A bound for the existence of invariant circles in a class of two-dimensional dissipative maps

Abstract

By generalizing results due to Mather we obtain an upper bound for the existence of smooth invariant circles in a class of dissipative two-dimensional maps. In particular the map x n+1 = x n + Ω + by n − (k/2π)sin 2πx n, y n+1 = by n − (k/2π) X sin 2πx n can have no smooth invariant circle for | k| > 2(1 + b)/(2 + b).