International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 383-392
doi:10.1155/S0161171200001484
One-sided Lebesgue Bernoulli maps of the sphere of degree n2 and 2n2
Julia A. Barnes1
and Lorelei Koss2
1Department of Mathematics and Computer Science, Western Carolina University, Cullowhee 28723, NC, USA
2Department of Mathematics, CB#3250, University of North Carolina at Chapel Hill, Chapel Hill 27599-3250, NC, USA
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.