Trellises formed by stable and unstable manifolds in the plane

Abstract

A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diffeomorphism of a 2 2 -manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones.