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

On growth rate and contact homology

It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of Reeb periodic orbits of any non-degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non-hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non-transverse to the fibers on a circle bundle

REEB PERIODIC ORBITS AFTER A BYPASS ATTACHMENT

Abstract. On a 3-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori. ensl-00768825, version 1- 24 Dec 2012 1

Étude dynamique des champs de Reeb et propriétés de croissance de l'homologie de contact

We study contact geometry, and focus on the study of periodic orbits of the Reeb vector field. It is a conjecture of Colin and Honda that for universally tight contact structures on hyperbolic manifolds, the number of Reeb periodic orbits grows exponentially with respect to the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic manifolds. Along the lines of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of periodic orbits of any non degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non transverse to the fibers on a circle bundle. Finally we study consequences on Reeb periodic orbits of a bypass attachment, an elementary change of the contact structure consisting in attachment of half an overtwisted disc along a Legendrian arc. We describe new periodic orbits in terms of Reeb chords of the attachment arc, we compute contact homology of a product neighborhood of convex surfaces after a bypass attachment and we compute contact homology for some contact structures on solid tori.Le sujet de cette thèse est la géométrie de contact, en particulier l'étude des orbites périodiques du champ de Reeb. Colin et Honda ont conjecturé que sur une variété hyperbolique munie d'une structure de contact universellement tendue, le nombre d'orbites périodiques de Reeb croit exponentiellement avec la période. Dans les cas non hyperboliques, ils prédisent un comportement polynomial de l'homologie de contact. On montre dans ce texte qu'une variété possédant une composante hyperbolique qui fibre sur le cercle porte une infinité de structures de contact non isomorphes pour lesquelles le nombre d'orbites périodiques de tout champ de Reeb non dégénéré croit exponentiellement avec la période. Ce résultat s'obtient grâce à un résultat de croissance de l'homologie de contact. De plus, on calcule l'homologie de contact et sa croissance dans un cas non hyperbolique : celui des structures universellement tendues non transversales aux fibres sur un fibré en cercles. Enfin, on étudie l'effet d'un recollement de rocade sur les orbites périodiques de Reeb. Cette opération décrit une modification élémentaire de la structure de contact. Elle consiste en l'attachement d'un demi-disque vrillé le long d'un arc legendrien contenu dans le bord de la variété. On montre que les orbites de Reeb créées s'expriment comme mots en les cordes de Reeb de l'arc d'attachement. On calcule l'homologie de contact d'un voisinage produit d'une surface convexe après recollement de rocade ainsi que de certaines structures sur le tore plein

Étude dynamique des champs de Reeb et propriétés de croissance de l'homologie de contact

Le sujet de cette thèse est la géométrie de contact, en particulier l étude des orbites périodiques du champ de Reeb. Colin et Honda ont conjecturé que sur une variété hyperbolique munie d une structure de contact universellement tendue, le nombre d orbites périodiques de Reeb croit exponentiellement avec la période. Dans les cas non hyperboliques, ils prédisent un comportement polynomial de l homologie de contact. On montre dans ce texte qu une variété possédant une composante hyperbolique qui fibre sur le cercle porte une infinité de structures de contact non isomorphes pour lesquelles le nombre d orbites périodiques de tout champ de Reeb non dégénéré croit exponentiellement avec la période. Ce résultat s obtient grâce à un résultat de croissance de l homologie de contact. De plus, on calcule l homologie de contact et sa croissance dans un cas non hyperbolique : celui des structures universellement tendues non transversales aux fibres sur un fibré en cercles. Enfin, on étudie l effet d un recollement de rocade sur les orbites périodiques de Reeb. Cette opération décrit une modification élémentaire de la structure de contact. Elle consiste en l attachement d un demi-disque vrillé le long d un arc legendrien contenu dans le bord de la variété. On montre que les orbites de Reeb créées s expriment comme mots en les cordes de Reeb de l arc d attachement. On calcule l homologie de contact d un voisinage produit d une surface convexe après recollement de rocade ainsi que de certaines structures sur le tore plein.We study contact geometry, and focus on the study of periodic orbits of the Reeb vector field. It is a conjecture of Colin and Honda that for universally tight contact structures on hyperbolic manifolds, the number of Reeb periodic orbits grows exponentially with respect to the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic manifolds. Along the lines of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of periodic orbits of any non degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non transverse to the fibers on a circle bundle. Finally we study consequences on Reeb periodic orbits of a bypass attachment, an elementary change of the contact structure consisting in attachment of half an overtwisted disc along a Legendrian arc. We describe new periodic orbits in terms of Reeb chords of the attachment arc, we compute contact homology of a product neighborhood of convex surfaces after a bypass attachment and we compute contact homology for some contact structures on solid toriNANTES-BU Sciences (441092104) / SudocSudocFranceF

CROISSANCE DES ORBITES PÉRIODIQUES ET COMPLEXITÉ POUR DES STRUCTURES DE CONTACT CONSTRUITES PAR CHIRURGIE

International audienceThis work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology. We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields

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

Impact of Pore Architecture on the Hydroconversion of Long Chain Alkanes over Micro and Mesoporous Catalysts

International audiencen-Hexadecane hydroconversion has been investigated in a series of bifunctionnal metal/acid cata-lysts featuring distinct well defined pore architectures. The acidic component were prepared from dealumi-nated Y zeolites with Si/Al of 15 and 30 post treated in alkaline medium to generate ordered or non-orderedsecondary networks of mesopores and from aluminated ordered mesoporous materials MCM-41, MCM-48,KIT-6 type materials and amorphous silica gel. Activity relates linearly to the strength and number of strongBrönsted acid sites, while selectivity, more precisely the yields in isomerization products, scales directly withthe mesopore volume of the catalysts. The architecture of the mesoporous network, namely the ordering,interconnectivity, homogeneity of the mesopores, affects little catalytst behavior. Confrontation of catalyticdata with diffusion measurements suggests the existence of an optimal mesopore size above which the numberof strong Brönsted sites and the mesopore volume are the only parameters governing catalytic performance