New examples of Bernoulli algebraic actions

Douglas Lind,Klaus Schmidt

doi:10.1017/etds.2021.56

Douglas Lind, Klaus Schmidt

Open Access

https://doi.org/10.1017/etds.2021.56

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Abstract

AbstractWe 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.

Highlights

Halmos [8] first observed that a continuous automorphism of a compact group automatically preserves Haar measure, providing a rich class of examples in ergodic theory
The study of the joint action of several commuting automorphisms of a compact abelian group was initiated by Bruce Kitchens and the second author [14]
This has led to a detailed understanding of such actions, called algebraic Zd -actions, as described in [24]

Summary

Introduction

Halmos [8] first observed that a continuous automorphism of a compact group automatically preserves Haar measure, providing a rich class of examples in ergodic theory. Little is known about when algebraic actions of sofic groups are measurably isomorphic to Bernoulli actions. We give an alternative argument, showing that the image of the 3-shift measure is invariant under translations by all elements in the dense homoclinic group, and it must be Haar measure.

Results

Conclusion