New York Journal of Mathematics
Volume 15 (2009) 451-483


Henk Bruin and Jane Hawkins

Rigidity of smooth one-sided Bernoulli endomorphisms

Published: October 31, 2009
Keywords: Bernoulli shifts, one-sided Bernoulli, noninvertible maps, interval maps, rational maps
Subject: 37A05, 37A10, 37C40, 37E05

A measure-preserving endomorphism is one-sided Bernoulli if it is isomorphic to a noninvertible Bernoulli shift. We show that in piecewise smooth settings this property is very strong and far more subtle than the weak Bernoulli property, by extending of results of W. Parry and P. Walters and proving new results based on continuity of the Radon-Nikodym derivative. In particular, we provide tests which work for noninvariant measures if an invariant measure equivalent to a natural measure exists but its density function is not known. Examples of families of interval maps and complex maps on the Riemann sphere illustrate the results.


This work was partly funded by the LMS (Scheme 2 grant 2603) and by EPSRC (grant GR/S91147/01)

Author information

Henk Bruin:
Department of Mathematics, University of Surrey, Guildford Surrey, GU2 7XH United Kingdom

Jane Hawkins:
Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina 27599-3250