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.
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Location: B660 University Hall Web page: http://www.cs.uleth.ca/~nathanng/ntcoseminar/ Patrick Ingram, Colorado State University