19 problems

A billiard orbit in a polygon is called perpendicular periodic when it begins perpendicular to one side and later hits some side perpendicularly, so that it retraces its path infin…

Consider the outer length billiard associated with a square, and call an orbit diverging if it is unbounded. Diverging-orbit conjecture. There exists an open ball of diverging orbi…

Let $Q$ be a polygon, and consider its outer length billiard, including its periodic orbits and their lengths. 3-periodic orbit conjecture. Every polygonal outer length billiard ad…

Non-periodic connection-point conjecture. For $n\geq 9$ odd, there are no non-periodic connection points on the double regular $n$-gon.

Heptagon connection-point conjecture. There exist non-periodic connection points on the double regular heptagon.

Let a convex polygon billiard be a billiard in a convex polygonal table, and let $W_t(P)$ denote the billiard wave front at time $t$ from a point $P$ in the table. The wave front b…

A triangular billiard table is a planar polygonal table whose boundary is a triangle, and a periodic billiard path is a billiard trajectory that, after finitely many collisions, re…

A combinatorial period length is the number of line segments in a periodic trajectory on the regular pentagon; a trajectory is asymmetric when it does not have 5-fold rotational sy…

Polygonal-billiard conjecture. For almost all simply connected polygonal tables, the billiard flow is ergodic and weakly mixing.

Let $P$ be a polygon, and let the counting function of generalized diagonals of $P$ count generalized diagonals up to a given length. Polynomial-growth conjecture. Polynomial bound…

Let $P$ be a polygonal table with a fixed number of vertices, and let its billiard flow be the flow obtained by elastic reflection at the sides of $P$. Gutkin–Katok conjecture. The…

A special octagon is a lattice octagon whose opposite sides are parallel with slopes $0$, $\pm1$, and $\infty$, and which has an axis of symmetry parallel to a pair of sides. Speci…

A Hexhouse is the lattice polygon obtained by placing a trapezoid on top of a square, with its moduli described by three integral parameters. Hexhouse periodicity conjecture. Every…

Let $A_{i,j}$ be the planar groups of pairs $(x,y)$ appearing in the conjectured period formula for periodic orbits on the regular hexagon, where $i$ and $j$ index the congruence c…

Position a regular hexagon so that a subdivision of its edge tessellation is the edge tessellation generated by a $120^\circ$-isosceles triangle, and restrict the initial angle to…

Let $P$ be a polygon, and let $F(n)$ denote the full billiard complexity, namely the number of codes of billiard orbits of combinatorial length $n$. Polynomial complexity conjectur…

Let $P$ be an irrational $n$-gon with phase space $\Phi$, and let $\Phi_i\subset\Phi$ be the set of phase points whose base points lie on side $a_i$. An invariant set…

Let $M$ be a polygon, let $P_n$ denote the number of generalized diagonals of $M$ whose discrete length is at most $n$, and let $\epsilon>0$. Katok's conjecture. For every polygon…

Let $P$ be a planar polygon, and let $f_P(n)$ denote the number of words of length $n$ generated by coding billiard orbits according to the domains of regularity they visit. Gutkin…