Intersecting self-similar Cantor sets

We define a self-similar set as the (unique) invariant set of an iterated function system of certain contracting affine functions. A topology on them is obtained (essentially) by inducing theC ¹-topology of the function space. We prove that the measure function is upper semi-continuous and give examples of discontinuities. We also show that the dimension is not upper semicontinuous. We exhibit a class of examples of self-similar sets of positive measure containing an open set. IfC ₁ andC ₂ are two self-similar setsC ₁ andC ₂ such that the sum of their dimensionsd(C ₁)+d(C ₂) is greater than one, it is known that the measure of the intersection setC ₂–C ₁ has positive measure for almost all self-similar sets. We prove that there are open sets of self-similar sets such thatC ₂–C ₁ has arbitrarily small measure.     

Waring-Hilbert problem on Cantor sets

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set $C$. It is shown that, for each $m\ge 3$, every real number in the unit interval $[0,1]$ is the sum $x_1^m+x_2^m+?+x_n^m$ with each $x_j$ in $C$ and some $n\le 6^m$. Furthermore, every real number $x$ in the interval $[0,8]$ can be written as $x=x_1^3+x_2^3+?+x_8^3$, the sum of eight cubic powers with each $x_j$ in $C$. Another Cantor set $C\times C$ is also considered. More specifically, when $C\times C$ is embedded into the complex plane $?$, the Waring–Hilbert problem on $C\times C$ has a positive answer for powers less than or equal to 4.     

Diophantine approximation and Cantor sets

We provide an explicit construction of elements of the middle third Cantor set with any prescribed irrationality exponent. This answers a question posed by Kurt Mahler.     

IFS attractors and Cantor sets

We build a metric space which is homeomorphic to a Cantor set but cannot be realized as the attractor of an iterated function system. We give also an example of a Cantor set K in R³ such that every homeomorphism f of R³ which preserves K coincides with the identity on K.     

Quasisymmetrically minimal uniform Cantor sets

The uniform Cantor set E(n,c) of Hausdorff dimension 1, defined by a bounded sequence n of positive integers and a gap sequence c, is shown to be minimal for 1-dimensional quasisymmetric maps.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

     

Cantor sets arising from continued radicals

If a ₁,a ₂,a ₃,… are nonnegative real numbers and $f_{j}(x) = \sqrt{a_{j}+x}$ , then lim _n→∞ f ₁°f ₂°?°f _n (0) is a continued radical with terms a ₁,a ₂,a ₃,…. The set of real numbers representable as a continued radical whose terms a _i are all from a set S={a,b} of two natural numbers is a Cantor set. We investigate the thickness, measure, and sums of such Cantor sets.     

Embeddings of self-similar ultrametric Cantor sets

We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R[Hdim(C)]+1, where [Hdim(C)] denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural sequence of refining coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of Rd is bi-Lipschitz embeddable in Rd+1.We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry.     

On Cantor sets and doubling measures

For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.     

Random intersections of thick Cantor sets

Let , be Cantor sets embedded in the real line, and let , be their respective thicknesses. If , then it is well known that the difference set is a disjoint union of closed intervals. B. Williams showed that for some , it may be that is as small as a single point. However, the author previously showed that generically, the other extreme is true; contains a Cantor set for all in a generic subset of . This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if , then contains a Cantor set for almost all in .

     

On arithmetic properties of Cantor sets

Science China Mathematics - In this paper, we study three types of Cantor sets. For any integer m ? 4, we show that every real number in [0, k] is the sum of at most k m-th powers of...     

Ultrametric Cantor sets and growth of measure

A class of ultrametric Cantor sets (C, d _u ) introduced recently (S. Raut and D. P. Datta, Fractals 17, 45–52 (2009)) is shown to enjoy some novel properties. The ultrametric d _u is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrization invariant, is identified with the Cantor function associated with a Cantor set $ \tilde C $ \tilde C , where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps of $ \tilde C $ \tilde C . The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrization invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension log₃ 2 and thickness 1 are constructed explicitly.     

Affine embeddings and intersections of Cantor sets

Let E,F⊂R^d$E,F\subset {\mathbb{R}}^d$ be two self-similar sets. Under mild conditions, we show that F   can be C¹$C^1$-embedded into E if and only if it can be affinely embedded into E; furthermore if F cannot be affinely embedded into E  , then the Hausdorff dimension of the intersection E∩f(F)$E\cap f(F)$ is strictly less than that of F   for any C¹$C^1$-diffeomorphism f   on R^d${\mathbb{R}}^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of E and F if F can be affinely embedded into E  . As an application, we show that _dimH?E∩f(F)<min?{_dimH?E,_dimH?F}${\dim }_H?E\cap f(F)<\min ?\{{\dim }_H?E,{\dim }_H?F\}$ when E is any Cantor-p set and F any Cantor-q   set, where p,q?2$p,q?2$ are two integers with log?p/log?q∉Q$\log ?p/\log ?q\notin \mathbb{Q}$. This is related to a conjecture of Furstenberg about the intersections of Cantor sets.     

Intersecting nonhomogeneous Cantor sets with their translations

A scheme is given to compute the Hausdorff dimensions for the intersection of a class of nonhomogeneous Cantor sets with their translations.     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].