Hausdorff dimension and distance sets

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

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 Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion
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