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arXiv · 1808.07218

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

Masayuki Asaoka·Katsutoshi Shinohara·Dmitry Turaev
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Abstract

We consider C^r-diffeomorphisms of a compact smooth manifold having a pair of robust heterodimensional cycles where r is a positive integer or infinity. We prove that if certain conditions about the signatures of non-linearities and Schwarzian derivatives of the transition maps are satisfied, then by giving C^r arbitrarily small perturbation, we can produce a periodic point at which the first return map in the center direction is C^r-flat. As a consequence, we will prove that C^r-generic diffeomorphisms in the neighborhood of the initial diffeomorphism exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions on non-linearities and the Schwarzian derivatives.

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Masayuki Asaoka, Katsutoshi Shinohara, Dmitry Turaev. 2018-08-22. Fast growth of the number of periodic points arising from heterodimensional connections. https://arxiv.org/abs/1808.07218

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