The Lyapunov exponent plays a key role in the
The Lyapunov exponent, plays a key role in the spectral analysis of the family . Here, S denotes the so-called Szegő transfer matrix, defined by Because the matrices so defined satisfy for , we have for . We define By general considerations, there is a compact set with for μ-a.e. . Moreover, by the Combes–Thomas estimate, one has . We are interested in spectral approximation of CMV operators, so the following questions are natural: In particular, helpful answers to both questions can allow one to bootstrap information about spectra of periodic approximants into information about the spectra of more exotic ergodic operators, using as an intermediary. Additionally, provides information about the absolutely continuous spectrum: one has for almost every ω by Kotani Theory, where denotes the essential closure of .
Spectral approximation of ergodic CMV matrices Let us start off by defining a version of the Gesztesy–Zinchenko (GZ) formalism . We remark that in our paper, we follow Simon's conventions ,  regarding how depends on the 's. Note that these conventions differ from the notation used in . We will use that any CMV operator enjoys a factorization into direct sums of unitaries of the form That is, we have where and the Θ matrix corresponding to acts on coordinates n and . Given a solution u of , we define . One can check that and hence (since each is real-symmetric). Using the equation , we have which can be rearranged to yield Similarly, using , we can deduce and get Thus, we have where For later use, we notice that (2.3) can also be inverted to yield The first technical lemma is a formula for the derivative of the nth band function in terms of associated Bloch wave solutions. We first introduce suitable truncations of whose eigenvectors can be used to generate Bloch waves. Suppose that is q-periodic; throughout this Protionamide section, we also assume that q is even. We define and as in [30, Equation (11.2.7)], that is: Then, we define and the dual operator and let denote an enumeration of the eigenvalues of . Observe that and have the same set of eigenvalues. This is evident by looking at the proof of Lemma 2.2 of ; one can also see that they are unitarily equivalent viz. . The eigenvectors of generate Bloch wave solutions to the difference equation . Concretely, let denote a normalized eigenvector of corresponding to the eigenvalue . We may extend to all , obtaining a solution of with We then define . One can check that solves the dual equation . As with u, we can extend to a globally defined solution of with
With Lemma 2.2 in hand, the main technical challenges have been dealt with. At this point, one can prove Theorem 1.1 in the same way that Last proves [23, Theorem 1]. We provide a short sketch for the reader's benefit. Please consult Section 11.2 of  for a helpful discussion of Floquet theory for CMV operators.
Spectral approximation of limit-periodic CMV matrices In this section, we will collect a handful of facts about the spectra of limit-periodic CMV matrices and their periodic approximants. Throughout, for a subset , we denote The Hausdorff distance between two compact sets is defined by The fundamental fact driving the analysis in this section is the following By way of Lemma 3.1, we can use the rate function of a limit-periodic operator to control the rate of spectral convergence of the periodic approximants. The following lemmata are classical, but their proofs are short, so we reproduce them for the sake of completeness:
Recall that for the special almost-periodic class of ergodic CMV matrices, the underlying probability space Ω is a compact monothetic group with translation T by a topological generator. In this case the action of the transformation T manifests as a shift on the Verblunsky coefficients α; since T is a topological generator for Ω, we call Ω the “shift-hull” of the almost-periodic CMV matrix in this case.