Zip Shift Maps in Symbolic Dynamics

• Zip shift maps are finite-to-1 local homeomorphisms that encode both forward evolution and backward preimage structures in noninvertible systems.
• They exhibit S-expansiveness and the shadowing property, enabling robust symbolic coding and topological conjugacy with complex dynamical systems.
• Their construction provides a universal framework for modeling chaotic dynamics through symbolic representations of noninvertible maps.

A zip shift map is a finite-to-1 local homeomorphism defined on a symbolic dynamical space organized over two alphabets, designed to encode both forward evolution and backward (preimage) structure for noninvertible endomorphisms. These maps form extended symbolic models supporting topological conjugacy with systems such as N-to-1 uniformly hyperbolic horseshoe maps, and they possess fundamental properties including S-expansiveness and the pseudo-orbit tracing (shadowing) property. Recent research has formalized their construction, dynamical properties, and symbolic coding power, clarifying their utility in modeling noninvertible dynamics.

1. Definition and Construction of Zip Shift Maps

Let $S = \{0, 1, ..., N-1\}$ and $Z = \{a_1, ..., a_k\}$ be finite alphabets ($N \geq k$), and let $\tau: S \to Z$ be a surjective transition map. Consider bi-infinite sequences $\bar{t} = (t_i)_{i \in \mathbb{Z}}$ over $S$. Construct an associated sequence $\bar{x} = (x_i)_{i \in \mathbb{Z}}$ in $S \cup Z$ by

$$x_i = \begin{cases} t_i, & \text{if } i \geq 0, \ \tau(t_i), & \text{if } i < 0. \end{cases}$$

The zip shift space $\Sigma$ is the collection of all such sequences $\bar{x}$.

The zip shift map $\sigma_\tau: \Sigma \to \Sigma$ is defined coordinate-wise by

$$(\sigma_\tau(\bar{x}))_i = \begin{cases} x_{i+1}, & \text{if } i \neq -1, \ \tau(x_0), & \text{if } i = -1. \end{cases}$$

A natural metric $d$ on $\Sigma$ is given by

$$M(x, y) = \begin{cases} \infty, & x = y, \ \min\{|i| : x_i \neq y_i\}, & x \neq y, \end{cases} \qquad d(x, y) = \frac{1}{\lambda^{M(x, y)}}$$

with $\lambda = |S|$, making $\Sigma$ a Cantor set.

2. Local Homeomorphism and Finite-to-1 Covering Structure

Zip shift maps are finite-to-1 local homeomorphisms: although globally noninvertible (due to the non-injectivity of the transition map $\tau$ and the loss of past information), locally, in the topology induced by the product structure and metric $d$, the map behaves as a homeomorphism with each point having finitely many preimages. Typically, the preimage structure is dictated by the fiber cardinality of $\tau$ (the number of $s \in S$ mapping to a given $z \in Z$ at negative indices).

This property has significant implications for symbolic representations of noninvertible dynamical systems, as it equips the system with robust topological features (e.g., cylinder sets and expansiveness) while faithfully encoding the ambiguity inherent in backward orbits.

3. S-Expansiveness and Shadowing in Zip Shift Maps

A map $f:X \to X$ is S-expansive if there exists $\gamma > 0$ such that for any distinct $x, y \in X$, some $n \in \mathbb{Z}$ satisfies $d(f^n(x), f^n(y)) > \gamma$ (with backward iterates defined via local homeomorphic branches).

Zip shift maps $\sigma_\tau$ on $\Sigma$ are S-expansive for $\gamma = 1/2$. Given any two distinct points, their earliest divergence (forward or backward coordinate) is brought forward to the present under the iterated action, producing a separation exceeding $\gamma$.

The shadowing property (pseudo-orbit tracing) is satisfied for zip shift maps as well. For $\epsilon > 0$, one selects $m \geq 1$ with $\lambda^{-m} < \epsilon$; any $\delta$-pseudo orbit with $\delta = \lambda^{-(m+1)}$ can be concatenated (through overlapping blocks) into a genuine orbit within $\epsilon$ of the pseudo orbit, as formalized in the cited construction.

