Critically separable rational maps in families

Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich's theorem for elliptic curves. We also define the minimal critical discriminant, a global object which can be viewed as a measure of arithmetic complexity of a rational map. We formulate a conjectural bound on the minimal critical discriminant, which is analogous to Szpiro's conjecture for elliptic curves, and we prove that a special case of our conjecture implies Szpiro's conjecture in the semistable case.Comment: In this version, some notation and terminology has changed. In particular, this results in a slight change in the title of the paper. Many small expository changes have been made, a reference has been added, and a remark/example has been added to the end of section

On the distribution of orbits in affine varieties

Given an affine variety $X$, a morphism $\phi:X\to X$, a point $\alpha\in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $\phi$-orbit of $\alpha$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces