arXiv subjects

Patrick Foulon

Publications and source records attributed to Patrick Foulon.

7 recordsLinked to original sources

arXiv2023

Simplicity of Lyapunov spectra and boundaries of non-conical strictly convex divisible sets

Let $Ω$ be a strictly convex divisible subset of the $n$-dimensional real projective space which is not an ellipsoid. Even though $\partialΩ$ is not $C^2$, Benoist showed that it is $C^{1+α}$ for some $α>0$, and Crampon established that $\partialΩ$ actually possesses a sort of anisotropic Hölder regularity -- described by a list $α_1\leq\dots\leqα_{n-1}$ of positive real numbers -- at almost all of its points. In this article, we show that $\partialΩ$ is maximally anisotropic in the sense that this list of approximate regularities of $\partialΩ$ does not contain repetitions. This result is a consequence of the simplicity of the Lyapunov spectrum of the Hilbert geodesic flow for every equilibrium measure associated to a Hölder potential.

math.DS↗

arXiv2022

Sinai-Ruelle-Bowen measure Entropy of geodesic flow on Convex Projective Surfaces

We study the entropy of Sinai-Ruelle-Bowen measure of the geodesic flow on convex real projective surfaces, and shows that the Hilbert area tends to infinity if the entropy tends to zero. For the Blaschke metric, the area tends to infinity if and only if the entropy tends to zero.

math.GT↗

arXiv2019

Orbit Growth Of Contact Structures After Surgery

Investigation of the effects of a contact surgery construction and of invariance of contact homology reveals a rich new field of inquiry at the intersection of dynamical systems and contact geometry. We produce contact 3-flows not topologically orbit-equivalent to any algebraic flow, including examples on many hyperbolic 3-manifolds, and we show how the surgery produces dynamical complexity for any Reeb flow compatible with the resulting contact structure. This includes exponential complexity when neither the surg-ered flow nor the surgered manifold are hyperbolic. We also demonstrate the use in dynamics of contact homology, a powerful tool in contact geometry.

math.DS↗

arXiv2019

Continuity of the Sinai-Ruelle-Bowen measure entropy

The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai-Ruelle-Bowen measure and show that it is a continuous function.

math.GT↗

arXiv2017

Zermelo deformation of Finsler metrics by Killing vector fields

We show how geodesics, Jacobi vector fields and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.

math.DG↗

arXiv2016

Topological Entropy and bulging deformation of real projective structures on surface

In this paper we study the deformation of strictly convex real projective structures on a closed surface. Specially we study the deformation in terms of the entropy on bulging deformations. As a byproduct we construct a sequence of divergent structures whose topological entropy converges to a designated number between 0 and 1.

math.GT↗

arXiv2010

Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian manifold and a uniformly quasiconformal transversely symplectic Anosov flow we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the longitudinal KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form. Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.

math.DS↗