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Since its inception in the 1970s at the hands of Feigenbaum and, independently, Coullet and Tresser the study of renormalization operators in dynamics has been very successful at explaining universality phenomena observed in certain families of dynamical systems. The first proof of existence of a hyperbolic fixed point for renormalization of area-preserving maps was given by Eckmann et al. (Mem Am Math Soc 47(289):vi+122, 1984). However, there are still many things that are unknown in this setting, in particular regarding the invariant Cantor sets of infinitely renormalizable maps. In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in a Lipschitz curve but never contained in a smooth curve. This extends previous results by de Carvalho, Lyubich and Martens about strongly dissipative maps of the plane close to unimodal maps to the area-preserving setting. The method used for constructing the Lipschitz curve is very similar to the method used in the dissipative case but proving the nonexistence of smooth curves requires new techniques. [ABSTRACT FROM AUTHOR]

We consider a minimal equicontinuous action of a finitely generated group G on a Cantor set X with invariant probability measure μ, and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup H of G such that the set of points in X whose stabilizers are conjugate to H has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions. [ABSTRACT FROM AUTHOR]

Let N ⩾ 2 and ρ ∈ (0, 1/N2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system Let s = dimH E be the Hausdorff dimension of E, and let be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density and upper s-density Θ∗s(μ, x) of μ at x. This extends some early results of Feng et al (2000 J. Math. Anal. Appl. 250 692–705). Furthermore, we determine two critical values ac and bc for the sets respectively, such that dimH E*(a) > 0 if and only if a < ac, and that dimH E*(b) > 0 if and only if b > bc. We emphasize that both values ac and bc are related to the Thue–Morse type sequences, and our strategy to find them relies on ideas from open dynamical systems and techniques from combinatorics on words. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$ -Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$ -graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by $$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$ where $\unicode[STIX]{x1D6EF}$ is the space of choice functions on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$. There are two ultrametrics, $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ , on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ which make the infinite path space $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of the Laplace–Beltrami operator $\unicode[STIX]{x1D6E5}_{s}$ associated to $Q_{s}$ , and the latter $d_{w_{\unicode[STIX]{x1D6FF}}}$ is associated to a weight function $w_{\unicode[STIX]{x1D6FF}}$ on ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , where $\unicode[STIX]{x1D6FF}\in (0,1)$. We show that the Perron–Frobenius measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ has the volume-doubling property with respect to both $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ and we study the asymptotic behavior of the heat kernel associated to $Q_{s}$. Moreover, we show that the Dirichlet form $Q_{s}$ coincides with a Dirichlet form ${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$ which is associated to a jump kernel $J_{s}$ and the measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ , and we investigate the asymptotic behavior and moments of displacements of the process. [ABSTRACT FROM AUTHOR]

Given a positive integer M and a real number x ∈ (0 , 1 ] , we call q ∈ (1 , M + 1 ] a univoque simply normal base of x if there exists a unique simply normal sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i. Similarly, a base q ∈ (1 , M + 1 ] is called a univoque irregular base of x if there exists a unique sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i and the sequence (d i) has no digit frequency. Let U S N (x) and U I r (x) be the sets of univoque simply normal bases and univoque irregular bases of x , respectively. In this paper we show that for any x ∈ (0 , 1 ] both U S N (x) and U I r (x) have full Hausdorff dimension. Furthermore, given finitely many rationals x 1 , x 2 , ... , x ℓ ∈ (0 , 1 ] so that each x i has a finite expansion in base M + 1 , we show that there exists a full Hausdorff dimensional set of q ∈ (1 , M + 1 ] such that each x i has a unique expansion in base q. [ABSTRACT FROM AUTHOR]

