Rational maps are $d$-adic Bernoulli

Abstract

Freire, Lopes and Mane proved that for any rational map f there exists a natural invariant measure \mu_f [5]. Mane showed there exists an n>0 such that (f^n, \mu_f) is measurably conjugate to the one-sided $d^n$-shift, with Bernoulli measure $(\frac 1{d^n},... ,\frac 1{d^n})$ \[15]. In this paper we show that (f,\mu_f)is conjugate to the one-sided Bernoulli $d$-shift. This verifies a conjecture of Freire, Lopes and Mane [5] and Lyubich [11].