**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 2*D*-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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