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On the Ergodicity of Rotation Extensions of Hyperbolic Endomorphisms

Fernando Micena, Raúl Ures

TL;DR

This work advances the ergodic theory of noninvertible, partially hyperbolic endomorphisms by analyzing rotation extensions over Anosov bases. Employing inverse-limit dynamics, SRB theory, and a Brin-style accessibility framework, it proves that nonintegrability of the stable–unstable splitting drives ergodicity and topological mixing for skew products on M × S^1, and that accessibility yields topological transitivity and ergodicity under broad conditions. It also establishes robust transitivity and stable ergodicity in low-center-dimensional settings and provides concrete examples with rotation extensions illustrating nonintegrability and ergodicity. The results extend Pesin-type arguments to noninvertible systems, offering practical criteria for ergodicity and stable ergodicity of rotation extensions and related skew products in a variety of base dynamics, including expanding maps. Overall, the paper broadens the Pugh–Shub program to endomorphisms, highlighting the central role of accessibility and the geometry of the stable/unstable splitting in driving ergodic behavior.

Abstract

We study the ergodicity of partially hyperbolic endomorphisms, focusing on skew products where the base dynamics are governed by Anosov endomorphisms. For this family, we establish ergodicity and prove that accessibility holds for an open and dense subset. By analyzing the topological implications of accessibility, we demonstrate that conservative accessible partially hyperbolic endomorphisms are topologically transitive. Leveraging accessibility, we further show ergodicity for skew products with $\mathbb{S}^1$-fibers. Finally, although out the context of rotation extensions, we prove ergodic stability results for partially hyperbolic endomorphisms with $\dim(E^c) = 1.$

On the Ergodicity of Rotation Extensions of Hyperbolic Endomorphisms

TL;DR

This work advances the ergodic theory of noninvertible, partially hyperbolic endomorphisms by analyzing rotation extensions over Anosov bases. Employing inverse-limit dynamics, SRB theory, and a Brin-style accessibility framework, it proves that nonintegrability of the stable–unstable splitting drives ergodicity and topological mixing for skew products on M × S^1, and that accessibility yields topological transitivity and ergodicity under broad conditions. It also establishes robust transitivity and stable ergodicity in low-center-dimensional settings and provides concrete examples with rotation extensions illustrating nonintegrability and ergodicity. The results extend Pesin-type arguments to noninvertible systems, offering practical criteria for ergodicity and stable ergodicity of rotation extensions and related skew products in a variety of base dynamics, including expanding maps. Overall, the paper broadens the Pugh–Shub program to endomorphisms, highlighting the central role of accessibility and the geometry of the stable/unstable splitting in driving ergodic behavior.

Abstract

We study the ergodicity of partially hyperbolic endomorphisms, focusing on skew products where the base dynamics are governed by Anosov endomorphisms. For this family, we establish ergodicity and prove that accessibility holds for an open and dense subset. By analyzing the topological implications of accessibility, we demonstrate that conservative accessible partially hyperbolic endomorphisms are topologically transitive. Leveraging accessibility, we further show ergodicity for skew products with -fibers. Finally, although out the context of rotation extensions, we prove ergodic stability results for partially hyperbolic endomorphisms with

Paper Structure

This paper contains 10 sections, 23 theorems, 28 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic Preliminaries of Pesin Theory
  3. Ergodic Decomposition
  4. Proof of Theorem \ref{['mainA']}
  5. Some Consequences of Theorem \ref{['mainA']}
  6. Proof of Theorem \ref{['accBrin']}
  7. Proof of Theorem \ref{['mainB']}
  8. Proof of Theorem \ref{['mainD']}
  9. Examples
  10. Proof of Theorem \ref{['mainC']}

Key Result

Theorem 1.6

Let $f:M\rightarrow M$ be a transitive partially hyperbolic endomorphism. For each $\sigma \in \{u,c\}$ the following dichotomy holds:

Figures (4)

  • Figure 1: The leaves $K_1$ and $K_2$ connected by an unstable leaf.
  • Figure 2: A $su-$path connecting $x$ and $y$.
  • Figure 3: Situation 1: Brin's quadrilateral configuration.
  • Figure 4: Situation 2: Two $su-$leaves from the same stable leaf.

Theorems & Definitions (40)

  • Conjecture 1.1: Pugh-Shub conjecture
  • Conjecture 1.2: Essential Accessibility Implies Ergodicity
  • Conjecture 1.3: Density of Stable Accessibility
  • Definition 1.4: (absolute) Partially Hyperbolic Endomorphism
  • Definition 1.5: Special Partially Hyperbolic Endomorphism
  • Theorem 1.6: micsa
  • Proposition 1.7: micsa
  • Corollary 1.8
  • Theorem 1: Integrability of the Stable-Unstalbe Splitting
  • Corollary 1.9
  • ...and 30 more