A sequence $a_0,a_1,a_2,\ldots $ satisfies a linear recurrence over the complex numbers if there is some $d\ge 1$ and constants $c_1,\ldots c_d$ such that $a_n=c_1 a_{n-1}+\cdots + c_d a_{n-d}$ for $n$ sufficiently large. A celebrated result of Skolem, Mahler, and Lech says that if $a_n$ is a sequence that satisfies a linear recurrence then the set of $n$ for which $a_n=0$ is a finite union of infinite arithmetic progressions along with a finite set. We show how this can be recast in a purely dynamical framework and use this to give a geometric generalization. We also show that these results no longer hold in positive characteristic and how one can use finite-state machines to get an analogue of these results in positive characteristic. This is joint work with Dragos Ghioca and Tom Tucker and also joint work with Boris Adamczewski.