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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K\subset \mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> be non-polar, compact and polynomially convex. We study the limits of equilibrium measures on preimages of compact sets, under <i>K</i>-regular sequences of polynomials, that center on <i>K</i> 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 <i>K</i>. From this we derive convergence of the equilibrium distributions on the Julia sets of the sequence of polynomials and their derivatives of all orders.</p>

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

  • 期刊论文

  • 发表日期：

    2025年12月8日

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  • Christian Henriksen,
  • Carsten Lunde Petersen,
  • Eva Uhre

摘要

Let \(K\subset \mathbb {C}\) 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.