Local instability of orbits in polygonal and polyhedral billiards

Abstract

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.

On the mathematical transport theory in microporous media: The billiard approach

“…The proof of the above results can be found in [80][81][82][83]. In [80] is also proved that in finite polygons every non periodic orbit must have an accumulation point at a vertex and thus billiards in finite polygons always have zero entropy with respect to any invariant measure. The ergodic properties of the billiard flow depend on the shape of the polygon.…”

λ-stability of periodic billiard orbits

“…The first return map of the billiard flow to the boundary is called the billiard map. Polygonal billiards are known to have zero topological entropy, thus zero Lyapunov exponents [9,10,11].…”

Escape orbits and ergodicity in infinite step billiards

“…(iii) [G3,GKT] For any given polygon, the metric entropy with respect to any flow-invariant measure is zero. Furthermore, the topological entropy is also zero.…”