Fast growth of the number of periodic points arising from heterodimensional connections. (2nd September 2021)

• Record Type: Journal Article
• Authors: Asaoka, Masayuki; Shinohara, Katsutoshi; Turaev, Dmitry
• Is Part Of: Compositio mathematica. Volume 157:Number 9(2021)
• Page Start: 1899
• Page End: 1963
• Publication Date: 2021-09-02
• Subjects: partially hyperbolic diffeomorphisms -- growth of periodic points -- heterodimensional cycles -- blenders; 37C35 -- 37D30 -- 37G25; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007405
• Languages: English
• ISSNs: 0010-437X

Abstract

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.