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 f1, . . . , fn is a finite sequence of real analytic functions fi : (−1, 1)→ (−1, 1) for which fi(0) = 0 and | f ′ i (0)| ≤ 1 (possibly zero), a = (a1, . . . , an) is an n-tuple of real numbers close enough to the origin and H(x1, . . . , xn) is a real analytic function of n variables, then the set {m ∈N : H( f ◦m 1 (a1), . . . , f ◦m n (an)) = 0} is either all of N, all of the odd numbers, all of the even numbers, or is finite.