Cantor Set Research Articles

On the homology of big mapping class groups

AbstractWe prove that the mapping class group of the one‐holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once‐punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite‐type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite‐type surfaces.

Applications of Cantor Set to Fractal Geometry

Fractal geometry is a subfield of mathematics that allows us to explain many of the complexities in nature. Considering this remarkable feature of fractal geometry, this study examines the Cantor set, which is one of the most basic examples of fractal geometry. First of all for the Cantor set, which is one of the basic example and important structure of it. Firstly, generalization of Cantor set in on ${\mathbb{R}}$, ${\mathbb{R}}^2$ and ${\mathbb{R}^3}$ are taken into consideration. Then the given structures are examined over curve and surface theory. This approach enables to given a relationship between fractal geometry and differential geometry. Finally, some examples are established.

On the hyperspaces of meager and regular continua

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite.In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that | D ( X ) | ∉ { 2,3,4,5,8 , 9 } h t ) for any continuum X. Also, we prove that D(X) is never homeomorphic to ℕ . On the other hand, given a point p ∈ X , we consider the meager composant and the filament composant of p in X, denoted by MXp and FcsX(p), respectively, and we study some relations between MXp and FcsX(p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs ( X ) = { FcsX ( p ) : p ∈ X } can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.

Characterization of the algebraic difference of special affine Cantor sets

We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \N$, $l+r< p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.

On transversal Hölder regularity for flat Wieler solenoids

Abstract This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids. Among the aspects studied are different types of cohomologies, the rates of mixing given by the Ruelle spectrum of the hyperbolic map acting on this space, and solutions of the cohomological equation in primitive substitution subshifts for Hölder functions. The overarching theme is that considerations of $\alpha $ -Hölder regularity on Cantor sets go a long way.

Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Given ρ∈(0,1/4], the four corner Cantor set E⊂R2 is a self-similar set generated by the iterated function system {(ρx,ρy),(ρx,ρy+1−ρ),(ρx+1−ρ,ρy),(ρx+1−ρ,ρy+1−ρ)}. For θ∈[0,π) let Eθ be the orthogonal projection of E onto a line with an angle θ to the x-axis. In principle, Eθ is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection Eθ is totally self-similar. We also study the spectrum of Eθ, which turns out that the spectrum achieves its maximum value if and only if Eθ is totally self-similar. Furthermore, when Eθ is totally self-similar, we calculate its Hausdorff dimension and study the subset Uθ which consists of all x∈Eθ having a unique coding. In particular, we show that dimHUθ=dimHEθ for Lebesgue almost every θ∈[0,π). Finally, for ρ=1/4 we prove that the possibility for Eθ to contain an interval is strictly smaller than that for Eθ to have an exact overlap.

Non-realizability of some big mapping class groups

In this note, we prove that the compactly supported mapping class group of a surface containing a genus 3 3 subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order 6 6 symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.

Lebesgue Constants for Cantor Sets

We evaluate the values of the Lebesgue constants in polynomial interpolation for three types of Cantor sets. In all cases, the sequences of Lebesgue constants are not bounded. This disproves the statement by Mergelyan.

The cantor set and inverse limits of upper semi-continuous functions

The cantor set and inverse limits of upper semi-continuous functions

Fourier transform and expanding maps on Cantor sets

abstract: We study the Fourier transforms $\widehat{\mu}(\xi)$ of non-atomic Gibbs measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on $[0,1]$ or Cantor sets with strong separation. When $T$ is totally non-linear, then $\widehat{\mu}(\xi)\to 0$ at a polynomial rate as $|\xi|\to\infty$.

