The dynamical Mordell-Lang problem

Given sets $Y \subseteq X$, a self-map $f : X \to X$ and $a \in X$, consider the return set $E := \{ n \in \mathbb{N} ~:~ f^n(a) \in Y \}$.  At this level of generality, the return set may be very complicated, but the dynamical Mordell-Lang conjecture predicts that if $X$ and $Y$ are algebraic varieties over a field of characteristic zero, then $E$ must be a finite union of points and arithmetic progressions. Already in the 1930s Skolem introduced a $p$-adic analytic method which may be used to solve this problem in many instances and a flurry of work over the past decade has resulted in many positive results, though the conjecture remains open. When the problem is transposed to characteristic $p > 0$, more complicated sets may appear as return sets, including, for example, the set of all natural numbers expressible as the sum of two powers of $p$. One might propose a positive characteristic dynamical Mordell-Lang conjecture taking these sets into account. In this lecture, I will survey the general sweep of the dynamical Mordell-Lang conjecture and then focus on some recent work joint with Corvaja, Ghioca and Zannier in which we both prove some cases of the positive characteristic dynamical Mordell-Lang conjecture and also show that the problem in general is equivalent to a class of difficult exponential Diophantine equations over the rational integers.