Self-similar Cantor Set Research Articles

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.

The Buffon's needle problem for random planar disk-like Cantor sets

We consider a model of randomness for self-similar Cantor sets of finite and positive 1-Hausdorff measure. We find the sharp rate of decay of the probability that a Buffon needle lands δ-close to a Cantor set of this particular randomness. Two quite different models of randomness for Cantor sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order of decay for the Buffon needle probability: clog⁡1δ. In this note, we prove the same rate of decay for a third model of randomness, which asserts a vague feeling that any “reasonable” random Cantor set of positive and finite length will have Favard length of order clog⁡1δ for its δ-neighbourhood. The estimate from below was obtained long ago by Mattila.

A new simple family of Cantor sets in [formula omitted] all of whose projections are one-dimensional

In 1994, J. Cobb described a Cantor set in R3 each of whose projections into 2-planes is one-dimensional. A series of works by other authors developing this field followed. We present another very simple series of Cantor sets in R3 all of whose projections are connected and one-dimensional. These are self-similar Cantor sets which go back to the work of Louis Antoine, and we celebrate their centenary birthday in 2020–2021.

Sums of two self-similar Cantor sets

We show that for any pair of self-similar Cantor sets with sum of Hausdorff dimensions greater than 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of self-similar Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis' conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis' conjecture.

SPECTRAL CLASSES AND THE PARAMETER DISTRIBUTION SET

Abstract. The natural projection of a parameter lower (upper) distri-bution set for a self-similar measure on a self-similar set satisfying theopen set condition is the cylindrical lower or upper local dimension setfor the Legendre self-similarmeasure which is derived from the self-similarmeasure and the self-similar set. 1. IntroductionRecently, we [1] investigated the relation between spectral classes of a self-similar Cantor set in a set theoretical sense. More recently, using the parameterdistribution, we find the parallel results for the self-similar set (attractor of theIFS consisting of n(≥ 2) similitudes satisfying the OSC (open set condition))instead of the self-similar Cantor set (attractor of the IFS consisting of 2 simil-itudes satisfying the SSC (strong separation condition)), which leads to a gen-eralization of [1]. In this paper, we define the Legendre self-similar measureson the self-similar set which is derived from the self-similar measure and theself-similar set. Using the Legendre self-similar measures on the self-similar set,we give full relationship between the natural projection of a parameter lower(upper) distribution set for a self-similar measure on a self-similar set and thecylindrical lower or upper local dimension set for the Legendre self-similar mea-sures.2. PreliminariesLet N and R be the set of positive integers and the set of real numbersrespectively. An attractor K in the d-dimensional Euclidean space R

Buffon's needle landing near Besicovitch irregular self-similar sets

In this paper we get an estimate of Favard length of an arbitrary neighbourhood of an arbitrary self-similar Cantor set. Consider $L$ closed disjoint discs of radius $1/L$ inside the unit disc. By using linear maps of smaller disc onto the unit disc we can generate a self-similar Cantor set $G$. Then $\G=\bigcap_n\G_n$. One may then ask the rate at which the Favard length - the average over all directions of the length of the orthogonal projection onto a line in that direction - of these sets $\G_n$ decays to zero as a function of $n$. The quantitative results for the Favard length problem were obtained by Peres-Solomyak and Tao; in the latter paper a general way of making a quantitative statement from the Besicovitch theorem is considered. But being rather general, this method does not give a good estimate for self-similar structures such as $\G_n$. Indeed, vastly improved estimates have been proven in these cases: in the paper of Nazarov-Peres-Volberg, it was shown that for 1/4 corner Cantor set one has $p 0$. In the present work we give an estimate that works for {\it any} Besicovitch set which is self-similar. However estimate is worse than the power one. The power estimate still appears to be related to a certain regularity property of zeros of a corresponding linear combination of exponents (we call this property {\it analytic tiling}).

Transverse Laplacians for Substitution Tilings

Pearson and Bellissard recently built a spectral triple - the data of Riemanian noncommutative geometry - for ultrametric Cantor sets. They derived a family of Laplace-Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to implement it. We show that the abscissa of convergence of the zeta-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace-Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.

On multifractal of Cantor dust

We consider quasi-self-similar measures with respect to all real numbers on a Cantor dust. We define a local index function on the real numbers for each quasi-self-similar measure at each point in a Cantor dust. The value of the local index function at the real number zero for all the quasi-self-similar measures at each point is the weak local dimension of the point. We also define transformed measures of a quasi-self-similar measure which are closely related to the local index function. We compute the local dimensions of transformed measures of a quasi-self-similar measure to find the multifractal spectrum of the quasi-self-similar measure. Furthermore we give an essential example for the theorem of local dimension of transformed measure. In fact, our result is an ultimate generalization of that of a self-similar measure on a self-similar Cantor set. Furthermore the results also explain the recent results about weak local dimensions on a Cantor dust.

