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Dimension and measure for typical random fractals

Published online by Cambridge University Press:  27 September 2013

JONATHAN M. FRASER
JONATHAN M. FRASER*
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland email jon.fraser32@gmail.com
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Abstract

We define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 854 - 882
Copyright
© Cambridge University Press, 2013 

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