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Sabot to other fractals with blow-ups and the resulting associated renormalization map is then a multi-variable rational function on a complex projective space. The decimation method has been generalized by C. In the case of the bounded Sierpinski gasket, the renormalization map is a polynomial of one variable on the complex plane. In particular, we discuss the decimation method, which introduces a renormalization map whose dynamics describes the spectrum of the operator. more In this survey article, we investigate the spectral properties of fractal differential operators on self-similar fractals. In this survey article, we investigate the spectral properties of fractal differential operators. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. A closely related tool is the class of `tube zeta functions', defined using the tube function of a fractal set. more In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. In 2009, the first author introduced a class of zeta functions, called `distance zeta functions. Appealing to an analysis of these zeta functions allows for the development of theories of complex dimensions for bounded sets in Euclidean space, extending techniques and results regarding (ordinary) fractal strings obtained by the first author and van Fr. Specifically, we define box-counting zeta functions of infinite bounded subsets of Euclidean space and discuss results pertaining to distance and tube zeta functions.
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more We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of some Euclidean space of any dimension, including: the box-counting dimension and equivalent definitions based on various box-counting functions the similarity dimension via the Moran equation (at least in the case of self-similar sets) the order of the (box-)counting function the classic result on compact subsets of the real line due to Besicovitch and Taylor, as adapted to the theory of fractal strings and the abscissae of convergence of new classes of zeta functions. We discuss a number of techniques for determining the Minkowski dimension of bounded subsets of s. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise C^1-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets.
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Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the SierpiĆski is the limit of finite graphs consisting of various affine images of an equilateral triangle. more Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Noncommutative geometry provides a framework, via the construction of spectral triples, for the s.