TY - JOUR

T1 - On the preperiodic points of an endomorphism of ℙ1 × ℙ1 which lie on a curve

AU - Mimar, Arman

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - Abstract. Let X be a projective curve in ℙ 1 × ℙ 1 and φ{symbol} be an endomorphism of degree ≥ 2 of ℙ 1 × ℙ 1, given by two rational functions by φ{symbol}(z,w) = (f(z), g(w)) (i.e., φ{symbol} = f × g), where all are defined over Q. In this paper, we prove a characterization of the existence of an infinite intersection of X(ℚ) with the set of φ{symbol}-preperiodic points in ℙ 1×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 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 J (f) and J (g) as well. We then find various sufficient conditions on the pair (X, φ{symbol}) and often on φ{symbol} alone, for the finiteness of the set of φ{symbol}-preperiodic points of X(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.

AB - Abstract. Let X be a projective curve in ℙ 1 × ℙ 1 and φ{symbol} be an endomorphism of degree ≥ 2 of ℙ 1 × ℙ 1, given by two rational functions by φ{symbol}(z,w) = (f(z), g(w)) (i.e., φ{symbol} = f × g), where all are defined over Q. In this paper, we prove a characterization of the existence of an infinite intersection of X(ℚ) with the set of φ{symbol}-preperiodic points in ℙ 1×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 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 J (f) and J (g) as well. We then find various sufficient conditions on the pair (X, φ{symbol}) and often on φ{symbol} alone, for the finiteness of the set of φ{symbol}-preperiodic points of X(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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U2 - 10.1090/S0002-9947-2012-05557-7

DO - 10.1090/S0002-9947-2012-05557-7

M3 - Article

AN - SCOPUS:84867804332

VL - 365

SP - 161

EP - 193

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -