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
DOI:
https://doi.org/10.1090/S0002-9947-2011-05350-X
Published electronically:
November 10, 2011
MathSciNet review:
2869188
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Given two rational maps and
on
of degree at least two, we study a symmetric, nonnegative real-valued pairing
which is closely related to the canonical height functions
and
associated to these maps. Our main results show a strong connection between the value of
and the canonical heights of points which are small with respect to at least one of the two maps
and
. Several necessary and sufficient conditions are given for the vanishing of
. We give an explicit upper bound on the difference between the canonical height
and the standard height
in terms of
, where
denotes the squaring map. The pairing
is computed or approximated for several families of rational maps
.
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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:
https://doi.org/10.1090/S0002-9947-2011-05350-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.