Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich  [ Deviations of ergodic averages, Teichmüller flow and Kontsevich-Zorich cocycle ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 927, p. 59-93

Given a regular function with zero mean on the 2-dimensional torus, it is not difficult to see that its ergodic averages above a “generic” translation of the torus are bounded. A decade ago, A. Zorich observed numerically a power law growth of these ergodic averages above generic non exact hamiltonian flows on higher genus surfaces and Kontsevich and Zorich proposed an explanation of this phenomenon by analysing the Teichmüller flow on the moduli space of abelian differentials. We intend to give in the present talk an exposition of some of the ideas of Giovanni Forni’s proof of the Kontsevich-Zorich conjecture.

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension 2, il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à 2, et Kontsevich et Zorich ont proposé une explication (conjecturelle) de ce phénomène par l’analyse du flot de Teichmüller sur l’espace des modules des différentielles abéliennes. Le but de l’exposé est de présenter quelques idées de la preuve que G. Forni a donnée de la conjecture de Kontsevich et Zorich.

Classification:  37F30,  32G15,  37H15,  58A14,  14D07
Keywords: ergodic means, moduli spaces, quadratic differentials, Oseledec theorem, Lyapunov exponents, currents, Hodge theory, period matrix
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     author = {Krikorian, Rapha\"el},
     title = {D\'eviations de moyennes ergodiques, flots de Teichm\"uller et cocycle de Kontsevich-Zorich},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:927},
     pages = {59-93},
     zbl = {1094.37021},
     mrnumber = {2167202},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__59_0}
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Krikorian, Raphaël. Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 927, pp. 59-93. http://www.numdam.org/item/SB_2003-2004__46__59_0/

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