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$S$-integral preperiodic points for monomial semigroups over number fields

Marley Young

Abstract

We consider semigroup dynamical systems defined by several monnomials over a number field $K$. We prove a finiteness result for preperiodic points of such systems which are $S$-integral with respect to a non-preperiodic point $β$, which is uniform as $β$ varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap, and verifies a special case of a natural generalisation of a conjecture of Ih.

$S$-integral preperiodic points for monomial semigroups over number fields

Abstract

We consider semigroup dynamical systems defined by several monnomials over a number field . We prove a finiteness result for preperiodic points of such systems which are -integral with respect to a non-preperiodic point , which is uniform as varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap, and verifies a special case of a natural generalisation of a conjecture of Ih.
Paper Structure (10 sections, 13 theorems, 97 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of rational maps
  4. Heights of algebraic numbers
  5. Lower bounds on linear forms in logarithms
  6. The Berkovich projective line and adelic measures
  7. Equilibrium measures for bounded sequences and quantitative equidistribution
  8. Structure of preperiodic points for monomial semigroups
  9. Bounds on distance to nearest preperiodic point
  10. Proof of Theorem \ref{['thm:main']}

Key Result

Proposition 1.3

If $\cG = \langle f_i \rangle_{i \in I}$ has two elements $f$ and $g$ with distinct Julia sets (when viewed as complex dynamical systems), then $\cG$ has only finitely many strongly preperiodic points, and so Conjecture conj:StrongSemigroupIh holds vacuously.

Figures (1)

  • Figure :

Theorems & Definitions (24)

  • Conjecture 1.1: Ih
  • Conjecture 1.2
  • Proposition 1.3
  • proof
  • Conjecture 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 14 more