Milena Radnović

We study billiard systems within an ellipsoid in the 4-dimensional pseudo-Euclidean spaces. We provide an analysis and description of periodic and weak periodic trajectories in algebro-geometric and functional-polynomial terms.

We derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted.

A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case systematically studied in the present paper, it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular, we connect the search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which, as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics. As a byproduct, an intriguing connection between Cayley-type conditions and discriminantly separable polynomials has been observed for all those small periods.

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

It is well known that changes in the Liouville foliations of the isoenergy surfaces of an integrable system imply that the bifurcation set has singularities at the corresponding energy level.We formulate certain genericity assumptions for two degrees of freedom integrable systems and we prove the opposite statement: the essential critical points of the bifurcation set appear only if the Liouville foliations of the isoenergy surfaces change at the corresponding energy levels. Along the proof, we give full classification of the structure of the isoenergy surfaces near the critical set under our genericity assumptions and we give their complete list using Fomenko graphs. This may be viewed as a step towards completing the Smale program for relating the energy surfaces foliation structure to singularities of the momentum mappings for non-degenerate integrable two degrees of freedom systems.