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 sens of Arnold-Kolmogorov. More precisely, we proved: Theorem: For every $infty>rge 1$, for every $kge 0$, 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 $(f_a)_ain hat U$, 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. For this end, we will develop the theory of Para-Dynamics, by giving a negative answer to the following problem of Arnold (1992): Theorem: For every $infty>rge 1$, for every $kge 0$, 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 $(f_a)_ain hat U$, for every $|a|le 1$, the map $f_a$ displays a fast increasing number of periodic points: $$limsup frac{log Card ; Per_n , f_a}n = infty$$ We also give a negative answer to questions asked by Smale 1967, Bowen in 1978 and by Arnold in 1989, for manifolds of any dimension $ge 2$: Theorem: For every $inftyge rge 1$, for every manifold of dimension $ge 2$, there exists an open set $U$ of $C^r$-diffeomorphisms, so that a generic $fin U$ displays a fast growth of the number of periodic points. The proof involves a new object, the $lambda$-$C^r$-parablender, the Renormalization for hetero-dimensional cycles, the Hirsh-Pugh-Shub theory, the parabolic renormalization for parameter family, and the KAM theory.
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