scholarly journals Projections of fractal percolations

2013 ◽  
Vol 35 (2) ◽  
pp. 530-545 ◽  
Author(s):  
MICHAŁ RAMS ◽  
KÁROLY SIMON
Keyword(s):  
Hausdorff Dimension ◽  
Orthogonal Projection ◽  
Cantor Sets ◽  
Orthogonal Projections ◽  
Radial Projection ◽  
Random Cantor Sets ◽  
Distance Set ◽  
Distance Sets

AbstractIn this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset { \mathbb{R} }^{2} $, which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than $1$ then the orthogonal projection to every line, the radial projection with every centre, and the distance set from every point contain intervals.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

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

2009 ◽  
Vol 20 (1) ◽  
pp. 131-149 ◽  
Author(s):  
Péter Móra ◽  
Károly Simon ◽  
Boris Solomyak
Keyword(s):  
Lebesgue Measure ◽  
Cantor Sets ◽  
Random Cantor Sets ◽  
Algebraic Difference

Quasisymmetrically Minimal Moran Sets

2013 ◽  
Vol 56 (2) ◽  
pp. 292-305 ◽  
Author(s):  
Mei-Feng Dai
Keyword(s):  
Hausdorff Dimension ◽  
Cantor Sets ◽  
Equal Length

AbstractM. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor sets of Hausdorff dimension 1, where at the k-th set one removes from each interval I a certain number nk of open subintervals of length ck|I|, leaving (nk + 1) closed subintervals of equal length. Quasisymmetrically Moran sets of Hausdorff dimension 1 considered in the paper are more general than uniform Cantor sets in that neither the open subintervals nor the closed subintervals are required to be of equal length.

Hausdorff dimension and distance sets

1994 ◽  
Vol 87 (1-3) ◽  
pp. 193-201 ◽  
Author(s):  
Jean Bourgain
Keyword(s):  
Hausdorff Dimension ◽  
Distance Sets

The volume of an isotropic random parallelotope

1979 ◽  
Vol 16 (1) ◽  
pp. 84-94 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Multivariate Statistics ◽  
Orthogonal Projection ◽  
Actual Distribution ◽  
Linear Subspaces ◽  
Radial Projection ◽  
Isotropic Random

The p-content of the p-parallelotope ∇p, n determined by p independent isotropic random points z1, …, zp in ℝn (1 < p ≦ n) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇p, n. This is done explicitly in several cases. The results also have interest in multivariate statistics.

Void distribution of random Cantor sets

1989 ◽  
Vol 40 (9) ◽  
pp. 5377-5381 ◽  
Author(s):  
R. R. Tremblay ◽  
A. P. Siebesma
Keyword(s):  
Cantor Sets ◽  
Void Distribution ◽  
Random Cantor Sets

Dimensions of measures under orthogonal projections

1996 ◽  
Vol 28 (2) ◽  
pp. 344-345
Author(s):  
Martina Zähle
Keyword(s):  
Hausdorff Dimension ◽  
Packing Dimension ◽  
Packing Measure ◽  
Orthogonal Projections

Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝn given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.

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