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New examples of Bernoulli algebraic actions
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 42 / Issue 9 / September 2022
- Published online by Cambridge University Press:
- 17 May 2021, pp. 2923-2934
- Print publication:
- September 2022
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- Article
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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.
