Smooth times of a flow in dimension 1

Let $\alpha$ be an irrational number and $I$ an interval of $\mathbb{R}$. If $\alpha$ is diophantine, we show that any one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ is in fact the flow of a $C^\infty$ vector field. If $\alpha$ is Liouville on the other hand, we construct a one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ but which is not the flow of a $C^2$ vector field (though, if $I$ has boundary, we explain that the hypotheses force it to be the flow of a $C^1$ vector field). We extend both results to families of irrational numbers, the critical arithmetic condition in this case being simultaneous "diophantinity". For one-parameter groups defining a free action of $(\mathbb{R},+)$ on $I$, these results follow from famous linearization theorems for circle diffeomorphisms. The novelty of this work concerns non-free actions.Comment: 35 pages, 8 figure

On the connectedness of the space of codimension one foliations on a closed 3-manifold

We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late 60's that every plane field can be deformed continuously to an integrable one, so the above inclusion induces a surjective map between connected components. We prove that this map is actually a bijection.Comment: 47 pages, 22 figure

Connectedness of the space of smooth actions of Zn\Z^n on the interval

We prove that the spaces of \Cinf orientation-preserving actions of $\Z^n$ on $[0,1]$ and nonfree actions of $\Z^2$ on the circle are connected

Arc-connectedness for the space of smooth Zd\mathbb{Z}^d-actions on 1-dimensional manifolds

We show that the space of $\mathbb{Z}^d$ actions by $C^2$ orientation-preserving diffeomorphisms of a compact 1-manifold is $C^{1+\mathrm{ac}}$ arcwise connected

On the failure of linearization for germs of C1C^1 hyperbolic vector fields in dimension 1

We investigate conjugacy classes of germs of hyperbolic 1-dimensional vector fields at the origin in low regularity. We show that the classical linearization theorem of Sternberg strongly fails in this setting by providing explicit uncountable families of mutually non-conjugate flows with the same multipliers, where conjugacy is considered in the bi-Lipschitz, $C^1$ and $C^{1+ac}$ settings.Comment: 20 page

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C^{1+bv} conjugacy class if and only if it is the time-1 map of a C^1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval interval diffeomorphisms. Several related results in other regularity classes are obtained, and many open questions are addressed.Comment: 60 pages, 2 picture

On the centralizer of diffeomorphisms of the half-line

International audienc

Sur deux questions connexes de connexité concernant les feuilletages et leurs holonomies

The two connectedness questions we are interested in refer to : – the space of codimension 1 foliations on a 3-manifold; – the space of representations of Z^2 into the group of smooth diffeomorphisms of the interval. The main result, which is proved in the second part of the dissertation, is the following : if two codimension 1 foliations on a closed 3-manifold have homotopic tangent subbundles, they can be linked by a continuous path of foliations. This statement hides a subtlety : if the given foliations are smooth, the path we construct can contain near its bounderies foliations which are only C^1. This is because we don't know whether the space of representations of Z^2 into the diffeomorphisms of the interval is connected or not. In an attempt to answer this very question, we pointed out the following phenomenon which is discussed in the first part of the dissertation : many smooth diffeomorphisms of R+ fixing only the origin have a C^infinite centralizer which is uncountable and dense in their C^1 centralizer (which is itself a one-parameter group). We also study the arithmetic properties of this subgroup.Les deux questions de connexité auxquelles on s'intéresse concernent : – l'espace des feuilletages de codimension 1 sur une variété de dimension 3 ; – l'espace des représentations du groupe Z^2 dans le groupe des difféomorphismes lisses de l'intervalle. Le résultat principal, qu'on démontre dans la seconde partie de la thèse, est le suivant : si deux feuilletages de codimension 1 sur une variété close de dimension 3 ont des sous-fibrés tangents homotopes, on peut les relier par un chemin de feuilletages. Cet énoncé cache une subtilité : si les feuilletages donnés sont lisses, le chemin obtenu peut contenir, près de ses extrémités, des feuilletages qui ne sont que C^1. Cela vient de ce qu'on ne sait pas si l'espace des représentations de Z^2 dans les difféomorphismes de l'intervalle est connexe ou non. En tentant de répondre à cette question, on a montré le phénomène suivant qui fait l'objet de la première partie de la thèse : de nombreux difféomorphismes lisses de R+, sans autre point fixe que l'origine, ont un centralisateur C^infini non dénombrable et dense dans leur centralisateur C^1, lequel est un groupe à un paramètre. On discute également les propriétés arithmétiques de ce sous-groupe

On the connectedness of the space of codimension one foliations on a closed 3-manifold

International audienc