Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918458973749248 |
|---|---|
| author | Tsujii, Masato Zhang, Zhiyuan |
| author_facet | Tsujii, Masato Zhang, Zhiyuan |
| contents | Let $s > 1$ be a large integer, and let $f$ be a diffeomorphism sufficiently close in the $C^{s}$-topology to the time-1 map of a $C^{s}$ generic volume-preserving Anosov flow on a $3$-dimensional compact manifold. We show that for any probability measure $μ$ with smooth density, $f^n_* μ$ converges exponentially fast to a common limit measure with full support.
As corollaries, we show the following: $f$ is topologically mixing; $f$ has a unique physical measure with basin of full Lebesgue measure, which is also the unique u-Gibbs state; if $f$ is volume preserving, then $f$ is exponentially mixing with respect to the volume form.
As applications, we give a class of time-1 maps of transitive Anosov flows non-approximable in $C^{s}$ by Axiom A maps, giving negative answer to a question of Palis-Pugh (1974); the first example of a $C^{s}$-stably transitive time-1 map of Anosov flow, a question mentioned in Bonatti-Guelman (2010), Rodriguez Hertz (2010); as well as the first example of a $C^{s}$-stably transitive diffeomorphism without periodic points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18891 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow Tsujii, Masato Zhang, Zhiyuan Dynamical Systems Let $s > 1$ be a large integer, and let $f$ be a diffeomorphism sufficiently close in the $C^{s}$-topology to the time-1 map of a $C^{s}$ generic volume-preserving Anosov flow on a $3$-dimensional compact manifold. We show that for any probability measure $μ$ with smooth density, $f^n_* μ$ converges exponentially fast to a common limit measure with full support. As corollaries, we show the following: $f$ is topologically mixing; $f$ has a unique physical measure with basin of full Lebesgue measure, which is also the unique u-Gibbs state; if $f$ is volume preserving, then $f$ is exponentially mixing with respect to the volume form. As applications, we give a class of time-1 maps of transitive Anosov flows non-approximable in $C^{s}$ by Axiom A maps, giving negative answer to a question of Palis-Pugh (1974); the first example of a $C^{s}$-stably transitive time-1 map of Anosov flow, a question mentioned in Bonatti-Guelman (2010), Rodriguez Hertz (2010); as well as the first example of a $C^{s}$-stably transitive diffeomorphism without periodic points. |
| title | Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2604.18891 |