Convergence of Equilibrium Measures under $K$-regular Polynomial Sequences and their Derivatives

Let $K\subset\mathbb{C}$ be non-polar, compact and polynomially convex. We study the limits of equilibrium measures on preimages of compact sets, under $K$-regular sequences of polynomials, that center on $K$ and under the sequences of derivatives of all orders of such sequences. We show that under mild assumptions such limits always exist and equal the equilibrium measure on $K$. From this we derive convergence of the equilibrium distributions on the Julia sets of the sequence of polynomials and their derivatives of all orders.

💡 Research Summary

This paper establishes a general framework for studying the convergence of equilibrium measures associated with sequences of polynomials that exhibit specific asymptotic properties relative to a compact set in the complex plane. The central object is a non-polar, compact, and polynomially convex set (K \subset \mathbb{C}), with (\Omega = \mathbb{C}\setminus K) its unbounded complement and (\omega_K) its equilibrium measure.

The authors focus on sequences of polynomials ((q_k)) of degree (n_k \to \infty) that satisfy two key properties: K-centering and K-regularity. K-centering means that, for large (k), all zeros of (q_k) lie within a fixed disk, and the number of zeros outside any neighborhood of (K) is uniformly bounded. K-regularity means the normalized logarithmic potentials ((1/n_k)\log|q_k(z)|) converge locally uniformly to the Green function (g_\Omega(z)) on the complement of a large disk. These properties are known to hold for many important sequences, including orthogonal polynomials, Fekete polynomials, and iterates of a fixed polynomial.

The first major result (Theorem 1) is that both properties are hereditary. If a sequence ((q_k)) is K-regular and K-centering, then the sequence of its derivatives ((q’_k)) also possesses these properties. By induction, this inheritance extends to the sequences of derivatives of all orders ((q_k^{(m)})). The proof leverages complex analysis and potential theory, showing that the asymptotic behavior of the log-derivative ((q’k/q_k)) is controlled by the derivative of the Green function (g’\Omega).

The core technical achievement is Theorem 4. It states that for any sequence of compact sets ((L_k)) whose logarithmic capacity grows slower than the degrees ((\log(\text{Cap}(L_k)) = o(n_k))), the equilibrium measures on the preimages (\omega_{q_k^{-1}(L_k)}) converge weakly-* to (\omega_K). This theorem provides a very general condition for convergence, requiring only mild control on the “size” (capacity) of the target sets (L_k).

From this powerful general theorem, two significant corollaries are derived. Theorem 2 follows by taking (L_k) to be a fixed disk (D(r)), proving (\omega_{q_k^{-1}(D(r))} \overset{w*}{\to} \omega_K). Theorem 3 applies the theory to dynamical objects, showing that the equilibrium measure on the filled Julia set (K(q_k)) of the polynomial (q_k) itself converges to (\omega_K). This connects the limiting statistical distribution of the dynamical system to the external potential-theoretic object (K).

A direct consequence of the hereditary nature of the properties (Corollaries 1 & 2) is that all these convergence results—for preimages of disks and for Julia set measures—hold equally true for the sequences of derivatives ((q_k^{(m)})) of any order (m \ge 0).

The paper illustrates the theory with concrete examples spanning different fields: orthogonal polynomials (extremal theory), Chebyshev polynomials, Fekete polynomials, and iterated polynomials (holomorphic dynamics). A notable example is the parameterized sequence (q_k(c) = P_c^k(0)) where (P_c(z)=z^2+c), which is shown to be M-regular and M-centering for the Mandelbrot set (M). Thus, the theory implies the convergence of the equilibrium measures on the Julia sets of both (q_k) and its derivatives to the equilibrium measure of the Mandelbrot set.

In summary, this work provides a unified potential-theoretic framework that explains and generalizes observed convergence phenomena for equilibrium measures across extremal polynomial theory and complex dynamics. By identifying the hereditary properties of K-regularity and K-centering, it establishes that such convergence is robust and preserved under the operation of taking derivatives, yielding a coherent set of limit theorems for a broad class of polynomial sequences.