Thurston equivalence to a rational map is decidable. Bonnot, S., Braverman, M., & Yampolsky, M. Mosc. Math. J., 12(4):747–763, 884, 2012.
bibtex   
@ARTICLE{BBY12,
  author = {Bonnot, Sylvain and Braverman, Mark and Yampolsky, Michael},
  title = {Thurston equivalence to a rational map is decidable},
  journal = {Mosc. Math. J.},
  year = {2012},
  volume = {12},
  pages = {747--763, 884},
  number = {4},
  fjournal = {Moscow Mathematical Journal},
  issn = {1609-3321},
  keywords = {bib,cca},
  mrclass = {37F20 (03B25 37E30)},
  mrnumber = {3076853},
  owner = {Vasco},
  timestamp = {2013.12.01}
}

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