Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Cyclicity, hypercyclicity and randomness in self-similar groups

Fariña-Asategui, Jorge LU (2025) In Monatshefte fur Mathematik 207(2). p.275-292
Abstract

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.

Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Automata, Ergodic, Fractal, Hypercyclic, Self-similar, Strongly mixing
in
Monatshefte fur Mathematik
volume
207
issue
2
pages
18 pages
publisher
Springer
external identifiers
  • scopus:85218182844
ISSN
0026-9255
DOI
10.1007/s00605-025-02061-6
language
English
LU publication?
yes
additional info
Publisher Copyright: © The Author(s) 2025.
id
f6b654e5-d2b0-49c2-9060-751421bc2bb6
date added to LUP
2025-07-04 13:59:01
date last changed
2025-07-04 13:59:25
@article{f6b654e5-d2b0-49c2-9060-751421bc2bb6,
  abstract     = {{<p>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.</p>}},
  author       = {{Fariña-Asategui, Jorge}},
  issn         = {{0026-9255}},
  keywords     = {{Automata; Ergodic; Fractal; Hypercyclic; Self-similar; Strongly mixing}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{275--292}},
  publisher    = {{Springer}},
  series       = {{Monatshefte fur Mathematik}},
  title        = {{Cyclicity, hypercyclicity and randomness in self-similar groups}},
  url          = {{http://dx.doi.org/10.1007/s00605-025-02061-6}},
  doi          = {{10.1007/s00605-025-02061-6}},
  volume       = {{207}},
  year         = {{2025}},
}