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arXiv · 1207.1292

Geometrization of sub-hyperbolic semi-rational branched coverings

Tao Cheng·Yunping Jiang
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Abstract

Given a sub-hyperbolic semi-rational branched covering which is not CLH-equivalent a rational map, it must have the non-empty canonical Thurston obstruction. By using this canonical Thurston obstruction, we decompose this dynamical system in this paper into several sub-dynamical systems. Each of these sub-dynamical systems is either a post-critically finite type branched covering or a sub-hyperbolic semi-rational type branched covering. If a sub-dynamical system is a post-critically finite type branched covering with a hyperbolic orbifold, then it has no Thurston obstruction and is combinatorially equivalent to a unique post-critically finite rational map (up to conjugation by an automorphism of the Riemann sphere) and, more importantly, if a sub-dynamical system is a sub-hyperbolic semi-rational type branched covering with hyperbolic orbifold, we prove in this paper that it has no Thurston obstruction and is CLH-equivalent to a unique geometrically finite rational map (up to conjugation by an automorphism of the Riemann sphere).

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Tao Cheng, Yunping Jiang. 2012-07-05. Geometrization of sub-hyperbolic semi-rational branched coverings. https://arxiv.org/abs/1207.1292

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