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On the preperiodic points of an endomorphism of $ \mathbb{P}^{1} \times\mathbb{P}^{1}$ which lie on a curve


Author: Arman Mimar
Journal: Trans. Amer. Math. Soc. 365 (2013), 161-193
MSC (2010): Primary 14G40; Secondary 37F10, 37P05, 37P30
DOI: https://doi.org/10.1090/S0002-9947-2012-05557-7
Published electronically: August 23, 2012
MathSciNet review: 2984056
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Abstract: Let $ X$ be a projective curve in $ \mathbb{P}^{1} \times \mathbb{P}^{1}$ and $ \varphi $ be an endomorphism of degree $ \geq 2$ of $ \mathbb{P}^{1} \times \mathbb{P}^{1}$, given by two rational functions by $ \varphi (z,w)=(f(z),g(w))$ (i.e., $ \varphi =f \times g$), where all are defined over $ \overline {\mathbb{Q}}$. In this paper, we prove a characterization of the existence of an infinite intersection of $ X(\overline {\mathbb{Q}})$ with the set of $ \varphi $-preperiodic points in $ \mathbb{P}^{1} \times \mathbb{P}^{1}$, by means of a binding relationship between the two sets of preperiodic points of the two rational functions $ f$ and $ g$, in their respective $ \mathbb{P}^{1}$-components. In turn, taking limits under the characterization of the Julia set of a rational function as the derived set of its preperiodic points, we obtain the same relationship between the respective Julia sets $ \mathcal {J}(f)$ and $ \mathcal {J}(g)$ as well. We then find various sufficient conditions on the pair $ (X,\varphi )$ and often on $ \varphi $ alone, for the finiteness of the set of $ \varphi $-preperiodic points of $ X(\overline {\mathbb{Q}})$. The finiteness criteria depend on the rational functions $ f$ and $ g$, and often but not always, on the curve. We consider in turn various properties of the Julia sets of $ f$ and $ g$, as well as their interactions, in order to develop such criteria. They include: topological properties, symmetry groups as well as potential theoretic properties of Julia sets.


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Additional Information

Arman Mimar
Affiliation: Department of Mathematics, Polytechnic University of New York, 6 Metrotech Center, Brooklyn, New York 11201
Email: amimar@poly.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05557-7
Received by editor(s): April 17, 2010
Received by editor(s) in revised form: August 10, 2010, September 27, 2010, and January 11, 2011
Published electronically: August 23, 2012
Additional Notes: This work consists of the author’s dissertation at Columbia University (1997), previously unpublished. The author takes this opportunity to extend his thanks to advisors L. Szpiro and S. Zhang for their help and support during that endeavor.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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