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

We consider C^r-diffeomorphisms (1 ≤ r ≤ +∞) 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.