Integrable billiard systems realize toric foliations on lens spaces and the 3-torus

Abstract

An integrable billiard system on a book, a complex of several billiard sheets glued together along the common spine, is considered. Each sheet is a planar domain bounded by arcs of confocal quadrics; it is known that a billiard in such a domain is integrable. In a number of interesting special cases of such billiards the Fomenko-Zieschang invariants of Liouville equivalence (marked molecules ) turn out to describe nontrivial toric foliations on lens spaces and on the 3-torus, which are isoenergy manifolds for these billiards. Bibliography: 18 titles.

Billiards and integrable systems

“…Vedyushkina [43] constructed books with Q 3 belonging to the complement of the class of Seifert manifolds to the class of Waldhausen graph manifolds [44], [45], so that for billiard systems the class of isoenergy surfaces Q 3 is not confined to Seifert manifolds. A number of recent results was also discussed in [46]- [48].…”

“…Integrable flows on a sphere and a torus were modelled by Vedyushkina and Fomenko [55], [46] by means of integrable circular topological billiards (for a linear first integral) and confocal topological billiards and billiard books (for a quadratic first integral). With each such flow defined by a non-trivial Riemannian metric they associated a piecewise planar table with plane metric inside 2-cells, and with these planar parts glued isometrically along their boundaries.…”

Billiard with slipping by an arbitrary rational angle

“…Fomenko, Vedyushkina and Kibkalo investigated the 'local' version of the general conjecture (see [21]): they showed that arbitrary values of the marks r and ε on an edge of a molecule (see [22]) and the integer mark n on a family subgraph (see [23]) can be realized, as well as certain combinations of marks (see [24]). Also, an isoenergy surface Q 3 of a billiard book was shown to be homeomorphic to a 3-manifold [25], and billiards were constructed for which Q 3 is homeomorphic to a 3-torus, an arbitrary lens space L(q, p) for coprime p and q, 0 < p < q (see [26]), or a connected sum of arbitrary lens spaces L(q i , p i ) and products S 1 × S 2 (see [27]). As a result, for a billiard system Q 3 is not necessarily a Seifert manifold: one example is the connected sum of three lens spaces L(2, 1) (which are homeomorphic to the three-dimensional projective space).…”