Descriptive set theory is a classical branch of mathematics formed at the beginning of the last century. We offer the basics of descriptive chaos theory. It is shown that if the topological entropy is positive, then a dynamical system: (1) has many different trajectory attractors, namely, a continuum of attractors; (2) the basins of most attractors have an overly complex structure, namely, are sets of the third class according to the terminology of descriptive set theory; (3) the basins of different attractors are too strongly intertwined and they cannot be separated from each other by any open or closed sets, but only by sets of the second complexity class, and (4) the set of all attractors of the dynamical system forms an attractor network (grid) in the space of closed sets of the state space (with the Hausdorff metric), the cells of which are created by Cantor sets from the attractors themselves. [ABSTRACT FROM AUTHOR]

In this paper we establish a topological property of geometric objects (lines, surfaces and solids) called Euler-Poincaré characteristic. Since the paper is intended for a large profile of mathematics teachers, our approach is entirely intuitive and majority of readers can omit two addenda whose understanding requires a solid knowledge of topology. E-P characteristic is an integer which we calculate here decomposing lines into fibers being finite sets of points, surfaces into fibers being lines and solids into fibers being surfaces. When an object is subjected to a "plastic" deformation its shape and size changes as well as the alternating sums resulting from the method of calculation, but the value of these sums stays unchanged. This fact serves to convince the reader that E-P characteristic is a stable topological property. The same fact gives this approach an advantage over usually practiced ones which require a triangulation of geometric objects. [ABSTRACT FROM AUTHOR]

We study the topological μ-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability, and finite model property over general topological spaces, as well as over T0 and TD spaces. We also investigate the relational μ-calculus, providing general completeness results for all natural fragments of the μ-calculus over many different classes of relational frames. Unlike most other such proofs for μ-calculi, ours is model theoretic, making an innovative use of a known method from modal logic (the 'final' submodel of the canonical model), which has the twin advantages of great generality and essential simplicity. [ABSTRACT FROM AUTHOR]

In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymptotic discriminant can arise. [ABSTRACT FROM AUTHOR]

In this paper, we structure the idea of the fractal memristor to simulate the nonlinear self-similarity dopant drift effect for the first time. We investigate the fractal memristor device with the use of the local fractional Laplace transform, and the dynamic fractal behavior of the fractal HP TiO2 memristor model is simulated by changing the integer order of traditional memristor model to ln 2 ln 3 . Compared with the traditional model, the fractal memristor can be well-presented in describing the non-differentiable dopant drift effect. [ABSTRACT FROM AUTHOR]

In 1998 Kleinbock conjectured that any set of weighted badly approximable d × n real matrices is a winning subset in the sense of Schmidt's game. In this paper we prove this conjecture in full for vectors in R d in arbitrary dimensions by showing that the corresponding set of weighted badly approximable vectors is hyperplane absolute winning. The proof uses the Cantor potential game played on the support of Ahlfors regular absolutely decaying measures and the quantitative nondivergence estimate for a class of fractal measures due to Kleinbock, Lindenstrauss and Weiss. To establish the existence of a relevant winning strategy in the Cantor potential game we introduce a new approach using two independent diagonal actions on the space of lattices. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectral property of the self-affine measure μ R , 𝒟 generated by an expanding real matrix R = diag (b , b) and the four-element digit set 𝒟 = { 0 0 , 1 0 , 0 1 , − 1 − 1 }. We show that μ R , 𝒟 is a spectral measure, i.e. there exists a discrete set Λ ⊆ ℝ 2 such that the collection of exponential functions { e − 2 π i 〈 λ , x 〉 : λ ∈ Λ } forms an orthonormal basis for L 2 (μ) , if and only if b = 2 k for some k ∈ ℕ. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math.231(3) (2012) 1681–1693], over which b = 2 k. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of μ R , 𝒟 by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math.242 (2013) 187–208]. We also extend these results to the more general self-affine measures. [ABSTRACT FROM AUTHOR]

In this paper we are exploring application of fusion lemma - a result about perfect trees, having its origin in forcing theory - to some special cases of a problem proposed by Leo Harrington in a book Analytic Sets. In all generality the problem ask whether given a sequence of functions from E" to [0; 1] one can find a subsequence of it that is pointwise convergent on a product of perfect subsets of R. We restrict our attention mainly to binary functions on the Cantor set as well as outline the possible direction of generalization of result to other topological spaces and different notions of measurablity. [ABSTRACT FROM AUTHOR]

