Pointwise dimension, entropy and Lyapunov exponents for $C^1$ maps
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- by Wen Huang and Pengfei Zhang PDF
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Abstract:
Let $f$ be a $C^1$ self-map on a smooth Riemannian manifold $M$, $\mu$ be an $f$-invariant ergodic Borel probability measure with a compact support $\Lambda$ and $\chi ^1_\mu >\cdots >\chi ^s_\mu$ be the Lyapunov exponents of $\mu$ with respect to $f$. If $\chi ^1_\mu >0$, then we give a lower bound of the lower pointwise dimension of $\mu$ in terms of $\chi ^1_\mu$ and of the entropy $h_\mu (f)$. Moreover, if $Df_{\{\cdot \}}$ is non-degenerate on $\Lambda$ and $\chi ^s_\mu >0$, then we give an upper bound of the upper pointwise dimension of $\mu$ in terms of $\chi ^s_\mu$ and of the entropy $h_\mu (f)$. Furthermore, if $f$ is $C^{1+\alpha }$ for some $\alpha >0$, then the non-degeneracy condition can be removed.
As direct applications of the above results, we also give the corresponding lower and upper bounds of some classical characteristics of dimensional type of $\mu$ in terms of the Lyapunov exponents and of the entropy $h_\mu (f)$.
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Additional Information
- Wen Huang
- Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei Anhui 230026, People’s Republic of China
- MR Author ID: 677726
- Email: wenh@mail.ustc.edu.cn
- Pengfei Zhang
- Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei Anhui 230026, People’s Republic of China
- Email: pfzhang5@mail.ustc.edu.cn
- Received by editor(s): August 30, 2009
- Received by editor(s) in revised form: November 30, 2010
- Published electronically: July 11, 2012
- Additional Notes: The first author was supported by the NSFC, the Fok Ying Tung Education Foundation, FANEDD (Grant 200520) and the Fundamental Research Funds for the Central Universities.
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 6355-6370
- MSC (2010): Primary 37C45; Secondary 37D25
- DOI: https://doi.org/10.1090/S0002-9947-2012-05527-9
- MathSciNet review: 2958939