A spectral cocycle for substitution systems and translation flows
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ALEXANDER I. BUFETOV AND BORIS SOLOMYAK Dedicated to Larry Zalcman, with admiration and gratitude Abstract. For substitution systems and translation flows, a new cocycle, which we call the 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.
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Introduction
This paper is devoted to the spectral theory of substitution systems and translation flows and continues the work started in [17, 18]. We focus on the local properties of spectral measures, such as local dimension, H¨older property, and closely related questions of singularity and absolute continuity. Our main construction is that of a new cocycle, which we call the spectral cocycle. This spectral cocycle is related to our earlier work, in particular, to the matrix Riesz products, used in [17] to obtain H¨older continuity of spectral measures for typical suspension flows over non-Pisot substitution systems. Our cocycle is defined over a skew product whose base is a shift transformation on a symbolic space arising from the realization of the Teichm¨uller flow and the fibre is a torus of dimension equal to the number of intervals in the associated interval exchange. Our main result, see (3.7), (3.13) below, is a formula relating the pointwise dimension of the spectral measure and the pointwise Lyapunov exponent of the cocycle. As a corollary, we obtain an inequality for the Lyapunov exponent at almost every point and a sufficient condition for singularity of the spectrum. Analogous results are obtained for suspension flows over S-adic systems, including classical substitution systems as a special case. 165 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0127-2
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A. I. BUFETOV AND B. SOLOMYAK
In the Appendix, which is independent of the rest of the paper, we explain how a modification of the argument from [18] proves H¨older continuity of spectral measures for typical translation flows in the stratum H(1, 1). Added in Proof. Further progress occurred after the paper was completed. Recently Forni [26], motivated by our work [18], obtained H¨older estimates for spectral measures in the case of surfaces of arbitrary genus. While Forni does not use the symbolic formalism, the main idea of his approach can also be formulated in symbolic terms: namely, that instead of the scalar estimates of [18] and the ˝s–Kahane argument in vector Appendix of the current paper, we can use the Erdo form. Following the idea of Forni and directly using the vector form of the ˝s–Kahane argument yields a considerable simplification of our initial proof Erdo and allowed us to prove the H¨older property for a general class of random Markov compacta, cf. [15], and, in particular, for almost all translation flows on surfaces of arbitrary genus, see [19]. In a different direction, our preprint [20] gives a sufficient condition for a substitution Z-act
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