The local topological complexity of Cr-diffeomorphisms with homoclinic tangency

Abstract

Let F be a Cr-diffeomorphism of a two-dimensional manifold M into itself with a saddle periodic point p and the property that branches of the stable and unstable manifolds of p exhibit a homoclinic tangency. Barge and Diamond (1999 Subcontinua of the closure of the unstable manifold at a homoclinic tangency Ergod. Theory Dynam. Sys. 19 289–307) have shown that, generically, the closure of the branch of the unstable manifold of p is nowhere locally the product of a Cantor set and an arc. This paper shows that Cr-close to F is an F̃ such that each non-empty relatively open set of the closure of the branch of the unstable manifold of p contains homeomorphic copies of all chainable continua. A non-local result is also included to illustrate that these chainable continua are quite large in this closure.