Let \(\mathcal{A}\) be an alphabet with \(d\geq2\) symbols. Let us denote by \(\mathcal{A}^+\) the set of non-empty finite words with letters in \(\mathcal{A}\).
A substitution \(\zeta\) is a self-map of \(\mathcal{A}^+\) that satisfies \(\zeta(a_1\dots a_n)=\zeta(a_1)\dots\zeta(a_n)\) for any word \(a_1\dots a_n\), where the \(a_i\)’s are letters in \(\mathcal{A}\). In particular, such a map is completely determined by the images of words with one letter.
The substitution space \(X_\zeta\) is the subspace of \(\mathcal{A}^\mathbb{Z}\) given by bi-infinite sequences such that any subword appears as a subword of some \(\zeta^n(a)\) for some \(a\in\mathcal{A}\) and \(n\in\mathbb{N}\). This subspace is shift-invariant and this gives the substitution dynamical system \((X_\zeta,T)\), where \(T\) is the shift.
The substitution matrix \(S = (S(i, j))\) is the \(d\times d\) matrix, such that \(S(i, j)\) is the number of symbols \(i\) in \(\zeta(j)\) where \(\mathcal{A}=\{1,\dots,d\}\). The substitution is primitive if this matrix has all entries strictly positive for some \(n\in\mathbb{N}\).
It is known that if the substitution is primitive, the system \((X_\zeta,T)\) is uniquely ergodic (see [M. Queffélec, Substitution dynamical systems. Spectral analysis. 2nd ed. Dordrecht: Springer (2010; Zbl 1225.11001)]). So one can consider the unitary operator of \(L^2(X_\zeta,\mu)\) associated to \(T\) where \(\mu\) is the spectrum of the unique invariant measure on \(X_\zeta\).
The main result of this paper is that the system has pure singular spectrum when the substitution is primitive aperiodic, \(S\) is irreducible and the logarithm of its Perron-Frobenius eigenvalue is strictly larger than twice the top Lyapunov exponent of the spectral cocycle.