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.
Bibliographical noteFunding Information:
We are deeply grateful to Mike Hochman for fruitful discussions of equidistribution results, to Adam Kanigowski for helpful comments, to Nir Lev for the reference to the paper by R. Goldberg, to Misha Sodin, Rotem Yaari, and the anonymous referee for valuable remarks which helped improve the article. The research of A. Bufetov has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 647133 (ICHAOS). A. Bufetov has also been funded by ANR grant ANR-18-CE40-0035 REPKA. The research of B. Solomyak was supported by the Israel Science Foundation (Grant 911/19). B.S. gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, where some of the research for this paper was done.
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
- Singular spectrum
- Spectral cocycle
- Substitution dynamical system