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

One-sided Lebesgue Bernoulli maps of the sphere of degreen2and2n2

One-sided Lebesgue Bernoulli maps of the sphere of degreen²and2n²