Knotted periodic orbits in suspensions of Smale's horseshoe: period multiplying and cabled knots
Following earlier work on knotted periodic orbits in a suspension of \textit{S. Smale}'s [Differ. and Combinat. Topology, Sympos. Marston Morse, Princeton, 63-80 (1965; Zbl 0142.411)] horseshoe diffeomorphism [\textit{P. J. Holmes} and \textit{R. F. Williams}, Arch. Rat. Mech. Anal. 90, 115-194 (1985; Zbl 0593.58027)], we define a notion of iterated horseshoe knots. We show that appoximately half the horseshoe knots are cabled and that, among these cablings, are infinitely many, nonisotopic, iterated torus knots. We obtain uniqueness results for certain families of resonant torus knots, related to cascades of period-doubling and period- multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the Hénon maps. We give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots.
scientific article; zbMATH DE number 4011475
scientific article