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

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.