Infinitely Many Periodic Attractors for Holomorphic Maps of 2 Variables

Abstract

An important development in the study of discrete dynamical systems was Newhouse's use of persistent homoclinic tangencies to show that a set of C2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors, or sinks [6], where large refers to a residual subset of an open set of diffeomorphisms. In the present paper, we obtain this result for various spaces of holomorphic maps of two variables. Newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency [7]. More recently, Palis and Viana extended this latter result to higher dimensions when the stable manifold has codimension one [10], and Romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks [12]. In each case, however, the construction reduces to the study of intersecting Cantor sets in the line: under an appropriate projection, the stable and unstable manifolds of a basic set are mapped to Cantor sets in the line, and a tangency between these manifolds corresponds to a point of intersection of the Cantor sets. The generic unfolding of tangencies then gives rise to periodic attractors or saddles. This reduction to linear Cantor sets depends heavily on the fact that there is only one expanding eigenvalue. Even Romero's result for higher codimension stable manifolds involves a reduction to this case. In the holomorphic setting, eigenvalues come in conjugate pairs (from a real point of view), so this reduction is not possible. Instead, stable and unstable manifolds are Riemann surfaces, and after extending the stable and unstable manifolds of a basic set to foliations, these foliations will be tangent in a (real) 2-dimensional disk, and the stable and unstable manifolds correspond to Cantor sets in this disk. Hence, persistent tangencies between basic sets correspond to the stable intersection of two Cantor sets in the plane.