Related papers: On the finite blocking property

A planar polygonal billiard $\P$ is said to have the finite blocking property if for every pair $(O,A)$ of points in $\P$ there exists a finite number of ``blocking'' points $B_1, ..., B_n$ such that every billiard trajectory from $O$ to…

A translation surface S is said to have the finite blocking property if for every pair (O,A) of points in S there exists a finite number of "blocking" points B_1,...,B_n such that every geodesic from O to A meets one of the B_i's. S is said…

We show that for a rational polygonal billiard, the set of pairs of points that do not illuminate each other (not connected by a billiard trajectory) is finite, and use the same method to extend the results of Leli\`evre, Monteil and Weiss,…

Over the course of studying billiard dynamics, several questions were raised. One of the questions was, which surfaces satisfy the following property (which is called Veech's dichotomy): Any direction is either completely periodic or…

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of…

The blocking number of a manifold is the minimal number of points needed to block out lights between any two given points in the manifold. It has been conjectured that if the blocking number of a manifold is finite, then the manifold must…

We show that the billiard in a regular polygon is weak mixing in almost every invariant surface, except in the trivial cases which give rise to lattices in the plane (triangle, square and hexagon). More generally, we study the problem of…

We consider various metric and analytic notions of finiteness on translation surfaces. The Veech group of a surface is discrete if the surface has finite area or is totally bounded.

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 prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one…

Let x and y be points in a billiard table M that is bounded by a curve sigma. We assume that sigma is a simple closed C^r curve with positive curvature, where r is at least 2. A subset B of M\{x,y} is called a blocking set for the pair…

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of…

Consider a collection of finitely many polygons in $\mathbb C$, such that for each side of each polygon, there exists another side of some polygon in the collection (possibly the same) that is parallel and of equal length. A translation…

A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or…

Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set…

Let $T\subset \R^{m+1}$ be a strictly convex domain bounded by a smooth hypersurface $X=\partial T$. In this paper we find lower bounds on the number of billiard trajectories in $T$ which have a prescribed intial point $A\in X$, a…

We identify all translation covers among triangular billiard surfaces. Our main tools are the holonomy field of Kenyon and Smillie and a geometric property of translation surfaces, which we call the fingerprint of a point, that is preserved…

We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument,…

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result…

We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if,…