Dynamical characterization of initial segments of the Markov and Lagrange spectra

TL;DR

This paper characterizes the initial segments of Markov and Lagrange spectra for all k ≥ 4, showing they match classical spectra intersections with specific intervals, revealing nuanced differences at k=3.

Contribution

It establishes a precise dynamical description of the initial segments of Markov and Lagrange spectra for k ≥ 4, clarifying their relation to classical spectra.

Findings

For k ≥ 4, M(k) and L(k) coincide with classical spectra intersections.

The statement holds for k=2 but fails for k=3.

Provides new insights into the structure of Markov and Lagrange spectra.

Abstract

We prove that, for every $k\ge 4$, the sets $M(k)$ and $L(k)$, which are Markov and Lagrange dynamical spectra related to conservative horseshoes and associated to continued fractions with coefficients bounded by $k$ coincide with the intersections of the classical Markov and Lagrange spectra with $(-\infty, \sqrt{k^2+4k}]$. We also observe that, despite the corresponding statement is also true for $k = 2$, it is false for $k = 3$.