[Paper Review] The core entropy for polynomials of higher degree

Yan Gao, Giulio Tiozzo|arXiv (Cornell University)|Mar 25, 2017

Mathematical Dynamics and Fractals29 references3 citations

TL;DR

This paper establishes the continuous variation of core entropy for polynomials of degree $d \geq 2$ as a function of both combinatorial data (primitive majors) and polynomial coefficients. By developing a theory of core entropy in higher-degree dynamics and proving continuity via convergence of critical portraits and primitive majors, the authors confirm a conjecture by W. Thurston on the continuity of core entropy in the general polynomial setting.

ABSTRACT

As defined by W. Thurston, the core entropy of a polynomial is the entropy of the restriction to its Hubbard tree. For each d >= 2, we study the core entropy as a function on the parameter space of polynomials of degree d, and prove it varies continuously both as a function of the combinatorial data and of the coefficients of the polynomials. This generalizes a conjecture of Thurston for quadratic polynomials.

Motivation & Objective

• To extend the theory of core entropy beyond quadratic polynomials to polynomials of arbitrary degree $d \geq 2$.
• To prove that core entropy varies continuously with respect to the combinatorial data of primitive majors in the parameter space $\mathrm{PM}(d)$.
• To establish the continuity of core entropy as a function of the coefficients of postcritically finite polynomials in the space $\mathcal{P}_d$.
• To confirm a conjecture by W. Thurston on the continuity of core entropy in higher-degree polynomial dynamics.

Proposed method

• Define the core entropy of a polynomial as the topological entropy of its restriction to the Hubbard tree.
• Use the space $\mathrm{PM}(d)$ of primitive majors as a combinatorial parameter space generalizing the Mandelbrot set's boundary for higher degrees.
• Prove that core entropy extends continuously to the closure of rational primitive majors in $\mathrm{PM}(d)$, using convergence of critical portraits and ray dynamics.
• Apply Hausdorff convergence of critical markings and convergence of induced primitive majors to establish continuity of entropy values.
• Leverage Rouché's theorem and ray-shrinking arguments to track the convergence of critical points and their dynamical behavior under perturbation.
• Use the equality $h(f) = h(m)$, where $m$ is the primitive major associated to $f$, to reduce continuity to the convergence of majors in $\mathrm{PM}(d)$.

Experimental results

Research questions

• RQ1 Does the core entropy of a polynomial of degree $d \geq 2$ vary continuously with respect to its combinatorial data (i.e., the primitive major)?
• RQ2 Is the core entropy a continuous function of the coefficients of postcritically finite polynomials in $\mathcal{P}_d$?
• RQ3 Can the core entropy be continuously extended over the entire combinatorial parameter space $\mathrm{PM}(d)$, including irrational majors?
• RQ4 How does the convergence of critical portraits and ray dynamics under perturbation affect the continuity of entropy?
• RQ5 To what extent can core entropy serve as a tool for understanding the hierarchical structure of the connectedness locus in higher-degree polynomial dynamics?

Key findings

• The core entropy function $h(m)$ extends continuously to the entire space $\mathrm{PM}(d)$ of primitive majors of degree $d$, confirming Thurston's conjecture.
• Core entropy varies continuously with respect to the coefficients of postcritically finite polynomials in $\mathcal{P}_d$, establishing continuity in the analytic parameter space.
• Convergence of critical portraits $\Theta_n$ to a weak critical marking $\Theta$ implies convergence of entropy values: $h(f_n) \to h(f)$ as $n \to \infty$.
• The entropy $h(f)$ is equal to the entropy $h(m)$ associated with the primitive major $m$ induced by the polynomial’s combinatorics.
• The convergence of primitive majors $m_n \to m$ in $\mathrm{PM}(d)$ implies $h(m_n) \to h(m)$, which is sufficient to prove continuity of core entropy.
• The proof relies on ray dynamics and the convergence of external rays and their landing points under perturbation, ensuring topological consistency in the limit.