Unitary groups and spectral sets

We study spectral theory for bounded Borel subsets of \br and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator \Ds = \frac1{2\pi i}\frac{d}{dx} with domain consisting of $C^\infty$ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding selfadjoint extensions of \Ds and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets $\Omega$ in \br^k such that $L^2(\Omega)$ has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to $\Omega$. In the general case, we characterize Borel sets $\Omega$ having this spectral property in terms of a unitary representation of (\br, +) acting by local translations. The case of $k = 1$ is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the selfadjoint extensions of the minimal operator \Ds. This allows for a direct and explicit interplay between geometry and spectra. As an application, we offer a new look at the Universal Tiling Conjecture and show that the spectral-implies-tile part of the Fuglede conjecture is equivalent to it and can be reduced to a variant of the Fuglede conjecture for unions of integer intervals.Comment: We improved the paper and partition it into several independent part

Martingales, endomorphisms, and covariant systems of operators in Hilbert space

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in symbolic dynamics. In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.Comment: 44 pages, LaTeX2e ("jotart" document class); v2: A few opening paragraphs were added to the paper; an addition where a bit of the history is explained, and where some more relevant papers are cited. Corrected a typographical error in Proposition 8.1. v3: A few minor additions: More motivation and explanations in the Intro; Remark 3.3 is new; and eleven relevant references/citations are added; v4: corrected and updated bibliography; v5: more bibliography updates and change of LaTeX document clas

Monic representations of the Cuntz algebra and Markov measures

We study representations of the Cuntz algebras \O_N. While, for fixed $N$, the set of equivalence classes of representations of \O_N is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of \O_N, that have a cyclic vector for the canonical abelian subalgebra. We show that \O_N has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic $S_i^*$-invariant space

Fourier duality for fractal measures with affine scales

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in \br^d, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$ in \br^d, and the other by a related subset $L$. Among other things, we show that there is then a pair of infinite discrete sets $\Gamma(L)$ and $\Gamma(B)$ in \br^d such that the $\Gamma(L)$-Fourier exponentials are orthogonal in $L^2(\mu_B)$, and the $\Gamma(B)$-Fourier exponentials are orthogonal in $L^2(\mu_L)$. These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line. Our duality pairs do not always yield orthonormal Fourier bases in the respective $L^2(\mu)$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.Comment: v