Abstract
We consider the problem of classifying the dynamics of complex polynomials f : C ! C restricted to the basins of innity 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 innity, 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 dene 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 ner invariant than both
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