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Thomas Barthelmé

Publications and source records attributed to Thomas Barthelmé.

30 records · Page 2Linked to original sources
arXiv2019

Centralizers of partially hyperbolic diffeomorphisms in dimension 3

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu (arXiv:1902.05201) who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in (arXiv:1908.06227).

math.DS↗
arXiv2017

Anomalous Anosov flows revisited

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In the second part, we give a new surgery type construction of Anosov flows, which yields non-transitive Anosov flows in all odd dimensions.

math.DS↗
arXiv2017

Flexibility of geometrical and dynamical data in fixed conformal classes

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $σ$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area $A$ conformally equivalent to $σ$. For such metrics, we show that the diameter is bounded above and the Laplace spectrum is bounded below away from zero by constants which depend on $σ$. On the other hand, we prove that the metric entropy of the geodesic flow with respect to the Liouville measure is flexible. Consequently, we also provide the first known example showing that the bottom of the $L^2$-spectrum of the Laplacian cannot be bounded from above by a function of the metric entropy. We also provide examples showing that our conditions are essential for the established bounds.

math.DS↗
arXiv2016

Entropy rigidity of Hilbert and Riemannian metrics

In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.

math.DG↗
arXiv2015

On deformations of the spectrum of a Finsler--Laplacian that preserve the length spectrum

In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4λ_1 > h^2$, where $λ_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow.

math.DG↗
arXiv2015

Counting periodic orbits of Anosov flows in free homotopy classes

The main result of this article is that if a $3$-manifold $M$ supports an Anosov flow, then the number of conjugacy classes in the fundamental group of $M$ grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen's version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class. In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only $\mathbb{R}$-covered Anosov flows up to orbit equivalence that do not admit at least one infinite free homotopy class of periodic orbits.

math.DS↗
arXiv2012

Laplacian and spectral gap in regular Hilbert geometries

We study the spectrum of the Finsler--Laplace operator for regular Hilbert geometries, defined by convex sets with $C^2$ boundaries. We show that for an $n$-dimensional geometry, the spectral gap is bounded above by $(n-1)^2/4$, which we prove to be the infimum of the essential spectrum. We also construct examples of convex sets with arbitrarily small eigenvalues.

math.DG↗
arXiv2012

Knot theory of R-covered Anosov flows: homotopy versus isotopy of closed orbits

In this article, we study the knots realized by periodic orbits of R-covered Anosov flows in compact 3-manifolds. We show that if two orbits are freely homotopic then in fact they are isotopic. We show that lifts of periodic orbits to the universal cover are unknotted. When the manifold is atoroidal, we deduce some finer properties regarding the existence of embedded cylinders connecting two given homotopic orbits.

math.DS↗
arXiv2012

Eigenvalues control for a Finsler--Laplace operator

Using the definition of a Finsler--Laplacian given by the first author, we show that two bi-Lipschitz Finsler metrics have a controlled spectrum. We deduce from that several generalizations of Riemannian results. In particular, we show that the spectrum on Finsler surfaces is controlled above by a constant depending on the topology of the surface and on the quasireversibility constant of the metric. In contrast to Riemannian geometry, we then give examples of highly non-reversible metrics on surfaces with arbitrarily large first eigenvalue.

math.DG↗
arXiv2012

A new Laplace operator in Finsler geometry and periodic orbits of Anosov flows

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is Hölder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to $\mathbb{R}$. When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces a regulating flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.

math.DG↗
arXiv2012

A natural Finsler--Laplace operator

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--Ziller metrics that this Finsler--Laplace operator admits explicit representations and computations of spectral data.

math.DG↗