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems. [ABSTRACT FROM AUTHOR]

Quantization of a probability distribution is the process of estimating a given probability by a discrete probability that assumes only a finite number of levels in its support. Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. In this paper, we investigate the optimal quantization and the centroidal Voronoi tessellations with n generators for a Borel probability measure P on R supported by a dyadic Cantor set generated by two self-similar mappings with similarity ratios r, where 0 < r⩽ 5-- √17∕ 2 . [ABSTRACT FROM AUTHOR]

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set can be affinely embedded into a symmetric Cantor set , then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062-1079]. [ABSTRACT FROM AUTHOR]

An additive process is a stochastic process with independent increments and that is continuous in probability. In this paper, we study the almost sure Hausdorff and Fourier dimension of the graph of continuous additive processes with zero mean. Such processes can be represented as X t = B V (t) where B is Brownian motion and V is a continuous increasing function. We show that these dimensions depend on the local uniform Hölder indices. In particular, if V is locally uniformly bi-Lipschitz, then the Hausdorff dimension of the graph will be 3/2. We also show that the Fourier dimension almost surely is positive if V admits at least one point with positive lower Hölder regularity. It is also possible to estimate the Hausdorff dimension of the graph through the L q spectrum of V. We will show that if V is generated by a self-similar measure on R 1 with convex open set condition, the Hausdorff dimension of the graph can be precisely computed by its L q spectrum. An illustrating example of the Cantor Devil Staircase function, the Hausdorff dimension of the graph is 1 + 1 2 ⋅ log 2 log 3 . Moreover, we will show that the graph of the Brownian staircase surprisingly has Fourier dimension zero almost surely. [ABSTRACT FROM AUTHOR]

Let C(a),C(b)⊂[0,1] be the central Cantor sets generated by sequences a,b(0,1)N. The first main result of the paper gives a necessary and a sufficient condition for sequences a and b which inform when C(a)-C(b) is equal to [-1,1] or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets which algebraic difference is an interval may be arbitrarily small. We also show that there are sets C(a) and C(b) with the Hausdorff dimension equal to 0 such that their algebraic difference is an interval. Finally, we give a full characterization of the case, when C(a)-C(b) is equal to [-1,1] or is a finite union of closed intervals. [ABSTRACT FROM AUTHOR]

In this paper, we define and discuss the properties of Cantor one- third set and Stern-Brocot sequence withthe principal aim of extract- ing the former from the latter. [ABSTRACT FROM AUTHOR]

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group G ∗ . These results can further be generalized to arbitrary locally compact abelian groups. [ABSTRACT FROM AUTHOR]

This paper is concerned with the time periodic solutions for the Sturm–Liouville boundary value problem of a one-dimensional Kirchhoff equation in presence of a time periodic external forcing with period 2 π / ω and amplitude ϵ . Such a model arises from the forced vibrations of a bounded string in which the dependence of the tension on the deformation cannot be neglected. By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2 π / ω and order ϵ . Such results hold for parameters (ω , ϵ) in a positive measure Cantor set that has asymptotically full measure as ϵ goes to zero. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the Langevin and Brownian equations on fractal time sets using Fα-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets. [ABSTRACT FROM AUTHOR]

In this paper, we study hyperbolic Cantor sets on the line. We prove that the Lebesgue measure of a hyperbolic Cantor set generated by a degree two expanding map satisfying a certain Zygmund condition is zero. Moreover, we show that for any ρ ∈ (0 , 1) there exists a C 1 -smooth degree two expanding map f such that the Lebesgue measure of the hyperbolic Cantor set generated by f is ρ. [ABSTRACT FROM AUTHOR]

Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and f : X → X a continuous map. We demonstrate that f has shadowing if and only if the system (f , X) is (conjugate to) the inverse limit of a directed system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if (f, X) is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if (f , X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits. [ABSTRACT FROM AUTHOR]

