On substitution automorphisms with pure singular spectrum

Abstract

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.

Appendix: On the Mahler measure of a polynomial

Mahler [30] defined the measure of a polynomial in several variables as follows:

$$\begin{aligned} M(P):= \exp \int _{{{\mathbb {T}}}^n} \log |P(z_1,\ldots ,z_n)| \,d\mathbf{t}, \end{aligned}$$

where $\mathbf{t}= (t_1,\ldots ,t_n)$ and $z_j = e^{2\pi i t_j}$. Now M(P) is called the Mahler measure and

$$\begin{aligned} {{\mathfrak {m}}}(P) = \log M(P) \end{aligned}$$

(A.1)

is the logarithmic Mahler measure of the polynomial P. For a polynomial of a single variable p(z), Jensen’s formula yields

$$\begin{aligned} M(p) = \exp \int _0^1 \log |p(e^{2\pi i t}|\,dt = |a_0|\cdot \prod _{j\ge 1} \max \{|\alpha _j|,1\}, \end{aligned}$$

(A.2)

where $a_0$ is the leading coefficient and $\alpha _j$ are the complex zeros of p. Observe that $M(p)\ge |a_0|$, hence writing

$$\begin{aligned} P(z_1,\ldots ,z_n) = a_0(z_1,\ldots ,z_{n-1}) z_n^k + \cdots + a_k(z_1,\ldots ,z_{n-1}) \end{aligned}$$

and integrating $\log |P(z_1,\ldots ,z_n)|$ with respect to $t_n$, we obtain $M(P)\ge M(a_0)$, and then by induction it follows that

$$\begin{aligned} \textit{if}\ P(z_1,\ldots ,z_n)\ \textit{has integer coefficients, then}\ M(P)\ge 1, \end{aligned}$$

(A.3)

see e.g., [8, p. 117] and [21] for details and for additional information on the Mahler measure.