arXiv subjects

Javier Echevarría Cuesta

Publications and source records attributed to Javier Echevarría Cuesta.

4 recordsLinked to original sources

arXiv2026

On the creation of conjugate points for thermostats

Let $(M, g)$ be a closed oriented Riemannian surface, and let $SM$ be its unit tangent bundle. We show that the interior in the $\mathcal{C}^2$ topology of the set of smooth functions $λ:SM\to \mathbb{R}$ for which the thermostat $(M, g, λ)$ has no conjugate points is a subset of those functions for which the thermostat is projectively Anosov. Moreover, we prove that if a reversible thermostat is projectively Anosov, then its non-wandering set contains no conjugate points.

math.DS↗

arXiv2026

Thermostats without conjugate points

We generalize Hopf's theorem to thermostats: the total thermostat curvature of a thermostat without conjugate points is non-positive and vanishes only if the thermostat curvature is identically zero. We further show that, if the thermostat curvature is zero, then the flow has no conjugate points and the Green bundles collapse almost everywhere. Given a thermostat without conjugate points, we prove that the Green bundles are transverse everywhere if and only if it is projectively Anosov. Finally, we provide an example showing that Hopf's rigidity theorem on the 2-torus cannot be extended to thermostats. It is also the first example of a projectively Anosov thermostat which is not Anosov.

math.DS↗

arXiv2025

Emergence of quantum dynamics from chaos: The case of prequantum cat maps

Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott--Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the $2n$-dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case $n=1$ previously treated by Faure.

math.DS↗

arXiv2025

Smooth orbit equivalence rigidity for dissipative geodesic flows

Let $M$ be a smooth closed oriented surface. Gaussian thermostats on $M$ correspond to the geodesic flows arising from metric connections, including those with non-zero torsion. These flows may not preserve any absolutely continuous measure. We prove that if two Gaussian thermostats on $M$ with negative thermostat curvature are related by a smooth orbit equivalence isotopic to the identity, then the two background metrics are conformally equivalent via a smooth diffeomorphism of $M$ isotopic to the identity. We also give a relationship between the thermostat forms themselves. Finally, we prove the same result for Anosov magnetic flows.

math.DS↗