189 problems

Consider a strictly convex closed planar curve $C$ and the billiard in the bounded domain it encloses. The billiard is Birkhoff caustic-integrable if a topological annulus adjacent…

An outer length billiard is associated with a plane oval, meaning a closed strictly convex smooth curve, and acts on the exterior of the oval. Its periodic points are points lying…

A Birkhoff billiard is the billiard system inside a smooth strictly convex plane domain, with trajectories reflecting elastically at the boundary. Such a billiard is integrable whe…

Ergodicity conjecture for circular polygons. For certain values of the parameters, the billiard is ergodic.

Let $C$ be an ellipse and let $O$ be an interior point that is not a focus of $C$. For each $n\geq 1$, the $n$-th caustic by reflection from $O$ is the envelope of the family of bi…

Let $\gamma\subset\mathbb{RP}^2$ be a strictly convex planar curve equipped with a family $(\sigma_P)_{P\in\gamma}$ of projective involutions of the tangent lines $L_P$, each fixin…

Aperiodic-word complexity conjecture. For $n$ sufficiently large, the complexity of each individual aperiodic word is

Let $\Omega$ be a convex planar billiard with smooth boundary. It is polynomially integrable if its billiard flow near the unit tangent bundle of the boundary has a first integral…

A convex billiard table is absolutely focusing if its boundary dynamics satisfy the absolute focusing property. A convex table is typical in the sense of the usual notion of generi…

Let $B$ be a perfectly reflecting, possibly disconnected, closed bounded body in Euclidean space. An oriented ray is a ray of invisibility with $k$ reflections if, after finitely m…

Let $m_0,\ldots,m_{n-1}$ be the winding numbers of a periodic billiard trajectory inside an ellipsoid, as defined by the oscillations of its elliptic coordinates. Winding-number or…

Let ${\bf D}$ be a fat disk-polygon in ${\mathbb E}^2$, and let ${\bf D}(\varepsilon)$ denote its $\varepsilon$-rounded disk-polygon, where $\varepsilon$ is at most the inradius of…

The periodic-orbit count conjecture. The estimate in case (A) should be improvable to $n+m-2$; that is, the number of distinct $D_n$-orbits of $n$-periodic billiard trajectories in…

Let $s$ and $k$ be positive integers, let $\varepsilon_i\in\{\pm1\}$ for $1\leq i\leq s-1$, and consider the closure of the braid … A generalized rosette knot is this closed braid.…

Let ${\mathfrak a}0beta\betasubset{\mathfrak a}0\mathbb{R}^2$ be a planar domain, let ${\mathfrak a}0Psi={\mathfrak a}0Psi(\beta)$ denote the interior phase space of its billiard m…

Weyl's conjecture. The asymptotic expansion of Weyl's law continues, and its second term involves the boundary volume ${\mathfrak a}0text{vol}({\mathfrak a}0partial\beta)$. This co…

A billiard table is a planar domain in which a billiard ball moves along straight segments and reflects specularly at the boundary; a two-parameter family of periodic orbits is a f…

Let the full rotation set and the admissible rotation set be the rotation sets associated with the billiard system with one obstacle, as described in the paper. The admissible rota…

Billiard-flow deviation conjecture. For all rational-angle Euclidean polygons, the deviation of ergodic averages for the billiard flow is the same for almost all directions, and de…

Boshernitzan's conjecture. Given a rational triangle, every nonsingular orbit in the invariant surface containing a perpendicular direction is periodic.

A Turks head knot is, for example, the closure of the 3-string braid … A Lissajous knot is a knot represented by the standard Lissajous parametrization associated with billiard kno…

A polytope is a polyhedral billiard table, and a knot type is an ambient-isotopy class of knots represented by a ball trajectory in that table. Infinite-knot-type conjecture. Any p…

A knot type is an ambient-isotopy class of knots in three-dimensional space, and a polytope is a three-dimensional polyhedral billiard table in which a ball follows a billiard traj…

Independent-sum asymptotics conjecture. The sum $S_n^{\boldsymbol{\kappa}}$ asymptotically behaves the same way as the sum of independent copies of $\boldsymbol{\kappa}$.

Let $\Omega$ be the billiard domain, let $\hat{A}$ be multiplication by a test function $A$, and define its spatial average by … For eigenfunctions $\phi_n$ of the Laplacian with e…