Billiards inside a cusp
type: Item
instance of: scholarly article
title: Billiards inside a cusp
zbMATH Open document ID: 0827.58032
DOI: 10.1007/BF03024711
author: Jonathan L. King
published in: The Mathematical Intelligencer
publication date: 1995-02-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 716478
zbMATH Keywords: invariant measures
zbMATH Keywords: Poincaré section
zbMATH Keywords: recurrence
zbMATH Keywords: conservative flow
reviewed by: Dieter Erle
MaRDI profile type: Publication
cites work: Three Problems in Search of a Measure
cites work: Geodesic flow on the two-sphere, Part I: Positive measure entropy
cites work: Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics
full work available at URL: https://doi.org/10.1007/bf03024711
OpenAlex ID: W1987503045
description: scientific article; zbMATH DE number 716478