Critical heights on the moduli space of polynomials

Laura DeMarco*, Kevin Pilgrim

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)350-372
Number of pages23
JournalAdvances in Mathematics
Issue number1
StatePublished - Jan 15 2011


  • Escape rate
  • Moduli space
  • Monotone-light
  • Polynomial dynamics
  • Shift locus
  • Trees

ASJC Scopus subject areas

  • Mathematics(all)

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