arXiv subjects
Publications and source records attributed to Christian Bonatti.
Let f be an orientation-preserving homeomorphism of the plane such that f-Id is contracting. Under these hypotheses, we establish the existence, for every periodic orbit, of a fixed point which has nonzero linking number with this periodic orbit.
We prove that the spaces of C1 symplectomorphisms and of C1 volume-preserving diffeomorphisms of connected manifolds both contain residual subsets of diffeomorphisms whose centralizers are trivial. (Les difféomorphismes conservatifs C1-génériques ont un centralisateur trivial. Nous montrons que l'espace des symplectomorphismes de classe C1 et l'espace des difféomomorphismes de classe C1 préservant une forme volume contiennent tous deux des sous-ensembles résiduels de difféomorphismes dont le centralisateur est trivial.)
Given any compact manifold M, we construct a non-empty open subset O of the space of C^1-diffeomorphisms of M and a dense subset D of O such that the centralizer of every diffeomorphism in D is uncountable, hence non-trivial.
In this announcement, we describe the solution in the C1 topology to a question asked by S. Smale on the genericity of trivial centralizers: the set of diffeomorphisms of a compact connected manifold with trivial centralizer residual in Diff^1 but does not contain an open and dense subset.
On the one hand, we prove that the spaces of C^1 symplectomorphisms and of C^1 volume-preserving diffeomorphisms both contain residual subsets of diffeomorphisms whose centralizers are trivial. On the other hand, we show that the space of C^1 diffeomorphisms of the circle and a non-empty open set of C^1 diffeomorphisms of the two-sphere contain dense subsets of diffeomorphisms whose centralizer has a sub-group isomorphic to R.
The purpose of this paper is the study of the roots in the mapping class groups. Let $Σ$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $Σ$, and let $\MM (Σ, \PP)$ denote the mapping class group of $(Σ, \PP)$. We prove that, if $Σ$ is of genus 0, then each $f \in \MM (Σ)$ has at most one $m$-root for all $m \ge 1$. We prove that, if $Σ$ is of genus 1 and has non-empty boundary, then each $f \in \MM (Σ)$ has at most one $m$-root up to conjugation for all $m \ge 1$. We prove that, however, if $Σ$ is of genus $\ge 2$, then there exist $f,g \in \MM (Σ, \PP)$ such that $f^2=g^2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Afterwards, we focus our study on the roots of the pseudo-Anosov elements. We prove that, if $\partial Σ\neq \emptyset$, then each pseudo-Anosov element $f \in \MM(Σ, \PP)$ has at most one $m$-root for all $m \ge 1$. We prove that, however, if $\partial Σ= \emptyset$ and the genus of $Σ$ is $\ge 2$, then there exist two pseudo-Anosov elements $f,g \in \MM (Σ)$ (explicitely constructed) such that $f^m=g^m$ for some $m\ge 2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Furthermore, if the genus of $Σ$ is $\equiv 0 (\mod 4)$, then we can take $m=2$. Finally, we show that, if $Γ$ is a pure subgroup of $\MM (Σ, \PP)$ and $f \in Γ$, then $f$ has at most one $m$-root in $Γ$ for all $m \ge 1$. Note that there are finite index pure subgroups in $\MM (Σ, \PP)$.
We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set is totally disconnected, then f is topologically conjugate to a conformal transformation.
We prove a C^1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C^1-generic diffeomorphisms. For instance, C^1-generic conservative diffeomorphisms are transitive. Nous montrons un lemme de connexion C^1 pour les pseudo-orbites des diffeomorphismes des varietes compactes. Nous explorons alors les consequences pour les diffeomorphismes C^1-generiques. Par exemple, les diffeomorphismes conservatifs C^1-generiques sont transitifs.
This paper gives a survey of recent results on the maximal transitive sets of $C^1$-generic diffeomorphisms.
We construct an example of a Hölder continuous vector field on the plane which is tangent to all foliations in a continuous family of pairwise distinct $C^1$ foliations. Given any $1 \le r <\infty,$ the construction can be done in such a way that each leaf of each foliation is the graph of a $C^r$ function from $\R$ to $\R.$ We also show the existence of a continuous vector field $X$ on $\R^2$ and two foliations $\cal{F}$ and $\cal{G}$ on $\R^2$ each tangent to $X$ with a dense subset $\cal E$ of $\R^2$ such that at every point $x\in \cal E$ the leaves $F_x$ and $G_x$ of the foliation $\cal{F}$ and $\cal{G}$ through $x$ are topologically transverse.