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.