Preprint A416/2005
Expanding maps of the circle rerevisited: Positive Lyapunov exponents in a rich family

Mike Shub | Pujals, Enrique R. | Robert, Leonel

**Keywords: **

In this paper we revisit once again, see \cite{S-S}, 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 Blashke product $B$ with
the rotations about zero given by mulitplication 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
$\int ln|B'(z)|dz$ .