Collapsed indecomposable continua in area preserving two-dimensional dynamical systems

Abstract:

While invariant indecomposable continua can occur in two dimensional area preserving dynamical systems, it is often the case that processes that would normally produce these continua instead produce a collapsed version of the continua because of the area preserving constraints. The collapsed continuum and the dynamics on it have a strong relationship to an indecomposable continuum in a related dynamical system. We also prove that the presence of a homoclinic point of a saddle point $p$ in such a system has a branch $W^{u+}(p)$ of its unstable manifold that is inaccessible from the complement of the closure of $W^{u+}(p)$.