9 problems

The dynamics of the outer billiard is defined in the exterior of a convex boundary $\Gamma \in \mathbb{R}^2$. For a point $z_0=(x_0,y_0)\in\mathbb{R}^2$, draw a tangent line to…

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 let its singularity set be the set of points where the outer length billiard is singular. Let an orbit be a sequence of points generated by the billiard,…

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…

Outer billiards are dynamical systems defined from a convex plane curve by the outer-billiard reflection construction. A polynomial function is preserved when its value is invarian…

Let $p/q$ be the rational parameter, let $Gamma(p/q)$ be its arithmetic landscape, and let $T_{p/q}$ be the affine map carrying the plaid model's fundamental domain to the arithmet…

For a regular polygon and a parameter $\lambda\in(0,1)$, let $T_\lambda$ denote its dissipative outer billiard map. Asymptotic periodicity conjecture. For almost every…

Let $K$ be a convex domain, not necessarily origin-symmetric, whose critical triangles lie on a circle and have a fixed area. Genin and Tabachnikov's conjecture. Among all such con…

Recall that the sets $\widehat A$ and $\widehat B$ are partitioned into atoms on whose interiors the first-return map of $\widehat\Psi$ is defined. Boundary-orbit conjecture. No po…