Revealing the Essence of Differential and Integral Calculus based on Scientific Philosophy

As it is known in the way where the basic equations of mathematical analysis and mathematical physics were taken as a basis it was possible to come to the main results of the Cantor set theory. However, contradictions and paradoxes were inherent to the results obtained in this field. In the way where the basic equations of theoretical physics, as well as empirical and probabilistic physics were taken as a basis, it was possible to arrive at the basic results of quantum theory. The results obtained in this area proved to be very useful in describing the experimental data. This was a sign of their truth. Therefore, there is reason to come to the following conclusion. That taking as a basis the ideas and results obtained in the field of quantum theory, there is a possibility to come to the disclosure of the essence of differential and integral calculus. As well as the results obtained in the field of new set theory. The main purpose of this work is an attempt to solve this problem.

Cantor Set from a Nonstandard Viewpoint

Cantor Set from a Nonstandard Viewpoint

A Hybrid Approach Combining the Lie Method and Long Short-Term Memory (LSTM) Network for Predicting the Bitcoin Return

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z2 −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers.

Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

AbstractIn this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite‐by‐virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property and ‐stability for their crossed product ‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.

SEMI-DOMAIN SOLUTIONS TO THE FRACTAL (3+1)-DIMENSIONAL JIMBO–MIWA EQUATION

In this work, the variational direct method (VDM) based on the variational theory and Ritz-like method, combined with the fractal two-scale transformation, is employed to construct the semi-domain solutions of the (3+1)-dimensional fractal Jimbo–Miwa (JM) equation, which is derived by He’s fractal derivative. The applicability and effectiveness of the acquired solutions on the Cantor sets are presented through the numerical results in the form of 3-dimensional graphs. In addition, as a comparison, the exact solutions of the classic (3+1)-dimensional JM equation for [Formula: see text] are also illustrated. The obtained results in this work are expected to open new perspectives for the traveling wave theory.

Non-Hausdorff germinal groupoids for actions of countable groups

We study conditions under which the germinal groupoid associated to a minimal equicontinuous action of a countable group on a Cantor set has non-Hausdorff topology. We develop a new criterion, which serves as an obstruction for the étale topology on the groupoid to be non-Hausdorff. We use this and other criteria to study the topology of germinal groupoids for a few classes of actions. In particular, we give examples of families of contracting and non-contracting self-similar groups, which are amenable and whose actions have associated germinal groupoids with non-Hausdorff topology.

A beam that can only bend on the Cantor set

In this work, we address the following question: Is it possible for a one‐dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by , each one only able to bend on the set associated with the ‐th step in the construction of the Cantor set and compute the ‐limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension , which is the dimension of the Cantor set.

On the union of homogeneous symmetric Cantor set with its translations

On the union of homogeneous symmetric Cantor set with its translations

Numerical representations of AB-type copolymer complexes: analysis of 1H NMR chemical shift patterns in terms of a Smith–Cantor set

When considering the possibility of storing information in the sequence of monomer residues within an AB-type copolymer chain, it is constructive to model that sequence as a string of ones and zeros. The intramolecular environment around any given digit (say a “1”) can then be represented by another string of integers—a code—obtained by summing pairs of digits at equivalent positions, in both directions, from that digit. The code can include only integers 0, 1 and 2, and can represent a number in any base b higher than 2. In base b = 3 the resulting set of codes includes all numbers (because only digits 0, 1 and 2 occur in ternary expansions), but in any base b > 3 the codes define a limited set of numbers comprising a fractal we term a Smith–Cantor set. The 1H NMR spectrum of a random, AB-type co(polyester-imide) shows, on complexation with pyrene, a pattern of complexation shifts approximating very closely to the Smith–Cantor set for which b = 4. Other co(polyimide) complexes show a 1H NMR pattern corresponding to a specific sub-set of this fractal. The sub-set arises from a “stop-at-zero” limitation, whereby digits in the initial string are set to zero for code-generating purposes if they occur beyond a zero, as viewed from the central “1”. The limitation arises in copolymers where pyrene binds by intercalation between pairs of adjacent diimide residues. This numerical approach provides a complete, unifying theory to account for the emergence of fractal character in the 1H NMR spectra of AB-type copolymer complexes.

Hausdorff measure of two and three scales Cantor sets

This paper gives the exact Hausdorff measure of two and three scales Cantor sets.