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In this paper we consider the algebraic crossed product ${\mathcal{A}}:=C_{K}(X)\rtimes _{T}\mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field with involution and $C_{K}(X)$ denotes the $K$-algebra of locally constant $K$-valued functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on ${\mathcal{A}}$ by means of full ergodic $T$-invariant probability measures $\unicode[STIX]{x1D707}$ on $X$. To do so, we present a general construction of an approximating sequence of $\ast$-subalgebras ${\mathcal{A}}_{n}$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $\ast$-algebra ${\mathcal{A}}$ into ${\mathcal{M}}_{K}$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on ${\mathcal{A}}$ by restricting the unique one defined on ${\mathcal{M}}_{K}$. This process gives a way to obtain a Sylvester matrix rank function on ${\mathcal{A}}$, unique with respect to a certain compatibility property concerning the measure $\unicode[STIX]{x1D707}$, namely that the rank of a characteristic function of a clopen subset $U\subseteq X$ must equal the measure of $U$.

After establishing a completeness theorem for continuous logic, Ben Yaacov and Pedersen conclude that if T is a complete recursive L -theory in continuous logic, and v ( φ ) is the truth value of the L -sentence φ in models of T, then v ( φ ) is a recursive real uniformly recursive in φ. Some of the examples to which the latter result applies are theories of the following structures: atomless probability structures, the Urysohn space of diameter 1, Hilbert space, the lattice-ordered group or ring of real-valued continuous functions on the Cantor set, and the complex ⁎-algebra of continuous functions on the Cantor set. This paper will explain why these examples obey much stronger results, yielding (for example) decision procedures for the conditions true in these structures.

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 $$2\pi /\omega $$ and amplitude $$\epsilon $$. 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 $$2\pi /\omega $$ and order $$\epsilon $$. Such results hold for parameters $$(\omega ,\epsilon )$$ in a positive measure Cantor set that has asymptotically full measure as $$\epsilon $$ goes to zero.

Let C be the Cantor set. For each n ⩾ 3 we construct an embedding A : C × C → R n such that A ( C × { s } ) , for s ∈ C , are pairwise ambiently incomparable everywhere wild Cantor sets (generalized Antoine's necklaces). This serves as a base for another new result proved in this paper: for each n ⩾ 3 and any non-empty perfect compact set X which is embeddable in R n − 1 , we describe an embedding A : X × C → R n such that each A ( X × { s } ) , s ∈ C , contains the corresponding A ( C × { s } ) , and is “nice” on the complement A ( X × { s } ) − A ( C × { s } ) ; in particular, the images A ( X × { s } ) , for s ∈ C , are ambiently incomparable pairwise disjoint copies of X. This generalizes and strengthens theorems of J.R. Stallings (1960), R.B. Sher (1968), and B.L. Brechner–J.C. Mayer (1988).

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ , for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

We consider a class of compact positive operators $$L:X\rightarrow X$$ given by $$(Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds$$ , acting on the space X of continuous $$2\pi $$ -periodic functions x. Here $$\eta $$ is continuous with $$\eta (t)\le t$$ and $$\eta (t+2\pi )=\eta (t)+2\pi $$ for all $$t\in \mathbf{R}$$ . We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem $$\kappa x=Lx$$ exists for some $$\kappa >0$$ (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally $$\eta $$ is analytic, we study the set $${\mathcal {A}}\subseteq \mathbf{R}$$ of points t at which x is analytic; in general $${\mathcal {A}}$$ is a proper subset of $$\mathbf{R}$$ , although x is $$C^\infty $$ everywhere. Among other results, we obtain conditions under which the complement $${\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}$$ of $${\mathcal {A}}$$ is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map $$t\rightarrow \eta (t)$$ .

We consider a typical (in the sense of Baire categories) convex body K in R d + 1 . The set of feet of its double normals is a Cantor set, having lower box-counting dimension 0 and packing dimension d. The set of lengths of those double normals is also a Cantor set of lower box-counting dimension 0. Its packing dimension is equal to 1 2 if d = 1 , is at least 3 4 if d = 2 , and equals 1 if d ≥ 3 . We also consider the lower and upper curvatures at feet of double normals of K, with a special interest for local maxima of the length function (they are countable and dense in the set of double normals). In particular, we improve a previous result about the metric diameter.

It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.

