Cantor-like Sets Research Articles

Stochastic Wave Equations Defined by Fractal Laplacians on Cantor-Like Sets

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the resolvent density. Afterwards, we establish existence and uniqueness of mild solutions to stochastic wave equations provided some Lipschitz and linear growth conditions. We prove H¨older continuity in space and time and compute the Hölder exponents. Moreover, we are concerned with the phenomenon of weak intermittency.

LEBESGUE DENSITY AND STATISTICAL CONVERGENCE

The paper presents a generalization of the density point’s notion to the ideal-convergence framework. For an ideal I⊆P(ℕ) (with Fin⊆I), Lebesgue measurable set A⊆ℝ we introduce a definition of a density point of A with respect to I; we prove that the classical approach fits into this generalization (Theorem 4); we construct a family of Cantorlike sets showing that Lebesgue Density Theorem cannot be maximally improved in this direction (Theorem 8).

Hausdorff measures, dyadic approximations, and the Dobiński set

The Dobiński set D is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of D by means of the mass transference principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in D.

Fractal Stochastic Processes on Thin Cantor-Like Sets

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

Random Variables and Stable Distributions on Fractal Cantor Sets

In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results.

Sumudu transform in fractal calculus

The Cη-Calculus includes functions on fractal sets, which are not differentiable or integrable using ordinary calculus. Sumudu transforms have an important role in control engineering problems because of preserving units, the scaling property of domains, easy visualization, and transforming linear differential equations to algebraic equations that can be easily solved. Analogues of the Laplace and Sumudu transforms in Cη-Calculus are defined and the corresponding theorems are proved. The generalized Laplace and Sumudu transforms involve functions with totally disconnected fractal sets in the real line. Linear differential equations on Cantor-like sets are solved utilizing fractal Sumudu transforms. The results are summarized in tables and figures. Illustrative examples are solved to give more details.

Dirichlet sets vs characterized subgroups

A subset A of the circle group T is a Dirichlet set if there exists an increasing sequence u=(un)n∈N0 in N such that ‖unx‖→0 uniformly on A. In particular, A is contained in the subgroup tu(T):={x∈T:‖unx‖→0}, which is the subgroup of T characterized by u.Using strictly increasing sequences u in N such that un divides un+1 for every n∈N, we find in T a family of closed perfect D-sets that are also Cantor-like sets. Moreover, we write T as the sum of two closed perfect D-sets. As a consequence, we solve an open problem by showing that T can be written as the sum of two of its proper characterized subgroups, i.e., T is factorizable. Finally, we describe all countable subgroups of T that are factorizable and we find a class of uncountable characterized subgroups of T that are factorizable.

On spectra of probability measures generated by GLS-expansions

We study properties of distributions of random variables with independent identically distributed symbols of generalized L\"{u}roth series (GLS) expansions (the family of GLS-expansions contains L\"{u}roth expansion and $Q_{\infty}$- and $G_{\infty}^2$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[\mathit{GLS},V]$ consisting of real numbers whose GLS-expansions contain only symbols from some countable set $V\subset N\cup\{0\}$, and derive exact formulae for the determination of the Hausdorff--Besicovitch dimension of $C[\mathit{GLS},V]$. Based on these results, we get general formulae for the Hausdorff--Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.

Assouad dimensions of Moran sets and Cantor-like sets

We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set introduced in [J. Math. Anal. Appl., 2015, 432: 888–917], we also study the Assouad dimensions of Cantor-like sets.

On the equivalence of weak quasisymmetry and quasisymmetry on non-connected sets

This paper studies the equivalence of the quasisymmetric mappings on non-connected sets. We introduce a generalized form of weak quasisymmetry and prove that, on a uniformly perfect metric space, a generalized weakly quasisymmetric mapping is quasisymmetric. We further improve this result on Cantor-like sets satisfying the small gap condition.

Edges of Saturn's rings are fractal.

