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Pierre-Louis Blayac

Publications and source records attributed to Pierre-Louis Blayac.

11 recordsLinked to original sources
arXiv2026

On 4-dimensional convex projective domains invariant by a lattice of $\mathrm{SL}_2 (\mathbb{R})$

This paper is a sequel to the erratum by the authors to a paper by Crampon and Marquis (see arXiv:1202.5442). The main result of the Erratum was relating several notions of geometrical finiteness in round convex projective geometry and we prove here that our series of implications was sharp, by providing counterexamples to the implications that were not established. Our counterexamples are 4-dimensional convex domains $Ω$ acted on by $ρ(Γ)$ where $Γ$ is a lattice of $\mathrm{SL}_2 (\mathbb R)$ and $ρ$ is the irreducible representation of $\mathrm{SL}_2 (\mathbb R)$ of dimension $5$. We give a description of all $ρ(Γ)$-invariant convex domains, and in particular we construct one which is "close enough" to the convex hull $\mathcal C$ of the limit set of $ρ(Γ)$ so that the Hilbert volume $\mathrm{Vol}_{Ω/Γ}(\mathcal C/Γ)$ of the convex core is infinite. We include an appendix with a smoothing procedure in the spirit of Cooper, Long and Tillman (arXiv:1511.06206) and Danciger, Guéritaud and Kassel (arXiv:1704.08711).

math.GT↗
arXiv2025

Hitchin grafting representations I: Geometry

We give a geometric interpretation of Fock--Goncharov positivity and show that bending deformations of Fuchsian representations stabilize a uniform Finsler quasi-convex disk in the symmetric space $\mathrm{PSL}_d(\mathbb R)/\mathrm{PSO}(d)$.

math.DG↗
arXiv2025

Hitchin grafting representations II: Dynamics

The Hitchin component of the character variety of representations of a surface group $π_1(S)$ into $\mathrm{PSL}_d(\mathbb{R})$ for some $d \geq 3$ can be equipped with a pressure metric whose restriction to the Fuchsian locus equals the Weil--Petersson metric up to a constant factor. We show that if the genus of $S$ is at least $3$, then the Fuchsian locus contains quasi-convex subsets of infinite diameter for the Weil--Petersson metric whose diameter for the path metric of the pressure metric is finite. This is established through showing that biinfinite paths of bending deformations have controlled bounded length.

math.DG↗
arXiv2024

Finitude géométrique en géométrie de Hilbert + an erratum/addendum

The paper is divided in 2 parts. The first part is the original paper of the second and third authors arXiv:1202.5442v2. The second part is an erratum/addendum written in english and concatenated at the end of the former paper. In the erratum/addentum, we amend Theorems 1.3 and 1.11 of arXiv:1202.5442v2: Finitude géométrique en géométrie de Hilbert. We seize the opportunity to show that in round Hilbert geometry, geometrical finiteness (gf) is equivalent to cusp-uniform action and to fill some small gaps that appear in two other proofs of arXiv:1202.5442v2.

math.GT↗
arXiv2024

Patterson-Sullivan theory for coarse cocycles

In this paper we develop a theory of Patterson--Sullivan measures associated to coarse cocycles of convergence groups. This framework includes Patterson-Sullivan measures associated to the Busemann cocycle on the geodesic boundary of a Gromov hyperbolic metric spaces and Patterson-Sullivan measures on flag manifolds associated to Anosov (or more general transverse) subgroups of semisimple Lie groups, as well as more examples. Under some natural geometric assumptions on the coarse cocycle, we prove existence, uniqueness, and ergodicity results.

math.DS↗
arXiv2023

Divisible convex sets with properly embedded cones

In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

math.GT↗
arXiv2021

Patterson--Sullivan densities in convex projective geometry

For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the convex core, and that it is mixing. We use this to establish asymptotics for the number of closed geodesics. In order to construct the measure of maximal entropy, we develop a theory of Patterson--Sullivan densities for general rank-one convex projective manifolds. In particular, we establish a Hopf--Tsuji--Sullivan--Roblin dichotomy, and prove that, when it is finite, the measure on the unit tangent bundle induced by a Patterson--Sullivan density is mixing under the action of the geodesic flow.

math.DS↗
arXiv2021

Ergodicity and equidistribution in Hilbert geometry

We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT($-1$) or rank-one CAT($0$) spaces, also hold for rank-one properly convex projective structures, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective structures. More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.

math.DS↗
arXiv2021

Topological mixing of the geodesic flow on convex projective manifolds

We introduce a natural subset of the unit tangent bundle of a convex projective manifold, the biproximal unit tangent bundle; it is closed and invariant under the geodesic flow, and we prove that the geodesic flow is topologically mixing on it whenever the manifold is irreducible. We also show that, for higher-rank, irreducible, compact convex projective manifolds, the geodesic flow is topologically mixing on each connected component of the non-wandering set.

math.DS↗