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

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