Self-Similar Fractals in Arithmetic

Birkhäuser Boston eBooks, 2000

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P-Adic Numbers, Ultrametric Analysis, and Applications, 2009

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings.

SIAM Journal on Computing, 2008

Self-similar fractals arise as the unique attractors of iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying an open set condition. Each point x in such a fractal F arising from an IFS S is naturally regarded as the "outcome" of an infinite coding sequence T (which need not be unique) over the alphabet Σ k = {0,. .. , k − 1}, where k is the number of contracting similarities in S. A classical theorem of Moran (1946) and Falconer (1989) states that the Hausdorff and packing dimensions of a self-similar fractal coincide with its similarity dimension, which depends only on the contraction ratios of the similarities. The theory of computing has recently been used to provide a meaningful notion of the dimensions of individual points in Euclidean space. In this paper, we use (and extend) this theory to analyze the dimensions of individual points in fractals that are computably self-similar, meaning that they are unique attractors of IFSs that are computable and satisfy the open set condition. Our main theorem states that, if F ⊆ R n is any computably self-similar fractal and S is any IFS testifying to this fact, then the dimension identities dim(x) = sdim(F) dim π S (T)

Experimental Mathematics, 2003

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Proceedings of Symposia in Pure Mathematics, 2004

We present an overview of a theory of complex dimensions of selfsimilar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several strands to discuss a possible approach to establishing a cohomological interpretation of the complex dimensions.

Arxiv preprint arXiv:0910.5014, 2009

Abstract: Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic Cantor sets is introduced by means of properly defining operators for the addition, subtraction, multiplication, and division. The new operators have the usual properties of the corresponding operations with real numbers. The combination of an infinitesimal change of fractal dimension with these ...

arXiv (Cornell University), 2021

We define the "coronas", which are especially spikey paths in the Farey graph going from 0 " p1, 0q to 8 " p0, 1q. We show that for R ě 2, px, yq P N`ˆN`, gcdpx, yq " 1, x`y ď R (is a corona. Preface For me, the greatest mystery of mathematics was André Weil's "Roseta Stone" [15], [17]: the analogies between number fields and function fields. Regarding the Riemann hypothesis, initially I followed Weil's approach [21] to Tate's thesis [14], viewing it as the harmonic analysis of the action of A ‹ K {K ‹ on A K {K ‹ (or on distribution on A K that are K ‹-invariant), and on P 1 pA k q {K ‹. This suggested that an (real valued) index-theorem, analogue of the Riemann-Roch for the associated surface, will give a proof of the Riemann Hypothesis along the lines of Weil's proof, see [4]. The formula [3] for Weil's explicit-sumsdistribution [20] was also the starting point for the program of Alain Connes and collaborates (cf. [1] appendix, where the formula of [3] is stated in an assymptotic form). But note that there is still not even a proof of Weil's Riemann-Hypothesis for a function field K [18] using the "non-commutative" space A K {K ‹ ! The mysterious analogy between number fields and function fields is clarified by the concept of generalized-ring (see [6] for a quick introduction). The language of generalized-rings can be used as the foundation of algebraic geometry in the style of Grothendieck (see [8]): • The final object of geometry is the absolute-point specpFq, where F is the initial object of generalized-rings, the "Field with one element" • The real R and complex C numbers, when viewed as (topological) generalized-rings, have (maximal compact topological)-sub-generalizedrings Z R Ď R, and Z C Ď C, (analogous to Z p Ď Q p); and specpZq, and specpO K q, K a number field, have natural compactifications.

2008

3 Self-similar sets and (semi)group actions 8 3. 11 Finitely presented dynamical systems and semi-Markovian spaces 57 12 Spectra of Schreier graphs and Hecke type operators 59 12. The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of " renormalization group " , which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. The notions of fractal and multi-fractal, playing an important role in singular geometry, measure theory and holomorphic dynamics, are also related. Self-similarity also appears in the theory of C *-algebras (for example in the representation theory of the Cuntz algebras) and in many other branches of mathematics. Starting from 1980 the idea of self-similarity entered algebra and began to exert great influence on asymptotic and geometric group theory. The aim of this paper is to present a survey of ideas, notions and results that are connected to self-simil...

Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpi´nski, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.