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New examples of Bernoulli algebraic actions

Part of: Topological dynamics Ergodic theory

Published online by Cambridge University Press:  17 May 2021

DOUGLAS LIND Open the ORCID record for DOUGLAS LIND [Opens in a new window]  and
KLAUS SCHMIDT
DOUGLAS LIND*
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington98195, USA
KLAUS SCHMIDT
Affiliation:
Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090Vienna, Austria (e-mail: klaus.schmidt@univie.ac.at)
*
e-mail: lind@math.washington.edu
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Abstract

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We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

Keywords

Bernoulli actionhomoclinic pointsymbolic coversofic group

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 37A35: Entropy and other invariants, isomorphism, classification
Secondary: 37B10: Symbolic dynamics 37A20: Orbit equivalence, cocycles, ergodic equivalence relations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2923 - 2934
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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