Exponential Diophantine approximation and symbolic dynamics

TL;DR

This paper extends a fundamental formula linking the Markoff-Lagrange spectrum with symbolic dynamics, using complex analysis, and introduces a versatile method applicable to various polynomials with multiple roots.

Contribution

It generalizes the key formula connecting the spectrum and symbolic dynamics and develops a new complex analysis-based approach applicable to a broad class of polynomials.

Findings

Extended the key formula relating the spectrum and symbolic dynamics.

Developed a new method using complex analysis for polynomial problems.

Derived several new consequences for the Markoff-Lagrange spectrum.

Abstract

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected on the Markoff-Lagrange spectrum.