Exceptional Points in Arithmetic Dynamics

Abstract

Let be a morphism of PN defined over a field K. We prove three main results: When K is a number field, we prove that there is a bound B depending only on such that every twist of has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly. When K is a finite field, we find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely pure power maps (z) = zd and Chebyshev polynomials. When K is a number field or Qp for p 6= 3; and L=K is an extension we prove that K fails to be critically reducible at 3. Meanwhile, Q3 is critically reducible at 3.