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Article Contents

# Entropy of diffeomorphisms of line

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 $f$.

Mathematics Subject Classification: 37B40, 37E05.

 Citation:

•  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. 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. B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. 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. M. Misiurewicz  and  W. Szlenk , Entropy of piecewise monotone mappings, Studia Math., 67 (1980) , 45-63. S. Newhouse , Continuity properties of entropy, Ann. of Math., 129 (1989) , 215-235.  doi: 10.2307/1971492. 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. P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975. Y. Yomdin , Volume growth and entropy, Israel J. Math., 57 (1987) , 285-300.  doi: 10.1007/BF02766215.