p-adic interpolation of iterates

TL;DR

This paper proves that under certain valuation conditions, the iterates of a p-adic analytic self-map can be interpolated p-adically by an analytic function in both space and iteration count.

Contribution

It extends previous work by establishing conditions for p-adic interpolation of iterates of analytic maps on polydisks.

Findings

Existence of a p-adic interpolating function g(x,n) for iterates of f

Conditions on coefficients ensure the interpolation is analytic in both variables

Generalization of earlier results by Bell and Ghioca-Tucker

Abstract

Extending work of Bell and of Bell, Ghioca, and Tucker, we prove that for a p-adic analytic self-map f on a closed unit polydisk, if every coefficient of f(x)-x has valuation greater than that of p^{1/(p-1)}, then the iterates of f can be p-adically interpolated; i.e., there exists a function g(x,n) analytic in both x and n such that g(x,n) = f^n(x) whenever n is a nonnegative integer.