Stable and unstable manifold structures in the Hénon family

Abstract

For parameter values $a$ where the quadratic map $f_a(x) = 1-ax^2$ has an attracting periodic point, and small values of $b$, the Hénon map $H_{a,b}(x, y) = (1 + y - ax^2, bx)$ has a periodic attractor and an attracting set that is homeomorphic with the inverse limit of $f_a|_{[1-a, 1]}$. This attracting set consists of the collection of unstable manifolds of a hyperbolic invariant set together with the attracting periodic orbit. In the case in which $f_a$ is also not renormalizable with smaller period and $b < 0$, we give a symbolic description of the collection of stable manifolds of this hyperbolic set and show that this collection is homeomorphic with the collection of unstable manifolds precisely when the attracting periodic orbit is accessible from the complement of the attracting set, a condition that can be characterized in terms of the kneading sequence of the quadratic map. As an application, we answer a question raised by Hubbard and Oberste-Vorth by proving that the basin boundaries corresponding to three distinct period five sinks in the Hénon family are non-homeomorphic.