American Institute of Mathematical Sciences

Advanced
September  2008, 10(2&3, September): 597-608. doi: 10.3934/dcdsb.2008.10.597

Entropy estimates for a family of expanding maps of the circle

Rafael De La Llave 1, , Michael Shub 2, and  Carles Simó 3,

1. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States
2. 

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4
3. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona

Received  April 2006 Revised  May 2007 Published  June 2008

In this paper we consider the family of circle maps $f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when written mod 1 are of the form $f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$ and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while the maximum with respect to $\alpha$ is larger. In the case of the average the difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families of expanding Blaschke products depending on rotations where the averages are equal and for which the inequality for averages goes in the other direction when the expanding property does not hold, see [4]. A striking fact for both results is that the maximum of the entropies is greater than or equal to $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with [3], where similar questions are considered for a family of diffeomorphisms of the two sphere.
Keywords: entropy estimates, expanding circle maps..
Mathematics Subject Classification: Primary: 37C40, 37D20, 37E10 ; Secondary: 37M2.
Citation: Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597
[1]

Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[2]

Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039
[3]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[4]

Carlangelo Liverani. A footnote on expanding maps. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3741-3751. doi: 10.3934/dcds.2013.33.3741
[5]

Peter Haïssinsky, Kevin M. Pilgrim. An algebraic characterization of expanding Thurston maps. Journal of Modern Dynamics, 2012, 6 (4) : 451-476. doi: 10.3934/jmd.2012.6.451
[6]

Peter Haïssinsky, Kevin M. Pilgrim. Examples of coarse expanding conformal maps. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2403-2416. doi: 10.3934/dcds.2012.32.2403
[7]

Yushi Nakano, Shota Sakamoto. Spectra of expanding maps on Besov spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1779-1797. doi: 10.3934/dcds.2019077
[8]

José F. Alves. Stochastic behavior of asymptotically expanding maps. Conference Publications, 2001, 2001 (Special) : 14-21. doi: 10.3934/proc.2001.2001.14
[9]

Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501
[10]

Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099
[11]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[12]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalizable Expanding Baker Maps: Coexistence of strange attractors. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1651-1678. doi: 10.3934/dcds.2017068
[13]

Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749
[14]

Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
[15]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685
[16]

Yong Fang. On smooth conjugacy of expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 687-697. doi: 10.3934/dcds.2011.30.687
[17]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[18]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[19]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[20]

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

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences