New York Journal of Mathematics
Volume 24 (2018), 1111-1122


Xavier Buff

On postcritically finite unicritical polynomials

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Published: December 11, 2018.
Keywords: Preperiodic points, unicritical polynomial, post-critically finite, arithmetic dynamics, complex dynamics.
Subject: 37P05, 37P35, 37F45.

In this article, we first study arithmetical properties of postcritically finite unicritical polynomials fa:z → azD+1 with D ≥ 2. In particular, we answer a question of Milnor, showing that there exist non-Galois conjugate parameters a1 ∈ C and a2 ∈ C such that fa1 and fa2 have critical orbits periodic with the same period. We also answer a question of Baker and DeMarco, proving that the set of parameters a ∈ C such that 0 and 1 are simultaneously (pre)periodic for qa:w → w2+a is equal to {0,-1,-2}.


This research was supported in part by the ANR grant Lambda ANR-13-BS01-0002.

Author information

Xavier Buff:
Institut de Mathématiques de Toulouse
UMR5219, Université de Toulouse
CNRS, UPS, F-31062 Toulouse Cedex 9, France