Analytic relations on a dynamical orbit

TL;DR

This paper investigates the structure of analytic relations on orbits of nonconstant analytic maps over complete discretely valued fields, revealing specific forms of relations depending on whether the fixed point is superattracting or merely attracting.

Contribution

It characterizes the analytic relations on orbits in non-Archimedean dynamics, distinguishing between superattracting and attracting fixed points, and describes their algebraic and analytic structures.

Findings

Relations are defined by equations like $x_i = b$ or $x_j = f^ ext{ extellipsis}(x_k)$ for superattracting fixed points.

When fixed points are attracting but not superattracting, relations originate from algebraic tori.

Analytic subvarieties meeting dense orbits are constrained to specific algebraic forms.

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.