Braid forcing and star-shaped train tracks

Global results are proved about the way in which Boylandʹs forcing partial order organizes a set of braid types: those of periodic orbits of Smaleʹs horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in de Carvalho and Hall (Exp. Math. 11(2) (2002) 271), which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parameterized by homoclinic orbits to the fixed point of code 0.