In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by 4 disjoint subdisks (with equal distance to each other) of radius 1/4 inside and tangent to the unit disk. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set D(ω). Let Dn be the n-th generation in the construction, which is comparable to the 4 -n-neighborhood of D. We are interested in the decay rate of the Favard length of these sets Dn as n → ∞, which is the likelihood (up to a constant) that "Buffon's needle" dropped randomly will fall into the 4 -n-neighborhood of D. It is well known that the lower bound of the Favard length of Dn(ω) is a constant multiple of n -1. We show that the upper bound of the Favard length of Dn(ω) is Cn -1 for some C > 0 in the average sense. We also prove the a similar linear decay for the Favard length of Ddn (ω), which is the d -n-neighborhood of a self-similar random Cantor set with degree d greater than 4. Notice that in the non-random case where the self-similar set has degree greater than 4, the best known result for the decay rate of the Favard length is e -c √log n. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as 𝜃 tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as 𝜃 tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension. [ABSTRACT FROM AUTHOR]

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if E is any arithmetic progression, the set of primes, or the set of squares n2n ∈ N, then the continued fractions whose digits lie in E have full dimension spectrum, which we denote by DS(CFE). Moreover we prove that if E is an infinite set of consecutive powers, then the dimension spectrum DS(CFE) always contains a non-trivial interval. We also show that there exists some E ⊂ N and two non-trivial intervals I1, I2, such that DS(CFE) ∩ I1 = I1 and DS(CE) ∩ I2 is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets. [ABSTRACT FROM AUTHOR]

We study boundary value problems for harmonic functions on certain domains in the level-l Sierpinski gaskets SGl(l≥2) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of SGl and the upper and lower domains generated by horizontal cuts of SGl are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work. [ABSTRACT FROM AUTHOR]

The article discusses that Terry Marks-Tarlow, is attempting to relate a field of pure mathematics with transpersonal psychology. Topics discussed include the creative element of pure mathematics is often an exercise in the transpersonal realm, one could in fact argue that the scope of the Marks-Tarlow's paper could and should be expanded; and the idea of "the unreasonable effectiveness of mathematics in the natural sciences" is well-known.

Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in [19] who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of 1 for these Julia sets can be attained. The proof of the latter considers the Julia set as the limit set of a non-autonomous conformal iterated function system and we calculate the Hausdorff dimension using a version of Bowen's formula given in the paper by Rempe-Gillen and Urbánski [15]. [ABSTRACT FROM AUTHOR]

Let P be a Borel probability measure on ℝ such that where S1 and S2 are two similarity mappings on ℝ such that for all x ∈ ℝ. Such a probability measure P has support the Cantor set generated by S1 and S2. For this probability measure, in this paper, we give an induction formula to determine the optimal sets of n-means and the nth quantization errors for all n ≥ 2. We have shown that the same induction formula also works for the Cantor distribution supported by the Cantor set generated by and for all x ∈ ℝ, where ψ is the square root of the Golden ratio. In addition, we give a counter example to show that the induction formula does not work for all Cantor distributions. Using the induction formula we obtain some results and observations which are also given in this paper. [ABSTRACT FROM AUTHOR]

For the quadratic family fc(z) = z² + c with c in the exterior of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. Let ĉ be a semi-hyperbolic parameter in the boundary of the Mandelbrot set. In this paper we prove that for each z = z(c) in the Julia set, the derivative dz(c)/dc is uniformly O(1/√|c − ĉ|) when c belongs to a parameter ray that lands on ĉ. We also characterize the degeneration of the dynamics along the parameter ray. [ABSTRACT FROM AUTHOR]

