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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. |
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Abstract
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}. |
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| Acknowledgements
This research was supported in part by the ANR grant Lambda ANR-13-BS01-0002.
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| Author information
Xavier Buff:
Institut de Mathématiques de Toulouse
UMR5219, Université de Toulouse
CNRS, UPS, F-31062 Toulouse Cedex 9, France
xavier.buff@math.univ-toulouse.fr
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