Remarks on the period three cycles of quadratic rational maps

Abstract

Milnor showed that the moduli space of Mobius conjugacy classes of quadratic rational maps of the complex projective line to itself is canonically biholomorphic to C-2. Moreover, the affine structure on C-2 has dynamical relevance: the loci Per(n) (eta) consisting of maps with an n-cycle of multiplier eta are affine lines for n = 1 and n = 2. The loci Per(3) (eta) are cubic algebraic curves. In this work, we further the investigation of the family of cubics Per(3)(eta)