Triadic Cantor Set Research Articles

Mahler’s question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

Mahler’s question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

Defect mode properties of self-similar symmetry order in triadic Cantor photonic crystals

The discovery of quasi-crystal tells us that not only the translational symmetry order (TSO) but also other orders can produce Bragg diffraction. As far as we know, in one dimensional (1D) photonic crystals (PCs) there are two symmetry orders, TSO and self-similar symmetry order (SSO). There has been a lot of researches exploring the properties of defect of TSO in the 1D PCs, but until now we do not find any studies related to the defect of SSO. In this work we consider the triadic Cantor set (TCS) PCs to stage 6, and attempt to create the defect of SSO by changing the scale factor of stage 2 to 3 and stage 3 to 4, where for the high-refractive-index layer only the thickness of stage 3 changes. It's well known that the defect of TSO governs the frequency of a kind of transmission peaks, called defect mode. Our results show that the defect of SSO influences the background intensity of the transmittance spectrum rather than the transmission peaks. We note that the defect mode frequency and the transmittance background intensity interchange their roles for the two symmetry orders, and it follows that the long neglected transmittance background intensity should be a significant optical property.

Optical properties of symmetry breaking of the self-similar order in triadic Cantor photonic crystals.

In general, the photonic crystal (PC) is a periodical optical structure, but there are some studies considering aperiodic structures. If we insert a defect layer into a one-dimensional periodic PC to break its translational symmetry order (TSO), some peaks, called defect modes, appear in the transmittance spectrum. The defect layer thickness governs the frequencies of these defect modes but almost does not affect the other part of the spectrum. The discovery of quasi-crystals tells us that not only the TSO but also other orders can produce Bragg diffraction. It is well known that triadic Cantor set (TCS) PCs, which lack TSO but have a self-similar symmetry order (SSO), still exhibit narrow transmission peaks. In this work, we try to break the SSO in TCS PCs and find the resulting optical phenomena, where single-negative materials and dielectrics are chosen as the constituents of PCs. The study method is the transfer matrix method, and the calculation results show that the background intensity of the transmittance spectrum rather than the frequency of peaks obviously periodically changes with the break of SSO. It follows that the SSO does have physical meaning, and not only the transmission peaks but also the background should be treated as a significant optical property.

Negative Powers of Contractions Having a Strong AA + Spectrum

Abstract Zarrabi proved in 1993 that if the spectrum of a contraction T on a Banach space is a countable subset of the unit circle 𝕋, and if lim n → + ∞ log ( ‖ T − n ‖ ) n = 0 {\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0 , then T is an isometry, so that ‖Tn ‖ = 1 for every n ∈ ℤ. It is also known that if C is the usual triadic Cantor set then every contraction T on a Banach space such that Spec(T ) ⊂ 𝒞 satisfying lim s u p n → + ∞ log ( ‖ T − n ‖ ) n α < + ∞ \lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} < + \infty for some α < log ( 3 ) − log ( 2 ) 2 log ( 3 ) − log ( 2 ) \alpha < {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}} is an isometry. In the other direction an easy refinement of known results shows that if a closed E ⊂ 𝕋 is not a “strong AA +-set” then for every sequence (un)n ≥1 of positive real numbers such that lim inf n →+∞ un = + ∞ there exists a contraction T on some Banach space such that Spec(T )⊂ E, ‖T −n ‖ = O(u n) as n → + ∞ and supn ≥1 ‖T −n‖ = + ∞. We show conversely that if E ⊂ 𝕋 is a strong AA +-set then there exists a nondecreasing unbounded sequence (u n)n ≥1 such that for every contraction T on a Banach space satsfying Spec(T) ⊂ E and ‖T −n ‖ = O(u n) as n → + ∞ we have supn >0 ‖T −n ‖ ≤ K, where K < + ∞ denotes the “AA +-constant” of E (closed countanble subsets of 𝕋 and the triadic Cantor set are strong AA +-sets of constant 1).

One-way absorption properties in asymmetric single-negative-based Cantor photonic crystals

In this work we theoretically study the one-way optical properties in asymmetric triadic-Cantor-set (TCS) photonic crystals (PCs), air/TCSN/G/TCSM/air, where TCSN is the stage N TCS composed of a lossy epsilon-negative (ENG) material and a lossy mu-negative (MNG) material. In addition, the defect layer G is a dielectric. Our results show that the absorption spectra are different in forward and backward propagation. Specially, it is first discovered that the one-way properties will disappear if the layer thicknesses are large. Besides, the layer thickness limit is smaller when the TCS stage is larger. Additionally, comparing with previous studies, we find that the type of interaction between the defect and non-defect modes is decided by the layer materials of the PC structure.

