Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian

Abstract

We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the coupling constant.

20 pages

References in corpus (2)

• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Bers and Hénon, Painlevé and Schroedinger

Cited by in corpus (27)

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• Bers and Hénon, Painlevé and Schroedinger
• Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
• Spectra of Discrete Schrödinger Operators with Primitive Invertible Substitution Potentials
• The fractal dimensions of the spectrum of Sturm Hamiltonian
• Spectral analysis of tridiagonal Fibonacci Hamiltonians
• Continuum Schrödinger Operators Associated With Aperiodic Subshifts
• Quantum and Spectral Properties of the Labyrinth Model
• Dynamical Green Functions and Discrete Schrödinger Operators with Potentials Generated by Primitive Invertible Substitution
• Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian
• Properties of 1D classical and quantum Ising quasicrystals: rigorous results
• The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type
• On the Hausdorff dimension of the spectrum of Thue-Morse Hamiltonian
• On the spectrum of 1D quantum Ising quasicrystal
• Driven Non-Unitary Dynamics of Quantum Critical Systems
• The spectrum of period-doubling Hamiltonian
• Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian
• "Mixed spectral nature" of Thue--Morse Hamiltonian
• The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator
• Almost Ballistic Transport for the Weakly Coupled Fibonacci Hamiltonian
• Iteration of polynomial pair under Thue-Morse dynamic
• Newhouse phenomena in the Fibonacci trace map
• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• The Spectrum of the Weakly Coupled Fibonacci Hamiltonian
• The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian