Fourier bases and Fourier frames on self-affine measures

This paper gives a review of the recent progress in the study of Fourier bases and Fourier frames on self-affine measures. In particular, we emphasize the new matrix analysis approach for checking the completeness of a mutually orthogonal set. This method helps us settle down a long-standing conjecture that Hadamard triples generates self-affine spectral measures. It also gives us non-trivial examples of fractal measures with Fourier frames. Furthermore, a new avenue is open to investigate whether the Middle Third Cantor measure admits Fourier frames

Wavelets and graph C∗C^*-algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph $C^{\ast}$-algebras, including the higher-rank graph $C^*$-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In \cite{FGKP}, we introduced the "cubical wavelets" associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of \cite{hammond} to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs

Fractals based on Harmonic Wavelets

In this paper a simple algorithm based on harmonic wavelets is given for the generation of self similar functions. Due to their self similarity property and scale dependence, harmonic wavelets might offer a good approximation of fractals by a very few instances of the wavelet series and a more direct interpretation of the scale invariance for deterministic localized fractals

On the Exel crossed product of topological covering maps

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ∗-algebras C(X) ⋊α,L N introduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical imbedding of C(X) into C(X)⋊α,LN is a maximal abelian C ∗-subalgebra of C(X)⋊α,LN; any nontrivial two sided ideal of C(X) ⋊α,L N has non-zero intersection with the imbedded copy of C(X); a certain natural representation of C(X) ⋊α,L N is faithful. This result is a generalization to noninvertible dynamics of the corresponding results for crossed product C ∗-algebras of homeomorphism dynamical systems.