Heights on moduli space for post-critically finite dynamical systems

The purpose of this Research In Teams event was to consider the arithmetic properties of post-critically finite (PCF) rational maps. In the study of complex holomorphic dynamics, it is a general theme that the dynamical properties of a holomorphic map are largely determined by the behaviour of the critical points. In studying the dynamics of a rational map, then, one is lead to consider the orbits of the critical points, and maps for which these critical orbits are all finite gain special prominence. These are the PCF maps. LetMd denote the moduli space of degree-d endomorphisms of P, up to change of variables. If d = m, certain PCF elements ofMd stand out, namely the so-called Lattès examples. These are maps f : P → P such that there is an elliptic curve E and an integer m, such that f is the action induced from [m] by viewing P as the Kummer surface of E. If Ld ⊆Md is the locus of these Lattès examples, then one expects PCF maps to be somewhat sparse inMd \ Ld. A deep result of Thurston makes this more concrete.