Related papers: A Remark on Unique Ergodicity and the Contact Type…

For many classes of symplectic manifolds, the Hamiltonian flow of a function with sufficiently large variation must have a fast periodic orbit. This principle is the base of the notion of Hofer-Zehnder capacity and some other symplectic…

The ergodic properties of two uncoupled oscillators, a horizontal and vertical one, residing in a class of non rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are…

In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate…

We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov…

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical…

For a geodesic flow on a negatively curved Riemannian manifold $M$ and some subset $A\subset T^1M$, we study the limit $A$-exceptional set, that is the set of points whose $\omega$-limit do not intersect $A$. We show that if the topological…

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a…

An outstanding property of any Hamiltonian system is the symplecticity of its flow, namely, the continuous trajectory preserves volume in phase space. Given a symplectic but discrete trajectory generated by a transition matrix applied at a…

In this paper, we conduct a comprehensive study on ergodic properties of the geodesic flow on a $C^\infty$ uniform visibility manifold $M$ without conjugate points. If $M$ is a closed surface of genus at least two without conjugate points,…

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational…

We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2)…

Let $M$ be a closed manifold and consider the Hamiltonian flow associated to an autonomous Tonelli Hamiltonian $H:T^*M\rightarrow \mathbb R$ and a twisted symplectic form. In this paper we study the existence of contractible periodic orbits…

Given a $C^{1+\beta}$ flow $\varphi$ with positive speed on a closed smooth Riemannian manifold, we code two homoclinically related $\varphi$-invariant probabilities by an irreducible countable topological Markov flow. As an application, we…

Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we…

We prove the hypersymplectic flow of simple type on standard torus $\mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a…

We prove that any weakly symplectically fillable contact manifold is tight. Furthermore we verify the strong Weinstein conjecture for contact manifolds that appear as the concave boundary of a directed symplectic cobordism whose positive…

A hypersymplectic structure on a 4-manifold is a triple $\omega_1, \omega_2, \omega_3$ of 2-forms for which every non-trivial linear combination $a^1\omega_1 + a^2 \omega_2 + a^3 \omega_3$ is a symplectic form. Donaldson has conjectured…

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first…

Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi$. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson,…

We prove that every non-degenerate Reeb flow on a closed contact manifold $M$ admitting a strong symplectic filling $W$ with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive…