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

TL;DR

This paper proves that for generic smooth maps on manifolds, the number of periodic points can grow arbitrarily fast, providing a comprehensive answer to longstanding questions in dynamical systems.

Contribution

It establishes the existence of open sets of maps with arbitrarily fast growth of periodic points across all dimensions and smoothness levels, extending previous results and answering open problems.

Findings

Existence of open sets of maps with arbitrarily fast periodic point growth.

Generic maps in these sets display this growth for all dimensions and smoothness levels.

Negative answer to Arnold's 1992 problem in the finitely smooth case.

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.