Subdivision Rules and Virtual Endomorphisms

Suppose f : S → S is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting. This is a first step in a program to clarify the relationships among various notions of expansion for noninvertible dynamical systems with branching behavior. 0. Introduction Let T 2 = R/Z denote the real two-dimensional torus, equipped with the Euclidean Riemannian metric ds inherited from the usual metric on R, and suppose f : T 2 → T 2 is a continuous orientation-preserving covering map. It is well-known that a necessary and sufficient condition for f to be homotopic to a covering map g : T 2 → T 2 which is expanding with respect to ds is that the spectrum of the induced linear map f∗ : H1(T ,R) → H1(T ,R) lies outside the closed unit disk. Thus, there is a complete homotopy-theoretic invariant for detecting those homotopy classes of coverings which contain expanding maps. In this note, we take a first step toward a similar detection result for certain branched self-coverings of the 2-sphere to itself, called Thurston maps, which arise naturally in the classification of holomorphic dynamical systems in one complex variable [DH]. Our main result asserts that for certain Thurston maps, if one form of combinatorial expansion property is satisfied, then so is another. It is one part in a program to clarify the relationships between various notions of expansion for Thurston maps. Let S denote the 2-sphere equipped with an orientation. An orientation-preserving branched covering map f : S → S of degree d ≥ 2 has, by the Riemann-Hurwitz formula, a set Bf of 2d − 2 branch points, counted with multiplicity. By a branch point, we mean a point at which the local degree deg(f, x) of f at x is strictly larger than one. We denote by f the n-fold composition of f with itself. If the postcritical set Pf = ⋃