Analytic Relations on a Dynamical Orbit

Let (K, |·|) be a complete discretely valued field and f : B1(K, 1) → B(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 ∈ K with limn→∞ f(a) = 0 and consider the orbit Of (a) := {f (a) : n ∈ N}. We show that if 0 is a superattracting fixed point, then every irreducible analytic subvariety of Bn(K, 1) meeting Of (a) n in an analytically Zariski dense set is defined by equations of the form xi = b and xj = f(xk). When 0 is an attracting, non-superattracting point, we show that all analytic relations come from algebraic tori.