Phase Transitions on the Markov and Lagrange Dynamical Spectra

TL;DR

This paper studies phase transition phenomena in the dynamical spectra associated with hyperbolic sets, revealing a sharp change in spectral structure around a transition point, with implications for classical spectra conjectures.

Contribution

It demonstrates the existence of a transition point in the spectra of horseshoes where the structure changes from Cantor sets to sets with interior, suggesting parallels with classical spectra.

Findings

Existence of a transition point in dynamical spectra of horseshoes.

Spectral intersection below the transition point is a union of Cantor sets with small Hausdorff dimension.

Spectral intersection above the transition point has non-empty interior.

Abstract

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.