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

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)161-193
Number of pages33
JournalTransactions of the American Mathematical Society
Volume365
Issue number1
DOIs
StatePublished - Oct 29 2012

Fingerprint

Rational functions
Endomorphism
Julia set
Curve
Rational function
Finiteness
Topological Properties
Symmetry Group
Set of points
Intersection
Sufficient Conditions
Interaction

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

On the preperiodic points of an endomorphism of ℙ1 × ℙ1 which lie on a curve. / Mimar, Arman.

In: Transactions of the American Mathematical Society, Vol. 365, No. 1, 29.10.2012, p. 161-193.

Research output: Contribution to journalArticle

@article{e0559d86252747048b1716b56ce7cd39,
title = "On the preperiodic points of an endomorphism of ℙ1 × ℙ1 which lie on a curve",
abstract = "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.",
author = "Arman Mimar",
year = "2012",
month = "10",
day = "29",
doi = "10.1090/S0002-9947-2012-05557-7",
language = "English (US)",
volume = "365",
pages = "161--193",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "1",

}

TY - JOUR

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

AU - Mimar, Arman

PY - 2012/10/29

Y1 - 2012/10/29

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.

UR - http://www.scopus.com/inward/record.url?scp=84867804332&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867804332&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2012-05557-7

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

M3 - Article

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 -