We study C^r (5 ≦ r ≦ ∞) diffeomorphisms on closed manifolds of
dimension at least three with a heteroclinic cycle between two
hyperbolic periodic points.
At each point, the unstable direction is one dimensional, and the
stable and unstable eigenvalues closest to 1 in modulus are real and
simple. One heteroclinic connection is transverse and the other is
non-transverse, and the product of those two eigenvalues is less than
1 at one point and greater than 1 at the other. Arbitrarily close to
such a map, there are open sets in which a residual subset of
diffeomorphisms has infinitely many attracting normally hyperbolic
periodic circles.