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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.

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 $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.

We give several sufficients conditions for a 2-cycle of $B$ Diff$(S^1{)}_d$ (resp. $B$ Diff${}_K(\mathbf{R}{)}_d$) represented by a foliated $S^1$-(resp. $\mathbf{R}$-) 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. $\mathbf{R}$). If $f$, $g$ have fixed points, we construct decompositions: $f=\pi f_i$, $g=\pi g_i$, where the interiors of Supp$\left(f_i\right)\cup$ Supp$\left(g_i\right)$ are disjoint, and $f_i$ and $g_i$ belong either to $\{h_i^n;n\in \mathbf{Z}\}$ ($h_i\in$ Diff${}^{\infty }$) or to a one-parameter subgroup generated by a $C^1$-vectorfield ${\xi }_i$. Under some conditions on the norms...

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.

Let $ℱ$ 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 $ℱ$ is isomorphic to $Z$, then $ℱ$ is without holonomy. We also show that if ${\pi }_2\left(M\right)\cong 0$ and the fundamental group of each leaf of $ℱ$ is isomorphic to $Z^k$ ($k\in Z_{\ge 0}$), then $ℱ$ is without holonomy.