Minimal complexity of equidistributed infinite permutations

Sergei V. Avgustinovich (1) , Anna E. Frid (2) , Svetlana Puzynina (1, 3)

1 Sobolev Institute of Mathematics
2 I2M - Institut de Mathématiques de Marseille
3 UPD7 - Université Paris Diderot - Paris 7

An infinite permutation is a linear ordering of the set of natural numbers. An infinite permutation can be defined by a sequence of real numbers where only the order of elements is taken into account; such sequence of reals is called a representative of a permutation. In this paper we consider infinite permutations which possess an equidistributed representative on [0, 1] (i.e., such that the prefix frequency of elements from each interval exists and is equal to the length of this interval), and we call such permutations equidistributed. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its subpermutations of length n. We show that, unlike for permutations in general, the minimal complexity of an equidistributed permutation alpha is p(alpha)(n) = n, establishing an analog of Morse and Hedlund theorem. The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

Mots clés: MAXIMAL PATTERN COMPLEXITY; THUE-MORSE WORD; STURMIAN WORDS; ENTROPY; NUMBER

Domaines: Combinatoire [math.CO]; Systèmes dynamiques [math.DS]

Sergei V. Avgustinovich, Anna E. Frid, Svetlana Puzynina. Minimal complexity of equidistributed infinite permutations. European Journal of Combinatorics, 2017, 65, pp.24 - 36. ⟨10.1016/j.ejc.2017.05.003⟩. ⟨hal-01622607⟩