Dynamical covering problems on the triadic Cantor set

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K. More precisely, let Tx=3x(mod1) be the natural map on K, μ the standard Cantor measure and x0∈K a given point. We consider the size of the set of points in K which can be well approximated by the orbit {Tnx0}n≥1 of x0, namely the setD(x0,φ):={y∈K:|Tnx0−y|<φ(n)for infinitely manyn∈N}, where φ is a positive function defined on N. It is shown that for μ almost all x0∈K, the Hausdorff measure of D(x0,φ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K.

Design of multichannel filters based on the use of periodic Cantor dielectric multilayers.

A fractal multilayer structure made of two dielectric materials can exhibit photonic bandgap (PBG). In this work, with the use of this PBG, we study the transmission properties of periodic triadic Cantor set structures. The results indicate that the structure can be used to design multichannel filters with channel number equal to N-1 for a given number of periods, N. In addition, the channel frequencies can be designed at will. The considered structure provides another new type of design for a tunable multichannel filter.

P-adic coverage method in fractal analysis of showers

Self-similarity in multiple processes at high energies is considered. It is assumed that a parton cascade transforms into a hadron shower with a fractal structure. The box counting (BC) method used to calculate the fractal dimension is analyzed. The parton shower with permissible 1/3 parts of pseudorapidity space, which corresponds to a triadic Cantor set, was used as a test fractal. It was found that there is an optimal set of bins (a parameter of the BC method) that allows one to find the fractal dimension with maximal accuracy. The optimal set of bins is shown to depend on the fractal generation law. The P-adic coverage (PaC) method is proposed and used in the fractal analysis. This method makes it possible to determine the fractal dimension of a shower as accurately as possible, the number of fractal levels and partons at each branching point during the parton shower evolution, the type of cascade (either random or regular), and its structure. It is shown to be applicable to an analysis of the regular and random N-ary cascades with permissible 1/k parts of the space studied.

Undergraduate experiment with fractal diffraction gratings

We present a simple diffraction experiment with fractal gratings based on the triadic Cantor set. Diffraction by fractals is proposed as a motivating strategy for students of optics in the potential applications of optical processing. Fraunhofer diffraction patterns are obtained using standard equipment present in most undergraduate physics laboratories and compared with those obtained with conventional periodic gratings. It is shown that fractal gratings produce self-similar diffraction patterns which can be evaluated analytically. Good agreement is obtained between experimental and numerical results.

An analogue of Cobham’s theorem for fractals

We introduce the notion of k-self-similarity for compact subsets of ℝ n and show that it is a natural analogue of the notion of k-automatic subsets of integers. We show that various well-known fractals such as the triadic Cantor set, the Sierpinski carpet or the Menger sponge turn out to be k-self-similar for some integers k. We then prove an analogue of Cobham's theorem for compact sets of ℝ that are self-similar with respect to two multiplicatively independent bases k and l. Namely, we show that X is both a k- and an l-self-similar compact subset of ℝ if and only if it is a finite union of closed intervals with rational endpoints.

Scattering from generalized Cantor fractals

A fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set, is considered. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that, for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.

Polyadic devil's lenses

Devil's lenses (DLs) were recently proposed as a new kind of kinoform lens in which the phase structure is characterized by the "devil's staircase" function. DLs are considered fractal lenses because they are constructed following the geometry of the triadic Cantor set and because they provide self-similar foci along the optical axis. Here, DLs are generalized allowing the inclusion of polyadic Cantor distributions in their design. The lacunarity of the selected polyadic fractal distribution is an additional design parameter. The results are coined polyadic DLs. Construction requirements and interrelations among the different parameters of these new fractal lenses are also presented. It is shown that the lacunarity parameter affects drastically the irradiance profile along the optical axis, appodizing higher-order foci, and these features are proved to improve the behavior of conventional DLs under polychromatic illumination.

Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets

An analogue of the Riemannian Geometry for an ultrametric Cantor set (C,d) is described using the tools of Noncommutative Geometry. Associated with (C,d) is a weighted rooted tree, its Michon tree [29]. This tree allows to define a family of spectral triples (\mathcal{C}_{\rm Lip}(C),\mathcal{H},D) using the ℓ^2 -space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here \mathcal{C}_{\rm Lip}(C) denotes the space of Lipschitz continuous functions on (C,d) . The family of spectral triples is indexed by the space of choice functions , which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the family of these spectral triples allows to recover the metric on C . The corresponding ζ -function is shown to have abscissa of convergence, s_0 , equal to the upper box dimension of (C,d) . Taking the residue at this singularity leads to the definition of a canonical probability measure on C , which in certain cases coincides with the Hausdorff measure at dimension s_0 . This measure in turn induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be interpreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace–Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C . This construction is applied to the simplest case, the triadic Cantor set, where: (i) the spectrum and the eigenfunctions of the Laplace–Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimension s_0 , (iii) the corresponding Markov process is shown to have an anomalous diffusion with \mathbb{E}(d(X_t,X_{t + δt})^2) ≃ δt \ln(1/δt) as δt ↓ 0 .

