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Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family

Published online by Cambridge University Press:  07 September 2006

ENRIQUE R. PUJALS ,
LEONEL ROBERT  and
MICHAEL SHUB
ENRIQUE R. PUJALS
Affiliation:
IMPA Estrada Dona Castorina 110, Rio de Janeiro, Brazil 22460-320 (e-mail: [email protected])
LEONEL ROBERT
Affiliation:
Math Dept, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada (e-mail: [email protected], [email protected])
MICHAEL SHUB
Affiliation:
Math Dept, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada (e-mail: [email protected], [email protected])
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Abstract

In this paper we revisit once again, see Shub and Sullivan (Ergod. Th. & Dynam. Sys.5 (1985), 285–289), a family of expanding circle endomorphisms. We consider a family $\{B_\theta\}$ of Blaschke products acting on the unit circle, $\mathbb{T}$, in the complex plane obtained by composing a given Blaschke product $B$ with the rotations about zero given by multiplication by $\theta \in \mathbb{T}$. While the initial map $B$ may have a fixed sink on $\mathbb{T}$, there is always an open set of $\theta$ for which $B_\theta$ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of $\int \ln|B'(z)|\,{\it dz}$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 26 , Issue 6 , December 2006 , pp. 1931 - 1937
Copyright
2006 Cambridge University Press

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