On the modulus of continuity for spectral measures in substitution dynamics

TL;DR

This paper provides quantitative estimates on the continuity properties of spectral measures in substitution dynamical systems, revealing their fractal structure and implications for translation flows, with new technical tools and decay estimates.

Contribution

It introduces novel Hoelder and log-Hoelder estimates for spectral measures in substitution flows, and develops an arithmetic-Diophantine tool with broader applications.

Findings

Hoelder estimate for spectral measure of almost all suspension flows

Log-Hoelder estimate for self-similar suspension flows

Hoelder asymptotic expansion of spectral measure at zero

Abstract

The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the "fractal" structure of these measures. The main results are, first, a Hoelder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hoelder estimate for self-similar suspension flows; and, third, a Hoelder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hoelder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an "arithmetic-Diophantine" proposition, which has other applications. In the appendix this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.