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

2023/12/22 by Henriksen, Christian, Petersen, Carsten Lunde, Uhre, Eva
#31A05 (Primary) 37F10 #42C05 (Secondary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2312.14740

Abstract

Let K⊂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.

Related

• Value Distributions of Derivatives of K-regular Polynomial Families
• Value distribution of the sequences of the derivatives of iterated polynomials
• Equilibrium measures on Julia sets of random quadratic polynomials
• Value distribution of derivatives in polynomial dynamics
• Note on the Equilibrium Measures of Julia sets of Exceptional Jacobi Polynomials