We study charge transport in the Peierls-Harper model with a quasi-periodic cosine potential. We compute the Landauer-type conductance of the wire. Our numerical results show the following: (i) When the Fermi energy lies in the absolutely continuous spectrum that is realized in the regime of the weak coupling, the conductance is quantized to the universal conductance. (ii) For the regime of localization that is realized for the strong coupling, the conductance is always vanishing irrespective of the value of the Fermi energy. Unfortunately, we cannot make a definite conclusion about the case with the critical coupling. We also compute the conductance of the Thue-Morse model. Although the potential of the model is not quasi-periodic, the energy spectrum is known to be a Cantor set with zero Lebesgue measure. Our numerical results for the Thue-Morse model also show the quantization of the conductance at many locations of the Fermi energy, except for the trivial localization regime. Besides, for the rest of the values of the Fermi energy, the conductance shows a similar behavior to that of the Peierls-Harper model with the critical coupling., Comment: 29 pages, 19 figures, v3: minor corrections, and references added

Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $\epsilon$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. We also use the potential game to provide a new proof of an important lemma in the classical proof of the existence of Hall's Ray.

In this paper, we investigate the dynamics of the following family of rational maps \begin{document}$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $\end{document} with one parameter \begin{document}$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $\end{document} , where \begin{document}$ n\geq 2 $\end{document} . This family of rational maps is viewed as a singular perturbation of the bi-critical map \begin{document}$ P_{-n}(z) = z^{-n} $\end{document} if \begin{document}$ \lambda \neq 0 $\end{document} is small. It is proved that the Julia set \begin{document}$ J(f_\lambda) $\end{document} is either a quasicircle, a Cantor set of circles, a Sierpinski carpet or a degenerate Sierpinski carpet provided the free critical orbits of \begin{document}$ f_\lambda $\end{document} are attracted by the super-attracting cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Furthermore, we prove that there exists suitable \begin{document}$ \lambda $\end{document} such that \begin{document}$ J(f_\lambda) $\end{document} is a Cantor set of circles but the dynamics of \begin{document}$ f_{\lambda} $\end{document} on \begin{document}$ J(f_{\lambda}) $\end{document} is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of \begin{document}$ J(f_{\lambda}) $\end{document} is also proved if the free critical orbits are not attracted by the cycle \begin{document}$ 0\leftrightarrow\infty $\end{document} . Finally we give an estimate of the Hausdorff dimension of the Julia set of \begin{document}$ f_\lambda $\end{document} in some special cases.

The Cantor Set C, that familiar oddity of real analysis, has many fascinating properties: C is an uncountable subset of [0,1] , yet has Lebesgue measure zero; C can be created by repeatedly removin...

Bending vibrations of thin beams and plates may be described by nonlinear Euler-Bernoulli beam equations with $x$-dependent coefficients. In this paper we investigate existence of families of time-periodic solutions to such a model using Lyapunov-Schmidt reduction and a differentiable Nash-Moser iteration scheme. The results hold for all parameters $(\epsilon,\omega)$ in a Cantor set with asymptotically full measure as $\epsilon\rightarrow0$.

In this paper, we define a new notion of "freely tracing property by free chains" on $G$-like continua and we prove that a positive topological entropy homeomorphism on a $G$-like continuum admits a Cantor set $Z$ such that every tuple of finite points in $Z$ is an $IE$-tuple of $f$ and $Z$ has the freely tracing property by free chains. Also, by use of this notion, we prove the following theorem: If $G$ is any graph and a homeomorphism $f$ on a $G$-like continuum $X$ has positive topological entropy, then there is a Cantor set $Z$ which is related to both the chaotic behaviors of Kerr and Li [18] in dynamical systems and composants of indecomposable continua in topology. Our main result is Theorem 3.3 whose proof is also a new proof of [6]. Also, we study dynamical properties of continuum-wise expansive homeomorphisms. In this case, we obtain more precise results concerning continuum-wise stable sets of chaotic continua and IE-tuples., Comment: 23 pages

In this work, we establish relations between DNA sequences with missing subsequences (the forbidden words) and the generalized Cantor sets. Various examples associated with some generalized Cantor sets, including Hao’s frame representation and the generalized Sierpinski Set, along with their fractal graphs, are also presented in this work.

