TY - JOUR

T1 - Limit periodic Jacobi matrices with a prescribed p-adic hull and a singular continuous spectrum

AU - Peherstorfer, F.

AU - Volberg, A.

AU - Yuditskii, P.

PY - 2006

Y1 - 2006

N2 - For all hyperbolic polynomials we proved in [9] a Lipschitz estimate 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 (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. 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. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.

AB - For all hyperbolic polynomials we proved in [9] a Lipschitz estimate 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 (with real Julia set). Here we prove that for all sufficiently hyperbolic polynomials this estimate becomes exponentially better when the dimension of the Jacobi matrix grows. In fact, our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. 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. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.

KW - Almost periodic Jacobi matrices

KW - Harmonic measure

KW - Hyperbolic polynomials

KW - Singular continuous spectrum

UR - http://www.scopus.com/inward/record.url?scp=33746134906&partnerID=8YFLogxK

U2 - 10.4310/MRL.2006.v13.n2.a4

DO - 10.4310/MRL.2006.v13.n2.a4

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AN - SCOPUS:33746134906

SN - 1073-2780

VL - 13

SP - 215

EP - 230

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 2-3

ER -