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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 [email protected]
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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.

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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References

Balka, R., Farkas, Á., Fraser, J. M. and Hyde, J. T.. Dimension and measure for generic continuous images. Ann. Acad. Sci. Fenn. Math. 38 (2013), 389–404.Google Scholar
Barnsley, M. F.. Superfractals. Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Barnsley, M. F., Hutchinson, J. E. and Stenflo, O.. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 13 (2005), 111–146.Google Scholar
Barnsley, M. F., Hutchinson, J. E. and Stenflo, O.. V-variable fractals: fractals with partial self similarity. Adv. Math. 218 (2008), 2051–2088.Google Scholar
Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Dissertation, University of Warwick, 1984.Google Scholar
Falconer, K. J.. Techniques in Fractal Geometry. John Wiley, Chichester, 1997.Google Scholar
Falconer, K. J.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley, Chichester, 2003.Google Scholar
Falconer, K. J.. Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986), 559–582.Google Scholar
Feng, D.-J., Hua, S. and Wen, Z.-Y.. Some relationships between packing premeasure and packing measure. Bull. Lond. Math. Soc. 31 (1999), 665–670.Google Scholar
Fraser, J. M. and Hyde, J. T.. The Hausdorff dimension of graphs of prevalent continuous functions. Real. Anal. Exchange 37 (2011/2012), 333–352.CrossRefGoogle Scholar
Fraser, J. M. and Olsen, L.. Multifractal spectra of random self-affine multifractal Sierpiński sponges in ${ \mathbb{R} }^{d} $. Indiana Univ. Math. J. 60 (2011), 937–984.CrossRefGoogle Scholar
Gatzouras, D. and Lalley, S. P.. Statistically self-affine sets: Hausdorff and box dimensions. J. Theoret. Probab. 7 (1994), 437–468.Google Scholar
Gui, Y. and Li, W.. A random version of McMullen–Bedford general Sierpinski carpets and its application. Nonlinearity 21 (2008), 1745–1758.CrossRefGoogle Scholar
Gui, Y. and Li, W.. Multiscale self-affine Sierpinski carpets. Nonlinearity 23 (2010), 495–512.Google Scholar
Haase, H.. A survey on the dimension of measures. Topology, Measures, and Fractals, Warnemünde, 1991 (Mathematics Research, 66). Akademie, Berlin, 1992, pp. 66–75.Google Scholar
Hambly, B. M.. Brownian motion on a random recursive Sierpinski gasket. Ann. Probab. 25 (1997), 1059–1102.Google Scholar
Joyce, H. and Preiss, D.. On the existence of subsets of finite positive packing measure. Mathematika 42 (1995), 15–24.Google Scholar
Lalley, S. P.. The packing and covering functions of some self-similar fractals. Indiana Univ. Math. J. 37 (1988), 699–710.Google Scholar
Liu, Y.-Y. and Wu, J.. Dimensions for random self-conformal sets. Math. Nachr. 250 (2003), 71–81.Google Scholar
Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.Google Scholar
Olsen, L.. Random Geometrically Graph Directed Self-Similar Multifractals. Longman, Harlow, 1994.Google Scholar
Olsen, L.. Fractal and multifractal dimensions of prevalent measures. Indiana Univ. Math. J. 59 (2010), 661–690.Google Scholar
Olsen, L.. Random self-affine multifractal Sierpiński sponges in ${ \mathbb{R} }^{d} $. Monatsh. Math. 162 (2011), 245–266.Google Scholar
Oxtoby, J. C.. Measure and Category, 2nd edn. Springer, New York, 1996.Google Scholar
Rand, D. A.. The singularity spectrum $f(\alpha )$ for cookie-cutters. Ergod. Th. & Dynam. Sys. 9 (1989), 527–541.Google Scholar
Rogers, C. A.. Hausdorff Measures. Cambridge University Press, Cambridge, 1998.Google Scholar
Šalát, T.. A remark on normal numbers. Rev. Roumaine Math. Pures Appl. 11 (1966), 53–56.Google Scholar
Schief, A.. Separation properties for self-similar sets. Proc. Amer. Math. Soc. 122 (1994), 111–115.Google Scholar
Schief, A.. Self-similar sets in complete metric spaces. Proc. Amer. Math. Soc. 124 (1996), 481–490.CrossRefGoogle Scholar
Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.Google Scholar
Wen, S.-Y. and Wen, Z.-Y.. Some properties of packing measure with doubling gauge. Studia Math. 165 (2004), 125–134.Google Scholar