The images recently sent by the Cassini spacecraft mission (on the NASA website http://saturn.jpl.nasa.gov/photos/halloffame/) show the complex and beautiful rings of Saturn. Over the past few decades, various conjectures were advanced that Saturn’s rings are Cantor-like sets, although no convincing fractal analysis of actual images has ever appeared. Here we focus on four images sent by the Cassini spacecraft mission (slide #42 “Mapping Clumps in Saturn’s Rings”, slide #54 “Scattered Sunshine”, slide #66 taken two weeks before the planet’s Augus’t 200’9 equinox, and slide #68 showing edge waves raised by Daphnis on the Keeler Gap) and one image from the Voyager 2’ mission in 1981. Using three box-counting methods, we determine the fractal dimension of edges of rings seen here to be consistently about 1.63 ~ 1.78. This clarifies in what sense Saturn’s rings are fractal.

Multifractal behavior of polynomial Fourier series

We study the spectrum of singularities of a family of Fourier series with polynomial frequencies, in particular we prove that they are multifractal functions. The case of degree two was treated by S. Jaffard in 1996. Higher degrees require completely different ideas essentially because harmonic analysis techniques (Poisson summation) are useless to study the oscillation at most of the points. We introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the Hölder exponent in thin Cantor-like sets.

THE CORRELATION DIMENSION OF GENERALIZED CANTOR-LIKE SETS

In the paper, a symbolic construction is considered to define generalized Cantor-like sets. Lower and upper bounds for the correlation dimension of the sets with a regular condition are obtained with respect to a probability Borel measure. Especially, for some special cases of the sets, the exact formulas of the correlation dimension are established and we show that the correlation dimension and the Hausdorff dimension of some of them are the same. Finally, we find a condition which guarantees the positive correlation dimension of the generalized Cantor-like sets.

The Structure of Disjoint Groups of Continuous Functions

Let I be an open interval. We describe the general structure of groups of continuous self functions on I which are disjoint, that is, the graphs of any two distinct elements of them do not intersect. Initially the class of all disjoint groups of continuous functions is divided in three subclasses: cyclic groups, groups the limit points of their orbits are Cantor‐like sets, and finally those the limit points of their orbits are the whole interval I. We will show that (1) each group of the second type is conjugate, via a specific homeomorphism, to a piecewise linear group of the same type; (2) each group of the third type is a subgroup of a continuous disjoint iteration group. We conclude the Zdun′s result on the structure of disjoint iteration groups of continuous functions as special case of our results.

Badly approximable numbers and vectors in Cantor-like sets

We show that a large class of Cantor-like sets of R-d, d >= 1, contains uncountably many badly approximable numbers, respectively badly approximable vectors, when d >= 2. An analogous result is also proved for subsets of R-d arising in the study of geodesic flows corresponding to (d+1)-dimensional manifolds of constant negative curvature and finite volume, generalizing the set of badly approximable numbers in R. Furthermore, we describe a condition on sets, which is fulfilled by a large class, ensuring a large intersection with these Cantor-like sets.

Cantor Groups

SummaryThe Cantor subset of the unit interval [0, 1) is large in cardinality and also large algebraically, that is, the smallest subgroup of [0, 1) generated by the Cantor set (using addition mod 1 as the group operation) is the whole of [0, 1). In this paper, we show how to construct Cantor-like sets which are large in cardinality but small algebraically. In fact for the set we construct, the subgroup of [0, 1) that it generates is, like the Cantor set itself, nowhere dense in [0, 1).

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set Ef of B-points is everywhere dense and has the maximal Hausdorff dimension, dim H(Ef ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles.

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar N-adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first N quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed.

Quantization dimension of probability measures supported on Cantor-like sets

Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied.

The Ostrogradsky series and related Cantor-like sets

We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}= &=\sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)...(g_1+g_2+...+g_n)}\equiv \bO1(g_1,g_2,...,g_n,...), \end{align*} where $q_{n+1}>q_n\in\N$, $g_1=q_1$, $g_{k+1}=q_{k+1}-q_k$. We compare this representation with the corresponding one in terms of continued fractions. We establish basic metric relations (equalities and inequalities for ratios of the length of cylindrical sets). We also compute the Lebesgue measure of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the $\bO1$-representation. In particular, the conditions for the set $\Cset{V}$, consisting of real numbers whose $\bO1$-symbols take values from the set $V \subset N$, to be of zero resp. positive Lebesgue measure are found. For a random variable $\xi$ with independent $\bO1$-symbols $g_n(\xi)$ we prove the theorem establishing the purity of the distribution. In the case of singularity the conditions for such distributions to be of Cantor type are also found.