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On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors
Entropy of diffeomorphisms of line
Mathematics and Science College, Shanghai Normal University, Shanghai 200433, China |
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 $f$.
References:
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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. |
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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. |
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B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. |
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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. |
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M. Misiurewicz and W. Szlenk,
Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
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S. Newhouse,
Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.
doi: 10.2307/1971492. |
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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. |
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P. Walters,
Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975. |
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Y. Yomdin,
Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.
doi: 10.1007/BF02766215. |
show all references
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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