Journal of Fractal Geometry


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Volume 4, Issue 3, 2017, pp. 257–303
DOI: 10.4171/JFG/51

Published online: 2017-09-12

On the dimensions of attractors of random self-similar graph directed iterated function systems

Sascha Troscheit[1]

(1) University of Waterloo, Canada

In this paper we propose a new model of random graph directed fractals that extends the current well-known model of random graph directed iterated function systems, $V$-variable attractors, and fractal and Mandelbrot percolation. We study its dimensional properties for similarities with and without overlaps. In particular we show that for the two classes of $1$-variable and $\infty$-variable random graph directed attractors we introduce, the Hausdorff and upper box counting dimension coincide almost surely, irrespective of overlap. Under the additional assumption of the uniform strong separation condition we give an expression for the almost sure Hausdorff and Assouad dimension.

Keywords: Self-similar, Graph directed attractor, Hausdorff dimension, Assouad dimension, Random set

Troscheit Sascha: On the dimensions of attractors of random self-similar graph directed iterated function systems. J. Fractal Geom. 4 (2017), 257-303. doi: 10.4171/JFG/51