Recently we showed that some degenerate bifurcations can occur robustly. Such a phenomena enables ones to prove that some pathological dynamics are not negligible and even typical in the sense of Arnold-Kolmogorov. In particular, we proved:

Theorem. For every $\infty>r\ge 1$, for every $k\ge 0$, for every manifold of dimension $\ge 2$, there exists an open set $\hat U$ of $C^r$-$k$-parameters families of self-mappings, so that for every topologically generic family $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the mapping $f_a$ displays infinitely many sinks. 

We will introduce the concept of Emergence which quantifies how wild is the dynamics from the statistical viewpoint, and we will conjecture the local typicality of super-polynomial ones in the space of differentiable dynamical systems.