On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929387437293568 |
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| author | Mohammadpour, Reza |
| author_facet | Mohammadpour, Reza |
| contents | In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds of any dimension that admit a dominated splitting $E^{cs} \oplus E^{cu}.$ We prove that if the Lyapunov exponents along $E^{cu}$ are positive for Lebesgue almost every point, then a map $f$ is non-uniformly expanding along $E^{cu}$ under the assumption that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. As a result, there exists a physical SRB measure for a $C^{1+α}$ diffeomorphism map $f$ that admits a dominated splitting $E^{s} \oplus E^{cu}$ under assumptions that $f$ has non-zero Lyapunov exponents for Lebesgue almost every point and that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_11149 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems Mohammadpour, Reza Dynamical Systems Mathematical Physics Differential Geometry In this paper we consider $C^{1}$ diffeomorphisms on compact Riemannian manifolds of any dimension that admit a dominated splitting $E^{cs} \oplus E^{cu}.$ We prove that if the Lyapunov exponents along $E^{cu}$ are positive for Lebesgue almost every point, then a map $f$ is non-uniformly expanding along $E^{cu}$ under the assumption that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. As a result, there exists a physical SRB measure for a $C^{1+α}$ diffeomorphism map $f$ that admits a dominated splitting $E^{s} \oplus E^{cu}$ under assumptions that $f$ has non-zero Lyapunov exponents for Lebesgue almost every point and that the cocycle $Df_{|E^{cu}(f)}^{-1}$ has a dominated splitting with index 1 on the support of an ergodic Lyapunov maximizing observable measure. |
| title | On positive Lyapunov exponents along $E^{cu}$ and non-uniformly expanding for partially hyperbolic systems |
| topic | Dynamical Systems Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2112.11149 |