Generic family displaying robustly a fast growth of the number of periodic points

Abstract

For any $ 2 \le r \le \infty$, $n\ge2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $n$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Kaloshin, Bonatti-D\' iaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore for any $1\le r<\infty$ and any $k\ge 0$, we prove the existence of an open set $\hat U$ of $k$-parameter families in $U$ so that for a generic $(f_p)_p\in \hat U$, for every $\|p\|\le 1$, the map $f_p$ displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.