10 problems

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 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.

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 $M$ 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…