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<p>This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for degree <i>d</i>, real analytic, unimodal functions fixed by <i>m</i>-th order Feigenbaum–Coullet–Tresser–Cvitanović renormalization operators using Chebyshev series approximation. Here the order <i>m</i> of the operator refers to the number of function compositions involved in its definition, and the degree <i>d</i> of the fixed function is the number of derivatives vanishing at its (unique) critical point. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>. This facilitates the development of a functional analytic approach independent of order and degree. The main disadvantage of working with Chebyshev series is that the operations of rescaling and composition, essential to the definition of the renormalization operators, are less straightforward for Chebyshev than for Taylor series. These difficulties are overcome via a combination of a-priori information about decay rates in Banach spaces of rapidly decaying coefficient sequences, with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. We illustrate the utility of the method by proving existence, local uniqueness, global non-uniqueness results, and interval enclosures of universal constants for fixed points of varying orders and degrees.</p>

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Validated Enclosure of Renormalization Fixed Points Via Chebyshev Series and the DFT

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Maxime Breden,
  • Jorge Gonzalez,
  • J. D. Mireles James

摘要

This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for degree d, real analytic, unimodal functions fixed by m-th order Feigenbaum–Coullet–Tresser–Cvitanović renormalization operators using Chebyshev series approximation. Here the order m of the operator refers to the number of function compositions involved in its definition, and the degree d of the fixed function is the number of derivatives vanishing at its (unique) critical point. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in \(\mathbb {C}\) C . This facilitates the development of a functional analytic approach independent of order and degree. The main disadvantage of working with Chebyshev series is that the operations of rescaling and composition, essential to the definition of the renormalization operators, are less straightforward for Chebyshev than for Taylor series. These difficulties are overcome via a combination of a-priori information about decay rates in Banach spaces of rapidly decaying coefficient sequences, with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. We illustrate the utility of the method by proving existence, local uniqueness, global non-uniqueness results, and interval enclosures of universal constants for fixed points of varying orders and degrees.