A Euclidean Skolem-Mahler-Lech-Chabauty method

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 $|f_i^{\prime }(0)|\le 1$ (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.