Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian

William Yessen

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Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian

William Yessen

Mathematical Physics · Submitted 12 Oct 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$.

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Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$. Our proof relies on the dynamics of the Fibonacci trace map in combination with the recent result of M. Hochman and P. Shmerkin on the Hausdorff dimension of sums of Cantor sets which are attractors of regular iterated function systems (Local entropy averages and projections of fractal measures, Ann. Math. 175 (2012), 1001--1059).

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$.	result 0.99	0.95	0.95	asserted 1.00
s1	Our proof relies on the dynamics of the Fibonacci trace map in combination with the recent result of M. Hochman and P.	method 1.00	0.53	0.03	asserted 0.93
s2	Shmerkin on the Hausdorff dimension of sums of Cantor sets which are attractors of regular iterated function systems (Local entropy averages and projections of fractal measures, Ann.	method 0.80	0.06	0.02	unclear 0.74
s3	Math. 175 (2012), 1001--1059).	background 0.85	0.03	0.01	unclear 0.99

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Original abstract

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$. Our proof relies on the dynamics of the Fibonacci trace map in combination with the recent result of M. Hochman and P. Shmerkin on the Hausdorff dimension of sums of Cantor sets which are attractors of regular iterated function systems (Local entropy averages and projections of fractal measures, Ann. Math. 175 (2012), 1001--1059).

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3 arXiv papers cite this in 3 sentences

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The study of the structure and the properties of sums of Cantor sets is motivated by applications in dynamical systems [ref] , number theory [ref] , harmonic analysis [ref] , and spectral theory [ref] .

Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian · Sums of dynamically defined Cantor sets · 0.95
Questions on the structure and properties of sums of Cantor sets appear naturally in dynamical systems [ref] , number theory [ref] , harmonic analysis [ref] , and spectral theory [ref] .

On Sums of Nearly Affine Cantor Sets · Introduction and Main results · 0.99

What it cites · 9 sentences without a confident label

Cites as background 2 papers

This operator has been widely studied in the context of electronic transport properties of quasicrystals for the past thirty years (see [ref] and references therein for details).

Spectral Properties of Schr\"odinger Operators Arising in the Study of Quasicrystals · Introduction · 1.00
This operator has been widely studied in the context of electronic transport properties of quasicrystals for the past thirty years (see [ref] and references therein for details).

The Fibonacci Hamiltonian · Introduction · 1.00

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