Remark to the paper Describing the set of words generated by interval exchange transformation, posted 15 November 2007

TL;DR

This paper explores the properties of words generated by interval exchange transformations, focusing on good subdivisions where symbol sets are convex and discontinuities lead to different symbols, establishing a framework for analyzing such words.

Contribution

It introduces the concept of good subdivisions and demonstrates that any subdivision can be refined into a good subdivision, linking superwords generated by piecewise-continuous transformations to those from interval exchange transformations.

Findings

Good subdivisions are convex and lead to consistent word generation.

Any subdivision can be refined into a good subdivision.

The set of words generated by piecewise-continuous transformations is equivalent to those from interval exchange transformations.

Abstract

Let us call subdivision {\it good}, if 1) set corresponding to each symbol is convex (i.e. interval or (semi)closed interval). 2) If points $A$ and $B$ corresponds to the some color and interval $(A,B)$ has discontinuity point, then $f(A)$ and $f(B)$ has different color. Every subdivision can be further divided into good subdivision, old superword can be obtained from new one by gluing letters. Hence in the section ``Equivalence of the set of uniformly recurrent words generated by piecewise-continuous transformation to the set of words generated by interval exchange transformation'' one can consider only good subdivision.