A Euclidean Skolem-Mahler-Lech-Chabauty method

TL;DR

This paper adapts the p-adic Skolem-Mahler-Lech-Chabauty method using o-minimality to analyze the dynamical Mordell-Lang conjecture for certain real analytic systems, classifying the zero sets of iterated functions.

Contribution

It introduces a novel approach combining o-minimality with p-adic methods to study dynamical systems and the Mordell-Lang conjecture in a real analytic context.

Findings

The zero set of a certain real analytic function composed with iterates is either all natural numbers, all odds, all evens, or finite.

The method applies to functions with derivatives at zero bounded by one, including zero.

Provides a classification of the zero sets for specific real analytic dynamical systems.

Abstract

Using the theory of o-minimality we show that the $p$-adic method of Skolem-Mahler-Lech-Chabauty may be adapted to prove instances of the dynamical Mordell-Lang conjecture for some real analytic dynamical systems. For example, we show that if $f_1,...,f_n$ is a finite sequence of real analytic functions $f_i:(-1,1) \to (-1,1)$ for which $f_i(0) = 0$ and $|f_i'(0)| \leq 1$ (possibly zero), $a = (a_1,...,a_n)$ is an $n$-tuple of real numbers close enough to the origin and $H(x_1,...,x_n)$ is a real analytic function of $n$ variables, then the set $\{m \in {\mathbb N} : H (f_1^{\circ m} (a_1),...,f_n^{\circ m}(a_n)) = 0 \}$ is either all of ${\mathbb N}$, all of the odd numbers, all of the even numbers, or is finite.