Higher dimensional expanding maps and toral extensions

Author:
Eugen Mihailescu

Journal:
Proc. Amer. Math. Soc. **141** (2013), 3467-3475

MSC (2010):
Primary 37D20, 37A35, 37C40

DOI:
https://doi.org/10.1090/S0002-9939-2013-11597-2

Published electronically:
June 12, 2013

MathSciNet review:
3080169

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Abstract: We prove that expanding endomorphisms on arbitrary tori are 1-sided Bernoulli with respect to their corresponding measure of maximal entropy and are thus, measurably, as far from invertible as possible. This applies in particular to expanding linear toral endomorphisms and their smooth perturbations. Then we study toral extensions of expanding toral endomorphisms, in particular probabilistic systems on skew products, and prove that under certain not too restrictive conditions on the extension cocycle, these skew products are 1-sided Bernoulli too. We also give a large class of examples of group extensions of expanding maps in higher dimensions, for which we check the conditions on the extension cocycle.

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Additional Information

**Eugen Mihailescu**

Affiliation:
Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania

Email:
Eugen.Mihailescu@imar.ro

DOI:
https://doi.org/10.1090/S0002-9939-2013-11597-2

Keywords:
Toral endomorphisms,
expanding maps,
1-sided Bernoullicity,
measures of maximal entropy,
group extensions,
cohomological conditions

Received by editor(s):
November 14, 2011

Received by editor(s) in revised form:
December 10, 2011

Published electronically:
June 12, 2013

Additional Notes:
This work was supported by CNCS - UEFISCDI, project PN II - IDEI PCE 2011-3-0269

Communicated by:
Bryna Kra

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.