Let α : G X be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action a is topologically free and has no G-invariant regular Borel probability measure on X, dynamical comparison implies that the reduced crossed product of a is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action a is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of a is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized Cantor sequence c , which is an ℓ -automatic sequence. We prove that the abelian complexity of the 2-block sequence of c is ℓ -regular if the factor set of the sequence c is mirror invariant. As a consequence, we show that the 2-abelian complexity of a generalized Cantor sequence satisfying certain conditions is ℓ -regular. • We prove that abelian complexity of 2-block coding of generalized Cantor sequences satisfying certain conditions is regular. • We build a bridge between 2-abelian complexity of generalized Cantor sequences and abelian complexity of its 2-block coding. • We prove that 2-abelian complexity of generalized Cantor sequences satisfying certain conditions is regular. [ABSTRACT FROM AUTHOR]

In this paper, we have generalized theFα-calculus by suggesting Fourier and Laplace transformations of the function with support of the fractals set which are the subset of the real line. Using this generalization, we have found the diffraction fringes from the fractal grating Cantor sets. [ABSTRACT FROM AUTHOR]

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of the following family of rational maps fλ(z) = z2n−λ3n+1/zn(z2n−λn−1) with one parameter λ ∈ C∗ − {λ : λ2n+2 = 1}, where n ≥ 2. This family of rational maps is viewed as a singular perturbation of the bi-critical map P−n(z) = z−n λ ≠ 0 is small. It is proved that the Julia set J(fλ) is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of fλ are attracted by the super-attracting cycle 0 ↔ ∞. Furthermore, we prove that there exists suitable λ such that J(fλ) is a Cantor set of circles but the dynamics of fλ on J(fλ) is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of J(fλ) is also proved if the free critical orbits are not attracted by the cycle 0 ↔ ∞. Finally we give an estimate of the Hausdorff dimension of the Julia set of fλ in some special cases. [ABSTRACT FROM AUTHOR]

In this paper, we present examples of one-parametric iterated function systems converging to the standard middle-third Cantor set. The main goal is the study of the continuous growth of the attractor of the corresponding parametrization. [ABSTRACT FROM AUTHOR]

In this paper, we are presenting a secure, compact and directly printable resonator for chipless RFID tag based on multifractal planar shape. The multifractal Cantor set shows an interesting performance of the monostatic radar cross section for chipless RFID tag. Unlike the classical RFID tag, where for each frequency a resonator is needed, so the number of resonators increases when the identification range increases, by using the Iterated Function System (IFS) the obtained resonator has many resonant frequencies leading to decreasing the number of resonators in chipless RFID tag. In addition, due to their auto similarities and multiscale resolutions resonant frequencies generated by the multifractal Cantor set are not independents: all resonant frequencies are correlated, so we cannot map a frequency peak to only one resonator, because the resonator itself represents a multi frequencies signature. These lead to a more secure tag, which is a challenge for many growing applications. [ABSTRACT FROM AUTHOR]

E. Bach showed that factorization of an integer n can be reduced in probabilistic polynomial time to the problem of computing exponents of elements in ℤ n * (in particular the group order of ℤ n * ). It is also known that factorization of square-free integer n can be reduced to the problem of computing the group order of an elliptic curve E/ℤn. In this paper we describe the analogous reduction for computing the orders of Jacobians over ℤn of hyperelliptic curves C over ℤn using the Mumford representation of divisor classes and Cantor's algorithm for addition. These reductions are based on the group structure of the Jacobian. We also propose other reduction of factorization to the problem of determining the number of points |C(ℤn)|, which makes use of elementary properties of twists of hyperelliptic curves. [ABSTRACT FROM AUTHOR]

Iterated function systems provide the most fundamental framework to create many fascinating fractal sets. They have been extensively studied when the functions are affine transformations of Euclidean spaces. This paper investigates the iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that the basics results of the classical Hutchinson–Barnsley theory can be carried over to construct fractal sets on hyperbolic number plane as its unique fixed point. We also discuss about the notion of hyperbolic derivative of an hyperbolic-valued function and then we use this notion to get some generalization of cookie-cutter Cantor sets in the real line to the hyperbolic number plane. [ABSTRACT FROM AUTHOR]

