Pacific Institute for the Mathematical Sciences - PIMS

The Arithmetic of Post-critically Finite Morphisms

Let f be an endomorphism of N-dimensional projective space. In complex dynamics, it has been known for a century (at least when N = 1) that the orbits of the critical points determines much of the dynamics of f. Morphisms for which all of these critical orbits are finite (so-called PCF maps) turn out to be an important class to understand. Thurston proved, when N = 1, that there are no algebraic families of PCF maps, except for a small number of easy-to-understand examples. I will discuss some recent research into the arithmetic properties of these maps, as well as a partial extension of Thurston's result to arbitrary dimension.