Self-similar structure on intersections of triadic Cantor sets

Let C be the triadic Cantor set. We characterize the all real number α such that the intersection C ∩ ( C + α ) is a self-similar set, and investigate the form and structure of the all iterated function systems which generate the self-similar set.

Diffraction by Cantor fractal zone plates

The paper reports studies, both experimental and using numerical simulation, of the Fresnel diffraction by recently introduced fractal zone plates associated with triadic and quintic Cantor sets. The evolution of the intensity patterns at planes transversal to the propagation direction is presented. A series of conventional and doughnut-like foci are observed around the principal focus. The position, depth and size of the foci depend on similarity dimensions and the fractal level of the encoded fractal structures, both directly related to the number of the corresponding Fresnel zones.

Resonance Analysis of Multilayered Filters with Triadic Cantor-Type One-Dimensional Quasi-Fractal Structures

Multilayered filters with a dielectric distribution along their thickness forming a one-dimensional quasi-fractal structure are theoretically analyzed, focusing on exposing their resonant properties in order to understand a dielectric Menger's sponge resonator [4], [5]. Quasi-fractal refers to the triadic Cantor set with finite generation. First, a novel calculation method that has the ability to deal with filters with fine fractal structures is derived. This method takes advantage of Clifford algebra based on the theory of thin-film optics. The method is then applied to classify resonant modes and, especially, to investigate quality factors for them in terms of the following design parameters: a dielectric constant, a loss tangent, and a stage number. The latter determines fractal structure. Finally, behavior of the filters with perfect fractal structure is considered. A crucial finding is that the high quality factor of the modes is not due to the complete self-similarity, but rather to the breaking of such a fractal symmetry.

Nontrivial temporal scaling in a Galilean stick-slip dynamics

We examine the stick-slip fluctuating response of a rough massive nonrotating cylinder moving on a rough inclined groove which is submitted to weak external perturbations and which is maintained well below the angle of repose. The experiments presented here, which are reminiscent of Galileo's works with rolling objects on inclines, have brought in the last years important insights into the friction between surfaces in relative motion and are of relevance for earthquakes, differing from classical block-spring models by the mechanism of energy input in the system. Robust nontrivial temporal scaling laws appearing in the dynamics of this system are reported, and it is shown that the time-support where dissipation occurs approaches a statistical fractal set with a fixed value of dimension. The distribution of periods of inactivity in the intermittent motion of the cylinder is also studied and found to be closely related to the lacunarity of a random version of the classic triadic Cantor set on the line.

Electronic properties of a Cantor lattice

We present results of some investigations on the electronic properties of a non-periodic self-similar structure based on the triadic Cantor set sequence. In particular, we examine the eigenvalue spectrum of a Cantor lattice using the trace map technique widely used for the conventional quasi-periodic sequences. In addition, we show that a Cantor lattice possesses a number of resonant extended wave functions, some of which exhibit a distribution that resembles the lattice itself. We present results for the transmission coefficient of finite, but arbitrarily large Cantor lattices. We also examine in some detail, the cycles of the matrix map for such a sequence, and the variety of cyclic behaviour that one may encounter depending on the specific choices of the Hamiltonian parameters.

Intersection of triadic Cantor sets with their translates. II. Hausdorff measure spectrum function and its introduction for the classification of Cantor sets

Initiated by the purpose of classification of sets having the same fractal dimension, we continue, in this second paper of a series of two, our investigation of intersection of triadic Cantor sets and their use in the classification of fractal sets. We exploit the infinite tree structure of translation elements to give the exact expressions of these elements. We generalize this result to a family of uniform Cantor sets for which we also give the Hausdorff measure spectrum function (HMSF). We develop three algorithms for the construction of HMSF of triadic Cantor sets. Then, we introduce a new method based on HMSF as a way for tracing the geometrical organization of a fractal set. The HMSF does carry a huge amount of information about the set to likely be explored in a chosen way. To extract this information, we develop a one by one step method and apply it to typical fractal sets. This results in a complete identification of fractals.

Intersection of triadic Cantor sets with their translates— I. Fundamental properties

Motivated by Mandelbrot's [The Fractal Geometry of Nature, Freeman, San Francisco, 1983] idea of referring to lacunarity of Cantor sets in terms of departure from translation invariance, we study the properties of these translation sets and show how they can be used for a classification purpose. This first paper of a series of two will be devoted to set up the fundamental properties of Hausdorff measures of those intersection sets. Using the triadic expansion of the shifting number, we determine the fractal structure of intersection of triadic Cantor sets with their translates. We found that the Hausdorff measure of these sets forms a discrete spectrum whose non-zero values come only from those shifting numbers with a finite triadic expansion. We characterize this set of shifting numbers by giving a partition expression of it and the steps of its construction from a fundamental root set. Finally, we prove that intersection of Cantor sets with their translates verify a measure-conservation law with scales. The second paper will take advantage of the properties exposed here in order to utilize them in a classification context. Mainly, it will deal with the use of the discrete spectrum of measures to distinguish two Cantor-like sets of the same fractal dimension.