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The quasi-Assouad dimension of stochastically self-similar sets

Part of: Classical measure theory Set functions and measures on spaces with additional structure Markov processes

Published online by Cambridge University Press:  24 January 2019

Sascha Troscheit
Sascha Troscheit*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Wien, Austria (saschatro@gmail.com)
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Abstract

The class of stochastically self-similar sets contains many famous examples of random sets, for example, Mandelbrot percolation and general fractal percolation. Under the assumption of the uniform open set condition and some mild assumptions on the iterated function systems used, we show that the quasi-Assouad dimension of self-similar random recursive sets is almost surely equal to the almost sure Hausdorff dimension of the set. We further comment on random homogeneous and V -variable sets and the removal of overlap conditions.

Keywords

stochastically self-similarAssouad dimensionGalton–Watson processrandom fractalsAssouad spectrumquasi-Assouad dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 28C15: Set functions and measures on topological spaces (regularity of measures, etc.) 60J85: Applications of branching processes

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 261 - 275
Copyright
Copyright © Royal Society of Edinburgh 2019

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