Dynamical characterization of initial segments of the 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 $$k=3$$ .