Diffeomorphisms with a heteroclinic cycle between two hyperbolic periodic points with specified eigenvalue and connection conditions admit perturbations containing infinitely many attracting normally hyperbolic periodic circles.
Devaney.An Introduction to Chaotic Dynamical Systems
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Infinitely Many Attracting Periodic Circles in Higher Dimensions
Diffeomorphisms with a heteroclinic cycle between two hyperbolic periodic points with specified eigenvalue and connection conditions admit perturbations containing infinitely many attracting normally hyperbolic periodic circles.