Mean dimension and spaces of pseudo-holomorphic maps

Mean dimension and spaces of pseudo-holomorphic maps
[Dimension moyenne et espaces d'applications pseudo-holomorphes]

Gournay, Antoine

Directeur de thèse : Pansu, Pierre

Membres du jury : Viterbo, Claude ; Gromov, Mikhail ; Pansu, Pierre ; Salamon, Dietmar ; Ball, Keith

Thèses d'Orsay, no. 757 (2008) , 128 p.

Résumé
Abstract

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 $l^{p}$ (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 $l^{p} (Γ; ℝ)$ 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, ${dim}_{l^{p}}$ , qui est reliée à la dimension de Von Neumann est aussi introduite s'inspirant de résultats de Gromov. Quelques unes des propriétés de ${dim}_{l^{p}}$ 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 $l^{p}$ 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 $l^{p} (Γ; ℝ)$ 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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     title = {Mean dimension and spaces of pseudo-holomorphic maps},
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     year = {2008},
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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/

Bibliographie
Cité par

[1] T. Aubin. Nonlinear analysis on manifolds. Monge-Ampère equations, volume 252 of Grundlehren der Mathematischen Wissenschaft en [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1982. | MR | Zbl

[2] M. Audin and J. Lafontaine, editors. Holomorphic curves in symplectic geometry, volume 117 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1994. | MR | Zbl | DOI

[3] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2000. | MR | Zbl

[4] J. Cheeger and M. Gromov. $L_{2}$ -cohomology and group cohomology. Topology, 25(2):189-215, 1986. | MR | Zbl | DOI

[5] S. Y. Cheng and P. Li. Heat kernel estimates and lower bound of eigenvalues. Comment. Math. Helv., 56(3):327-338, 1981. | MR | Zbl | DOI

[6] M. Coornaert. Dimension topologique et systèmes dynamiques, volume 14 of Cours Spécialisés. Société Mathématique de France, Paris, 2005. | MR | Zbl

[7] M. Coornaert and F. Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete Contin. Dyn. Syst., 13(3):779-793, 2005. | MR | Zbl | DOI

[8] S. K. Donaldson. The approximation of instantons. Geom. Funct. Anal., 3(2):179-200, 1993. | MR | Zbl | DOI

[9] S. K. Donaldson and P. B. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1990., Oxford Science Publications. | MR | Zbl

[10] D. L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289-1306, 2006. | MR | Zbl | DOI

[11] A. N. Dranishnikov. Homological dimension theory. Uspekhi Mat. Nauk, 43(4(262): 11-55, 255, 1988. | Zbl | MR

[12] J. Dydak. Cohomological dimension theory. In Handbook of geometric topology, pages 423-470. North-Holland, Amsterdam, 2002. | MR | Zbl

[13] H. Federer. Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969. | MR | Zbl

[14] R. Fenn. Embedding polyhedra. Bull. London Math. Soc., 2:316-318, 1970. | MR | Zbl | DOI

[15] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. | MR | Zbl

[16] F. P. Greenleaf. Invariant means on topological groups and their applications. Van Nostrand Mathematical Studies, No. 16. Van Nostrand Reinhold Co., New York, 1969. | MR | Zbl

[17] M. Gromov. Filling Riemannian manifolds. J. Differential Geom., 18(1):1-147, 1983. | MR | Zbl | DOI

[18] M. Gromov. Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), volume 182 of London Math. Soc. Lecture Note Ser., pages 1-295. Cambridge Univ. Press, Cambridge, 1993. | MR | Zbl

[19] M. Gromov. Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom., 2(4):323-415, 1999. | MR | Zbl | DOI

[20] L. Hörmander. The analysis of linear partial differential operators. I. Classics in Mathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition. | MR | Zbl | DOI

[21] C. Hummel. Gromov's compactness theorem for pseudo-holomorphic curves, volume 151 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1997. | MR | Zbl | DOI

[22] W. Hurewicz and H. Wallman. Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, Princeton, N. J., 1941. | MR | Zbl | JFM

[23] V. I. Ivanov and S. A. Pichugov. Jung constants in $l_{p}^{n}$ -spaces. Mat. Zametki, 48(4):37-47, 158, 1990. | Zbl | MR

