Conformal equivalence of measures and dynamics of orthogonal polynomials

ABSTRACT

We introduce a notion of asymptotically orthonormal polynomials for a Borel measure μ with compact non-polar support in $\mathbb{C}$. Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials, see also Christiansen et al. [Julia sets of orthogonal polynomials, Potential Anal. 50(3) (2019), pp. 401–413]. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study the dynamics of sequences of asymptotically orthonormal polynomials for the pullback measure ${\phi }^{\ast }\left(\mu \right)$ under affine mappings φ. We prove that the sequences of Julia sets and filled Julia sets of affine deformations of sequences of asymptotically orthonormal polynomials for μ also have the same convergence properties as the Julia sets and filled Julia sets of the orthonormal polynomials. This leads to theorems on the convergence properties of affine deformations of the family of iterates of any fixed monic centred polynomial and, in the case the polynomial is hyperbolic, on the corresponding family of affine parameter spaces.

Keywords:

• Orthogonal polynomials
• Julia sets
• polynomial dynamics

Acknowledgments

The first author would like to thank Laura DeMarco for helpful conversations. This paper is inspired by the master's thesis written by the first author at IMFUFA, Department of Science and Environment at Roskilde University. The authors would like to thank IMFUFA for its hospitality during the conception and initial writing of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author would like to thank the Danish Council for Independent Research | Natural Sciences for support via the grant DFF – 4181-00502.