Related papers: Local rigidity for symplectic billiards

We prove that a bounded domain $\Omega$ in $\R^n$ with smooth boundary has a periodic billiard trajectory with at most $n+1$ bounce times and of length less than $C_n r(\Omega)$, where $C_n$ is a positive constant which depends only on $n$,…

We introduce a method to find differential equations for functions which define tables, such that associated billiard systems admit a local first integral. We illustrate this method in three situations: the case of (locally) integrable wire…

Billiards in ellipses have a confocal ellipse or hyperbola as caustic. The goal of this paper is to prove that for each billiard of one type there exists an isometric counterpart of the other type. Isometry means here that the lengths of…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number $1/q$ is polynomial sense in the smooth…

We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis…

In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.

We develop a framework for dealing with smooth approximations to billiards with corners in the two-dimensional setting. Let a polygonal trajectory in a billiard start and end up at the same billiard's corner point. We prove that smooth…

We provide a lower bound for the embedding capacity of higher-dimensional symplectic ellipsoids, formulated in terms of the Lagrangian capacity of ellipsoids. Our approach relies on examining the Borman--Sheridan class of a Weinstein…

In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian…

We consider the dual billiard map with respect to a smooth strictly convex closed hypersurface in linear 2m-dimensional symplectic space and prove that it has at least 2m distinct 3-periodic orbits.

In this paper we are interested in the motion of a ball inside a billiard table bounded by a particular smooth curve. This table belongs to a family of billiards which can all be drawn by a common process: the so-called gardener's string…

We consider billiards with several possibly non-isometric and asymmetric cusps at flat points; the case of a single symmetric cusp was studied previously in Zhang (2017) and Jung & Zhang (2018). In particular, we show that properly…

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross…

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

The billiard dynamics inside an ellipse is integrable. It has zero topological entropy, four separatrices in the phase space, and a continuous family of convex caustics: the confocal ellipses. We prove that the curvature flow destroys the…

We prove a rigidity property in non-Archimedean dynamics, reminiscent of Zdunik theorem in complex dynamics: every rational map whose equilibrium measure charges an interval in the Berkovich projective line is affine Bernoulli. Our proof is…

A class of Hamiltonian impact systems exhibiting smooth near integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded…

We show that many toric domains $X$ in $R^4$ admit symplectic embeddings $\phi$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $\phi(X)$ to $X$. For instance $X$…