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

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. $\mathit{\text{Ergodic}}\mathit{\text{Theory}}\mathit{\text{Dynam.}}\mathit{\text{Systems.}}$ $\mathbf{\text{26}}(6)$ $(2006),$ $1931$-$1937$] 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.