Dynamical Shafarevich results for rational maps

Given a number field K and a finite set S of places of K, this dissertation studies rational maps with prescribed good reduction at every place v ∉ S. The first result shows that the set of all quadratic rational maps with the standard notion of good reduction outside S is Zariski dense in the moduli space $M2$. The second result shows that if the notion of good reduction is strengthened by requiring a double unramified fixed point structure or an unramified two cycle, then one obtains a non-Zariksi density statement. The next result proves the existence of global minimal models of endomorphisms on $Pn$ defined over the fractional field of principal ideal domain. This result is used to prove the last main theorem—the finiteness of twists of a rational maps on $Pn$ over K with good reduction outside S..