  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.