In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is given by arboreal representations of absolute Galois groups of fields. We show how to associate to an arboreal representation an action of a discrete group, and give examples of arboreal representations with stable and wild asymptotic discriminant. [ABSTRACT FROM AUTHOR]

In this paper, we address the free (uncontrolled) dynamics of a snakeboard consisting of two wheel pairs fastened to a platform. The snakeboard is one of the well-known sports vehicles on which the sportsman executes necessary body movements. From the theoretical point of view, this system is a direct generalization of the classical nonholonomic system of the Chaplygin sleigh. We carry out a topological and qualitative analysis of trajectories of this dynamical system. An important feature of the problem is that the common level set of first integrals is a compact two-dimensional surface of genus 5. We specify conditions under which the reaction forces infinitely increase during motion and the so-called phenomenon of nonholonomic jamming is observed. In this case, the nonholonomic model ceases to work and it is necessary to use more complex mechanical models incorporating sliding, elasticity, etc. [ABSTRACT FROM AUTHOR]

The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics. [ABSTRACT FROM AUTHOR]

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we use a local fractional Laplace decomposition method to solve the diffusion equation with the diffusive parameter variable in fractal media. The obtained result illustrates the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives. [ABSTRACT FROM AUTHOR]

Self-similarity and Lipschitz equivalence are two basic and important properties of fractal sets. In this paper, we consider those properties of the union of Cantor set and its translate. We give a necessary and sufficient condition that the union is a self-similar set. Moreover, we show that the union satisfies the strong separation condition if it is of the self-similarity. By using the augment tree, we characterize the Lipschitz equivalence between Cantor set and the union of Cantor set and its translate. [ABSTRACT FROM AUTHOR]

This paper is motivated by questions concerning the discrete dynamical system on the 3-adic integers ℤ3 given by multiplication by 2. The exceptional set εℤ3/ is defined to be the set of all elements of ℤ3 whose forward orbits under this action intersect the 3-adic Cantor set Σ3,... (of 3-adic integers whose expansions omit the digit 2) infinitely many times. It has been shown that this set has Hausdorff dimension at most ½, and it is conjectured that it has Hausdorff dimension 0. Upper bounds on its Hausdorff dimension can be obtained with sufficient knowledge of Hausdorff dimensions of intersections of multiplicative translates of Cantor sets by powers of 2. This paper studies more generally the structure of finite intersections of general multiplicative translates S = Σ 3,... ∩ 1/M1 Σ3,... ∩ ⋯ ∩ 1/Mn Σ 3,... by integers 1 < M1 < M2 < ⋯ < Mn. These sets are describable as sets of 3-adic integers whose 3-adic expansions have one-sided symbolic dynamics given by a finite automaton. This paper gives a method to determine the automaton for given data (M1,..., Mn) and to compute the Hausdorff dimension, which is always of the form log3 (β) where β is an algebraic integer. Computational examples indicate that in general the Hausdorff dimension of such sets depends in a very complicated way on the integers M1,..., Mn. Exact answers are obtained for certain infinite families, which show as a corollary that a relaxed notion of generalized exceptional set has a positive Hausdorff dimension. Mathematics Subject Classification (2010). Primary: 11K55; Secondary: 11S82, 28A80, 28D20, 37B10. [ABSTRACT FROM AUTHOR]

Suppose that $${{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}$$ and x = E. E E · · · is the P-Cantor series expansion of $${x \in \mathbb{R}}$$ . We define The functions $${\psi_{P,Q}}$$ are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of $${\psi_{P,Q}}$$ restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [] is improved. Properties of the functions $${\psi_{P,Q}}$$ are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of $${\psi_{P,Q}}$$ on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures $${\mu_1}$$ and $${\mu_2}$$ on $${{\mathbb{N}_2^\mathbb{N}}}$$ , we study which of these properties $${\psi_{P,Q}}$$ satisfies for $${\mu_1 \times \mu_2}$$ -almost every ( P, Q) $${{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}$$ . Related classes of random fractals are also studied. [ABSTRACT FROM AUTHOR]