Limit periodic Jacobi matrices with a singular continuous spectrum and the renormalization of periodic matrices

Details Limit periodic Jacobi matrices with a singular continuous spectrum and the renormalization of periodic matrices Limit periodic Jacobi matrices with a singular continuous spectrum and the renormalization of periodic matrices

2005-10-06

arxiv.org/abs/math-ph/0510019v1

Physics

Mathematical Physics

Scientific paper

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.

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