We prove that there are families of rational maps of the sphere of degree n 2 ( n = 2 , 3 , 4 , … ) and 2 n 2 ( 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.