A census of quadratic post-critically finite rational functions defined over
David Victorious Lukas, Michelle A. Manes, Diane Yap · LMS Journal of Computation and Mathematics · 2014
Abstract We find all quadratic post-critically finite (PCF) rational functions defined over $\mathbb{Q}$ , up to conjugation by elements of $\mathop{\rm PGL}_2(\overline{\mathbb{Q}})$ . We describe an algorithm to search for possibly PCF functions. Using the algorithm, we eliminate all but 12 rational functions, all of which are verified to be PCF. We also give a complete description of all possible rational preperiodic structures for quadratic PCF functions defined over $\mathbb{Q}$ .