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Portraits of preperiodic points for rational maps

Published online by Cambridge University Press:  08 May 2015

DRAGOS GHIOCA ,
KHOA NGUYEN  and
THOMAS J. TUCKER
DRAGOS GHIOCA
Affiliation:
Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada. e-mail: dghioca@math.ubc.edu
KHOA NGUYEN
Affiliation:
Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada. e-mail: dknguyen@math.ubc.ca
THOMAS J. TUCKER
Affiliation:
Department of Mathematics University of Rochester Rochester, NY 14627, USA. e-mail: thomas.tucker@rochester.edu
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Abstract

Let K be a function field over an algebraically closed field k of characteristic 0, let ϕ ∈ K(z) be a rational function of degree at least equal to 2 for which there is no point at which ϕ is totally ramified and let α ∈ K. We show that for all but finitely many pairs (m, n) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ there exists a place $\mathfrak{p}$ of K such that the point α has preperiod m and minimum period n under the action of ϕ. This answers a conjecture made by Ingram–Silverman [13] and Faber–Granville [8]. We prove a similar result, under suitable modification, also when ϕ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple (c0, . . ., cd−2) ∈ kd−1 and for almost all pairs (mi, ni) ∈ $\mathbb{Z}$⩾0 × $\mathbb{N}$ for i = 0, . . ., d − 2, there exists a polynomial f ∈ k[z] of degree d in normal form such that for each i = 0, . . ., d − 2, the point ci has preperiod mi and minimum period ni under the action of f.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 165 - 186
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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