Mean dimension and spaces of pseudo-holomorphic maps
[Dimension moyenne et espaces d'applications pseudo-holomorphes]
Gournay, Antoine
Directeur de thèse : Pansu, Pierre
Thèses d'Orsay, no. 757 (2008) , 128 p.

Le présent texte s’articule en deux thèmes. Le premier commence par l’évaluation des largeurs de boules unités dans des espaces de Banach. Ces évaluations peuvent être perçues comme un problème de compression: on s’intéresse à des applications non linéaires de fibre aussi petites que possible qui envoient ces boules unités vers des polyèdres de dimension fixée. Des bornes pour ces quantités sont obtenues, le cas des boules lp (de dimension finie ou infinie) avec leur métrique y est plus particulièrement étudié. Les largeurs interviennent aussi dans la définition de la dimension moyenne, une adaptation de l'entropie à des cas où elle est infinie. Cependant, cet invariant dynamique est insuffisant pour différencier les systèmes donnée par la boule unité de lp(Γ;ℝ) avec action naturelle de Γ où p est fini et Γ est un groupe (typiquement ℤ). Une modification de la dimension moyenne est ainsi introduite pour s'occuper de ces cas, elle n’est cependant plus un invariant topologique mais est Hôlder covariante. Ceci est encore suffisant pour obtenir des obstructions. Une autre variante, dimlp, qui est reliée à la dimension de Von Neumann est aussi introduite s'inspirant de résultats de Gromov. Quelques unes des propriétés de dimlp sont démontrées (requérant une généralisation du lemme d’Ornstein-Weiss). Cependant, des propriétés importantes restent en suspens.

Le second thème traite des courbes pseudo-holomorphes. Un résultat sur le recollement de deux courbes pseudo-holomorphes est d’abord démontré. Celui-ci permet d’avoir une idée plus précise du comportement de la courbe recollée près du point où les deux courbes d’origines se touchent. Ensuite, nous nous intéressons à former des cylindres pseudo-holomorphes depuis une chaîne de courbes pseudo-holomorphes, et sous de fortes hypothèses, un résultat d’interpolation est obtenu. L’interpolation permet entre autres de montrer que les cylindres obtenus sont simples, d’images distinctes, et forment une famille de dimension infinie. Les deux thèmes se rejoignent étant donné que la famille d’applications obtenue est de dimension moyenne positive.

Un appendice contient une adaptation de la "boîte à outils" de Taubes (des méthodes d’analyse elliptique introduite dans "The existence of anti-self-dual structures") au cas de dimension 2. Cependant, à cause de la spécificité du noyau de Green en dimension 2, celles-ci n’a pu être appliquée à la démonstration d’un théorème de Runge pour les courbes pseudo-holomorphes.

This thesis covers two themes. The first begins by evaluating the width of unit balls in Banach spaces. Evaluation of width can be seen as a problem arising from compressed sensing: we look at nonlinear maps with small fiber diameters that send these unit balls to polyhedra of given dimension. Bounds for these quantities are found, focusing on the case of lp balls (of finite or infinite dimension) with their proper metric. Widths are also related to mean dimension, an adaptation of entropy to cases where it would be infinite. However, this dynamical invariant turns out to be inefficient if one wishes to distinguish between the dynamical systems given by the unit ball of lp(Γ;ℝ) with natural action of Γ for finite p and Γ a discrete group (typically ℤ). A alteration of mean dimension is thus introduced to deal with this case, but it is no longer a topological invariant but Hölder covariant. This is still sufficient to obtain obstructions. Another variant which relates to Von Neumann dimension is also introduced, follow ing Gromov, and some properties are then proved (requiring in particular an extension of the Orstein-Weiss lemma). However, important properties are left unproven.

The second theme deals with pseudo-holomorphic curves. We first modify a result on the gluing of two pseudo-holomorphic curves so as to have a precise behaviour of the glued curve close to the point of intersection of the two curves it comes from. Then pseudo-holomorphic cylinders are constructed from a chain of pseudo-holomorphic curves. Under strong assumptions, we obtain an interpolation result on these cylinders. This interpolation result has many consequences, in particular, that thedifferent cylinders obtained are simple, have different images, and form a family of infinite dimension. This theme is reunited with the first as this family has also positive mean dimension.

An appendix contains an adaptation of "Taubes toolbox" (methods of elliptic analysis developed by Taubes in "The existence of anti-self-dual structures") to the 2-dimensional case. However, due to the specificity of Green’s kernel in dimension 2, these could not be applied to the proof of a Runge theorem for pseudo-holomorphic curves.

Mot clés : largeur, dimension moyenne, lemme d’Ornstein-Weiss, dimension de Von Neumann, applications pseudo-holomorphe, chirurgie dénombrable, interpolation, analyse elliptique à la Taubes 
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Gournay, Antoine. Mean dimension and spaces of pseudo-holomorphic maps. Thèses d'Orsay, no. 757 (2008), 128 p. http://numdam.org/item/BJHTUP11_2008__0757__P0_0/

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