The ergodicity of 1-Lipschitz transformations on 2-adic spheres

Abstract

In this paper we present results about ergodicity of dynamical systems on $2$-adic spheres for 1-Lipschitz maps $f:{\mathbb{Z}}_2\to {\mathbb{Z}}_2$ announced in [8], and extension of Theorem 3 from [8] for the case of spheres of radii greater than $\frac{1}{8}.$ We propose a new approach to study ergodic properties of 1-Lipschitz transformations of $2$-adic spheres. We use a representation of continuous functions $f$ via its van der Put series. This technique allows us to go beyond the classes of smooth 1-Lipschitz transformations which were studied earlier.