Billiards inside a cusp
Can an ideal (i.e. point shaped) billiard ball running between \(\text{graph} (f)\) and \(\text{graph}(-f)\) escape to infinity in a different way than exactly along the \(x\)-axis? As a warm-up, we are introduced to an elementary solution of this problem for \(f(x) = 1/x\), or any smooth eventually convex function asymptotic to the \(x\)-axis. The author aims at convincing us that invariant measures come in naturally and solve the billiard problem in fairly general circumstances. If the billiard area is finite, invariance of the volume measure under the billiard flow leads to an escape set of measure zero, which is shown to be empty by invoking Poincaré's recurrence theorem. For the infinite area case, the flux as a measure on a Poincaré section helps to conclude that if \(\lim_{x \to \infty} \inf f(x) = 0\), then a billiard ball cannot escape to infinity.
scientific article; zbMATH DE number 716478