Fourier decay of equilibrium states and the Fibonacci Hamiltonian

Abstract

We show power Fourier decay for equilibrium states of nonlinear, area preserving, smooth Axiom-A diffeomorphisms on surfaces. This implies positivity of the lower Fourier dimension for self-conformal measures under $C^{1+}$ iterated function systems that are factors of hyperbolic diffeomorphisms, which is the first result of this kind in this low-regularity setting. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of a temporal distance function for a well chosen suspension flow, behaving like a 3-dimensional Axiom A 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. To illustrate the applications, we prove two corollaries. We first establish a spectral gap, proving exponential mixing for generic circle extensions over hyperbolic maps on surfaces. As a second application, we prove power Fourier decay for the density of states measure of the Fibonacci Hamiltonian. This implies phase-averaged escape-of-mass estimates, which is the first result of this type in a quasicrystal.