Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2016
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| _version_ | 1866914803264520192 |
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| author | Wang, Xiaodong Zhang, Jinhua |
| author_facet | Wang, Xiaodong Zhang, Jinhua |
| contents | We prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ contains two hyperbolic periodic orbits of indices $i$ and $i+k$ respectively and $H(P)$ has no domination of index $j$ for any $j\in\{i+1,\cdots,i+k-1\}$, then there exists a non-hyperbolic ergodic measure whose $(i+l)^{th}$ Lyapunov exponent vanishes for any $l\in\{1,\cdots, k\}$, and whose support is the whole homoclinic class.
We also prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ has a dominated splitting of the form $E\oplus F\oplus G$, such that the center bundle $F$ has no finer dominated splitting, and $H(p)$ contains a hyperbolic periodic orbit $Q_1$ of index $\dim(E)$ and a hyperbolic periodic orbit $Q_2$ whose absolute Jacobian along the bundle $F$ is strictly less than $1$, then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle $F$ all vanish and whose support is the whole homoclinic class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1604_03342 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes Wang, Xiaodong Zhang, Jinhua Dynamical Systems We prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ contains two hyperbolic periodic orbits of indices $i$ and $i+k$ respectively and $H(P)$ has no domination of index $j$ for any $j\in\{i+1,\cdots,i+k-1\}$, then there exists a non-hyperbolic ergodic measure whose $(i+l)^{th}$ Lyapunov exponent vanishes for any $l\in\{1,\cdots, k\}$, and whose support is the whole homoclinic class. We also prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ has a dominated splitting of the form $E\oplus F\oplus G$, such that the center bundle $F$ has no finer dominated splitting, and $H(p)$ contains a hyperbolic periodic orbit $Q_1$ of index $\dim(E)$ and a hyperbolic periodic orbit $Q_2$ whose absolute Jacobian along the bundle $F$ is strictly less than $1$, then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle $F$ all vanish and whose support is the whole homoclinic class. |
| title | Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1604.03342 |