Non-Hyperbolic Chaotic Dynamics: Renormalization and More

Abstract

In two-dimensional unfoldings of homoclinic tangencies, the parameter space contains codimension-1 laminations whose leaves consist of maps with invariant non-hyperbolic Cantor sets. These sets are wild and unstable in the sense of Newhouse and contain Collet-Eckmann points with dense orbits. Thus, wild and non-hyperbolic chaotic dynamics can coexist on a single invariant set, while persisting along codimension-1 manifolds. Even more strikingly, each leaf of the lamination contains a map with infinitely many sinks accumulating on the invariant Cantor set carrying the Collet-Eckmann dynamics. In particular, the occurrence of infinitely many sinks is compatible with this type of non-hyperbolic chaotic behavior, with both phenomena organized around the same invariant Cantor set. To analyze these phenomena, we introduce a generalized renormalization scheme for two-dimensional systems, which describes the dynamics at successive scales and reveals a finite-scale hyperbolic structure within these non-hyperbolic regimes.