# American Institute of Mathematical Sciences

September  2017, 37(9): 4753-4766. doi: 10.3934/dcds.2017204

## Entropy of diffeomorphisms of line

 Mathematics and Science College, Shanghai Normal University, Shanghai 200433, China

Received  February 2017 Revised  April 2017 Published  June 2017

Fund Project: The author is supported by National Natural Science Foundation of China (grant no. 11501371,11671025), and Science Foundation of Shanghai Normal University (SK201502).

For diffeomorphisms of line, we set up the identities between their length growth rate and their entropy. Then, we prove that there is $C^0$-open and $C^r$-dense subset $\mathcal{U}$ of $\text{Diff}^r (\mathbb{R})$ with bounded first derivative, $r=1,2,\cdots$, $+\infty$, such that the entropy map with respect to strong $C^0$-topology is continuous on $\mathcal{U}$; moreover, for any $f \in \mathcal{U}$, if it is uniformly expanding or $h(f)=0$, then the entropy map is locally constant at $f$.

Also, we construct two examples:

1. there exists open subset $\mathcal{U}$ of $\text{Diff}^{\infty} (\mathbb{R})$ such that for any $f \in \mathcal{U}$, the entropy map with respect to strong $C^{\infty}$-topology, is not locally constant at $f$.

2. there exists $f \in \text{Diff}^{\infty}(\mathbb{R})$ such that the entropy map with respect to strong $C^{\infty}$-topology, is neither lower semi-continuous nor upper semi-continuous at $f$.

Citation: Baolin He. Entropy of diffeomorphisms of line. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
##### References:
 [1] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity (A Global Geometric and Probabilistic Perspective), Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005. [2] S. Crovisier, Birth of homoclinic intersections: A model for the central dynamics of partially hyperbolic systems, Ann. of Math., 172 (2010), 1641-1677.  doi: 10.4007/annals.2010.172.1641. [3] B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. [4] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems, 1342 (1988), Springer Lecture Note in Mathematics, 465-563.  doi: 10.1007/BFb0082847. [5] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. [6] S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492. [7] R. Saghin and J. Yang, Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows, Israel J. Math., 215 (2016), 857-875.  doi: 10.1007/s11856-016-1396-4. [8] P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975. [9] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.

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##### References:
 [1] C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity (A Global Geometric and Probabilistic Perspective), Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005. [2] S. Crovisier, Birth of homoclinic intersections: A model for the central dynamics of partially hyperbolic systems, Ann. of Math., 172 (2010), 1641-1677.  doi: 10.4007/annals.2010.172.1641. [3] B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. [4] J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems, 1342 (1988), Springer Lecture Note in Mathematics, 465-563.  doi: 10.1007/BFb0082847. [5] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. [6] S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492. [7] R. Saghin and J. Yang, Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows, Israel J. Math., 215 (2016), 857-875.  doi: 10.1007/s11856-016-1396-4. [8] P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975. [9] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.
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