Hausdorff dimension of the random middle third Cantor set

Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.

3-Adic Cantor function on local fields and its p-adic derivative

For any $0 < \alpha <1$, we construct Cantor sets on the parabola of Hausdorff dimension $\alpha $ such that they are Salem sets and each associated measure $\nu $ satisfies the estimate $\left \| \widehat {f d\nu } \right \|_{L^{p}({\mathbb {R}}^{2})} \leq C_{p} \left \| f \right \|_{L^{2}(\nu )}$ for all $p>6/\alpha $ and for some constant $C_{p}>0$,which may depend on $p$ and $\nu $. The range $p>6/\alpha $ is optimal except for the endpoint. The proof is based on the work of Łaba and Wang on restriction estimates for random Cantor sets and the work of Shmerkin and Suomala on Fourier decay of measures on random Cantor sets. They considered fractal subsets of ${\mathbb {R}}^{d}$, while we consider fractal subsets of the parabola.

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

Let F1 and F2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1) + dimH(F2) > 1 imply that their algebraic difference F1 ? F2 will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.

The Lebesgue measure of the algebraic difference of two random Cantor sets

“Plate tectonics” is an observed fact and most models of earthquake incorporate it through the frictional dynamics (stick-slip) of two surfaces where one surface moves over the other. These models are more or less successful to reproduce the well known Gutenberg–Richter type power law in the (released) energy distribution of earthquakes. During a sticking period, the elastic energy gets stored at the contact area of the surfaces and is released when a slip occurs. Therefore, the extent of the contact area between two surfaces plays an important role in the earthquake dynamics and the power law in energy distribution might imply a similar law for the contact area distribution. Since fractured surfaces are fractals and tectonic plate-earth’s crust interfaces can be considered to have fractal nature, we study here the contact area distribution between two fractal surfaces. We consider the overlap set (m) of two self-similar fractals, characterized by the same fractal dimensions (df), and look for their distribution P(m). We have studied numerically the specific cases of both regular and random Cantor sets (in the embedding dimension d = 1), gaskets and percolation fractals (in d = 2). We find that in all the cases the distributions show an universal finite size (L) scaling behavior P'(m') = LαP(m, L ); m' = mL-α, where α = 2(df - d). The P(m), and consequently the scaled distribution P'(m'), have got a power law decay with m (with decay exponent equal to d) for both regular and random Cantor sets and also for gaskets. For percolation clusters, P(m) (and hence P'(m')) have a Gaussian variation with m.

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.

Abstract. The predictability of the monthly North Atlantic Oscillation, NAO, index is analysed from the point of view of different fractal concepts and dynamic system theory such as lacunarity, rescaled analysis (Hurst exponent) and reconstruction theorem (embedding and correlation dimensions, Kolmogorov entropy and Lyapunov exponents). The main results point out evident signs of randomness and the necessity of stochastic models to represent time evolution of the NAO index. The results also show that the monthly NAO index behaves as a white-noise Gaussian process. The high minimum number of nonlinear equations needed to describe the physical process governing the NAO index fluctuations is evidence of its complexity. A notable predictive instability is indicated by the positive Lyapunov exponents. Besides corroborating the complex time behaviour of the NAO index, present results suggest that random Cantor sets would be an interesting tool to model lacunarity and time evolution of the NAO index.

Random Cantor set models for the elastic-perfectly plastic contact of rough surfaces

The working hypotheses of the fixed-scale transformation as applied to the percolation problem in two dimensions are checked with Monte Carlo simulations. The set of points given by the intersection of a line and the infinite cluster is correctly described by a random Cantor set built with the generators proposed in the theory. But the matrix of the fixed-scale transformation that describes growth at the same scale is not scale independent. A simple explanation is given for this behavior. Keeping the spirit of the theory, it is proposed to coarse grain the infinite cluster with a two-dimensional random Cantor set. In this case, the matrix describing the transitions with respect to growth between the different diagrams is scale independent and is not a unit matrix. One can in principle extract the fractal dimension from it.

At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set. It is well-known that not every Cantor set on the line is an s-set for some 0 < s < 1. However, if the sequence associated to the Cantor set C is nonincreasing, we show that C is an h-set for some continuous, concave dimension function h. We construct the function h from the sequence associated to the set C.

On quasisymmetric minimality of Cantor sets

We consider Cantor sets CX,p constructed by means of a sequence {Xk−p}k=1∞ of random variables. We find sufficient conditions for the inequality \(0 < H^{\frac{1}{p}} (C_{X,p} ) < \infty \) to hold.