Mapped Exponent and Asymptotic Critical Exponent of Words

2025-06-04

AI Summary

This study investigates the impact of bijective morphisms on word repetitiveness, focusing on how fractional exponents of finite words and asymptotic critical exponents of infinite words change under such mappings. Using combinatorial word theory and morphic analysis, we fully characterize the structure of finite words that can attain arbitrarily high fractional exponents via injective morphisms—a first complete structural description. We further prove that applying any bijective morphism to an infinite word increases its asymptotic critical exponent by at most a factor equal to the alphabet size. Tight upper and lower bounds on exponent changes are established for binary and general alphabets, resolving a long-standing gap in the structural understanding of repetitiveness measures. These results provide foundational tools and a new paradigm for repetitiveness theory in formal languages.

Technology Category

Natural Language Processing: Lexical Semantics and MorphologyKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSecurity and Privacy: Large-scale security measurements

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.

Problem

• Study injective morphisms' impact on word repetitiveness
• Characterize finite words with high fractional exponents
• Bound asymptotic critical exponent growth under morphisms

Innovation

• Injective morphisms analyze word repetitiveness
• Characterize finite words with high exponents
• Bound asymptotic critical exponent growth