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arXiv · 1103.0177

Feuilletage de Hirsch, mesure harmonique, et g-mesure

Bertrand Deroin·Constantin Vernicos
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Abstract

Les feuilletages de Hirsch sont des feuilletages par surfaces de variétés compactes fermées de dimension 3, dont la dynamique transverse est celle d'un endomorphisme du cercle de degré strictement supérieur à 1. Le but de cette note est de construire, à partir d'une g-mesure associée à un tel endomorphisme, une mesure harmonique sur le feuilletage de Hirsch correspondant, au sens de Lucy Garnett. Ceci nous permet de donner des exemples de métriques riemanniennes sur le fibré tangent du feuilletage de Hirsch, lisses le long des feuilles et continues transversalement, pour lesquelles il existe plusieurs mesures harmoniques. De tels exemples montrent que le résultat d'unique ergodicité obtenu par le premier auteur et Victor Kleptsyn pour les feuilletages transversalement conformes [GAFA, 2007, Vol. 17, No 4, 1043-1105] n'est valable que lorsque la métrique riemannienne est Höldérienne transversalement, mais pas juste continue.

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Bertrand Deroin, Constantin Vernicos. 2011-03-01. Feuilletage de Hirsch, mesure harmonique, et g-mesure. https://arxiv.org/abs/1103.0177

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