Equilibrium measures on Julia sets of random quadratic polynomials

Abstract

We consider sequences of compositions of quadratic polynomials $f_{c_n}(z)=z^2+c_n$. For such sequences one can naturally generalize the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence $\omega =(c_0,c_1,...)$. We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ''scale of randomness''.