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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Pointwise dimension, entropy and Lyapunov exponents for $ C^1$ maps


Authors: Wen Huang and Pengfei Zhang
Journal: Trans. Amer. Math. Soc. 364 (2012), 6355-6370
MSC (2010): Primary 37C45; Secondary 37D25
Published electronically: July 11, 2012
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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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05527-9
PII: S 0002-9947(2012)05527-9
Keywords: Pointwise dimension, local entropy, Lyapunov exponents, ergodic measures
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.
Article copyright: © Copyright 2012 American Mathematical Society