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Uniform boundedness on rational maps with automorphisms

Minsik Han

Abstract

In this paper, we study the dynamical uniform boundedness principle over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree $d\ge 2$ whose automorphism group contains the cyclic group of order $d$. We prove that a subfamily of this family satisfies the dynamical uniform boundedness principle.

Uniform boundedness on rational maps with automorphisms

Abstract

In this paper, we study the dynamical uniform boundedness principle over a family of rational maps with certain nontrivial automorphisms. Specifically, we consider a family of rational maps of an arbitrary degree whose automorphism group contains the cyclic group of order . We prove that a subfamily of this family satisfies the dynamical uniform boundedness principle.
Paper Structure (2 sections, 11 theorems, 72 equations)

This paper contains 2 sections, 11 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 3

Let $b\in\mathbb{Q}^*$. Then any rational periodic cycle of $\psi_{a,1}$ or $\psi_{a,-1}$ has length at most $4$. Furthermore, for any $a \in \mathbb{Q}^*$, we have

Theorems & Definitions (22)

  • Conjecture 1: Morton-Silverman, 1994, ubc
  • Conjecture 2
  • Theorem 3: Levy-Manes-Thompson, 2014, manes
  • Theorem 4: Manes, manes2
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 12 more