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Some Remarks on Anosov Families

Jeovanny de Jesus Muentes Acevedo, Raquel Ribeiro

TL;DR

The paper extends classical hyperbolic theory to non-stationary dynamics by developing a rigorous framework for Anosov families: sequences of diffeomorphisms with a $D\mathcal{F}$-invariant hyperbolic splitting $E^s\oplus E^u$ on a disjoint union $\mathcal{M}$. For the class $\mathcal{A}^2_b(\mathcal{M})$, it establishes canonical coordinates via exponential charts, proves expansiveness, and proves a Shadowing Lemma, while in the constant-base case it constructs Markov partitions, enabling non-autonomous symbolic coding. These results generalize fundamental autonomous hyperbolic phenomena to non-stationary systems, facilitating stability analysis and numerical shadowing in time-varying contexts and guiding future extensions to flows and broader base manifolds. Overall, the work provides a robust non-autonomous hyperbolic theory linking geometric, symbolic, and dynamical properties of Anosov families.

Abstract

We study Anosov families which are sequences of diffeomorphisms along compact Riemannian manifolds such that the tangent bundle split into expanding and contracting subspaces. In this paper we prove that a certain class of Anosov families: (i) admit canonical coordinates (ii) are expansive, (iii) satisfy the shadowing property, and (iv) exhibit a Markov partition.

Some Remarks on Anosov Families

TL;DR

The paper extends classical hyperbolic theory to non-stationary dynamics by developing a rigorous framework for Anosov families: sequences of diffeomorphisms with a -invariant hyperbolic splitting on a disjoint union . For the class , it establishes canonical coordinates via exponential charts, proves expansiveness, and proves a Shadowing Lemma, while in the constant-base case it constructs Markov partitions, enabling non-autonomous symbolic coding. These results generalize fundamental autonomous hyperbolic phenomena to non-stationary systems, facilitating stability analysis and numerical shadowing in time-varying contexts and guiding future extensions to flows and broader base manifolds. Overall, the work provides a robust non-autonomous hyperbolic theory linking geometric, symbolic, and dynamical properties of Anosov families.

Abstract

We study Anosov families which are sequences of diffeomorphisms along compact Riemannian manifolds such that the tangent bundle split into expanding and contracting subspaces. In this paper we prove that a certain class of Anosov families: (i) admit canonical coordinates (ii) are expansive, (iii) satisfy the shadowing property, and (iv) exhibit a Markov partition.

Paper Structure

This paper contains 9 sections, 13 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Anosov Families and Definitions
  3. Stable and Unstable Manifolds for Anosov Families
  4. Canonical Coordinates for Anosov Families
  5. Expansiveness and Shadowing
  6. Expansiveness
  7. Shadowing Lemma
  8. Markov partition for Anosov family
  9. Further Generalizations

Key Result

Theorem 3.3

For each $p\in \mathcal{M}$, $\mathcal{W}^{u}(p,\epsilon)$ is a differentiable submanifold of $\mathcal{M}$ and there exists $K^{u}>0$ such that:

Figures (1)

  • Figure 4.1: $z=\text{exp}_{p}^{-1}(q)$; the vertical cone is $\text{exp}_{p}^{-1}(q)+D (\text{exp}_{p}^{-1})_{q}( K_{\alpha,q}^{u})$; the horizontal cone is $K_{\alpha,p}^{u}$; the curve inside $K_{\alpha,p}^{u}$ is $\mathcal{W} ^{s}(p,\epsilon)$.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 30 more