Mapped Exponent and Asymptotic Critical Exponent of Words

TL;DR

This paper investigates how injective morphisms affect the repetitiveness of words, focusing on fractional exponents and asymptotic critical exponents, providing characterizations and bounds for finite and infinite words.

Contribution

It characterizes finite words that can have arbitrarily high fractional exponents under injective morphisms and bounds the growth of asymptotic critical exponents for infinite words.

Findings

Finite words with arbitrarily high fractional exponents are characterized.

Asymptotic critical exponent grows at most by a constant factor under injective morphisms.

Binary case is better understood than the general case.

Abstract

We study how much injective morphisms can increase the repetitiveness of a given word. This question has a few possible variations depending on the meaning of ``repetitiveness''. We concentrate on fractional exponents of finite words and asymptotic critical exponents of infinite words. We characterize finite words that, when mapped by injective morphisms, can have arbitrarily high fractional exponent. For infinite words, alongside other results, we show that the asymptotic critical exponent grows at most by a constant factor (depending on the size of the alphabet) when mapped by an injective morphism. For both finite and infinite words, the binary case is better understood than the general case.