Ergodic infinite permutations of minimal complexity

Sergey V. Avgustinovich, Anna E. Frid, Svetlana Puzynina

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

An infinite permutation can be defined as a linear ordering of the set of natural numbers. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its factors of length n. For infinite words, a classical result of Morse and Hedlund, 1940, states that if the complexity of an infinite word satisfies p(n) ≤ n for some n, then the word is ultimately periodic. Hence minimal complexity of aperiodic words is equal to n + 1, and words with such complexity are called Sturmian. For infinite permutations this does not hold: There exist aperiodic permutations with complexity functions of arbitrarily slow growth, and hence there are no permutations of minimal complexity. In the paper we introduce a new notion of ergodic permutation, i.e., a permutation which can be defined by a sequence of numbers from [0, 1], such that the frequency of its elements in any interval is equal to the length of that interval. We show that the minimal complexity of an ergodic permutation is p(n) = n, and that the class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

Original languageEnglish
Title of host publicationDevelopments in Language Theory - 19th International Conference, DLT 2015, Proceedings
EditorsIgor Potapov
PublisherSpringer Nature
Pages71-84
Number of pages14
ISBN (Print)9783319214993
DOIs
StatePublished - 1 Jan 2015
Event19th International Conference on Developments in Language Theory, DLT 2015 - Liverpool, United Kingdom
Duration: 27 Jul 201530 Jul 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9168
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference19th International Conference on Developments in Language Theory, DLT 2015
CountryUnited Kingdom
CityLiverpool
Period27/07/1530/07/15

Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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