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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
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