Metamorphoses: Sudden jumps in basin boundaries (Q1182046)

In some invertible maps of the plane (for example the Hénon map) that depend on a parameter, the boundaries of attraction are extremely sensitive to small changes in the parameter. The boundary can jump suddenly or even change from being smooth to being fractal. The authors prove a theorem (under certain non-degeneracy conditions) that a metamorphosis occurs when the stable and the unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency. The underlying idea is that with increasing number of iterates an infinite family of (local) horseshoe maps and sequences of Newhouse saddle point arises.