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Cyclicity, hypercyclicity and randomness in self-similar groups

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Názov: Cyclicity, hypercyclicity and randomness in self-similar groups
Autori: Fariña-Asategui, Jorge
Prispievatelia: Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematics (Faculty of Sciences), Lunds universitet, Naturvetenskapliga fakulteten, Matematikcentrum, Matematik (naturvetenskapliga fakulteten), Originator
Zdroj: Monatshefte fur Mathematik. 207(2):275-292
Predmety: Natural Sciences, Mathematical Sciences, Mathematical Analysis, Naturvetenskap, Matematik, Matematisk analys, Discrete Mathematics, Diskret matematik
Popis: 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.
Prístupová URL adresa: https://doi.org/10.1007/s00605-025-02061-6
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  Data: Cyclicity, hypercyclicity and randomness in self-similar groups
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  Data: <i>Monatshefte fur Mathematik</i>. 207(2):275-292
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  Data: 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.
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