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A gap principle for dynamics

Part of: Arithmetic problems. Diophantine geometry Complex dynamical systems Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  26 April 2010

Robert L. Benedetto ,
Dragos Ghioca ,
Pär Kurlberg  and
Thomas J. Tucker
Robert L. Benedetto
Affiliation:
Department of Mathematics, Amherst College, Amherst, MA 01002, USA (email: rlb@math.amherst.edu)
Dragos Ghioca
Affiliation:
Department of Mathematics & Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB T1K 3M4, Canada (email: dragos.ghioca@uleth.ca)
Pär Kurlberg
Affiliation:
Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden (email: kurlberg@math.kth.se)
Thomas J. Tucker
Affiliation:
Department of Mathematics, Hylan Building, University of Rochester, Rochester, NY 14627, USA (email: ttucker@math.rochester.edu)
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Abstract

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Let f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the logfunction. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.

Keywords

p-adic dynamicsMordell–Lang conjecture

MSC classification

Secondary: 37P55: Arithmetic dynamics on general algebraic varieties 14G25: Global ground fields 37F10: Polynomials; rational maps; entire and meromorphic functions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 1056 - 1072
Copyright
Copyright © Foundation Compositio Mathematica 2010

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