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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A dynamical pairing between two rational maps


Authors: Clayton Petsche, Lucien Szpiro and Thomas J. Tucker
Journal: Trans. Amer. Math. Soc. 364 (2012), 1687-1710
MSC (2010): Primary 11G50, 14G40, 37P15
Published electronically: November 10, 2011
MathSciNet review: 2869188
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Abstract: Given two rational maps $ \varphi $ and $ \psi $ on $ \mathbb{P}^1$ of degree at least two, we study a symmetric, nonnegative real-valued pairing $ \langle \varphi ,\psi \rangle $ which is closely related to the canonical height functions $ h_\varphi $ and $ h_\psi $ associated to these maps. Our main results show a strong connection between the value of $ \langle \varphi ,\psi \rangle $ and the canonical heights of points which are small with respect to at least one of the two maps $ \varphi $ and $ \psi $. Several necessary and sufficient conditions are given for the vanishing of $ \langle \varphi ,\psi \rangle $. We give an explicit upper bound on the difference between the canonical height $ h_\psi $ and the standard height $ h_{\mathrm {st}}$ in terms of $ \langle \sigma ,\psi \rangle $, where $ \sigma (x)=x^2$ denotes the squaring map. The pairing $ \langle \sigma ,\psi \rangle $ is computed or approximated for several families of rational maps $ \psi $.


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Additional Information

Clayton Petsche
Affiliation: Department of Mathematics and Statistics, Hunter College, 695 Park Avenue, New York, New York 10065
Address at time of publication: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
Email: cpetsche@hunter.cuny.edu, petschec@math.oregonstate.edu

Lucien Szpiro
Affiliation: Ph.D. Program in Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
Email: lszpiro@gc.cuny.edu

Thomas J. Tucker
Affiliation: Department of Mathematics, Hylan Building, University of Rochester, Rochester, New York 14627
Email: ttucker@math.rochester.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05350-X
PII: S 0002-9947(2011)05350-X
Keywords: Arithmetic dynamical systems, canonical heights, equidistribution of small points
Received by editor(s): November 13, 2009
Received by editor(s) in revised form: March 6, 2010
Published electronically: November 10, 2011
Additional Notes: The first author was partially supported by NSF Grant DMS-0901147.
The second author was supported by NSF Grants DMS-0854746 and DMS-0739346.
The third author was supported by NSF Grants DMS-0801072 and DMS-0854839.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.