The classification of polynomial basins of infinity

Details The classification of polynomial basins of infinity The classification of polynomial basins of infinity

2011-07-06

arxiv.org/abs/1107.1091v1

Mathematics

Dynamical Systems

68 pages, 16 figures

Scientific paper

We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. 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. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.

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