Phase transitions on the Markov and Lagrange dynamical spectra

The Markov and Lagrange dynamical spectra were introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case.

They can be a good source of natural conjectures about the classical spectra: it is natural to conjecture that some properties which hold for generic dynamical spectra associated to hyperbolic maps also hold for the classical Markov and Lagrange spectra.

In this paper, we show that, for generic dynamical spectra associated to horseshoes, there are transition points a and $\tilde{a}$ in the Markov and Lagrange spectra respectively, such that for any $\delta >0$, the intersection of the Markov spectrum with $(-\infty ,a-\delta )$ has Hausdorff dimension smaller than one, while the intersection of the Markov spectrum with $(a,a+\delta )$ has non-empty interior. Similarly, the intersection of the Lagrange spectrum with $(-\infty ,\tilde{a}-\delta )$ has Hausdorff dimension smaller than one, while the intersection of the Lagrange spectrum with $(\tilde{a},\tilde{a}+\delta )$ has non-empty interior. We give an open set of examples where $a\ne \tilde{a}$ and we prove that, in the conservative case, generically, $a=\tilde{a}$ and, for any $\delta >0$, the intersection of the Lagrange spectrum with $(a-\delta ,a)$ has Hausdorff dimension one.