[Paper Review] Galois conjugates of entropies of real unimodal maps

Giulio Tiozzo|arXiv (Cornell University)|Oct 28, 2013

Mathematical Dynamics and Fractals3 references4 citations

TL;DR

This paper investigates the algebraic and geometric structure of Galois conjugates of topological entropy growth rates for real unimodal maps, particularly superattracting quadratic polynomials. Using kneading theory and connections to power series with coefficients in {±1}, it proves that the entropy spectrum—the closure of all Galois conjugates of growth rates—is path-connected and locally connected, resolving a conjecture of Thurston on the fractal geometry of these algebraic numbers.

ABSTRACT

We investigate the set of Galois conjugates of growth rates of superattracting real quadratic polynomials, following W. Thurston. In particular, we prove that the closure of this set is path-connected and locally connected.

Motivation & Objective

• To understand the algebraic and geometric properties of Galois conjugates of growth rates of postcritically finite unimodal maps.
• To resolve Thurston's conjecture that the set of Galois conjugates of entropies of real quadratic polynomials forms a path-connected and locally connected fractal set.
• To establish a precise connection between the entropy spectrum and the zero sets of kneading determinants.
• To analyze the structure of the entropy spectrum both inside and outside the unit disk using combinatorial and dynamical systems techniques.

Proposed method

• Uses kneading theory to associate each real quadratic map $f_c(z) = z^2 + c$ with a kneading determinant $K_c(t)$, whose roots relate to the inverse of the growth rate $s(f_c)$.
• Defines the entropy spectrum $\Sigma$ as the closure of the union of all Galois conjugates of growth rates $s(f_c)$ for superattracting parameters $c \in M_0 \cap \mathbb{R}$.
• Analyzes the part of $\Sigma$ inside the unit disk by proving it coincides with the set $\Sigma_{\pm 1}$ of zeros of polynomials with coefficients in $\{\pm 1\}$, using a reflection and scaling argument.
• Proves that the intersection $\Sigma_{kn} \cap \mathbb{E}$ (outside the unit disk) is connected and locally connected by studying the structure of kneading determinants.
• Establishes that both $\Sigma$ and $\Sigma_{kn}$ contain a neighborhood of the unit circle, completing the proof of path-connectedness and local connectivity.
• Applies techniques from symbolic dynamics and renormalization theory to analyze irreducibility of kneading polynomials and their Galois conjugates.

Experimental results

Research questions

• RQ1 Is the set of Galois conjugates of growth rates of superattracting real quadratic polynomials path-connected and locally connected?
• RQ2 How does the entropy spectrum $\Sigma$ relate to the zero sets of power series with coefficients in $\{\pm 1\}$?
• RQ3 What is the geometric structure of the entropy spectrum both inside and outside the unit disk?
• RQ4 Can the full entropy spectrum $\Sigma$ be shown to be connected by analyzing its components in $\mathbb{D}$ and $\mathbb{E}$ separately?
• RQ5 How do renormalization and combinatorics of orbits influence the irreducibility of kneading polynomials and their Galois conjugates?

Key findings

• The entropy spectrum $\Sigma$ is path-connected and locally connected, confirming a conjecture of W. Thurston.
• The part of $\Sigma$ inside the unit disk coincides exactly with the set $\Sigma_{\pm 1}$ of zeros of all polynomials with coefficients in $\{\pm 1\}$.
• The intersection $\Sigma_{kn} \cap \mathbb{E}$, where $\Sigma_{kn}$ is the set of all zeros of kneading determinants, is connected and locally connected.
• Both $\Sigma$ and $\Sigma_{kn}$ contain a neighborhood of the unit circle in the complex plane.
• The proof relies on a combination of kneading theory, symbolic dynamics, and geometric covering arguments involving scaled and reflected lattices.
• The result establishes a deep link between the dynamics of unimodal maps and the fractal geometry of algebraic integers arising from their growth rates.