A Euclidean Skolem-Mahler-Lech-Chabauty method

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.