Study of Dynamics of Branched Coverings on the Sphere and Dynamical Zeta Function

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Among the ends of this research is to classify branched coverings on the 2-dimensional sphere up to "isotopy." In the 1-dimensional case, we have a good invariant, called a kneading sequence, which divides maps on the interval into "isotopy" classes. However, we face the difficulty that a kneading sequence has no standard extension in 2-dimension. Thus we consider all possible geometric semiconjugacy from a symbolic dynamics to the Julia set.We have the following results. Let f be a subhyperbolic rational map. Denote by J the Julia set of f, and by J^* the lift of J by the universal covering. Consider the set Cod(f) of codings of J obtained by geometric coding trees.Then1. If the attractor K of an IFS constructed by lifts of a collection of inverses of f has a positive measure, then K tiles J^*.2. A coding map is an n-to-one except on a null set, where n is an integer.3. The collapsing of a coding map is described by a finite directed graph.4. Cod(f) is isomorphic to the quotient of the set of trees by some action of a subgroup of the fundamental group. Moreover, the monoid of rational maps commuting with f naturally acts on Cod (f).Another direction of our study is to investigate nontrivial symmetries of fractal sets. The figure obtained by gluing two copies of Sierpinski's gasket at their "boundaries" has infinitely many automorphisms, while Sierpinski's gasket itself has the symmetry of the regular triangle. We show when a glued fractal has nontrivial automorphisms and how to construct such a fractal. Furthermore we describe the structure of the automorphism group, and proved that under some assumption,. the group can be realized by Moebius.transforms.