Related papers: Billiards with singular invariant curves

We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex…

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams $E_6, E_7$, $E_8$. These curves are…

A singular point of a smooth map F: M -> N of manifolds is a point in M at which the rank of the differential dF is less than the minimum of dimensions of M and N. The classical invariant of the set S of singular points of F of a given type…

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic…

The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ;…

Light propagation on a two-dimensional curved surface embedded in a three-dimensional space has attracted increasing attention as an analog model of four-dimensional curved spacetime in laboratory. Despite recent developments in modern…

In this paper, we consider deformations of singular complex curves on complex surfaces. Despite the fundamental nature of the problem, little seems to be known for curves on general surfaces. Let $C\subset S$ be a complete integral curve on…

We show that any sufficiently (finitely) smooth $\mathbb Z_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length…

We define a computable topological invariant $\mu(\gamma)$ for generic closed planar regular curves $\gamma$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we…

We study the pull-back of regular 1-forms on a complex irreducible plane curve singularity under the normalization morphism.

We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated…

We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we…

In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. We show that near an isolated invariant continuum the flow is topologically equivalent to a C1 flow. We deduce…

We consider the outer billiards map with contraction outside polygons. We construct a 1-parameter family of systems such that each system has an open set in which the dynamics is reduced to that of a piecewise contraction on the interval.…

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…

We define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality…

Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto…

In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will discuss the existence of…

The notion of a Radon transform is introduced for completely integrable billiard tables. In the case of Liouville billiard tables of dimension 3 we prove that the Radon transform is one-to-one on the space of continuous functions $K$ on the…