Maximal bifix decoding

TL;DR

This paper introduces a new class of word sets generalizing Sturmian and interval exchange sets, proving it is closed under maximal bifix decoding using properties of return words.

Contribution

It defines the class of uniformly recurrent tree sets and proves its closure under maximal bifix decoding, expanding understanding of word set transformations.

Findings

The class of uniformly recurrent tree sets generalizes known word sets.

Closure under maximal bifix decoding is established for this class.

The proof leverages the closure under decoding with respect to return words.

Abstract

We introduce a class of sets of words which is a natural common generalization of Sturmian sets and of interval exchange sets. This class of sets consists of the uniformly recurrent tree sets, where the tree sets are defined by a condition on the possible extensions of bispecial factors. We prove that this class is closed under maximal bifix decoding. The proof uses the fact that the class is also closed under decoding with respect to return words.