| Abstract: |
Given a polygon in the Euclidean or hyperbolic plane, its billiard
flow on the tangent bundle has trajectories that describe the paths of
particles in the polygon (the billiard trajectories) traveling along
straight lines and "bouncing" off the sides. Labeling the sides of
the polygon the billiard flow determines a symbolic coding we call the
bounce spectrum, which is the set of biinfinite sequences of labels
corresponding to the sides encountered by all trajectories. A natural
question asks the extent to which the bounce spectrum determines the
shape of the polygon. For both Euclidean and hyperbolic polygons,
there are nontrivial constructions of polygons with the same bounce
spectrum that are not isometric/similar. In this talk, I'll describe
these constructions, and then results of joint work with Duchin,
Erlandsson, and Sadanand stating that these are in fact the only ways
in which non-isometric/non-similar polygons can have the same bounce
spectrum.
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