The classification of polynomial basins of infinity

We consider the problem of classifying the dynamics of complex polynomials f: W C → C restricted to the basins of inFInity X(f). We synthesize existing combinatorial tools-tableaux, trees, and laminations-into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to inFInity, we obtain a complete description of the set of topological conjugacy classes with given pictograph. For arbitrary polynomials, we compute the total number of topological conjugacy classes of basins (f;X(f)) with a given pictograph. We also deFIne abstract pictographs and prove that every abstract pictograph is realized by a polynomial. Extra details are given in degree 3, and we provide examples that show the pictograph is a FIner invariant than both the tableau of [5] and the tree of [10].