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Thurston made several important contributions to complex dynamics, such as the topological characterization of rational maps among self branched coverings of 2-spheres and the theory of laminations for polynomials. The former was especially important in giving a systematic way to construct rational maps from given topological/combinatorial data. Also its essential use of Teichmüller theory changed the way we think about complex dynamics.
Mitsuhiro Shishikura of Kyoto University explains the role of this result in complex dynamics and related open questions, June 26, 2014 at the Bill Thurston Legacy Conference.
The conference, "What's Next? The mathematical legacy of Bill Thurston," held at Cornell June 23-27, 2014, brought together mathematicians from a broad spectrum of areas to describe recent advances and explore future directions motivated by Thurston's transformative ideas.