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
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
Received 24 November 1997; Revised 10 February 1998
Copyright © 2000 Julia A. Barnes and Lorelei Koss. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.