No abstract

No abstract

Let K be a global function field and let φ P Krxs. For all wandering basepoints b P K, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set Zpφ, bq that depends only on φ, the poles of the b, and K. Moreover, when we order b P O K,S by height, we show that Zpφ, bq is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of φpxq " x d`f is a finite index subgroup of an iterated wreath product of cyclic groups. Finally, we establish similar results on Zsigmondy sets when K is the field of rational numbers or K is a quadratic imaginary field subject to an added stipulation: either zero has finite orbit under iteration of φ or the Vojta conjecture for algebraic points on curves holds.

We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.