arXiv:1503.06188v2 [math.CO] 7 Dec 2015 Ergod. Th. & Dynam. Sys. (1111), 11,1 Printed in the United Kingdom c  1111 Cambridge University Press Minimal complexity of ergodic infinite permutations S. V. AVGUSTINOVICH†, A. E. FRID‡ and S. PUZYNINA§† † Sobolev Institute of Mathematics, Novosibirsk, Russia ‡ Aix-Marseille Universit´ e, France § LIP, ENS de Lyon, Universit´ e de Lyon, France (e-mail: anna.e.frid@gmail.com) (Received September 10, 2018 ) Abstract. In the paper we investigate a new natural notion of ergodic infinite permutations, that is, infinite permutations which can be defined by equidistributed sequences. We show that, unlike for permutations in general, the minimal complexity of an ergodic permutation α is p α (n)= n. The class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words. 1. Introduction Infinite permutations can be defined as equivalence classes of real sequences with distinct elements, such that only the order of elements is taken into account. In other words, an infinite permutation is a linear order on N. We consider it as an object close to an infinite word, but instead of symbols, we have transitive relations < or > between each pair of elements. So, many properties of such permutations can be considered in a way close to symbolic dynamics. Infinite permutations in the considered sense were introduced in [10]; see also a very similar approach coming from dynamics [7] and summarised in [2]. Since then, they were studied in two main directions: first, permutations directly constructed with the use of words are studied to reveal new properties of words used for their construction [9, 17, 18, 19, 21, 22, 23]. In the other approach, properties of infinite permutations are studied in comparison with those of infinite words, showing some resemblance and some difference. In particular, both for words and permutations, the (factor) complexity is bounded if and only if the word or the permutation is ultimately periodic [10, 20]. † Supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit´ e de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Prepared using etds.cls [Version: 1999/07/21 v1.0]