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

In this paper we revisit once again, see [ShSu], a family of expanding circle endomorphisms. We consider a family {Bθ} of Blaschke products acting on the unit circle, T, in the complex plane obtained by composing a given Blashke product B with the rotations about zero given by mulitplication by θ ∈ T. While the initial map B may have a fixed sink on T there is always an open set of θ for which Bθ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of R ln|B′(z)|dz .