Common preperiodic points for quadratic polynomials
Common preperiodic points for quadratic polynomials
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DOI：
10.3934/jmd.2022012
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发表时间：
2019-11
影响因子：
1.1
通讯作者：
Laura Demarco;Holly Krieger;Hexi Ye
中科院分区：
文献类型：
--
作者：
Laura Demarco;Holly Krieger;Hexi Ye
Let \begin{document}$ f_c(z) = z^2+c $\end{document} for \begin{document}$ c \in {\mathbb C} $\end{document}. We show there exists a uniform upper bound on the number of points in \begin{document}$ {\mathbb P}^1( {\mathbb C}) $\end{document} that can be preperiodic for both \begin{document}$ f_{c_1} $\end{document} and \begin{document}$ f_{c_2} $\end{document}, for any pair \begin{document}$ c_1\not = c_2 $\end{document} in \begin{document}$ {\mathbb C} $\end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in \begin{document}$ \overline{\mathbb{Q}} $\end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.
Let \begin{document}$ f_c(z) = z^2+c $\end{document} for \begin{document}$ c \in {\mathbb C} $\end{document}. We show there exists a uniform upper bound on the number of points in \begin{document}$ {\mathbb P}^1( {\mathbb C}) $\end{document} that can be preperiodic for both \begin{document}$ f_{c_1} $\end{document} and \begin{document}$ f_{c_2} $\end{document}, for any pair \begin{document}$ c_1\not = c_2 $\end{document} in \begin{document}$ {\mathbb C} $\end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in \begin{document}$ \overline{\mathbb{Q}} $\end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.