Critical exponent of infinite words coding beta-integers associated with non-simple Parry numbers

TL;DR

This paper analyzes the critical exponent of infinite words coding beta-integers for non-simple Parry numbers, revealing the maximal repetitions and long-term repetition behavior using bispecial factors and return words.

Contribution

It provides a method to compute the critical exponent for infinite words associated with non-simple Parry numbers, applicable to primitive substitution fixed points.

Findings

Calculated the critical exponent for $eta$-integer coding words.

Determined the ultimate critical exponent for long-factor repetitions.

Developed a general method using bispecial factors and return words.

Abstract

In this paper, we study the critical exponent of infinite words $\ubeta$ coding $\beta$-integers for $\beta$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the infinite word in question. We calculate also the ultimate critical exponent that expresses how long repetitions occur in the infinite word $\ubeta$ when the factors of length growing ad infinitum are considered. The basic ingredients of our method are the description of all bispecial factors of $\ubeta$ and the notion of return words. This method can be applied to any fixed point of any primitive substitution.