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International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 383-392
http://dx.doi.org/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, NC 28723, USA
2Department of Mathematics, CB#3250, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received 24 November 1997; Revised 10 February 1998

Copyright © 2000 Hindawi Publishing Corporation. 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.