## Tag Archives: absolutely focusing

### Billiards

9. Victor Ivrii conjecture. Let $Q$ be a strictly convex domain, $F$ be the billiard map on the phase space $\Omega=\partial Q\times(0,\pi)$. Let $\omega$ be the Lebesgue measure of $\Omega$, and $\ell$ be the Lebesgue measure on $Q$.

Conjecture 1. $\omega(\text{Per}(F))=0$ for all $Q$ with $C^\infty$ boundaries.

Remark. This is about a general domain $Q$, not a generic domain.

Definition. A point $q\in\partial Q$ is said to be an absolute looping point, if $\omega_q(\bigcup_{n\neq0}F^n\Omega_q)>0$. Let $L(Q)$ be the set of absolute looping points.

Conjecture 2. $\ell(L(Q))=0$ for all $Q$.

Question: When $L(Q)=\emptyset$?

8. In Boltzmann gas model, the identical round molecules are confined by a box. Sinai has replaced the box by periodic boundary conditions so that the molecules move on a flat torus.

On circular and elliptic billiard tables, for all $p\ge 3$, the $(p,q)$-periodic orbits forms a continuous family and hence all the trajectories have the same length.

An invariant noncontractible topological annulus, $A\subset\Omega$, whose interior contains no invariant circles, is a Birkhoff instability region. The dynamics in an instability region
has positive topological entropy. Hence Birkhoff conjecture implies
that any non-elliptical billiard has positive topological entropy.

How to construct a strictly convex $C^1$-smooth billiard table with metric positive entropy? b) How to construct a convex $C^2$-smooth billiard table with positive metric entropy?

Recall Bunimovich stadium is not $C^2$, and not strictly convex.

A periodic orbit of period $q$ corresponds to an (oriented) closed polygon with $q$ sides, inscribed in $Q$, and satisfying the condition on the angles it makes with the boundary. Birkhoff called these the harmonic polygons.

Then the maximal circumference of 2-orbit yields the diameter of $Q$. The minimax circumference of 2-orbit corresponds
to the width of $Q$.