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Martin Mion-Mouton

Publications and source records attributed to Martin Mion-Mouton.

8 recordsLinked to original sources
arXiv2026

Rigidity of the timelike marked length spectrum and length-twist coordinates of singular de-Sitter tori

In this paper, we study the closed timelike geodesics of de-Sitter tori with one singularity and prove their uniqueness in their free homotopy class. We introduce the notion of timelike marked length spectrum of such a torus, and establish its rigidity with respect to the lengths of two homotopy classes of intersection number one. We also construct length-twist coordinates on the deformation space of de-Sitter tori with one singularity.

math.DG↗
arXiv2025

Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation

In this paper, we introduce a natural notion of constant curvature Lorentzian surfaces with conical singularities, and provide a large class of examples of such structures. We moreover initiate the study of their global rigidity, by proving that de-Sitter tori with a single singularity of a fixed angle are determined by the topological equivalence class of their lightlike bi-foliation. While this is reminiscent of Troyanov's uniformization results on Riemannian surfaces with conical singularities, the rigidity will come from topological dynamics in the Lorentzian case.

math.DG↗
arXiv2025

Reductions of path structures and classification of homogeneous structures in dimension three

In this paper we show that if a path structure has non-vanishing curvature at a point then it has a canonical reduction to a Z/2Z-structure at a neighbourhood of that point (in many cases it has a canonical parallelism). A simple implication of this result is that the automorphism group of a non-flat path structure is of maximal dimension three (a result by Tresse of 1896). We also classify the invariant path structures on three-dimensional Lie groups.

math.DG↗
arXiv2023

Cartan connections and path structures with large automorphisms groups

We classify compact manifolds of dimension three equipped with a path structure and a fixed contact form (which we refer to as a strict path structure) under the hypothesis that their automorphism group is non-compact. We use a Cartan connection associated to the structure and show that its curvature is constant.

math.DG↗
arXiv2021

Geometrical compactifications of geodesic flows and path structures

In this paper, we construct a geometrical compactification of the geodesic flow of non-compact complete hyperbolic surfaces $Σ$ without cusps having finitely generated fundamental group. We study the dynamical properties of the compactified flow, for which we show the existence of attractive circles at infinity. The geometric structure of ${\mathrm{T}}^1Σ$ for which this compactification is realized is the pair of one-dimensional distributions tangent to the stable and unstable horocyles of ${\mathrm{T}}^1Σ$. This is a Kleinian path structure, that is a quotient of an open subset of the flag space by a discrete subgroup $Γ$ of ${\mathrm{PGL}}_3(\mathbb{R})$. Our study relies on a detailed description of the dynamics of ${\mathrm{PGL}}_3(\mathbb{R})$ on the flag space, and on the construction of an explicit fundamental domain for the action of $Γ$ on its maximal open subset of discontinuity in the flag space.

math.DS↗
arXiv2021

Partially hyperbolic diffeomorphisms and Lagrangian contact structures

In this paper, we classify the three-dimensional contact partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to the time-one map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil-manifold. The rigid geometric structure induced by the three invariant distributions plays a fundamental role in the proof.

math.DG↗