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Fast growth of the number of periodic points arising from heterodimensional connections

Published online by Cambridge University Press:  02 August 2021

Masayuki Asaoka
Affiliation:
Faculty of Science and Engineering, Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe610-0394, [email protected]
Katsutoshi Shinohara
Affiliation:
Graduate School of Commerce and Management, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo186-8601, [email protected]
Dmitry Turaev
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, [email protected] Higher School of Economics – Nizhny Novgorod, Russia

Abstract

We consider $C^{r}$-diffeomorphisms ($1 \leq r \leq +\infty$) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$-small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$, provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$-generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

Type
Research Article
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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