  The Markov and Lagrange dynamical spectra, was 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 holds for the
classical Markov and Lagrange spectra. In this paper, we show that, for generic
dynamical spectra associated to horseshoes, there is a transition point a in
the spectra such that for any $\delta > 0$, the intersection of the spectra
with $(a-\delta; a)$ is a countable union of Cantor sets with Hausdorff
dimension smaller than 1, while the intersection of the spectra with $(a; a +
\delta)$ have non-empty interior. This is still open for the classical Markov
and Lagrange spectra.
