The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids. (English) Zbl 1153.37023

Kin, Eiko

Osaka J. Math. 45, No. 3, 757-772 (2008).

Summary: Li-Yorke theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland’s dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics.

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
57M27	Invariants of knots and \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

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