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The quasi-Assouad dimension of stochastically self-similar sets

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

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 ([email protected])
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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
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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References

1Angelevska, J. and Troscheit, S.. A dichotomy of self-conformal subsets of ℝ with overlaps. preprint.Google Scholar
2Assouad, P.. Espaces métriques, plongements, facteurs. Ph.D. thesis, Univ, Paris XI Orsay, 1977.Google Scholar
3Assouad, 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), A731–A734.Google Scholar
4Athreya, K. B.. Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab. 4 (1994), 779–790.10.1214/aoap/1177004971CrossRefGoogle Scholar
5Athreya, K. B. and Ney, P. E.. Branching processes. Die Grundlehren der mathematischen Wissenschaften, Band 196 (New York-Heidelberg: Springer-Verlag, 1972).10.1007/978-3-642-65371-1CrossRefGoogle Scholar
6Barnsley, M., Hutchinson, J., Stenflo, Ö.. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 13 (2005), 111–146.10.1142/S0218348X05002799CrossRefGoogle Scholar
7Barnsley, M. F., Hutchinson, J. E. and Stenflo, Ö.. V -variable fractals: fractals with partial self similarity. Adv. Math. 218 (2008), 2051–2088.CrossRefGoogle Scholar
8Barnsley, M., Hutchinson, J. E. and Stenflo, Ö.. V -variable fractals: dimension results. Forum Math. 24 (2012), 445–470.CrossRefGoogle Scholar
9Berlinkov, A. and Järvenpää, E.. Porosities of Mandelbrot Percolation. ArXiv e-prints.Google Scholar
10Falconer, K. J.. Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986), 559–582.10.1017/S0305004100066299CrossRefGoogle Scholar
11Falconer, K. and Jin, X.. Dimension conservation for self-similar sets and fractal percolation. Int. Math. Res. Not. IMRN 24 (2015), 13260–13289.10.1093/imrn/rnv103CrossRefGoogle Scholar
12Fraser, J. M.. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366 (2014), 6687–6733.10.1090/S0002-9947-2014-06202-8CrossRefGoogle Scholar
13Fraser, J. M. and Yu, H.. Assouad type spectra for some fractal families. Indiana U. Math. J. 67 (2018), 2005–2043.CrossRefGoogle Scholar
14Fraser, J. M. and Yu, H.. New dimension spectra: finer information on scaling and homogeneity. Adv. Math. 329 (2018), 273–328.10.1016/j.aim.2017.12.019CrossRefGoogle Scholar
15Fraser, J. M., Miao, J.-J. and Troscheit, S.. The Assouad dimension of randomly generated fractals. Ergodic Theory Dynam. Systems 38 (2018), 982–1011.10.1017/etds.2016.64CrossRefGoogle Scholar
16Fraser, 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), 188–214.10.1016/j.aim.2014.12.026CrossRefGoogle Scholar
17García, I. and Hare, K.. Properties of Quasi-Assouad dimension. ArXiv e-prints.Google Scholar
18Garcia, I., Hare, K. and Mendivil, F.. Assouad dimensions of complementary sets. Proc. Royal Soc. Edinb. A 148 (2018), 517–540.10.1017/S0308210517000488CrossRefGoogle Scholar
19Graf, S.. Statistically self-similar fractals. Probab. Theory Related Fields 74 (1987), 357–392.10.1007/BF00699096CrossRefGoogle Scholar
20Hambly, B. M.. Brownian motion on a homogeneous random fractal. Probab. Theory Related Fields 94 (1992), 1–38.10.1007/BF01222507CrossRefGoogle Scholar
21Järvenpää, E., Järvenpää, M., Käenmäki, A., Koivusalo, H., Stenflo, O. and Suomala, V.. Dimensions of random affine code tree fractals. Ergodic Theory Dynam. Systems 34 (2014), 854–875.10.1017/etds.2012.168CrossRefGoogle Scholar
22Järvenpää, E., Järvenpää, M., Li, B., Stenflo, Ö.. Random affine code tree fractals and Falconer-Sloan condition. Ergodic Theory Dynam. Systems 36 (2016), 1516–1533.CrossRefGoogle Scholar
23Järvenpää, E., Järvenpää, M., Wu, M. and Wu, W.. Random affine code tree fractals: Hausdorff and affinity dimensions and pressure. Math. Proc. Cambridge Philos. Soc. 162 (2017), 367–382.10.1017/S0305004116000694CrossRefGoogle Scholar
24Käenmäki, A. and Rossi, E.. Weak separation condition. Assouad dimension, and Furstenberg homogeneity. Ann. Acad. Sci. Fenn. Math. 41 (2016), 465–490.10.5186/aasfm.2016.4133CrossRefGoogle Scholar
25Lü, F. and Xi, L.-F.. Quasi-Assouad dimension of fractals. J. Fractal Geom. 3 (2016), 187–215.CrossRefGoogle Scholar
26Luukkainen, J.. Assouad dimension: antifractal metrization. porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), 23–76.Google Scholar
27Mackay, J. M.. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. 15 (2011), 177–187.10.1090/S1088-4173-2011-00232-3CrossRefGoogle Scholar
28Olsen, L.. Random self-affine multifractal Sierpinski sponges in ℝd. Monatsh. Math. 162 (2011), 89–117.CrossRefGoogle Scholar
29Olson, E. J., Robinson, J. C. and Sharples, N.. Generalised Cantor sets and the dimension of products. Math. Proc. Cambridge Philos. Soc. 160 (2016), 51–75.CrossRefGoogle Scholar
30Troscheit, S.. Dimension theory of random self-similar and self-affine constructions. Ph.D. thesis, University of St Andrews, April (2017).Google Scholar
31Troscheit, S.. On the dimensions of attractors of random self-similar graph directed iterated function systems. J. Fractal Geom. 4 (2017), 257–303.10.4171/JFG/51CrossRefGoogle Scholar