Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to hyperbolic polynomials

F. Peherstorfer, A. Volberg, P. Yuditskii

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Abstract

We are going to prove a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operators associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. It is obtained using some sufficient conditions for two-weight boundedness of the Hilbert transform. It has been proved in [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230] for all polynomials with sufficiently big hyperbolicity and in the most symmetric case t = 0 that the Lipschitz estimate becomes exponentially better when the dimension of the Jacobi matrix grows. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. We suggest a scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the present paper and those of [F. Peherstorfer, A. Volberg, P. Yuditskii, Limit periodic Jacobi matrices with prescribed p-adic hull and a singular continuous spectrum, Math. Res. Lett. 13 (2-3) (2006) 215-230]. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron-Frobenius-Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron-Frobenius-Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.

Original languageEnglish
Pages (from-to)1-30
Number of pages30
JournalJournal of Functional Analysis
Volume246
Issue number1
DOIs
StatePublished - 1 May 2007
Externally publishedYes

Bibliographical note

Funding Information:
✩ Partially supported by NSF grant DMS-0200713, and the Austrian Science Fund FWF, project number: P16390–N04. * Corresponding author. E-mail address: volberg@math.msu.edu (A. Volberg). 1 Supported by Marie Curie International Fellowship within the 6th European Community Framework Program (Contract Number: MIF1-CT-2005-006966).

Keywords

  • Almost periodic Jacobi matrices
  • Bowen-Ruelle measures
  • Harmonic measure
  • Hyperbolic polynomials
  • Singular continuous spectrum

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