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<p>In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(z^2-\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> <mo>-</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> with large enough <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>; this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has empty point spectrum. To prove this, we introduce a matrix version of Ruelle operators.</p>

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On Point Spectrum of Jacobi Matrices Generated by Iterations of Quadratic Polynomials

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月5日

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  • Benjamin Eichinger,
  • Milivoje Lukić,
  • Peter Yuditskii

摘要

In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial \(z^2-\lambda \) z 2 - λ with large enough \(\lambda \) λ ; this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has empty point spectrum. To prove this, we introduce a matrix version of Ruelle operators.