The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator

2008/07/18 by David Damanik, Anton Gorodetski, Damanik, David +1
Materials Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #math.DS #math.SP

paper · pdf · doi:10.48550/arxiv.0807.3024

10 pages; corrected a few typos

openalex publication_date 2008/07/18 · arxiv created 2008/07/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In addition, if the two hopping parameters are distinct but sufficiently close to each other, we show that the spectrum is a dynamically defined Cantor set, which has a variety of consequences for its local and global fractal dimension.

Citations

• Metallic Phase with Long-Range Orientational Order and No Translational Symmetry

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