Growth of number of periodic orbits of one family of skew product maps

Esteves, Salete

doi:10.1080/14689367.2017.1290051

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2021

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“…Thus, we only need to construct such an example in such a way that it has a robust heterodimensional cycles. See also [Est18].…”

Section: Examplesmentioning

confidence: 99%

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

2021

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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.

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“…Thus, we only need to construct such an example in such a way that it has a robust heterodimensional cycles. See also [Est18].…”

Section: Examplesmentioning

confidence: 99%