A spectral cocycle for substitution systems and translation flows

TL;DR

This paper introduces a spectral cocycle for substitution systems and translation flows, linking Lyapunov exponents to the local spectral measure dimensions using symbolic representations and matrix Riesz products.

Contribution

It presents a novel spectral cocycle construction that connects Lyapunov exponents with spectral measure dimensions in substitution systems and translation flows.

Findings

Lyapunov exponents govern local spectral measure dimensions

Spectral cocycle constructed via symbolic representation and matrix Riesz products

Provides new insights into spectral properties of substitution systems

Abstract

For substitution systems and translation flows, a new cocycle, which we call {\em spectral cocycle}, is introduced, whose Lyapunov exponents govern the local dimension of the spectral measure for higher-level cylindrical functions. The construction relies on the symbolic representation of translation flows and the formalism of matrix Riesz products.