The growth rate for the number of singular and periodic orbits for a polygonal billiard
type: Item
instance of: scholarly article
title: The growth rate for the number of singular and periodic orbits for a polygonal billiard
zbMATH Open document ID: 0631.58020
DOI: 10.1007/BF01239021
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: Let P be a simply connected polygon in a plane. A broken (polygonal) line formed by the segments \([x_ 0,x_ 1]\), \([x_ 1,x_ 2],...,[x_{n- 1},x_ n]\) is called a generalized diagonal of P if it lies inside P except for the points \(x_ 0,...,x_ n\), the points \(x_ 0\) and \(x_ n\) are vertices of P, the points \(x_ 1,...,x_{n-1}\) lie on the sides of P, and for \(i=1,...,n-1\), the segments \([x_{i-1},x_ i]\) and \([x_ i,x_{i+1}]\) form the same angle with the side of P passing through \(x_ i\). Let \(N_ T(P)\) be the number of different generalized diagonals of P of length \(\leq T.\) The main purpose of this note is to prove that \[ \lim_{T\to \infty}\frac{\log (N_ T(P))}{T}=0. \] This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 4026394
zbMATH Keywords: polygonal billiard
zbMATH Keywords: singular and periodic orbits
zbMATH Keywords: generalized diagonal
zbMATH Keywords: isolated periodic orbits
zbMATH Keywords: topological entropy
author: Anatole Katok
reviewed by: Ding Tongren
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01239021
OpenAlex ID: W2072554153
description: scientific article; zbMATH DE number 4026394