We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a finite orbit, or a Cantor set, or a circle. If $X$ is a graph different from a circle, such a minimal $M$ is a finite orbit. These results extend those of the authors for group actions on dendrites. On the other hand, we show that, for any group $G$ acting on a local dendrite $X$ different from a circle, the following properties are equivalent: (1) ($G, X$) is pointwise almost periodic. (2) The orbit closure relation $R = \{(x, y)\in X\times X: y\in \overline{G(x)}\}$ is closed. (3) Every non-endpoint of $X$ is periodic. In addition, if $G$ is countable and $X$ is a local dendrite, then ($G, X$) is pointwise periodic if and only if the orbit space $X/G$ is Hausdorff., 16 pages

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

Let X be a local dendrite and let f : X → X be a monotone map. Denote by P ( f ) and Ω ( f ) the sets of periodic points and nonwandering points of f, respectively. We show that Ω ( f ) = P ( f ) ‾ , whenever P ( f ) is nonempty and Ω ( f ) is the unique minimal set included in a circle which is either a Cantor set or a circle, whenever P ( f ) is empty. In the case where the set of endpoints of X is countable, we show that Ω ( f ) = P ( f ) whenever P ( f ) is nonempty.

This work intends to publicize the already well-known Cantor set. The idea is to show a more detailed demonstration of some important properties that it has, in a certain way being not very common to find in texts in Portuguese. We will also see that, except for homeomorphism, the Cantor Set is the only one, as metric space, with all the indicated properties. Este trabalho tem por intenção divulgar o já conhecido Conjunto de Cantor. A ideia é exibir uma demonstração mais detalhada de algumas propriedades importantes que ele possui, não sendo, de certa forma, tão comum encontrá-las em textos em português. Também veremos que, a menos de homeomorfismo, o Conjunto de Cantor é o único, como espaço métrico, com todas as propriedades indicadas.

We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1\!-\!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $\alpha$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $\alpha$. We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time., Comment: 11 pages, 6 captioned figures each containing 2 subfigures

Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate in this paper the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions. We also provide shorter proofs of the main results of Baker in [B] by adapting the method given in [EJK] for the case $M=1$., Comment: 33 pages. arXiv admin note: text overlap with arXiv:math/0609708

Financial support from research projects 2009SGR-169, ECO2009-7616, and Consolider-Ingenio 2010 (Xavier Martinez-Giralt), 2009SGR-00600 and SEJ2008-01850/ECON (Rosella Nicoli) This paper discusses a novel argument to interpret the importance of thinking of collaborative partnerships in pre-competitive agreements. To do so, we adopt a dynamic iterative process to model technology diffusion between the partners of an agreement. We find that the success of an agreement of a given length hinges around identifying the suitable efficient combinations of the initial technological endowments of partners. As the time horizon of the agreement expands, the probability of identifying a suitable partner decreases, thus justifying the prevalence of short-horizon R&D agreements

The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory. For instance, Giordano, Putnam and Skau have shown in [29] that the dimension group (see [24] for an introduction about dimension groups) of a minimal Z-action on the Cantor set completely determines its strong orbit equivalence class. Furthermore, the topological full group of such a system, which is known from Juschenko and Monod [38] to be amenable, determines its flip-conjugacy class (see [6] and [30] for more details). On the other hand, the amenability of the topological full groups of minimal Z-actions together with their properties shown in [41] by Matui make them the first known examples of infinite groups which are at the same time amenable, simple and finitely generated. Recently, another algebraic invariant, the group of automorphisms of actions on the Cantor set, has caught the eye of several researchers working in the field [13, 15, 16, 17, 14, 19, 20]. In [5], Boyle, Lind and Rudolph focused their attention on the group of automorphisms of subshifts of finite type, showing that these groups are always countable and residually finite. At the same time, they gave an example of a minimal Z-action on the Cantor set whose group of automorphisms contains Q, which implies that the automorphism group of a minimal action may be a non-residually finite group (recall that the Z-subshifts of finite type are not minimal). This leads to the natural question about the relation between the algebraic properties of the group of automorphisms and the dynamics of the system. Indeed, the residually finite property of the group of automorphisms of the subshifts of finite type is a consequence of the existence of periodic points.

To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincare, Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered.

By means of sequences of random variables of controlled growth, that is, either rapidly decreasing or slowly increasing, we define a space of random tempered generalized functions—of Schwartz type—on a locally compact abelian separable group, and we characterize the set of those generalized functions that have a mean. We show that this approach covers both the case of the torus and the real line case and also allows us to define tempered random distributions over the Cantor set by means of the associated Cantor group.

