Self-similar p-adic fractal strings and their complex dimensions

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. The text was submitted by the authors in English.     

Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

In a previous paper by the first two authors, a tube formula for fractal sprays was obtained which also applies to a certain class of self-similar fractals. The proof of this formula uses distributional techniques and requires fairly strong conditions on the geometry of the tiling (specifically, the inner tube formula for each generator of the fractal spray is required to be polynomial). Now we extend and strengthen the tube formula by removing the conditions on the geometry of the generators, and also by giving a proof which holds pointwise, rather than distributionally. Hence, our results for fractal sprays extend to higher dimensions the pointwise tube formula for (1-dimensional) fractal strings obtained earlier by Lapidus and van Frankenhuijsen.Our pointwise tube formulas are expressed as a sum of the residues of the “tubular zeta function” of the fractal spray in Rd. This sum ranges over the complex dimensions of the spray, that is, over the poles of the geometric zeta function of the underlying fractal string and the integers 0,1,…,d. The resulting “fractal tube formulas” are applied to the important special case of self-similar tilings, but are also illustrated in other geometrically natural situations. Our tube formulas may also be seen as fractal analogues of the classical Steiner formula.     

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

Spectral zeta functions of fractals and the complex dynamics of polynomials

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

     

I.I.D. STATISTICAL CONTRACTION OPERATORS AND STATISTICALLY SELF－SIMILAR SETS

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.     

A note on fractal strings and spacetime

A short discussion is presented on the role of Mersenne primes and even perfect numbers in fractal strings, Cantorian-fractal spacetime, spin structures on Riemann surfaces and the classification of consistent string theories. Since in principle the number of Mersenne primes should be infinite this entails that the family of consistent quantum string theories in higher dimensions than 26 could be infinite as well and, consequently, this could be the physical reason why the number of quantum-spacetime dimensions is infinite. The Tsallis entropy of fractal strings is constructed followed by the exact expression for the fractal string mass spectrum.     

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.     

随机分形

本文概括了随机分形的主要结果，综述了随机分形的最新进展和目前的动态，提出了一些末解决的问题，全文共分为三部分：（１）由随机过程和随机场（如Ｌｅｖｙ过程，Ｇａｕｓｓ场，自相似过程等）产生的各种随机分形集（如象集、水平集、Ｋ重点集等）的Ｈａｕｓｄｏｒｆｆ维数、测度和ｐａｃｋｉｎｇ维数、测试；（２）随机Ｃａｎｔｏｒ型集和统计自相似集的维数和测试；（３）分形集（如Ｓｐｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ，ｎｅ     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.     

Fractal strings as an alternative justification for El Naschie''s cantorian spacetime and the fine structure constant

Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's cantorian-fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, p prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature.     

正实数维分形集的构造方法

给出了测量一类分形集维数的简单方法. 根据这种测量方法, 可以构造出任意实数维分形集,并且分形集可以不是自相似的.     

控制系统中的分形

本文将整数维与分形的Hausdorff测度引入并应用于控制系统,同时也介绍了准自相似集这个新概念,证明了这种集合的存在性与唯一性.并将计算自相似集维数的公式推广到准自相似集,在此基础上,说明了控制系统的可达集可以具有分数维.表明在分析非线性系统可控性与可观性时,分形几何学也将是一种有意义的工具.     

Microscopic model of a non-Debye dielectric relaxation: The Cole-Cole law and its generalization

Based on a self-similar spatial-temporal structure of the relaxation process, we construct a microscopic model for a non-Debye (nonexponential) dielectric relaxation in complex systems. In this model, we derive the Cole-Cole expression for the complex dielectric permittivity and show that the exponent ?? involved in that expression is equal to the fractal dimension of the spatial-temporal self-similar ensemble characterizing the structure of the medium and the relaxation process occurring in it. We find a relation between the macroscopic relaxation time and the micro- and mesoparameters of the system. We obtain a generalized Cole-Cole expression for the complex dielectric permittivity involving log-periodic corrections that occur because of a discrete scaling invariance of the fractal structure generating the relaxation process on the mesoscopic scale. The found expression for the dielectric permittivity can be used to interpret dielectric spectra in disordered dielectrics.     

