Analytic relations on a dynamical orbit

Abstract

Let $(K,|\cdot|)$ be a complete discretely valued field and $f:{\mathbb B}_1(K,1) \to {\mathbb B}_1(K,1)$ a nonconstant analytic map from the unit back to itself. We assume that 0 is an attracting fixed point of $f$. Let $a \in K$ with $\lim_{n \to \infty} f^n(a) = 0$ and consider the orbit ${\mathcal O}_f(a) := \{f^n(a) : n \in {\mathbb N} \}$. We show that if 0 is a \emph{superattracting} fixed point, then every irreducible analytic subvariety of ${\mathbb B}_n(K,1)$ meeting ${\mathcal O}_f(a)^n$ in an analytically Zariski dense set is defined by equations of the form $x_i = b$ and $x_j = f^\ell(x_k)$. When 0 is an attracting, non-superattracting point, we show that all analytic relations come from algebraic tori.