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Entropy of general diffeomorphisms on line

Part of: Topological dynamics Low-dimensional dynamical systems

Published online by Cambridge University Press:  18 February 2020

BAOLIN HE
BAOLIN HE*
Affiliation:
Shanghai Normal University, Mathematics, Shanghai, 200234, PR China email hebaolin@shnu.edu.cn
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Abstract

We study the continuity of topological entropy of general diffeomorphisms on line. First, we prove that the entropy map is continuous with respect to the strong $C^{0}$-topology on the union of uniformly topologically hyperbolic diffeomorphisms contained in $\text{Diff}_{0}^{r}(\mathbb{R})$ (whose first derivative is uniformly away from zero), which is a $C^{0}$-open and $C^{r}$-dense subset of $\text{Diff}_{0}^{r}(\mathbb{R})$, $r=1,2,\ldots ,\infty$, and $\unicode[STIX]{x1D714}$ (real analytic). Second, we give some examples where entropy map is not continuous. Finally, we prove some results on the continuity of entropy of general diffeomorphisms on the (real) line.

Keywords

dimension theorysmooth dynamicstopological dynamics

MSC classification

Primary: 37B40: Topological entropy 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1139 - 1155
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

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