Dynamical characterization of initial segments of the Markov and Lagrange spectra

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 $k = 3$.