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On Sums of Nearly Affine Cantor Sets  [PDF]
Anton Gorodetski,Scott Northrup
Mathematics , 2015,
Abstract: For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_\lambda\}_{\lambda\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ for all $\lambda\in J$, under natural technical conditions we prove that the sum $K+C_\lambda$ has positive Lebesgue measure for almost all values of the parameter $\lambda$. As a corollary, we show that generically the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one.
On the arithmetic difference of middle Cantor sets  [PDF]
M. Pourbarat
Mathematics , 2013,
Abstract: Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all triples (that are dense in $\mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$) has been determined such that $C_\alpha- \lambda C_\beta$ forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets $\mathcal{P}$ in a way that for each $(C_\alpha,~ C_\beta)\in\mathcal{P}$, there exists a dense subfield $F\subset \mathbb{R}$ such that for each $\mu \in F$, the set $C_\alpha- \mu C_\beta$ contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions $f, ~g$ and the pair $(C_\alpha,~C_\beta)$ is provided which $f(C_{\alpha})- g(C_{\beta})$ contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one. At the end, special middle Cantor sets $C_\alpha$ and $C_\beta$ are introduced. Then the iterated function system corresponding to the attractor $C_{\alpha}-\frac{2\alpha}{\beta}C_\beta$ is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that $\sqrt{C_{\alpha}}$ - $\sqrt{C_{\beta}}$ contains at least one interval.
A curious arithmetic of fractal dimension for polyadic Cantor sets  [PDF]
Francisco R. Villatoro
Mathematics , 2009,
Abstract: Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic Cantor sets is introduced by means of properly defining operators for the addition, subtraction, multiplication, and division. The new operators have the usual properties of the corresponding operations with real numbers. The combination of an infinitesimal change of fractal dimension with these arithmetic operators allows the manipulation of fractal dimension with the tools of calculus.
Moreira's Theorem on the arithmetic sum of dynamically defined Cantor sets  [PDF]
Pablo Shmerkin
Mathematics , 2008,
Abstract: We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller.
Stable intersection of middle-$α$ Cantor sets  [PDF]
M. Pourbarat
Mathematics , 2013,
Abstract: In the present paper, We introduce a pair of middle Cantor sets namely $(C_\alpha, C_\beta)$ having stable intersection, while the product of their thickness is smaller than one. Furthermore, the arithmetic difference $C_\alpha- \lambda C_\beta$ contains at least one interval for each nonzero number $\lambda$.
Resonance between Cantor sets  [PDF]
Yuval Peres,Pablo Shmerkin
Mathematics , 2007,
Abstract: Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where $\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\dim(K)+\dim(K') \le 1$ (``geometric resonance''), then there exists $r<1$ such that all contraction ratios of the similitudes defining $K$ and $K'$ are powers of $r$ (``algebraic resonance''). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.
Julia Sets and wild Cantor sets  [PDF]
Alastair Fletcher,Jang-Mei Wu
Mathematics , 2014,
Abstract: There exist uniformly quasiregular maps $f:\mathbb{R}^3 \to \mathbb{R}^3$ whose Julia sets are wild Cantor sets.
Some remarks about Cantor sets  [PDF]
Stephen Semmes
Mathematics , 2004,
Abstract: These informal notes briefly discuss various aspects of Cantor sets.
Visible and Invisible Cantor sets  [PDF]
Carlos Cabrelli,Udayan Darji,Ursula Molter
Mathematics , 2011,
Abstract: In this article we study for which Cantor sets there exists a gauge-function h, such that the h-Hausdorff-measure is positive and finite. We show that the collection of sets for which this is true is dense in the set of all compact subsets of a Polish space X. More general, any generic Cantor set satisfies that there exists a translation-invariant measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e. a Cantor set for which any translation invariant measure is either zero or non-sigma-finite, that enables us to show that the collection of these sets is also dense in the set of all compact subsets of a Polish space X.
On Cantor sets and doubling measures  [PDF]
Marianna Cs?rnyei,Ville Suomala
Mathematics , 2012,
Abstract: 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.
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