Related papers: $C^0$-gap between entropy-zero Hamiltonians and au…
Given the interest in relating the large $N$ limit of SU(N) to groups of area-preserving diffeomorphisms, we consider the topologies of these groups and show that both in terms of homology and homotopy, they are extremely different. Similar…
We show that if a closed manifold M admits an F-structure (possibly of rank 0) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial circle action. As a corollary we obtain that the simplicial volume…
This paper is concerned about the orbit equivalence types of $C^\infty$ diffeomorphisms of $S^1$ seen as nonsingular automorphisms of $(S^1,m)$, where $m$ is the Lebesgue measure. Given any Liouville number $\alpha$, it is shown that each…
A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit a canonical integration $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) groupoid structure. In this short note we show that when $\pi$ is not…
This paper deals with proper holomorphic self-maps of smoothly bounded pseudoconvex domains in $\C^2$. We study the dynamical properties of their extension to the boundary and show that their non-wandering sets are always contained in the…
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…
In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism,…
We show that, for any compact surface, there is a residual (dense $G_\delta$) set of $C^1$ area preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mane, but no proof was…
We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by…
We study the class of norms on the space of smooth functions on a closed symplectic manifold, which are invariant under the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such norm that is continuous with…
A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the…
Consider a closed coisotropic submanifold $N$ of a symplectic manifold $(M,\omega)$ and a Hamiltonian diffeomorphism $\phi$ on $M$. The main result of this article states that $\phi$ has at least the cup-length of $N$ many leafwise fixed…
Petrov and Pilyugin (2015) generalized a notion of $C^0$ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from ${\mathbb R}^n$ to a manifold, which is a purely topological one. They also…
The deformation problem for pseudoholomorphic curves and related geometrical properties of the total moduli space of pseudoholomorphic curves are studied. A sufficient condition for the saddle point property of the total moduli space is…
A perforated surface is the complement $\mathring\Sigma:=\Sigma\setminus A$ of a countable dense subset $A$ in a connected paracompact surface $\Sigma$. It is known that the topological type of $\Sigma\setminus A$ is independent of the…
We push forward the study of higher dimensional stable Hamiltonian topology by establishing two non-density results. First, we prove that stable hypersurfaces are not $C^3$-dense in any isotopy class of embedded hypersurfaces on any ambient…
We provide an infinite family of diffeomorphic symplectic forms on ruled surfaces, which are pairwise non-isotopic. This answers a uniqueness question regarding symplectic structures up to isotopy on closed symplectic four-manifolds.
We consider a convex solid cone $\mathcal{C}\subset\mathbb{R}^{n+1}$ with vertex at the origin and boundary $\partial\mathcal{C}$ smooth away from $0$. Our main result shows that a compact two-sided hypersurface $\Sigma$ immersed in…
This paper is a step towards the complete topological classification of {\Omega}-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically…
It is proved that the group of holomorphic automorphisms of holomorphically homogeneous nondegenerate (finite Bloom-Graham type + holomorphic nondegenaracy) model surface Q is a subgroup of the group of birational automorphisms of the…