An infinite permutation can be defined as a linear ordering of the set of natural numbers. Similarly to infinite words, a complexityp(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 \(p(n)\le n\) 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.
Conjugate Class Permutation Entropy Irrational Rotation Minimal Complexity Canonical Sequence
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