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On 2-cycles of B Diff ( S 1 ) which are represented by foliated S 1 -bundles over T 2

Takashi Tsuboi — 1981

Annales de l'institut Fourier

We give several sufficients conditions for a 2-cycle of B Diff ( S 1 ) d (resp. B Diff K ( R ) d ) represented by a foliated S 1 -(resp. R -) bundle over a 2-torus to be homologous to zero. Such a 2-cycle is determined by two commuting diffeomorphisms f , g of S 1 (resp. R ). If f , g have fixed points, we construct decompositions: f = π f i , g = π g i , where the interiors of Supp ( f i ) Supp ( g i ) are disjoint, and f i and g i belong either to { h i n ; n Z } ( h i Diff ) or to a one-parameter subgroup generated by a C 1 -vectorfield ξ i . Under some conditions on the norms...

Area functionals and Godbillon-Vey cocycles

Takashi Tsuboi — 1992

Annales de l'institut Fourier

We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.

On the group of real analytic diffeomorphisms

Takashi Tsuboi — 2009

Annales scientifiques de l'École Normale Supérieure

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the n -dimensional torus, its identity component is a simple group. For U ( 1 ) fibered manifolds, for manifolds admitting special semi-free U ( 1 ) actions and for 2- or 3-dimensional manifolds with nontrivial U ( 1 ) actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

Codimension one minimal foliations and the fundamental groups of leaves

Tomoo YokoyamaTakashi Tsuboi — 2008

Annales de l’institut Fourier

Let be a transversely orientable transversely real-analytic codimension one minimal foliation of a paracompact manifold M . We show that if the fundamental group of each leaf of is isomorphic to Z , then is without holonomy. We also show that if π 2 ( M ) 0 and the fundamental group of each leaf of is isomorphic to Z k ( k Z 0 ), then is without holonomy.

Différentiabilité des conjugaisons entre systèmes dynamiques de dimension 1

Étienne GhysTakashi Tsuboi — 1988

Annales de l'institut Fourier

Si deux systèmes dynamiques de dimension 1 et de classe C r sont C 1 -conjugués, dans quelles conditions sont-ils C r -conjugués ? Par “système dynamique de dimension 1”, nous entendons ici un feuilletage de codimension 1 ou une application du cercle dans lui-même. Nous donnons des conditions très faibles pour que la réponse à la question précédente soit positive.

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