A magnetic arc. (b) An example of two trajectories with the same value for 2 where χ and χ are supplementary.. .. .. .. 8 2.3 The behavior of the return map for fixed s 0 and varying u 0 when µ < ρ min. The Larmor centers are in orange and the dark purple points are P 0 and the corresponding P 1 , P 2 for each value of u 0. . 2.4 (a) A (2,4) periodic orbit for an ellipse with semi-major axis 3, semiminor axis 2, µ = 4/5 < ρ min = 4/3, and (s 0 , u 0) ≈ (1.3796, −0.491598);

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We provide periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions, classical Chebyshev polynomials, Akhiezer polynomials, and general extremal polynomials over unions of two intervals. These results are contrasted with the cases of billiards in the Minkowski and the Euclidean planes.

No abstract

The principal angles between binary collision subspaces in an N-billiard system in d-dimensional Euclidean space are computed. These angles are computed for equal masses and arbitrary masses. We then provide a bound on the number of collisions in the planar 3-billiard system problem. Comparison of this result with known billiard collision bounds in lower dimensions is discussed.

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We derive conditions for elliptic periodicity for such billiards. We describe the topology of these billiard systems in terms of Fomenko invariants. Then we provide periodicity conditions in terms of functional Pell equations and related extremal polynomials. Several examples are computed in terms of elliptic functions and classical Chebyshev and Zolotarev polynomials, as extremal polynomials over one or two intervals. These results are contrasted with the cases of billiards on the Minkowski and Euclidean planes. Dedicated to R. Baxter on the occasion of his 80th anniversary. Bibliography: 51 titles.