Fourier decay of equilibrium states and the Fibonacci Hamiltonian

TL;DR

This paper demonstrates the positivity of the lower Fourier dimension for equilibrium states of certain surface diffeomorphisms and applies this to prove Fourier decay properties of the Fibonacci Hamiltonian's density of states, linking dynamical systems and spectral theory.

Contribution

It introduces a method to establish Fourier decay for equilibrium states in nonlinear Axiom A systems and applies it to the Fibonacci Hamiltonian, a novel connection between dynamics and spectral analysis.

Findings

Positivity of lower Fourier dimension for equilibrium states.

Power Fourier decay for the density of states measure of the Fibonacci Hamiltonian.

Implications for phase-averaged dispersive estimates in quasicrystals.

Abstract

We show positivity of the lower Fourier dimension for equilibrium states of nonlinear, area preserving, Axiom A diffeomorphisms on surfaces. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of some temporal distance function for a well chosen suspension flow, whose mixing properties reflects the nonlinearity of our base dynamics. We then generalize in an Axiom A setting the methods of Tsujii-Zhang, dealing with exponential mixing of three-dimensional Anosov flows arXiv:2006.04293. The nonlinearity condition is generic and can be checked in concrete contexts. As a corollary, we prove power Fourier decay for the density of states measure of the Fibonacci Hamiltonian, which is related to the measure of maximal entropy of the Fibonacci trace map. This proves positivity of the lower Fourier dimension for the spectrum of the Fibonacci Hamiltonian, and suggest strong phase-averaged dispersive estimates in quasicrystals.