Beyond hyperbolicity: chaos and stability

What aspects of chaotic dynamics survive small perturbations when hyperbolicity is absent? Starting from Smale’s conjecture that hyperbolicity should describe typical dynamics, I will discuss this question with particular attention to the Hénon attractor.
To investigate persistence beyond hyperbolicity, we study unfoldings of homoclinic tangencies. We find codimension-one laminations of maps with invariant non-hyperbolic Cantor sets. These sets are wild in the sense of Newhouse and contain Collet–Eckmann points with dense orbits. Each leaf also contains a map with infinitely many sinks accumulating on the same non-hyperbolic chaotic Cantor set.
We establish these results using a generalized renormalization scheme that describes the hyperbolic structure present at finite scales. I will discuss how persistence along the leaves suggests weaker forms of stability, and how related stability questions arise from biological observations.