Markoff-Lagrange spectrum of one-sided shifts

TL;DR

This paper investigates the Markoff-Lagrange spectrum for one-sided shifts, identifying minimal accumulation points for binary and ternary sequences and characterizing their structure using substitutions and S-adic sequences.

Contribution

It introduces a comprehensive analysis of the Markoff-Lagrange spectrum for one-sided shifts, including new characterizations of minimal accumulation points via substitutions and S-adic sequences.

Findings

Identified the smallest accumulation point for binary sequences in shift orbits.

Characterized solutions as fixed points of substitutions for alternating lexicographic order.

Extended the analysis to symmetric ternary shifts related to the multiplicative Markoff-Lagrange spectrum.

Abstract

For the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are $S$-adic sequences, where $S$ is a certain infinite set of substitutions that includes Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff-Lagrange spectrum.