Heaviness in symbolic dynamics: Substitution and Sturmian systems

David Ralston

doi:10.3934/dcdss.2009.2.287

Abstract

Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in [11]. 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.

Highlights

Dynamical systems devotes much attention to the asymptotic behavior of points or other elements of a system
Finite observations do not lend themselves to discussion of limits, but any observer might be concerned with extremal behavior of the partial sums
In applying these notions to symbolic dynamics, we will note a distinction between systems with rational eigenvalues (§3) and a class of systems with no rational eigenvalues (§4)

Summary

Introduction

Dynamical systems devotes much attention to the asymptotic behavior of points or other elements of a system. Finite observations do not lend themselves to discussion of limits, but any observer might be concerned with extremal behavior of the partial sums (a motivation similar to, but distinct from, that in the study of large deviations) With this restriction and motivation in mind, we define the heavy set (subject to various restrictions to be outlined later) to be those points in a system for which these partial sums maintain a natural lower bound over a natural collection of finite ranges. In applying these notions to symbolic dynamics, we will note a distinction between systems with rational eigenvalues (§3) and a class of systems with no rational eigenvalues (§4).

Heaviness in symbolic dynamics

The Morse-Thue sequence and substitution systems

Sturmian sequences