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arXiv · 2504.18993

Genericity of ergodicity for Sobolev homeomorphisms

Assis Azevedo·Davide Azevedo·Mário Bessa·Maria Joana Torres
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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.

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Assis Azevedo, Davide Azevedo, Mário Bessa, Maria Joana Torres. 2025-04-26. Genericity of ergodicity for Sobolev homeomorphisms. https://arxiv.org/abs/2504.18993

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