9. Victor Ivrii conjecture. Let be a strictly convex domain, be the billiard map on the phase space . Let be the Lebesgue measure of , and be the Lebesgue measure on .

**Conjecture 1.** for all with boundaries.

*Remark.* This is about a general domain , not a generic domain.

**Definition.** A point is said to be an *absolute looping point,* if . Let be the set of absolute looping points.

**Conjecture 2.** for all .

*Question:* When ?

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 , the -periodic orbits forms a continuous family and hence all the trajectories have the same length.

An invariant noncontractible topological annulus, , 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 -smooth billiard table with metric positive entropy? b) How to construct a convex -smooth billiard table with positive metric entropy?

Recall Bunimovich stadium is not , and not strictly convex.

A periodic orbit of period corresponds to an (oriented) closed polygon with sides, inscribed in , 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 . The minimax circumference of 2-orbit corresponds

to the width of .