We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are uniformly bounded. Each of the recursion algorithms leads to a rational dynamical system whose formula, degree and the dimension of the space it acts upon depend on the specific algorithm. Nevertheless, we demonstrate that the qualitative behavior of the dynamics exhibit universal features that can be exploited to draw conclusions about the zero sets.