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December 2021 Generic family displaying robustly a fast growth of the number of periodic points
Pierre Berger
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Acta Math. 227(2): 205-262 (December 2021). DOI: 10.4310/ACTA.2021.v227.n2.a1

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.

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Pierre Berger. "Generic family displaying robustly a fast growth of the number of periodic points." Acta Math. 227 (2) 205 - 262, December 2021. https://doi.org/10.4310/ACTA.2021.v227.n2.a1

Information

Received: 9 January 2017; Accepted: 30 December 2020; Published: December 2021
First available in Project Euclid: 1 March 2023

Digital Object Identifier: 10.4310/ACTA.2021.v227.n2.a1

Vol.227 • No. 2 • December 2021
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