Enumeration formul{\ae} in neutral sets

TL;DR

This paper introduces new enumeration formulas for neutral sets of words, extending known results for Sturmian and tree sets, with applications to interval exchange transformations.

Contribution

It generalizes enumeration formulas for return words and bifix codes to a broader class of neutral sets, including geometric examples.

Findings

New enumeration formulas for neutral sets.

Extension of known formulas from Sturmian and tree sets.

Application to natural coding of interval exchange transformations.

Abstract

We present several enumeration results holding in sets of words called neutral and which satisfy restrictive conditions on the set of possible extensions of nonempty words. These formulae concern return words and bifix codes. They generalize formulae previously known for Sturmian sets or more generally for tree sets. We also give a geometric example of this class of sets, namely the natural coding of some interval exchange transformations.