The Almost Sure Essential Spectrum of the Doubling Map Model is Connected

Abstract

We consider discrete Schrödinger operators on the half line with potentials generated by the doubling map and continuous sampling functions. We show that the essential spectrum of these operators is always connected. This result is obtained by computing the subgroup of the range of the Schwartzman homomorphism associated with homotopy classes of continuous maps on the suspension of the standard solenoid that factor through the suspension of the doubling map and then showing that this subgroup characterizes the topological structure of the spectrum.

Appendix A. Strong Approximation and the Almost-Sure Spectrum

We end with a short appendix about approximating the almost-sure spectrum via periodic points. Since this is general and does not depend on using a particular base dynamical system, we formulate it in the general setting.

Suppose $\Omega$ is a compact metric space, $T:\Omega \rightarrow \Omega$ is a homeomorphism, $\mu$ is a T-ergodic Borel probability measure with , and $f \in C(\Omega ,{{\mathbb {R}}})$. For each $\omega \in \Omega$, define $V_\omega (n) = f(T^n \omega )$, and consider the operators $H_\omega = \Delta + V_\omega$. By general results, there is a fixed compact set $\Sigma \subseteq {{\mathbb {R}}}$ with $\Sigma = \sigma (H_\omega )$ for $\mu$-a.e. $\omega \in \Omega$ [16].

As usual, we say that $\omega \in \Omega$ is a periodic point of T if $T^p\omega = \omega$ for some $p \in \mathbb {N}$, and we denote by $\textrm{Per}(T)$ the set of periodic points of T.

Theorem A.1

Let $\Omega$, T, $\mu$, and f be as above.

• (a) $\Sigma = \sigma (H_\omega )$ for any $\omega$ with a dense T-orbit.

• (b) One has $\sigma (H_\omega ) \subseteq \Sigma$ for every $\omega \in \Omega$.

• (c) If $\textrm{Per}(T)$ is dense in $\Omega$, then

  $$\begin{aligned} \Sigma = \overline{\bigcup _{\omega \in \textrm{Per}(T)} \sigma (H_\omega )} \end{aligned}$$

  (A.1)

Proof

If $\omega ,\omega ' \in \Omega$ are such that the closure of the T-orbit of $\omega$ contains $\omega '$, then there are $n_k \in {{\mathbb {Z}}}$ with $T^{n_k}\omega \rightarrow \omega '$. Since f is continuous, it follows by strong approximation (e.g. [16, Corollary 1.4.22]) that

$$\begin{aligned} \sigma (H_{\omega '}) \subseteq \sigma (H_\omega ). \end{aligned}$$

(A.2)

The assumption implies that $\mu$-a.e. $\omega \in \Omega$ has a dense T-orbit, so (a) and (b) follow immediately. To prove (c), note that the inclusion $\supseteq$ follows immediately from (b) (and the fact that the spectrum is closed). The other inclusion follows by strong approximation in a similar way: this time, approximate a general $\omega$ by periodic points and then again apply [16, Corollary 1.4.22]. $\square$

Remark A.2

There is an analogous statement for half-line operators, which can be applied in the case in which T is merely continuous and not a homeomorphism, for example when T is the doubling map.