American Institute of Mathematical Sciences

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

Entropy of diffeomorphisms of line

Baolin He

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 $f$.

Keywords: Diffeomorphisms, line, topological entropy, length growth rate, entropy map.
Mathematics Subject Classification: 37B40, 37E05.
Citation: Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[3]

B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. Google Scholar

[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.  Google Scholar

[5]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.   Google Scholar

[6]

S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[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.  Google Scholar

[8]

P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[9]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

B. He, Entropy of diffeomorphisms with unbounded derivatives, In preparing. Google Scholar

[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.  Google Scholar

[5]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.   Google Scholar

[6]

S. Newhouse, Continuity properties of entropy, Ann. of Math., 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar

[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.  Google Scholar

[8]

P. Walters, Ergodic Theory-Introductory Lectures, Lecture Notes in Mathematics, Vol. 458. Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[9]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

[1]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[2]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[3]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[4]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789
[5]

O. Goubet, N. Maaroufi. Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1253-1267. doi: 10.3934/cpaa.2012.11.1253
[6]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[7]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[8]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[9]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[10]

Paweł G. Walczak. Expansion growth, entropy and invariant measures of distal groups and pseudogroups of homeo- and diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4731-4742. doi: 10.3934/dcds.2013.33.4731
[11]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[12]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[13]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[14]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[15]

Lorenzo J. Díaz, Todd Fisher, M. J. Pacifico, José L. Vieitez. Entropy-expansiveness for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4195-4207. doi: 10.3934/dcds.2012.32.4195
[16]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[17]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[18]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[19]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
[20]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (22)
  • HTML views (24)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences