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Proportion of periodic points in reduction of polynomials

Santiago Radi

Abstract

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given $K$ a number field and $f$ a rational function with coefficients in $K$, if $f_\mathfrak{p}$ denotes the reduction of $f$ modulo a prime ideal $\mathfrak{p}$ in the ring of integers of $K$, what is the limit inferior of the proportion of periodic points of $f_\mathfrak{p}$ when the norm of $\mathfrak{p}$ goes to infinity? Recent results of Fariña-Asategui and the author show that when $f$ is a polynomial of degree $d \geq 2$ non-linearly conjugate over $\mathbb{C}$ to a Chebyshev polynomial then the limit is zero. In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.

Proportion of periodic points in reduction of polynomials

Abstract

In 2014, Juul, Kurlberg, Madhu and Tucker asked the following: given a number field and a rational function with coefficients in , if denotes the reduction of modulo a prime ideal in the ring of integers of , what is the limit inferior of the proportion of periodic points of when the norm of goes to infinity? Recent results of Fariña-Asategui and the author show that when is a polynomial of degree non-linearly conjugate over to a Chebyshev polynomial then the limit is zero. In this article, we address the remaining cases to give a complete classification of the problem in the case of polynomials.
Paper Structure (9 sections, 21 theorems, 128 equations)

This paper contains 9 sections, 21 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iteration of maps and local degree
  4. Algebraic number theory
  5. Chebyshev polynomials
  6. Polynomials not linearly conjugate over $\mathbb C$ to $\pm T_d$
  7. Proportion of periodic points in Chebyshev polynomials
  8. Polynomials linearly conjugate over $\mathbb C$ to $\pm T_d$
  9. Analysis of $\mathop{\mathrm{Per}}\nolimits_{\inf}(T_d,K)$

Key Result

Theorem 1

Let $K$ be a number field and let $m_K$ denote the greatest positive number such that $K$ contains a $m_K$th primitive root of unity. Let $f$ be a polynomial with coefficients in $K$ and degree $d$.

Theorems & Definitions (41)

  • Theorem 1
  • Corollary 2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 31 more