The growth rate for the number of singular and periodic orbits for a polygonal billiard
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.
scientific article; zbMATH DE number 4026394