4. Dynamical Conjugacy and Modeling Noninvertible Systems

Zip shift maps provide symbolic models for noninvertible systems such as N-to-1 uniformly hyperbolic horseshoe maps (Lamei et al., 16 Feb 2025). For instance, one can construct a dynamical system $f: \Lambda \to \Lambda$ (e.g., the geometric horseshoe map) and build a Markov partition matching the forward dynamics to symbols in $S$, while the past is encoded via the "zipping" process using $\tau$.

This yields a topological conjugacy: $f = \nu^{-1} \circ \sigma_\tau \circ \nu$ where $\nu: \Lambda \to \Sigma$ is a coding homeomorphism. All dynamical properties (periodic points, entropy, preimage sets) are transferred to the symbolic structure. The finite-to-1 property reflects the noninvertibility; the backward ambiguity of preimage selection corresponds to the multiple histories in the zip shift space.

5. Stable/Unstable Sets and Orbit Structures

Stable ($W^s$) and unstable ($W^u$) sets are encoded in the zip shift space via tail agreement: $$W^s(s) = \{ t \in \Sigma : \exists N' \text{ s.t. } t_n = s_n \,\, \forall n \geq N' \}$$

$$W^u(s) = \{ r \in \Sigma : \exists N \text{ s.t. } r_{-n} = s_{-n} \,\, \forall n \geq N \}$$

Homoclinic points are defined by the requirement that their coordinates agree with those of a periodic point in both the forward and backward tails for sufficiently large $|n|$.

Preimage studies leverage the zip shift structure: for a given point $x \in \Sigma$, the set of preimages under $\sigma_\tau$ is determined by the possible inverse selections of $s \in S$ mapping to $z \in Z$ (and further steered by backward labeled graphs or sofic codes where relevant).

6. Factor and Universality Results for S-Expansive Local Homeomorphisms

A central theorem asserts that any S-expansive $m$-to-1 local homeomorphism $f: X \to X$ of a compact connected metric space is a factor of a zip shift map (Lamei et al., 14 Oct 2025). That is, there exist sets $S$, $Z$, an invariant subset $\Sigma \subset \Sigma_{Z,S}$, a zip shift map $\sigma_\tau: \Sigma \to \Sigma$, and a surjection $\pi: \Sigma \to X$ such that $\pi \circ \sigma_\tau = f \circ \pi$. This establishes zip shift maps as universal symbolic models for S-expansive local homeomorphisms. S-expansiveness is further characterized by the existence of (weak) generators, providing symbolic codings central to the paper of conjugacy and classification.

7. Connections to Sliding Block Codes, Covering Maps, and Generalizations

Zip shift maps can be interpreted as generalized sliding block codes with particular covering properties (Willis, 2010). For shift spaces over infinite alphabets or ultragraph contexts, analogous constructions and Curtis–Hedlund–Lyndon theorems confirm that continuous, shift-commuting maps—often encoding variable anticipation or zipping data—can always be described by finitely defined local rules or partitions (Gonçalves et al., 2015, Gonçalves et al., 2018). Within finite settings, zip shift maps often realize $k$-fold covering maps derived from regressive block maps, ensuring local homeomorphicity and a finite preimage structure.

Summary Table: Properties of Zip Shift Maps

Property	Description	Source Paper
Finite-to-1 covering	Each point has finitely many preimages	(Lamei et al., 16 Feb 2025, Willis, 2010)
S-expansiveness	Uniform separation in orbits	(Lamei et al., 14 Oct 2025)
Shadowing property	Tracing pseudo-orbits by genuine orbits	(Lamei et al., 14 Oct 2025)
Universal factorability	Every S-expansive l.h. is a zip shift factor	(Lamei et al., 14 Oct 2025)
Sliding block code form	Locally determined by symbolic rule	(Willis, 2010, Gonçalves et al., 2018)
Stable/unstable sets	Encoded by tail agreement in $\Sigma$	(Lamei et al., 16 Feb 2025)

Zip shift maps now form a foundational class for the symbolic modeling of noninvertible, S-expansive local homeomorphisms, bridging finite and infinite alphabet settings, supporting robust coding structures, and enabling explicit conjugacies with a broad array of chaotic dynamical phenomena.