Cyclicity, hypercyclicity and randomness in self-similar groups

We introduce the concept of cyclicity and hypercyclicity in self-similar groups as an analogue of cyclic and hypercyclic vectors for an operator on a Banach space. We derive a sufficient condition for cyclicity of non-finitary automorphisms in contracting discrete automata groups. In the profinite setting we prove that fractal profinite groups may be regarded as measure-preserving dynamical systems and derive a sufficient condition for the ergodicity and the mixing properties of these dynamical systems. Furthermore, we show that a Haar-random element in a super strongly fractal profinite group is hypercyclic almost surely as an application of Birkhoff’s ergodic theorem for free semigroup actions.