Helene Eynard-Bontemps

We study conjugacy classes of germs of nonflat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This may...

We show path-connectedness for the space of $\mathbb{Z}^d$ actions by $C^1$ diffeomorphisms with absolutely continuous derivative on both the closed interval and the circle. We also give a new and short proof of the connectedness of the space of $\mathbb{Z}^d$ actions by $C^2$ diffeomorphisms on the interval.

We study conjugacy classes of germs of non-flat diffeomorphisms of the real line fixing the origin. Based on the work of Takens and Yoccoz, we establish results that are sharp in terms of differentiability classes and order of tangency to the identity. The core of all of this lies in the invariance of residues under low-regular conjugacies. This ma...

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 b...

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...

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.

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. We show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that 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 i...

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 homeom...

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...

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.

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z^r_f its centralizer in the group of C^r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z^1_f is always a one-parameter group, naturally identified to \R, (with f identified to 1). On the other hand, Z^r_f, for r greater or equal to 2, can be sma...

Let D^r_+[0,1], r >= 1, denote the group of orientation-preserving C^r diffeomorphisms of [0,1]. We show that any two representations of Z^2 in D^r_+[0,1], r >= 2, are connected by a continuous path of representations of Z^2 in D^1_+[0,1]. We derive this result from the classical works by G. Szekeres and N. Kopell on the C^1 centralizers of the dif...

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 clos...

Let f be a smooth diffeomorphism of the closed half-line R+ with a single fixed point at the origin. In this article, we study the centralizer of f in the group D r of C r diffeomorphisms of R+, 1 ≤ r ≤ ∞, that is, the (closed) subgroup Z r f of Dr made up of all diffeomorphisms commuting with f. The first things to observe are that Z r decreases w...

Let f be a smooth diffeomorphism of the half-line fixing only the origin and Z r its centralizer in the group of C r diffeomorphisms. According to well-known results of Szekeres and Kopell, Z 1 is a one-parameter group. On the other hand, Sergeraert constructed an f whose centralizer Z ∞ reduces to the infinite cyclic group generated by f. We prese...

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 de...