On substitution automorphisms with pure singular spectrum
Alexander I. Bufetov, Boris Solomyak
TL;DR
This work develops a spectral-theoretic framework for substitution dynamical systems via the spectral cocycle ${\mathscr M}_\zeta$ and its top Lyapunov exponent $\chi({\mathscr M}_\zeta)$. By relating $\chi({\mathscr M}_\zeta)$ to the Perron–Frobenius eigenvalue $\theta_1$ of the substitution matrix and employing Host/Meiri equidistribution results, the authors derive a practical sufficient condition $\chi({\mathscr M}_\zeta) < \tfrac{1}{2}\log \theta_1$ guaranteeing pure singular spectrum for the substitution $\mathbb{Z}$-action. They extend this criterion to two-symbol, non-constant-length cases and demonstrate its applicability to non-Pisot substitutions and to a family of self-similar interval exchange transformations (Salem-type), providing explicit constructions and highlighting open questions about necessity. The paper thus connects renormalization cocycles, Lyapunov exponents, and equidistribution methods to produce new instances of purely singular spectra in the non-constant-length substitution setting, advancing understanding of spectral types in these systems.
Abstract
A sufficient condition for a substitution automorphism to have pure singular spectrum is given in terms of the top Lyapunov exponent of the associated spectral cocycle. As a corollary, singularity of the spectrum is established for an infinite family of self-similar interval exchange transformations.