Let f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.

We prove a dynamical version of the Mordell–Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to those employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

It is proved here that if $f : {\rm C\!\!\!I} \rightarrow \bar{{\rm C\!\!\!I}}$ is an elliptic function and $q$ is the maximal multiplicity of all poles of $f$, then the Hausdorff dimension of the Julia set of $f$ is greater than $2q/(q+1)$, and the Hausdorff dimension of the set of points that escape to infinity is less than or equal to $2q/(q+1)$. In particular, the area of this latter set is equal to 0.