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Dimensions of an overlapping generalization of Barański carpets

Published online by Cambridge University Press:  25 September 2017

LETICIA PARDO-SIMÓN
LETICIA PARDO-SIMÓN*
Affiliation:
University of Liverpool, Mathematical Sciences, Liverpool L69 7ZL, UK
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Abstract

We determine the Hausdorff, the packing and the box-counting dimensions of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the structure of the rows and columns. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulae will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and the Hausdorff dimension do not necessarily agree. These results rely on Hochman’s recent work on the dimensions of self-similar sets and measures, and can be seen as an extension of Fraser and Shmerkin [On the dimensions of a family of overlapping self-affine carpets. Ergod. Th. & Dynam. Sys.doi: 10.1017/etds.2015.21. Published online: 21 July 2015] results for Bedford–McMullen carpets with columns overlapping.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 733 - 763
Copyright
© Cambridge University Press, 2017 

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References

Barański, K.. Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210(1) (2007), 215–245.Google Scholar
Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
Falconer, K.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103(2) (1988), 339–350.Google Scholar
Falconer, K.. Generalized dimensions of measures on self-affine sets. Nonlinearity 12(4) (1999), 877.Google Scholar
Falconer, K.. Fractal Geometry, 3rd edn. John Wiley and Sons, Hoboken, NJ, 2014.Google Scholar
Feng, D. J. and Hu, H.. Dimension theory of iterated function systems. Comm. Pure Appl. Math. 62(11) (2009), 1435–1500.Google Scholar
Ferguson, A., Jordan, T. and Shmerkin, P.. The Hausdorff dimension of the projections of self-affine carpets. Fund. Math. 209(3) (2010), 193–213.Google Scholar
Fraser, J.. On the packing dimension of box-like self-affine sets in the plane. Nonlinearity 25(7) (2012), 2075–2092.Google Scholar
Fraser, J. and Shmerkin, P.. On the dimensions of a family of overlapping self-affine carpets. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2015.21. Published online: 21 July 2015.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
Hueter, I. and Lalley, S. P.. Falconer’s formula for the Hausdorff dimension of a self-affine set in ℝ2 . Ergod. Th. & Dynam. Sys. 15 (1995), 77–97.Google Scholar
Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2) 26(3) (2014), 773–822.Google Scholar
Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^{d}$ . Mem. Amer. Math. Soc., to appear, Preprint, 2015, arXiv:1503.09043v1.Google Scholar
Howroyd, J. D.. On Hausdorff and packing dimension of product spaces. Math. Proc. Cambridge Philos. Soc. 119 (1996), 715–727.Google Scholar
Hutchinson, J. E.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.Google Scholar
Jordan, T., Pollicott, M. and Simon, K.. Hausdorff dimension for randomly perturbed self-affine attractors. Comm. Math. Phys. 270(2) (2007), 519–544.Google Scholar
Käenmäki, A. and Shmerkin, P.. Overlapping self-affine sets of Kakeya type. Ergod. Th. & Dynam. Sys. 29 (2009), 941–965.Google Scholar
Lalley, S. P. and Gatzouras, D.. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41(2) (1992), 533–568.Google Scholar
Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge University Press, Cambridge, 1999, p. 44.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. Math. Proc. Cambridge Philos. Soc. 42(1) (1946), 15–23.Google Scholar
Peres, Y. and Solomyak, B.. Problems on self-similar sets and self-affine sets: an update. Fractal Geom. Stoch. II 46 (2000), 95–106.Google Scholar
Przytycki, F. and Urbański, M.. On the Hausdorff dimension of some fractal sets. Stud. Math. 93(2) (1989.), 155–186.Google Scholar
Shmerkin, P.. Overlapping self-affine sets. Indiana Univ. Math. J. 55 (2006), 1291–1331.Google Scholar
Solomayak, B.. Measure and dimension for some fractal families. Math. Proc. Cambridge Philos. Soc. 124 (1998), 531–546.Google Scholar