A note on spectral properties of random $S$-adic systems
Abstract
The paper is concerned with random $S$-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the Lyapunov exponent of the random walk on $SL(2,\mathbb{R})$ driven by the sequence of substitution matrices, then almost surely the spectrum of the $S$-adic $\mathbb{Z}$-action is singular with respect to any (fixed in advance) continuous measure. Finally, the appendix discusses the weak-mixing property for random $S$-adic systems associated to the family of substitutions introduced in Example 4.2.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.08884
- arXiv:
- arXiv:2403.08884
- Bibcode:
- 2024arXiv240308884S
- Keywords:
-
- Mathematics - Dynamical Systems;
- 37A30;
- 37B10
- E-Print:
- Appendix was added, written by Pascal Hubert and Carlos Matheus, in which they sketch a proof of almost sure weak-mixing for the random $S$-adic $\mathbb{Z}$-action under some natural assumptions, the key ones being two positive Lyapunov exponents and Zariski density of the associated group of matrices. This is shown to imply weak-mixing of the system in Example 4.2 under an arithmetic condition