Generalized $u$-Gibbs measures for $C^\infty$ diffeomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908839760101376 |
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| author | Ovadia, Snir Ben Burguet, David |
| author_facet | Ovadia, Snir Ben Burguet, David |
| contents | We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on smoothly embedded disks subordinated to unstable leaves. As an application, we prove a strong version of the Viana conjecture in any dimension.
Our methods include developing a quantitative approach to high-dimensional Yomdin theory which allows to control the geometry of disks, and introducing a notion of ``measured disks" in order to provide a disintegration by absolutely continuous conditionals. In particular, we provide also a new proof for the case of surfaces (a previous result by the second author) proving directly the absolute continuity of conditionals rather than mere entropy estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized $u$-Gibbs measures for $C^\infty$ diffeomorphisms Ovadia, Snir Ben Burguet, David Dynamical Systems We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on smoothly embedded disks subordinated to unstable leaves. As an application, we prove a strong version of the Viana conjecture in any dimension. Our methods include developing a quantitative approach to high-dimensional Yomdin theory which allows to control the geometry of disks, and introducing a notion of ``measured disks" in order to provide a disintegration by absolutely continuous conditionals. In particular, we provide also a new proof for the case of surfaces (a previous result by the second author) proving directly the absolute continuity of conditionals rather than mere entropy estimates. |
| title | Generalized $u$-Gibbs measures for $C^\infty$ diffeomorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.18238 |