Critical heights on the moduli space of polynomials

Let Md be the moduli space of one-dimensional, degree d⩾2, 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, we show that 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 Td⁎ obtained by collapsing each connected component of a fiber of G to a point. The space Td⁎ is a parameter-space analog of the polynomial tree T(f) associated to a polynomial f:C→C, studied in DeMarco and McMullen (2008) [6], and there is a natural projection from Td⁎ to the space of trees Td. We show that the projectivization PTd⁎ is compact and contractible; further, the shift locus in PTd⁎ has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one correspondence with topological conjugacy classes of structurally stable polynomials in the shift locus.