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Agafonov’s theorem for finite and infinite alphabets and probability distributions different from equidistribution

Part of: Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Hamiltonian and Lagrangian mechanics

Published online by Cambridge University Press:  21 May 2025

THOMAS SEILLER Open the ORCID record for THOMAS SEILLER [Opens in a new window]  and
JAKOB GRUE SIMONSEN
THOMAS SEILLER*
Affiliation:
CNRS, LIPN – UMR 7030 CNRS & Université Sorbonne Paris Nord, Villetaneuse 93439, France
JAKOB GRUE SIMONSEN
Affiliation:
IT University of Copenhagen, Rued Langgaards Vej 7, DK-2300 Copenhagen S, Denmark (e-mail: jags@itu.dk)
*
e-mail: thomas.seiller@cnrs.fr
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Abstract

An infinite sequence $\alpha $ over an alphabet $\Sigma $ is $\mu $-distributed with respect to a probability map $\mu $ if, for every finite string w, the limiting frequency of w in $\alpha $ exists and equals $\mu (w)$. We prove the following result for any finite or countably infinite alphabet $\Sigma $: every finite-state selector over $\Sigma $ selects a $\mu $-distributed sequence from every $\mu $-distributed sequence if and only if $\mu $ is induced by a Bernoulli distribution on $\Sigma $, that is, a probability distribution on the alphabet extended to words by taking the product. The primary—and remarkable—consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $\mu $-distributedness. As a consequence, the shift-invariant measures $\mu $ on $\Sigma ^{\omega }$, such that any finite-state selector preserves the property of genericity for $\mu $, are exactly the positive Bernoulli measures.

Keywords

equidistributionsymbolic dynamical systemsAgafonov’s theorem

MSC classification

Primary: 70H08: Nearly integrable Hamiltonian systems, KAM theory
Secondary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 34
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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