Lyapunov exponents for binary substitutions of constant length

Abstract

A method of confirming the absence of absolutely continuous diffraction via the positivity of Lyapunov exponents derived from the corresponding Fourier matrices is presented, which provides an approach that is independent of previous results on the basis of Dekking's criterion. This yields a positive result for all constant length substitutions on a binary alphabet which are primitive and aperiodic.Comment: 12 page

Binary Constant-Length Substitutions and Mahler Measures of Borwein Polynomials

“…As stated in Theorem 1.1 in the Introduction, the maximal Lyapunov exponent can be written as a logarithmic Mahler measure. We prove this as Theorem 4.2 below, which is a more detailed version of Theorem 1.1; compare [33]. In general, the Fourier matrix of ̺ satisfies…”

“…In particular, by measure-theoretic arguments, one can conclude that, if |χ B max | < log √ L, the diffraction measure, for an arbitrary choice of weights, never has an absolutely continuous component. We refer to the literature for further details, namely to [33] for the binary constant length case, and to [4,10] for an appropriate extension to a family of non-Pisot substitutions, via the corresponding inflation tiling. ♦…”

Renormalisation of Pair Correlation Measures for Primitive Inflation Rules and Absence of Absolutely Continuous Diffraction

“…where the first estimate follows from Jensen's inequality and is strict, as is the second because β j is not a monomial in u. The last equality is Parseval's identity; compare [43]. The diffraction measures of constant-length substitutions are closely related to the spectral measures of characteristic functions.…”

“…The IDA of the induced inflation rule is irreducible, which has an interesting consequence on the way the AC spectrum is encoded in the Fourier matrix cocycle. In particular, we do no longer have a k-independent subspace with unitary dynamics as in the original version [3,43], but a k-dependent equivariant family. This would deserve further exploration, in particular via extending some results in this direction from [21].…”

“…In this paper, we develop the renormalisation approach for the geometric setting of primitive substitutions in more generality, building on previous work on several classes of examples [3,43,8]. Moreover, we extend the approach to inflation tilings of finite local complexity (FLC) in Euclidean space of arbitrary dimension.…”