Critical heights on the moduli space of polynomials

Abstract. Let Md be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map G: Md → Rd−1. For generic values of G, each connected component of a fiber of G is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space T ∗d obtained by collapsing each connected component of a fiber of G to a point. The space T ∗d is a parameter-space analog of the polynomial tree T (f) associated to a polynomial f: C → C, studied in [DM], and there is a natural projection from T ∗d to the space of trees Td. We show that the projectivization PT ∗d is compact and contractible; further, the shift locus in PT ∗d has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one corespondence with topological conjugacy classes of structurally stable polynomials in the shift locus. 1