Related papers: Quantitative error term in the counting problem on…

Let $W$ be an irreducible affine Weyl group, and let $\mathsf{b}$ be a finite word over the alphabet of simple reflections of $W$. Fix a probability $p\in(0,1)$. For each integer $K\geq 0$, let $\mathsf{sub}_p(\mathsf{b}^K)$ be the random…

Periodic surface structures are nowadays standard building blocks of optical devices. If such structures are illuminated by aperiodic time-harmonic incident waves as, e.g., Gaussian beams, the resulting surface scattering problem must be…

Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level,…

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both…

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…

Numerical calculation and analysis of extremely high-lying energy spectra, containing thousands of levels with sequential quantum number up to 62,000 per symmetry class, of a generic chaotic 3D quantum billiard is reported. The shape of the…

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,…

A periodic orbit on a frictionless billiard table is a piecewise linear path of a billiard ball that begins and ends at the same point with the same angle of incidence. The period of a primitive periodic orbit is the number of times the…

We consider billiard trajectories in a smooth convex body in $\mathbb R^d$ and estimate the number of distinct periodic trajectories that make exactly $p$ reflections per period at the boundary of the body. In the case of prime $p$ we…

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and…

We estimate the error in the semiclassical trace formula for the Sinai billiard under the assumption that the largest source of error is due to Penumbra diffraction, that is diffraction effects for trajectories passing within a distance R…

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover…

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional polygonal billiards. In polygons, diffraction typically occurs at the boundary of a family of…

We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in…

In this paper, we consider the obstacle scattering problem for biharmonic equations with a Dirichlet boundary condition in both two and three dimensions. Some basic properties are first derived for the biharmonic scattering solutions, which…

We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same…

This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning,…

Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian…

Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices. It is demonstrated that in the semiclassical limit…

Rational polygonal billiards are one of the key models among the larger class of pseudo-integrable billiards. Their billiard flow may be lifted to the geodesic flow on a translation surface. Whereas such classical billiards have been much…