing forward and backward alphabets through a transition map $\varphi_n$ and a zip shift map $\sigma$. The paper develops the foundations, including higher block and sliding block code generalizations, and introduces labeled/backward-labeled graphs to compute pre-images in the sofic case, extending to the Curtis–Hedlund–Lyndon framework for zip shifts. A central result is the construction of an $N$-to-$1$ endomorphism horseshoe that is conjugate to a zip shift map, with a concrete code map $\varsigma$ demonstrating the correspondence and providing explicit $2$-to-$1$ and $4$-to-$1$ examples. The study further analyzes stable/unstable sets, periodic/pre-periodic points, and homoclinic/heteroclinic orbits within this zip-shift framework, yielding detailed combinatorial descriptions of orbit structures and pre-image multiplicities. Together, these results offer a finite-alphabet, two-sided symbolic framework for analyzing non-invertible hyperbolic dynamics and their ergodic properties, with potential applications to classification and dynamical systems theory.

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.

Figures (8)

• Figure 1: The right graph is the labeled graph $\mathcal{G}$ of the left graph.
• Figure 2: This graph represents a zip shift space $\Sigma_{\mathcal{A},\mathcal{A}'}$ with $\mathcal{A}'=\{1,2,3,4,5,6\}$ and $\mathcal{A}=\{a,b,c,d,e\}$. Walking in direction of edges and considering the numbers on edges, we get entries with non-negative indices and walking in opposite direction of edges and picking the letters written on edges, we get the entries with negative indices of a point $x\in \Sigma_{\mathcal{A},\mathcal{A}'}$.
• Figure 3: Graphs (a), (b), (c), and (e) represent the labeled or vertex-like graphs for the even shift $\Sigma_{\mathcal{A}'}$. Graphs (d) and (f) represent the labeled graphs for $\Sigma_{\mathcal{A},\mathcal{A}'}$ with the transition map described in Example \ref{['sofic eg']}.
• Figure 6: The rectangles $H_a$ and $H_b$ are fromed from the first iterate of $f$. The rectangles $H_{aa}$, $H_{ab}$, $H_{ba}$ and $H_{bb}$ are shown with labels $aa$, $ab$, $ba$ and $bb$.
• Figure 7: The first backward iterate of the horseshoe map $f$ for $N=2$.

Theorems & Definitions (35)

• Example 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• proof
• Corollary 2.9