Fractal space,cosmic strings and spontaneous symmetry breaking

The present paper is conceived within the framework of El Naschie's fractal-Cantorian program and proposes to develop a model of the fractal properties of spacetime. We show that, starting from the most fundamental level of elementary particles and rising up to the largest scale structure of the Universe, the fractal signature of spacetime is imprinted onto matter and fields via the common concept for all scales emanating from the physical spacetime vacuum fluctuations. The fractal structure of matter, field and spacetime (i.e. the nature and the Universe) possesses a universal character and can encompass also the well-known geometric structures of spacetime as Riemannian curvature and torsion and includes also, deviations from Newtonian or Einsteinian gravity (e.g. the Rössler conjecture). The leitmotiv of the paper is generated by cosmic strings as a fractal evidence of cosmic structures which are directly related to physical properties of a vacuum state of matter (VSM). We present also some physical aspects of a spontaneous breaking of symmetry and the Higgs mechanism in their relation with cosmic string phenomenology. Superconducting cosmic strings and the presence of cosmic inhomogeneities can induce to cosmic Josephson junctions (weak links) along a cosmic string or in connection with a cosmic string (self) interactions and thus some intermittency routes to a cosmic chaos can be explored. The key aspect of fractals in physics and of fractal geometry is to understand why nature gives rise to fractal structures. Our present answer is: because a fractal structure is a manifestation of the universality of self-organisation processes, as a result of a sequence of spontaneous symmetry breaking (SSB). Our conclusion is that it is very difficult to prescribe a certain type of fractal within an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter will decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not, but consider on the other hand a space far removed from usual matter, or a space containing isolated small particles in which a very low density matter can exist. Very low density matter might be influenced by a fractal structure of space, for example in the sense that it is subject also to fluctuations structured by random fractals. Diffraction and diffusion experiments in an empty space and very low density matter could provide evidence of a fractal structure of space. However, at very high (Planck) densities, and a spacetime in which fluctuations represent also the source of matter and fields (which is very resonable within the context of a quantum gravity), we can assert that Einstein's dream of geometrising physics and El Naschie's hope to prove the fractalisation (or Cantorisation) of spacetime are fully realised.     

FRACTAL PROPERTIES OF POLAR SETS OF RANDOM STRING PROCESSES

This paper studies fractal properties of polar sets for random string processes. We give upper and lower bounds of the hitting probabilities on compact sets and prove some sufficient conditions and necessary conditions for compact sets to be polar for the random string process. Moreover, we also determine the smallest Hausdorff dimensions of non-polar sets by constructing a Cantor-type set to connect its Hausdorff dimension and capacity.     

Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale. Project supported by the National Natural Science Foundation of China (Grant Nos. 59771050 and 59872004) and the Foundation Fund of Ministry of Metallurgical Industry.     

Upper Limitation of Kolmogorov Complexity and Universal P. Martin-Löf Tests

In this paper we study the Kolmogorov complexity of initial strings in infinite sequences (being inspired by [9]), and we relate it with the theory of P. Martin-Lof random sequences. Working with partial recursive functions instead of recursive functions we can localize on an apriori given recursive set the points where the complexity is small. The relation between Kolmogorov's complexity and P. Martin-Lof's universal tests enables us to show that the randomness theories for finite strings and infinite sequences are not compatible (e.g.because no universal test is sequential).We lay stress upon the fact that we work within the general framework of a non-necessarily binary alphabet.     

星积分形曲面及其维数

通过分形曲线定义了一类分形曲面(被称为星积分形曲面),讨论了这类分形曲面的分形维数,得出了分形曲线的维数与它们所构造出的分形曲面维数之间的关系。     

Generating hierarchial scale-free graphs from fractals

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabási, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal Λ. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal Λ we generate random graph sequence sharing similar properties.