New examples of Bernoulli algebraic actions

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.

Distortion of Dynamical Systems in the Context of Focusing the Chaos Around the Point

In the paper, we examine local aspects of chaos for self-functions on topological manifolds. For this purpose, issues related to focal entropy points and homoclinic points are used. We consider also distortion of dynamical systems which is a theoretical analogue to wave interference.