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Orthogonality properties of iterated polynomial mappings

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Abstract

We consider a measure defined on a complex contour and its associated orthogonal polynomials. The action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given. Iterating the transformation provides an invariant measure, whose support is the Julia set corresponding to the polynomial transformation. It appears that, up to a constant, the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials, which fulfill a three term recursion relation. An algorithm is given to compute the coefficients of this recursion relation, which can be interpreted as a linear extension of the iterative procedure.

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Authors and Affiliations

  1. Service de Physique Théorique, CEN-Saclay, F-91191, Gif-sur-Yvette, France

    D. Bessis & P. Moussa

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  1. D. Bessis
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  2. P. Moussa
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Communicated by O.E. Lanford

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Bessis, D., Moussa, P. Orthogonality properties of iterated polynomial mappings. Commun.Math. Phys. 88, 503–529 (1983). https://doi.org/10.1007/BF01211956

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  • DOI: https://doi.org/10.1007/BF01211956

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