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Genericity of ergodicity for Sobolev homeomorphisms

Assis Azevedo, Davide Azevedo, Mário Bessa, Maria Joana Torres

TL;DR

The paper develops a weak Lusin-type theorem in the Sobolev $(1,p)$ setting for volume-preserving Lipschitz homeomorphisms and proves that ergodicity is generic in $\mathcal{M}^{1,p}_\lambda(X)$ for $0<p<1$, providing a Sobolev analogue of the Oxtoby–Ulam theorem. It also shows that topological transitivity is generic when $1\le p<d-1$, using a novel perturbation mechanism based on pseudo-rings and localized mass relocation. A key technical contribution is a perturbation theorem that constructs volume-preserving diffeomorphisms with precise control of $\|\cdot\|_\infty$ and $W^{1,p}$ norms, enabling Lusin-type approximations and transitivity arguments. The work highlights sharp regime differences across Sobolev indices, includes an explicit example showing limits of the Sobolev functional structure, and opens avenues for further exploration of entropy and ergodicity in Sobolev conservative dynamics.

Abstract

In this paper we obtain a weak version of Lusin's theorem in the Sobolev-$(1,p)$ uniform closure of volume preserving Lipschitz homeomorphisms on closed and connected $d$-dimensional manifolds, $d \geq 2$ and $0<p<1$. With this result at hand we will be able to prove that the ergodic elements are generic. This establishes a version of Oxtoby and Ulam theorem for this Sobolev class. We also prove that, for $1\leq p<d-1$, the topological transitive maps are generic.

Genericity of ergodicity for Sobolev homeomorphisms

TL;DR

The paper develops a weak Lusin-type theorem in the Sobolev setting for volume-preserving Lipschitz homeomorphisms and proves that ergodicity is generic in for , providing a Sobolev analogue of the Oxtoby–Ulam theorem. It also shows that topological transitivity is generic when , using a novel perturbation mechanism based on pseudo-rings and localized mass relocation. A key technical contribution is a perturbation theorem that constructs volume-preserving diffeomorphisms with precise control of and norms, enabling Lusin-type approximations and transitivity arguments. The work highlights sharp regime differences across Sobolev indices, includes an explicit example showing limits of the Sobolev functional structure, and opens avenues for further exploration of entropy and ergodicity in Sobolev conservative dynamics.

Abstract

In this paper we obtain a weak version of Lusin's theorem in the Sobolev- uniform closure of volume preserving Lipschitz homeomorphisms on closed and connected -dimensional manifolds, and . With this result at hand we will be able to prove that the ergodic elements are generic. This establishes a version of Oxtoby and Ulam theorem for this Sobolev class. We also prove that, for , the topological transitive maps are generic.
Paper Structure (17 sections, 13 theorems, 79 equations, 2 figures)

This paper contains 17 sections, 13 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. To be or not to be ... ergodic
  3. The Sobolev class
  4. A volume preserving Sobolev Lusin theorem
  5. Towards to Oxtoby and Ulam theorem
  6. Genericity of topological transitivity
  7. Open problems
  8. Preliminaries
  9. Automorphisms and homeomorphisms of $(X,\lambda)$
  10. Sobolev maps
  11. A key perturbation theorem
  12. Pseudo-ring
  13. Proof of Theorem \ref{['key:pert2']}
  14. Proof of Volume preserving Sobolev weak Lusin theorem (Theorem \ref{['crucial']})
  15. Proof of Sobolev Oxtoby-Ulam theorem (Theorem \ref{['OU']})
  16. ...and 2 more sections

Key Result

Theorem A

Let $X$ be a closed connected $d$-dimensional manifold, $d \geq 2$ and $0<p<1$. Let $g \in \mathcal{G}_\lambda(X)$, $h\in\overline{\textrm{Lip}_\lambda(X)}^{\,p}$ and consider $\delta>0$. Then given any weak topology neighbourhood $\mathcal{W}$ of $g$, there exists $f \in\textrm{Lip}_\lambda(X)$ suc

Figures (2)

  • Figure 1: Graphic of $f_5$ (with $a=\frac{1}{25}$) and of the identity.
  • Figure 2: An illustration of the example.

Theorems & Definitions (36)

  • Theorem A: Volume preserving Sobolev weak Lusin theorem
  • Theorem B: Sobolev Oxtoby-Ulam theorem
  • Theorem C: Sobolev generic transitivity
  • Example 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 26 more