[Difféomorphismes partiellement hyperboliques de la nil-variété de Heisenberg et applications d'holonomie]
Dans cette note, nous démontrons que les automorphismes partiellement hyperboliques de la nil-variété non abélienne de dimension 3 peuvent tous être approchés dans la topologie par des difféomorphismes structurellement stables, chacun possédant un attracteur et un répulseur comme seuls ensembles récurrents par chaîne. Cela implique que ces automorphismes partiellement hyperboliques ne sont pas robustement transitifs. Comme corollaire, nous en déduisons que les holonomies des feuilletages stables et instables des difféomorphismes approximants sont des homéomorphismes quasi périodiquement forcés twistés du cercle, qui sont transitifs mais pas minimaux, qui satisfont à certaines propriétés de régularité dans les fibres.
In this note we show that all partially hyperbolic automorphisms on a 3-dimensional non-Abelian nilmanifold can be -approximated by structurally stable -diffeomorphisms, whose chain recurrent set consists of one attractor and one repeller. In particular, all these partially hyperbolic automorphisms are not robustly transitive. As a corollary, the holonomy maps of the stable and unstable foliations of the approximating diffeomorphisms are twisted quasiperiodically forced circle homeomorphisms, which are transitive but non-minimal and satisfy certain fiberwise regularity properties.
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@article{CRMATH_2014__352_9_743_0,
author = {Shi, Yi},
title = {Partially hyperbolic diffeomorphisms on {Heisenberg} nilmanifolds and holonomy maps},
journal = {Comptes Rendus. Math\'ematique},
pages = {743--747},
publisher = {Elsevier},
volume = {352},
number = {9},
year = {2014},
doi = {10.1016/j.crma.2014.07.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.07.002/}
}
TY - JOUR AU - Shi, Yi TI - Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps JO - Comptes Rendus. Mathématique PY - 2014 SP - 743 EP - 747 VL - 352 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.07.002/ DO - 10.1016/j.crma.2014.07.002 LA - en ID - CRMATH_2014__352_9_743_0 ER -
%0 Journal Article %A Shi, Yi %T Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps %J Comptes Rendus. Mathématique %D 2014 %P 743-747 %V 352 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.07.002/ %R 10.1016/j.crma.2014.07.002 %G en %F CRMATH_2014__352_9_743_0
Shi, Yi. Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps. Comptes Rendus. Mathématique, Tome 352 (2014) no. 9, pp. 743-747. doi : 10.1016/j.crma.2014.07.002. http://www.numdam.org/articles/10.1016/j.crma.2014.07.002/
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