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# Quelques propriétés géométriques et dynamiques globales des structures Lagrangiennes de contact

Abstract : In this PhD thesis, we study the interactions between some geometrical properties of Lagrangian contact structures, and some dynamical properties of their automorphisms. We study those three-dimensional partially hyperbolic diffeomorphisms, whose three invariant distributions are smooth, and whose stable and unstable distributions generate a contact distribution. These last two distributions define a Lagrangian contact structure, whose analysis allows us to classify the investigated partially hyperbolic diffeomorphisms. Our principal tool to study Lagrangian contact structures is their normal Cartan geometry, whose equivalence problem is described in detail. These Cartan geometries are modelled on the space of pointed projective lines of $\mathbb{R}\mathbf{P}^2$, homogeneous under the action of $\mathrm{PGL}_3(\mathbb{R})$. The study of the geometry of this model space, and of the dynamical patterns of the action of $\mathrm{PGL}_3(\mathbb{R})$, allow us to construct compactifications of some Kleinian Lagrangian contact structures, on which we obtain examples of non-conservatives Lagrangian contact automorphisms.
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https://tel.archives-ouvertes.fr/tel-03013231
Contributor : Martin Mion-Mouton <>
Submitted on : Monday, January 4, 2021 - 5:51:07 PM
Last modification on : Thursday, January 7, 2021 - 3:19:54 PM

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• HAL Id : tel-03013231, version 1

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Martin Mion-Mouton. Quelques propriétés géométriques et dynamiques globales des structures Lagrangiennes de contact. Géométrie différentielle [math.DG]. Université de Strasbourg, 2020. Français. ⟨tel-03013231⟩

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