Dynamical generalizations of the Lagrange spectrum

TL;DR

This paper explores the generalization of the classical Lagrange spectrum through the computation of invariants related to topological conjugacy in various symbolic dynamical systems, extending understanding beyond rotations.

Contribution

It introduces a new framework for analyzing symbolic systems via invariants that generalize the Lagrange spectrum, covering rotations, three-interval exchanges, and Arnoux-Rauzy systems.

Findings

Computed invariants for multiple symbolic systems

Generalized the Lagrange spectrum to new families of systems

Identified the set of invariant values for each family

Abstract

We compute two invariants of topological conjugacy, the upper and lower limits of the inverse of Boshernitzan's ne_n, where e_n is the smallest measure of a cylinder of length n, for three families of symbolic systems, the natural codings of rotations and three-interval exchanges and the Arnoux-Rauzy systems. The sets of values of these invariants for a given family of systems generalize the Lagrange spectrum, which is what we get for the family of rotations with the upper limit of 1/ne_n.