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

Abstract

For any $2 \leqslant r \leqslant \infty , n \geqslant 2$, 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, Kaloshin, Bonatti–Díaz–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 $1 \leqslant r \lt \infty$ and any $k \geqslant 0$, we prove the existence of an open set $\hat{U}$ of $k$-parameter families in $U$ such that for a generic $(f_p)^p \in \hat{U}$, for every $\lVert p \rVert \leqslant 1$, 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.