Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian

TL;DR

This paper proves that the trace map for the Fibonacci Hamiltonian exhibits hyperbolic dynamics at small coupling, leading to smooth, coinciding fractal dimensions of its spectrum as a function of the coupling.

Contribution

It establishes hyperbolicity of the trace map for small coupling constants and links this to smooth spectral dimension functions.

Findings

The non-wandering set of the trace map is hyperbolic at small coupling.

Spectral dimensions are smooth functions of the coupling constant.

Hausdorff and box counting dimensions coincide for small coupling.

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.