Minimal complexity of equidistributed infinite permutations

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 pα(n) = n. The class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.