Canonical Representatives of Morphic Permutations

TL;DR

This paper investigates when infinite permutations derived from morphic words have a canonical representative, linking this to unique ergodicity, and extends previous constructions to a broader class of words.

Contribution

It establishes the existence criteria for canonical representatives of ergodic permutations and generalizes prior constructions to morphic words.

Findings

Canonical representatives exist iff the word is uniquely ergodic

Extension of Makarov's construction to wider class of infinite words

Provides methods to construct canonical representatives for ergodic permutations

Abstract

An infinite permutation can be defined as a linear ordering of the set of natural numbers. In particular, an infinite permutation can be constructed with an aperiodic infinite word over $\{0,\ldots,q-1\}$ as the lexicographic order of the shifts of the word. In this paper, we discuss the question if an infinite permutation defined this way admits a canonical representative, that is, 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 a canonical representative exists if and only if the word is uniquely ergodic, and that is why we use the term ergodic permutations. We also discuss ways to construct the canonical representative of a permutation defined by a morphic word and generalize the construction of Makarov, 2009, for the Thue-Morse permutation to a wider class of infinite words.