Average measure theoretic entropy for a family of expanding on average random Blaschke products

TL;DR

This paper derives a formula for the average measure theoretic entropy of a family of random Blaschke products, extending previous work to include non-expanding maps and those with attracting fixed points.

Contribution

It provides a computable formula for entropy in a random setting, generalizing existing results to broader classes of Blaschke products.

Findings

Derived a formula for average measure theoretic entropy

Described the random invariant measure for admissible Blaschke products

Included non-expanding maps and maps with attracting fixed points

Abstract

This work gives a computable formula for the average measure theoretic entropy of a family of expanding on average random Blaschke products, generalizing work by Pujals, Roberts and Shub [Expanding maps of the circle revisited: positive Lyapunov exponents in a rich family. $\textit{Ergodic Theory Dynam. Systems.}$ $\textbf{26}(6)$ $(2006),$ $1931$-$1937$] to the random setting. In doing so, we describe the random invariant measure and associated measure theoretic entropy for a class of admissible random Blaschke products, allowing for maps which are not necessarily expanding and may even have an attracting fixed point.