Related papers: (Arc-)connectedness for the space of $\mathbb{Z}^d…

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…

We prove that the space of actions of Z^d by C^1 (orientation-preserving) diffeomorphisms of either the interval or the circle is connected by arcs. This is proved by showing that all such actions can be C^0 conjugated via a 1-parameter…

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 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.

Denote by $\DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^\infty$ manifold $M$. We show that if $\dim(M)\geq2$ and $r\neq \dim(M)+1$, then any homomorphism from $\DC(M)_0$ to…

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This…

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of…

We study smooth locally free actions of ${\mathbb R}^n$ on manifolds $M$ of dimension $n+1$. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact…

Denote by $\DC(M)_0$ the identity component of the group of compactly supported $C^\infty$ diffeomorphisms of a connected $C^\infty$ manifold $M$, and by $\HR$ the group of the homeomorphisms of $\R$. We show that if $M$ is a closed…

Let $G$ be a countable group with no finitely generated subgroup of exponential growth. We show that every action of $G$ on a countable set preserving a linear (respectively, circular) order can be realised as the restriction of some action…

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,\omega$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$.…

We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+\alpha}$ conformal) is robustly minimal (resp. ergodic with respect to Lebesgue measure). We also show how, locally, a…

In the first part of this text we give a survey of the properties satisfied by the C1-generic conservative diffeomorphisms of compact surfaces. The main result that we will discuss is that a C1-generic conservative diffeomorphism of a…

We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic…

We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs),…

We define boundedness properties on the contractible fixed points set of the time-one map of an identity isotopy on a closed oriented surface with genus $g\geq1$. In symplectic geometry, a classical object is the notion of action function,…

Let $M$ be a connected 1-manifold, i.e., $M = \R \cong (0, 1), [0, 1), [0, 1]$, or $S^1$, and let $\Homeo_+(M)$ (resp. $\Diff_+^1(M)$) be the group of orientation-preserving homeomorphisms (resp. $C^1$ diffeomorphisms) of $M$. It is a…

Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the…

A space $X$ is $n$-arc connected (respectively, $n$-circle connected) if for any choice of at most $n$ points there is an arc (respectively, a circle) in $X$ containing the specified points. We study $n$-arc connectedness and $n$-circle…

The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show…