Random Cantor Sets and Their Projections

Dekking, Michel

doi:10.1007/978-3-0346-0030-9_10

Cited by 11 publications

(8 citation statements)

References 18 publications

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“…The topological properties of Mandelbrot percolation have been extensively studied, see [3,6,34]. In particular, there is a critical probability p c with 1/M < p c < 1 such that if p > p c then, conditional on non-extinction, E p contains many connected components, so its projections onto all lines necessarily have positive Lebesgue measure.…”

Section: Projections Of Percolation Setsmentioning

confidence: 99%

Self-Similar Sets: Projections, Sections and Percolation

Falconer

Jin

2017

Trends in Mathematics

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We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.

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Section: Projections Of Percolation Setsmentioning

confidence: 99%

Self-Similar Sets: Projections, Sections and Percolation

Falconer

Jin

2017

Trends in Mathematics

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“…The topological properties of Mandelbrot percolation have been studied extensively, see [11,17,78] for surveys. In particular there is a critical probability p c with 1/M < p c < 1 such that if p > p c then, conditional on non-extinction, E contains many connected components, so projections onto all lines automatically have positive Lebesgue measure.…”

Section: Projections Of Random Setsmentioning

confidence: 99%

Fractal Geometry and Stochastics V

2015

Progress in Probability

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Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.

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“…The probability space corresponding to this random construction is best described by Dekking [De09]. For the convenience of the reader we repeat it here.…”

Section: The Corresponding Probability Space and Statistical Self-simmentioning

confidence: 99%

“…There is a very nice and more detailed survey of this field due to M. Dekking [De09]. In the previous section we studied the connectivity properties and the 90 • projections of random Cantor sets.…”

Section: The Arithmetic Sum/difference Of Two Fractal Percolationsmentioning

confidence: 99%

Geometry and Analysis of Fractals

Ding¹,

Lau²

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.More information about this series at

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Random Cantor Sets and Their Projections

Cited by 11 publications

References 18 publications

Self-Similar Sets: Projections, Sections and Percolation

Self-Similar Sets: Projections, Sections and Percolation

Fractal Geometry and Stochastics V

Geometry and Analysis of Fractals

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