Absolute continuity of stationary measures
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929516978372608 |
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| author | Brown, Aaron Lee, Homin Obata, Davi Ruan, Yuping |
| author_facet | Brown, Aaron Lee, Homin Obata, Davi Ruan, Yuping |
| contents | Let $f$ and $g$ be two volume preserving, Anosov diffeomorphisms on $\mathbb{T}^2$, sharing common stable and unstable cones. In this paper, we find conditions for the existence of (dissipative) neighborhoods of $f$ and $g$, $\mathcal{U}_f$ and $\mathcal{U}_g$, with the following property: for any probability measure $μ$, supported on the union of these neighborhoods, and verifying certain conditions, the unique $μ$-stationary SRB measure is absolutely continuous with respect to the ambient Haar measure. Our proof is inspired in the work of Tsujii for partially hyperbolic endomorphisms [Tsu05]. We also obtain some equidistribution results using the main result of [BRH17]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18252 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Absolute continuity of stationary measures Brown, Aaron Lee, Homin Obata, Davi Ruan, Yuping Dynamical Systems 37D20, 37C40, 37C85, 37H12 Let $f$ and $g$ be two volume preserving, Anosov diffeomorphisms on $\mathbb{T}^2$, sharing common stable and unstable cones. In this paper, we find conditions for the existence of (dissipative) neighborhoods of $f$ and $g$, $\mathcal{U}_f$ and $\mathcal{U}_g$, with the following property: for any probability measure $μ$, supported on the union of these neighborhoods, and verifying certain conditions, the unique $μ$-stationary SRB measure is absolutely continuous with respect to the ambient Haar measure. Our proof is inspired in the work of Tsujii for partially hyperbolic endomorphisms [Tsu05]. We also obtain some equidistribution results using the main result of [BRH17]. |
| title | Absolute continuity of stationary measures |
| topic | Dynamical Systems 37D20, 37C40, 37C85, 37H12 |
| url | https://arxiv.org/abs/2409.18252 |