Nandor Simanyi: Homotopical Complexity of Two Billiard Models

This is a preliminary report on some new results concerning
the homotopical complexity of the orbits of certain $3D$ cylindric
billiard flows. There are three models under investigation: The
billiard flow in the flat 3-torus $T^3$ minus two/three orthogonal,
mutually intersecting cylindric scatteres, and the billiard flow in
the flat 3-torus $T^3$ minus three orthogonal, disjoint cylindric
scatteres. The homotopical complexity of long orbit segments is
measured in the Cayley graphs of the fundamental groups of these
billiard tables, which groups by themselves are pretty intriguing
hyperbolic groups. We give lower and upper bounds for the radial sizes
of the arising homotopical rotation sets. The primary tool for the
construction of long orbit segments following a prescribed homotopical
itinerary is the length minimizing variational method. We make this
method work by introducing the proper notion of the so called
admissible orbit segments for both models.

This is an ongoing joint research project with my PhD student, Caleb
C. Moxley.namics seminar on 8 April, Friday