Heaviness in Symbolic Dynamics: Substitution and Sturmian Systems

TL;DR

This paper explores the concept of heaviness in symbolic dynamics, analyzing Morse and Sturmian sequences to understand their properties and introducing new conditions for Sturmian sequences.

Contribution

It introduces a new condition for Sturmian sequences and applies heaviness concepts to analyze Morse and Sturmian systems in symbolic dynamics.

Findings

Heaviness maintains a lower bound in symbolic sequences.

A new condition characterizes Sturmian sequences.

Morse sequences exhibit specific heaviness properties.

Abstract

Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in "Heaviness: and Extension of a Lemma of Y. Peres." After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial rational eigenvalues. To contrast, Sturmian sequences are examined, including a new condition for a sequence to be Sturmian.