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.

💡 Research Summary

The paper introduces the notions of cyclicity and hypercyclicity for self‑similar groups, mirroring the classical concepts of cyclic and hyper‑cyclic vectors for linear operators on Banach spaces. For a self‑similar subgroup G≤Aut T of the automorphism group of a rooted regular tree T, the authors consider the natural action of the free monoid T (identified with the set of finite words) on G via sections: for each vertex v∈T the operator T_v(g)=g|v sends an element g to its section at v. An element g∈G is called cyclic if the smallest self‑similar closed subgroup containing g (the self‑similar closure ⟨g⟩{SS}) equals G; it is hyper‑cyclic if the set of all its sections {g|_v : v∈T} already equals G. This definition is motivated by a concrete example: the elementary p‑abelian self‑similar profinite group A is identified with the F_p‑vector space V of infinite sequences, and the left‑shift operator τ on V corresponds exactly to taking sections at the first level. In this setting, cyclic vectors for τ are those whose τ‑orbit generates a dense subspace, while hyper‑cyclic vectors have dense τ‑orbits; the homeomorphism f:A→V transports these notions back to A.

The authors then develop sufficient conditions for the existence of cyclic automorphisms in discrete automata groups, focusing on non‑finitary automorphisms in contracting groups. The key hypothesis combines contracting behavior (sections eventually become trivial) with fractal self‑similarity (the image of vertex stabilizers under the section map recovers the whole group). Under these assumptions many classical self‑similar groups (e.g., Grigorchuk’s group, the Basilica group) admit cyclic elements.

The core of the paper consists of two main theorems.

Theorem A concerns profinite self‑similar groups. If G is fractal, then the action of the free monoid T on the probability space (G, μ_G) (where μ_G is the unique Haar measure) is measure‑preserving, turning (G, μ_G, T, T) into a dynamical system. If G is super‑strongly fractal—meaning that for every vertex v at any level n the section map ϕ_v sends the level‑n stabilizer St_G(n) onto the whole group—then this system is ergodic and strongly mixing. The proof shows that sections are measure‑preserving homomorphisms and that the free monoid’s action satisfies the mixing condition because sections at deeper levels become asymptotically independent. Proposition 2.3 is used to deduce ergodicity from strong mixing.

Theorem B addresses randomness. For a super‑strongly fractal profinite group G, a Haar‑random element g∈G is almost surely hyper‑cyclic; i.e., with probability one the set of its sections generates G. The argument relies on a generalized Birkhoff ergodic theorem for actions of free semigroups: the orbit of a typical point under the semigroup averages to the invariant measure, which in this context forces the sections of a random element to be dense in G. Consequently every super‑strongly fractal profinite group contains a hyper‑cyclic (hence cyclic) automorphism, answering the authors’ Question 1 in the affirmative for this class.

The paper also discusses implications for the Hausdorff spectrum of self‑similar groups. Since a Haar‑random element is hyper‑cyclic, random finite generating sets cannot produce closed self‑similar subgroups of full Hausdorff dimension, extending earlier results of Abert–Virág and the author’s own work. Moreover, Theorem A has already been applied (see reference