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The Assouad dimension of randomly generated fractals

Published online by Cambridge University Press:  22 September 2016

JONATHAN M. FRASER ,
JUN JIE MIAO  and
SASCHA TROSCHEIT Open the ORCID record for SASCHA TROSCHEIT [Opens in a new window]
JONATHAN M. FRASER
Affiliation:
School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK email jon.fraser32@gmail.com
JUN JIE MIAO
Affiliation:
Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, PR China email jjmiao@math.ecnu.edu.cn
SASCHA TROSCHEIT
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK email s.troscheit@st-andrews.ac.uk
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Abstract

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 982 - 1011
Copyright
© Cambridge University Press, 2016 

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References

Aikawa, H.. Quasiadditivity of Riesz capacity. Math. Scand. 69 (1991), 15–30.Google Scholar
Assouad, P.. Espaces métriques, plongements, facteurs. Thèse de doctorat d’état, Publ. Math. Orsay 223-7769, Université Paris XI, Orsay, 1977.Google Scholar
Assouad, P.. Étude d’une dimension métrique liée à la possibilité de plongements dans ℝ n . C. R. Acad. Sci. Paris Sér. A–B 288 (1979), 731–734.Google Scholar
Barański, K.. Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210 (2007), 215–245.Google Scholar
Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Dissertation, University of Warwick, 1984.Google Scholar
Bandt, C. and Graf, S.. Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc. 114 (1992), 995–1001.Google Scholar
Barnsley, M. F., Hutchinson, J. and Stenflo, Ö.. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 13 (2005), 111–146.Google Scholar
Barnsley, M. F., Hutchinson, J. E. and Stenflo, Ö.. V-variable fractals: fractals with partial self similarity. Adv. Math. 218 (2008), 2051–2088.Google Scholar
Barnsley, M., Hutchinson, J. E. and Stenflo, Ö.. V-variable fractals: dimension results. Forum Math. 24 (2012), 445–470.Google Scholar
Berlinkov, A. and Järvenpää, E.. Porosities in Mandelbrot percolation. Preprint, 2003, http://www.math.jyu.fi/research/pspdf/280.pdf.Google Scholar
Falconer, K. J.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley, London, 2003.Google Scholar
Falconer, K. J.. Techniques in Fractal Geometry. John Wiley, London, 1997.Google Scholar
Fraser, J. M., Henderson, A. M., Olson, E. J. and Robinson, J. C.. On the Assouad dimension of self-similar sets with overlaps. Adv. Math. 273 (2015), 188–214.Google Scholar
Falconer, K. J. and Jin, X.. Exact dimensionality and projections of random self-similar measures and sets. J. Lond. Math. Soc. 90 (2014), 388–412.Google Scholar
Fraser, J. M. and Olsen, L.. Multifractal spectra of random self-affine multifractal Sierpiński sponges in ℝ d . Indiana Univ. Math. J. 60 (2011), 937–984.Google Scholar
Fraser, J. M.. On the packing dimension of box-like self-affine sets in the plane. Nonlinearity 25 (2012), 2075–2092.Google Scholar
Fraser, J. M.. Dimension and measure for typical random fractals. Ergod. Th. & Dynam. Sys. 35 (2015), 854–882.Google Scholar
Fraser, J. M.. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366 (2014), 6687–6733.Google Scholar
Fraser, J. M. and Shmerkin, P.. On the dimensions of a family of overlapping self-affine carpets. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2014, arXiv:1405.4919.Google Scholar
Furstenberg, H.. Ergodic fractal measures and dimension conservation. Ergod. Th. & Dynam. Sys. 28 (2008), 405–422.Google Scholar
Feng, D.-J. and Wang, Y.. A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11 (2005), 107–124.Google Scholar
Gui, Y. and Li, W.. A random version of McMullen–Bedford general Sierpinski carpets and its application. Nonlinearity 21 (2008), 1745–1758.Google Scholar
Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, 2001.Google Scholar
Hyde, J. T., Laschos, V., Olsen, L., Petrykiewicz, I. and Shaw, A.. Iterated Cesàro averages, frequencies of digits, and Baire category. Acta Arith. 144 (2010), 287–293.Google Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
Järvenpää, E., Järvenpää, M. and Mauldin, R. D.. Deterministic and random aspects of porosities. Discrete Contin. Dyn. Syst. 8 (2002), 121–136.Google Scholar
Käenmäki, A., Lehrbäck, J. and Vuorinen, M.. Dimensions, Whitney covers, and tubular neighborhoods. Indiana Univ. Math. J. 62 (2013), 1861–1889.Google Scholar
Koskela, P. and Zhong, X.. Hardy’s inequality and the boundary size. Proc. Amer. Math. Soc. 131 (2003), 1151–1158.Google Scholar
Luukkainen, J.. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), 23–76.Google Scholar
Lalley, S. P. and Gatzouras, D.. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41 (1992), 533–568.Google Scholar
Lalley, S. P. and Gatzouras, D.. Statistically self-affine sets: Hausdorff and box dimensions. J. Theoret. Probab. 7 (1994), 437–468.Google Scholar
Li, W.-W., Li, W.-X., Miao, J. J. and Xi, L.-F.. Assouad dimensions of Moran sets and Cantor-like sets. Preprint, 2014, arXiv:1404.4409v3.Google Scholar
Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math. 141 (1999), 45–96.Google Scholar
Lehrbäck, J. and Tuominen, H.. A note on the dimensions of Assouad and Aikawa. J. Math. Soc. Japan 65 (2013), 343–356.Google Scholar
Mackay, J. M.. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. 15 (2011), 177–187.Google Scholar
Mandelbrot, B. B.. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62 (1974), 331–358.Google Scholar
McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.Google Scholar
Moran, P. A. P.. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15–23.Google Scholar
Mackay, J. M. and Tyson, J. T.. Conformal Dimension: Theory and Application (University Lecture Series, 54) . American Mathematical Society, Providence, RI, 2010.Google Scholar
Olsen, L.. On the Assouad dimension of graph directed Moran fractals. Fractals 19 (2011), 221–226.Google Scholar
Olson, E.. Bouligand dimension and almost Lipschitz embeddings. Pacific J. Math. 202 (2002), 459–474.CrossRefGoogle Scholar
Olson, E. J. and Robinson, J. C.. Almost bi-Lipschitz embeddings and almost homogeneous sets. Trans. Amer. Math. Soc. 362(1) (2010), 145–168.Google Scholar
Olson, E., Robinson, J. C. and Sharples, N.. Generalised Cantor sets and the dimension of products. Proc. Cambridge Philos. Soc. 160 (2016), 51–75.Google Scholar
Oxtoby, J. C.. Measure and Category, 2nd edn. Springer, New York, 1996.Google Scholar
Robinson, J. C.. Dimensions, Embeddings, and Attractors. Cambridge University Press, Cambridge, 2011.Google Scholar
Rams, M. and Simon, K.. The geometry of fractal percolation. Geometry and Analysis of Fractals (Springer Proceedings in Mathematics & Statistics, 88) . Eds. Feng, D.-J. and Lau, K.-S.. Springer, Berlin, 2014, pp. 303–324.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
Troscheit, S.. On the dimensions of random self-similar graph directed iterated function systems. J. Fractal Geom. to appear. Preprint, 2015, arXiv:1511.03461.Google Scholar
Troscheit, S.. The box dimension of random box-like self-affine sets. Preprint, 2015,arXiv:1512.07022.Google Scholar
Zerner, M. P. W.. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc. 124 (1996), 3529–3539.Google Scholar