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

Abstract

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.

References in corpus (5)

• Asymptotics of Chebyshev Polynomials, II. DCT Subsets of $\mathbb{R}$
• Asymptotics of Chebyshev Polynomials, I. Subsets of $\mathbb{R}$
• Weak limits of the measures of maximal entropy for Orthogonal polynomials
• On Dynamics of Asymptotically Minimal Polynomials
• Equidistribution of the zeros of higher order derivatives in polynomial dynamics