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

Abstract

For any $2\le r\le \infty$, $n\ge 2$, 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, Gochenko-Shil'nikov-Turaev, Kaloshin, Bonatti-Diaz-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 $2\le r<\infty$ 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 $(f_a)_a\in \hat U$, for every $\|a\|\le 1$, the map $f_a$ displays a fast growth of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.