A Euclidean Skolem–Mahler–Lech–Chabauty Method

Using the theory of o-minimality we show that the $p$-adicmethod of Skolem–Mahler–Lech–Chabauty may be adapted toprove instances of the dynamical Mordell–Lang conjecturefor some real analytic dynamical systems. For example, weshow that if $f_1, \ldots, f_n$ is a finite sequence ofreal analytic functions $f_i\,:\,(-1,1) \to (-1,1)$ forwhich $f_i(0) = 0$ and $|f_i'(0)| \leq 1$ (possibly zero),$a = (a_1,\ldots,a_n)$ is an $n$-tuple of real numbersclose enough to the origin and $H(x_1,\ldots,x_n)$ is areal analytic function of $n$ variables, then the set $\{ m\in \NN : H (f_1^{\circ m} (a_1), \ldots, f_n^{\circm}(a_n)) = 0 \}$ is either all of $\NN$, all of the oddnumbers, all of the even numbers, or is finite.

Full Text (PDF format)

Published 28 October 2011