A census of quadratic post-critically finite rational functions defined over $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Q}$. Issue Volume 17:Issue A(2014) (July 2014)

• Record Type: Journal Article
• Authors: Lukas, David; Manes, Michelle; Yap, Diane
• Is Part Of: LMS journal of computation and mathematics. Volume 17:Issue A(2014)
• Page Start: 314
• Page End: 329
• Publication Date: 2014-07
• Subjects: 37P05 (primary), -- 37P15, -- 11R99, -- 11Y99 (secondary)
• DOI: 10.1112/S1461157014000266
• Languages: English
• ISSNs: 1461-1570

Abstract

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 $\mathbb{Q}$ .