Tag Archives: non-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.

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