Period-Doubling Cascades Invariants: Braided Routes To Chaos

By a classical result of Kathleen Alligood and James Yorke we know that as we isotopically deform a map $f:ABCD\to {\mathbb{R}}^2$ to a Smale horseshoe map we should often expect the dynamical complexity to increase via a period–doubling route to chaos. Inspired by this fact and by how braids force the existence of complex dynamics, in this paper we introduce three topological invariants that describe the topology of period–doubling routes to chaos. As an application, we use our methods to ascribe symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and to study the dynamics of the Henon map.