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.