[24] M. Katz. The filling radius of two-point homogeneous spaces. J. Differential Geom., 18(3):505-511, 1983. | MR | Zbl | DOI

[25] S. Kobayashi. Differential geometry of complex vector bundles, volume 15 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1987. , Kano Memorial Lectures, 5. | MR | Zbl

[26] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol. II. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996. Reprint of the 1969 original, A Wiley- Interscience Publication. | MR

[27] A. N. Kolmogorov and S. V. Fomīn. Introductory real analysis. Dover Publications Inc., New York, 1975. Translated from the second Russian edition and edited by Richard A. Silverman, Corrected reprinting. | MR

[28] F. Krieger. Le lemme d'Ornstein-Weiss d'après Gromov. In Dynamics, ergodic theory, and geometry, volume 54 of Math. Sci. Res. Inst. Publ., pages 99-111. Cambridge Univ. Press, Cambridge, 2007. | MR | Zbl

[29] F. Krieger. Sous-décalages de Toeplitz sur les groupes moyennables résiduellement finis. J. Lond. Math. Soc. (2), 75(2):447-462, 2007. | MR | Zbl | DOI

[30] E. Lindenstrauss. Mean dimension, small entropy factors and an embedding theorem. Inst. Hautes Études Sci. Publ. Math., 89:227-262 (2000), 1999. | MR | Zbl | DOI | Numdam

[31] E. Lindenstrauss and B. Weiss. Mean topological dimension. Israel J. Math., 115:1-24, 2000. | MR | Zbl | DOI

[32] J. Lindenstrauss and L. Tzafriri. Classical Banach spaces. Springer-Verlag, Berlin, 1973. Lecture Notes in Mathematics, Vol. 338. | MR | Zbl

[33] W. Lück. $L^{2}$ -invariants of regular coverings of compact manifolds and CW-complexes. In Handbook of geometric topology, pages 735-817. North-Holland, Amsterdam, 2002. | MR | Zbl

[34] D. Mcduff and D. Salamon. Introduction to symplectic topology. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, second edition, 1998. | MR | Zbl

[35] D. Mcduff and D. Salamon. J-holomorphic curves and symplectic topology, volume 52 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. | MR | Zbl

[36] F. J. Murray. On complementary manifolds and projections in spaces $L_{p}$ and $l_{p}$ . Trans. Amer. Math. Soc., 41(1) : 138-152, 1937. | MR | JFM | Zbl

[37] A. Newlander and L. Nirenberg. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2), 65:391-404, 1957. | MR | Zbl | DOI

[38] D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math., 48:1-141, 1987. | MR | Zbl | DOI

[39] R. Paley. On orthogonal matrices. Journ. Math, and Phys. (MIT), 12:311-312, 1933. | Zbl | JFM | DOI

[40] P. Pansu. Introduction to $L^{2}$ Betti numbers. In Riemannian geometry (Waterloo, ON, 1993), volume 4 of Fields Inst. Monogr., pages 53-86. Amer. Math. Soc., Providence, RI, 1996. | MR | Zbl

[41] Y. B. Pesin. Dimension theory in dynamical systems. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. | MR

[42] L. S. Pontryagin. Application of combinatorial topology to compact metric spaces. Uspekhi Mat. Nauk, 39(5(239)):131-164, 1984. | MR

[43] W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, second edition, 1991. | MR | Zbl

[44] J.-C. Sikorav. Singularities of J-holomorphic curves. Math. Z., 226(3):359-373, 1997. | MR | Zbl | DOI

[45] S. Smale. An infinite dimensional version of Sard's theorem. Amer. J. Math., 87:861-866, 1965. | MR | Zbl | DOI

[46] A. Sobczyk. Projections in Minkowski and Banach spaces. Duke Math. J., 8:78-106, 1941. | MR | JFM | DOI

[47] C. H. Taubes. The existence of anti-self-dual conformal structures. J. Differential Geom., 36(1): 163-253, 1992. | MR | Zbl | DOI

[48] M. Tsukamoto. Macroscopic dimension of the $ℓ^{p}$ -ball with respect to the $ℓ^{q}$ -norm. http://www.math.kyoto-u.ac.jp/preprint/2007/10tsukamoto.pdf or arXiv:0801.4548, page 10, 2008. | MR | Zbl