The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

TL;DR

This paper studies the density of states measure for the Fibonacci Hamiltonian at small coupling, showing it is exact-dimensional, smoothly varying with coupling, and converges to full dimension as coupling vanishes, linking spectral theory with dynamical systems.

Contribution

It establishes the exact-dimensionality of the density of states measure for small coupling and connects it to the measure of maximal entropy via the Fibonacci trace map.

Findings

Density of states measure is exact-dimensional for small V.

The local scaling exponent varies smoothly with V.

The dimension converges to one as V approaches zero.

Abstract

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant $V$, this measure is exact-dimensional and the almost everywhere value $d_V$ of the local scaling exponent is a smooth function of $V$, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as $V$ tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the $V$-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.