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The Assouad spectrum of random self-affine carpets

Part of: Smooth dynamical systems: general theory Classical measure theory

Published online by Cambridge University Press:  15 October 2020

JONATHAN M. FRASER  and
SASCHA TROSCHEIT
JONATHAN M. FRASER
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK (e-mail: [email protected])
SASCHA TROSCHEIT
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria (e-mail: [email protected])
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Abstract

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We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal and non-conformal settings. In the conformal setting, the Assouad spectrum and quasi-Assouad dimension behave rather differently, tending to almost surely coincide with the upper box dimension. Here we investigate the non-conformal setting and find that the Assouad spectrum and quasi-Assouad dimension generally do not coincide with the box dimension or Assouad dimension. We provide examples highlighting the subtle differences between these notions. Our proofs combine deterministic covering techniques with suitably adapted Chernoff estimates and Borel–Cantelli-type arguments.

Keywords

Assouad spectrumquasi-Assouad dimensionrandom self-affine carpet

MSC classification

Primary: 28A80: Fractals
Secondary: 37C45: Dimension theory of dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 10 , October 2021 , pp. 2927 - 2945
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

REFERENCES

Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Dissertation, University of Warwick, 1984.Google Scholar
Falconer, K. J.. Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. John Wiley, Hoboken, NJ, 2003.10.1002/0470013850CrossRefGoogle Scholar
Fraser, 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
Fraser, J. M., Hare, K. E., Hare, K. G., Troscheit, S. and Yu, H.. The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra. Ann. Acad. Sci. Fenn. Math. 44(1) (2019), 379–387.10.5186/aasfm.2019.4419CrossRefGoogle Scholar
Fraser, J. M., Miao, J. J. and Troscheit, S.. The Assouad dimension of randomly generated fractals. Ergod. Th. & Dynam. Sys. 38 (2018), 982–1011.10.1017/etds.2016.64CrossRefGoogle Scholar
Fraser, 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
Fraser, J. M. and Yu, H.. Assouad type spectra for some fractal families. Indiana Univ. Math. J. 67 (2018), 2005–2043.10.1512/iumj.2018.67.7509CrossRefGoogle Scholar
García, I. and Hare, K.. Properties of quasi-Assouad dimension. Preprint, 2017, http://arxiv.org/abs/1703.02526. Ann. Acad. Sci. Fenn., to appear.Google Scholar
Gatzouras, D. and Lalley, S.. Statistically self-affine sets: Hausdorff and box dimensions. J. Theoret. Probab. 7 (1994), 437–468.10.1007/BF02214277CrossRefGoogle Scholar
Gui, Y. and Li, W.. A random version of McMullen–Bedford general Sierpinski carpets and its application. Nonlinearity 21 (2008), 1745–1758.10.1088/0951-7715/21/8/004CrossRefGoogle Scholar
Luukkainen, J.. Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc. 35 (1998), 23–76.Google Scholar
Mackay, J. M.. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. 15 (2011), 177–187.10.1090/S1088-4173-2011-00232-3CrossRefGoogle Scholar
McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.10.1017/S0027763000021085CrossRefGoogle Scholar
Robinson, J. C.. Dimensions, Embeddings, and Attractors (Cambridge Tracts in Mathematics, 186). Cambridge University Press, Cambridge, 2011.Google Scholar
Troscheit, S.. The box dimension of random box-like self-affine sets. Indiana Univ. Math. J. 67 (2018), 495–535.10.1512/iumj.2018.67.7295CrossRefGoogle Scholar
Troscheit, S.. The quasi-Assouad dimension of stochastically self-similar sets. Proc. Roy. Soc. Edinburgh Sect. A 150(1) (2020), 261–275.10.1017/prm.2018.112CrossRefGoogle Scholar