Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality

The Fatou-Shishikura Inequality bounds the number of nonrepelling cycles of a rational map by the number of infinite tails of critical orbits - in particular, any rational map of degree D has at most 2D-2 such cycles. The original proof involved hyperbolic geometry and quasiconformal mappings. Our argument makes use of elementary considerations about meromorphic quadratic differentials, as in Thurston's work on postcritically finite maps, and has a considerably more algebraic flavour.

Adam Epstein

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