Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rigidity of Lyapunov exponents for polynomials

Zhuchao Ji, Junyi Xie, Geng-Rui Zhang

Abstract

Let $f,g\in\overline{\mathbb{Q}}[z]$ be polynomials of degree $d\geq2$ with disconnected Julia sets. We prove that they have the same Lyapunov exponent $\mathcal{L}_f=\mathcal{L}_g$ if and only if either $f$ and $g$ are intertwined, or $f$ and $\overline{g}$ are intertwined. The analogous result for critical heights is also obtained. As an application, we provide a new proof of the theorem stating that the multiplier spectrum morphism on the moduli space of polynomials is generically injective.

Rigidity of Lyapunov exponents for polynomials

Abstract

Let be polynomials of degree with disconnected Julia sets. We prove that they have the same Lyapunov exponent if and only if either and are intertwined, or and are intertwined. The analogous result for critical heights is also obtained. As an application, we provide a new proof of the theorem stating that the multiplier spectrum morphism on the moduli space of polynomials is generically injective.
Paper Structure (10 sections, 10 theorems, 57 equations)

This paper contains 10 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. An analogue for critical heights
  4. Application to multiplier spectra
  5. Preliminaries
  6. Lyapunov exponents and Green functions
  7. Böttcher coordinates
  8. The intertwined relation for non-exceptional polynomials
  9. Algebraicity of Böttcher coordinates and multiplicative canonical heights
  10. Proof of the main results

Key Result

Theorem 1.2

Let $f,g\in\overline{\mathbb{Q}}[z]$ be polynomials of degree $d\geq2$. Assume that $\mathcal{L}_f>\log(d)$ (equivalently, $J(f)$ is disconnected). Then $\mathcal{L}_f=\mathcal{L}_g$ if and only if $f$ and $\hat{g}$ are intertwined for some $\hat{g}\in\{g,\overline{g}\}$.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6: JX23
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 18 more