We introduce a generalization of the McMullen family $f_{\lambda}(z)=z^n+\lambda/z^d$. In 1988, C. McMullen showed that the Julia set of $f_{\lambda}$ is a Cantor set of circles if and only if $1/n+1/d, Comment: 21 pages, 2 figures

Based on a recent generalization of Bloch's theorem, we present a Bloch ansatz for the Harper model with an arbitrary rational magnetic flux in various geometries, and solve the associated ansatz equations analytically. In the case of a cylinder and a particular boundary condition, we find that the energy spectrum of edge states has no dependence on the length of the cylinder, which allows us to construct a quasi-one-dimensional edge theory that is exact and describes two edges simultaneously. We prove that energies of bulk states, generating the so-called Hofstadter's butterfly, depend on a single geometry-dependent spectral parameter and have exactly the same functional form for the cylinder and the torus with general twisted boundary conditions, and argue that the (edge) bulk spectrum of a semi-infinite cylinder in an irrational magnetic field is (the complement of) a Cantor set. Finally, realizing that the bulk projection of the Harper Hamiltonian is a linear form over a deformed Weyl algebra, we introduce a Bethe ansatz valid for both cylinder and torus geometries., Comment: 13+3 pages, 2+1 figures

In this article, for any graph G we define a new notion of “free tracing property by free G-chains” on G-like continua and we show that a positive topological entropy homeomorphism f of a G-like continuum X admits a Cantor set Z in X such that any sequence \((z_1,z_2,...,z_n)\) of points in Z is an IE-tuple of f, Z has the free tracing property by free G-chains and the minimal continuum H containing Z in X is indecomposable. Moreover, we show that the similar result can be obtained for positive topological entropy “monotone” maps. Also we give characterization theorems of G-like continua containing indecomposable subcontinua.

We show that if a Cantor set $E$ as considered by Garnett in \cite{G2} has positive Hausdorff $h$-measure for a non-decreasing function $h$ satisfying $\int^1_0r^{-3}\,h(r)^2\,dr < \infty$, then the analytic capacity of $E$ is positive. Our tool will be the Menger three-point curvature and Melnikov's identity relating it to the Cauchy kernel. We shall also prove some related more general results.

A symmetric sparse array (SSA), recursively generated, followed by Cantor set is analyzed for direction finding. Like Many Sparse Arrays with non-uniform spacing, it detect more uncorrelated sources i.e. Degree Of Freedom (DOF). The properties such as maximally economic, hole-free, systematic and symmetric array configuration attract the use of this sparse array in some practical implementation. This paper analyzes the DOA estimation performances using this SSA both in ideal and non-idealities conditions. Various numerical simulations demonstrate better performance of SSA with reduced mutual coupling effect.

In this work, we show a generalization to the ternary Cantor set based on the beta-expantion of a number, futhermore, we present that, under appropriate hypotheses, this extension also corresponds to a constructive way of the definition of the ternary Cantor set . Finally, we prove that these sets that are, in effect, Cantor sets. En el presente artículo, se expone una generalización del conjunto ternario de Cantor basada en la beta-expansión de un número; además, se presenta que, bajo hipótesis adecuadas, esta extensión también corresponde a la manera constructiva de la definición del conjunto ternario de Cantor. Finalmente, se demuestra que los conjuntos que se obtienen son, en efecto, conjuntos de Cantor.

The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To give a pointfree description of the Cantor set, we specify the frame of $\mathbb{Z}_{p}$ by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of $\mathbb{Z}_{p}$ and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the Cantor set $\mathcal{L}(\mathbb{Z}_{p})$. We prove that $\mathcal{L}(\mathbb{Z}_{p})$ is a spatial frame whose space of points is homeomorphic to $\mathbb{Z}_{p}$. In particular, we show with pointfree arguments that $\mathcal{L}(\mathbb{Z}_{p})$ is $0$-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that \emph{every compact metric space is a continuous image of the Cantor space} (see, e.g. \cite{Alexandroff} and \cite{Hausdorff}). We prove the point-free analog: if $L$ is a compact metrizable frame, then there is an injective frame homomorphism from $L$ into $\mathcal{L}(\mathbb{Z}_{2})$., 1 figure

We supply two optimal lower bounds for the solution of the analytical traveling salesman problem in terms of Jones' β-numbers. We prove a linear lower bound for the length of every connecting curve in the general case. The four corners Cantor set shows that the linear bound is attained. For connected compact sets, we prove an exponential lower bound for their length. The estimate is optimal by the work of Bishop and Jones [2] . Finally, we bridge connected and disconnected cases by taking into account multiplicity of orthogonal projections. Applications to other areas of mathematics are briefly discussed.