Analytic relations on a dynamical orbit

Let $(K,\mathrm{\mid }\cdot \mathrm{\mid })$ 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 ${\lim }_{n\to \mathrm{\infty }}{f}^n(a)=0$ 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.