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

Part of: Local and nonlocal bifurcation theory Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  02 August 2021

Masayuki Asaoka ,
Katsutoshi Shinohara  and
Dmitry Turaev
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
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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.

Keywords

partially hyperbolic diffeomorphismsgrowth of periodic pointsheterodimensional cyclesblenders

MSC classification

Primary: 37C35: Orbit growth
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37G25: Bifurcations connected with nontransversal intersection
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 9 , September 2021 , pp. 1899 - 1963
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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