Semicocycle discontinuities for substitutions and reverse-reading automata

TL;DR

This paper introduces a semigroup-based automaton framework for analyzing substitution sequences, particularly focusing on semicocycle discontinuities in sequences generated by substitutions with coincidences.

Contribution

It constructs a minimal automaton from a substitution's semigroup that fully characterizes semicocycle discontinuities in reverse-reading automata.

Findings

Automaton fully describes semicocycle discontinuities for substitutions with coincidences.

Semigroup approach provides a new method for analyzing substitution sequences.

Automaton construction is minimal and automates the detection of discontinuities.

Abstract

In this article we define the semigroup associated to a substitution. We use it to construct a minimal automaton which generates a substitution sequence u in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the semicocycle discontinuities of u.