Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian

2016/01/07 by David Damanik, Anton Gorodetski, Damanik, David +1 · 1 citation
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Quasicrystal Structures and Properties #Spectral Theory (math.SP) #math-ph #math.DS #math.MP #math.SP #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1601.01639

13 pages

arxiv created 2016/01/07 · openalex publication_date 2016/01/07 · arxiv updated 2016/01/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We show that under natural technical conditions, the sum of a C² dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.

Citations

• The Fibonacci Hamiltonian
• Fractal Geometry: Mathematical Foundations and Applications.

Cited by

• Sums of two homogeneous Cantor sets

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