Minimal complexity of equidistributed infinite permutations

Abstract

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 α is pα(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.