ALGEBRAIC STABILITY OF MEROMORPHIC MAPS DESCENDED FROM THURSTON'S PULLBACK MAPS

Let phi : S-2 -> S-2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n. If phi has a fully ramified periodic point p(infinity) and satisfies certain additional conditions, then, by work of Koch, f induces a meromorphic self-map R-phi on the moduli space M-0,M-n; R-phi descends from Thurston's pullback map on Teichmuller space. Here, we relate the dynamics of R-phi on M-0,M- n to the dynamics of phi on S-2. Let l be the length of the periodic cycle in which the fully ramified point p(infinity) lies; we show that R-phi is algebraically stable on the heavy-light Hassett space corresponding to l heavy marked points and (n - l) light points.