One-sided Lebesgue Bernoulli maps of the sphere of degree n² and 2n²

Abstract

We prove that there are families of rational maps of the sphere of degree n2(n=2,3,4,…) and 2n2(n=1,2,3,…) which, with respect to a finite invariant measure equivalent to the surface area measure, are isomorphic to one-sided Bernoulli shifts of maximal entropy. The maps in question were constructed by Böettcher (1903--1904) and independently by Lattès (1919). They were the first examples of maps with Julia set equal to the whole sphere.