On the exact Hausdorff measure of a class of self-similar sets satisfying open set condition

In this paper, we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition (OSC). As applications, we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.

DIMENSIONS OF DISTRIBUTION SETS IN THE UNIT INTERVAL

The unit interval is not homeomorphic to a self-similar Cantor set in which we studied the dimensions of distribution subsets. However we show that similar results regarding dimensions of the distribution subsets also hold for the unit interval since the distribution subsets have similar structures with those in a self-similar Cantor set.

SIMPLE APPROACH TO MULTIFRACTAL SPECTRUM OF A SELF-SIMILAR CANTOR SET

We study the transformed measures with respect to the real parameters of a self-similar measure on a self-similar Can­tor set to give a simple proof for some result of its multifractal spectrum. A transformed measure with respect to a real parameter of a self-similar measure on a self-similar Cantor set is also a self­similar measure on the self-similar Cantor set and it gives a better information for multifractals than the original self-similar measure. A transformed measure with respect to an optimal parameter deter­mines Hausdorff and packing dimensions of a set of the points which has same local dimension for a self-similar measure. We compute the values of the transformed measures with respect to the real parameters for a set of the points which has same local dimension for a self-similar measure. Finally we investigate the magnitude of the local dimensions of a self-similar measure and give some correlation between the local dimensions.

PACKING DIMENSION OF MEASURES ON A RANDOM CANTOR SET

Packing dimension of a set is an upper bound for the packing dimensions of measures on the set. Recently the packing dimension of statistically self-similar Cantor set, which has uniform distributions for contraction ratios, was shown to be its Hausdorff dimension. We study the method to find an upper bound of packing dimensions and the upper Renyi dimensions of measures on a statistically quasi-self-similar Cantor set (its packing dimension is still unknown) which has non-uniform distributions of contraction ratios. As results, in some statistically quasi-self-similar Cantor set we show that every probability measure on it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely and it has its subset of full measure whose packing dimension is also its Hausdorff dimension almost surely for almost all probability measure on it.

Relation between spectral classes of a self-similar Cantor set

A self-similar Cantor set is completely decomposed as a class of the lower (upper) distribution sets. We give a relationship between the distribution sets in the distribution class and the subsets in a spectral class generated by the lower (upper) local dimensions of a self-similar measure. In particular, we show that each subset of a spectral class is exactly a distribution set having full measure of a self-similar measure related to the distribution set using the strong law of large numbers. This gives essential information of its Hausdorff and packing dimensions. In fact, the spectral class by the lower (upper) local dimensions of every self-similar measure, except for a singular one, is characterized by the lower or upper distribution class. Finally, we compare our results with those of other authors.

A note on Whitney's critical sets*

Abstract The critical arc is an arc which is the critical set for some differentiable function not constant along the arc. In literature on the topic the differentiable functions are always increasing along the arc but in this paper we provide a critical arc such that the corresponding differentiable function could not be increasing along the arc. It is also proved that any self-similar Cantor set of dimension greater than 1 is a generalized Whitney's critical set.

Scattering map for two black holes

We study the motion of light in the gravitational field of two Schwarzschild black holes, making the approximation that they are far apart, so that the motion of light rays in the neighborhood of one black hole can be considered to be the result of the action of each black hole separately. Using this approximation, the dynamics is reduced to a two-dimensional map, which we study both numerically and analytically. The map is found to be chaotic, with a fractal basin boundary separating the possible outcomes of the orbits (escape or falling into one of the black holes). In the limit of large separation distances, the basin boundary becomes a self-similar Cantor set, and we find that the box-counting dimension decays slowly with the separation distance, following a logarithmic decay law.

Self-Similarity and Probability: Parameters Describing the Geometry of Cantor Sets

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated. 1991 Mathematics Subject Classification 28A80, 60D05.

Removable singularities for hardy spaces

In this paper we study removable singularities for Hardy spaces of analytic funtions on general domains. Two different definitions are given. For compact sets they turn out to be equal and moreover independent of the surrounding domain, as was proved by D. A Hejhal For non-compact sets the difference between the definitions is studied. A survey of the present knowledge is given, except for the special cases of singularities lying on curves and singularities being self-similar Cantor sets, which the author deals with in other papers. Among the results is the non-removability for Hp of sets with dimension greater than ρ. 0 < ρ < 1. Many counterexamples are provided and the Hp capacities are introduced and studied.

Correlation dimension for self-similar Cantor sets with overlaps

Correlation dimension for self-similar Cantor sets with overlaps

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.

Self-similar Cantor sets and quasiregular mappings.

Self-similar Cantor sets and quasiregular mappings.