An N-to-1 Smale Horseshoe

TL;DR

This work extends Conley–Moser theory to $N$-to-$1$ local diffeomorphisms and encodes the resulting Smale horseshoes with a zip-shift symbolic system, achieving a topological conjugacy between the $N$-to-$1$ horseshoe and a two-symbol zip shift. It constructs an invariant Cantor set $\Lambda$ with chaotic symbolic dynamics and proves the existence of a conjugacy to a zip-shift space, both in topological and differentiable settings via sector bundles. The paper then establishes inverse-limit and $C^1$-structural stability for the non-invertible system, showing perturbations preserve the conjugate zip-shift dynamics on the invariant set. Overall, it provides a rigorous, unified framework for non-invertible hyperbolic endomorphisms, linking geometric hyperbolicity, symbolic coding, and stability analyses through the Zip Shift formalism.

Abstract

In this work we extend the Conley-Moser Theorem for $N$-to-1 local diffeomorphisms. By the aim of some extended symbolic dynamics we encode generalized $N$-to-1 horseshoe maps and as a corollary their structural stability is verified.

Figures (7)

• Figure 1: First and second iteration of a 2-to-1 horseshoe map $f$ on $Q$
• Figure 2: First pre-image of the $2$-to-1 horseshoe map $f$ on $Q$
• Figure 3: $Z=\{a\}$ and $S=\{0,1\}$.
• Figure 4: A 2-to-1 horizontal-vertical curve (HV-curve)
• Figure 5: $H_{lm}$ and $V_{ij}$ for $n = 2$.

Theorems & Definitions (44)

• Definition 2.1: An $N$-to-1 local homeomorphism
• Example 2.2
• Lemma 3.1
• Definition 3.2: Full zip shift map
• Example 3.3
• Example 3.4
• Definition 3.5: Expansivity
• Definition 3.6: Periodic and pre-periodic points
• Remark 3.7
• Example 3.8