Generalised Cantor sets and the dimension of products. (English) Zbl 1371.28024
Summary: In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘\(s\)-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any \(\alpha\in (0, 1)\) and any \(\beta,\gamma\in (0, 1)\) such that \(\beta+\gamma\geq 1\) we can construct two generalised Cantor sets \(C\) and \(D\) such that \(\dim_{B}C = \alpha\beta\), \(\dim_{B}D = \alpha\gamma\), and \(\dim_{A}C = \dim_{A}D = \dim_{A}(C\times D) = \dim_{B}(C\times D) = \alpha\).
MSC:
| 28A80 | Fractals |
References:
| [1] | [1]P.AssouadÉtude d’une dimension métrique liéea la possibilité de plongements dans <![CDATA \([\mathbb{R}]]\)>^n. C.R. Acad. Sci. Paris Sér. A-B288 (1979), A731-A734. · Zbl 0409.54020 |
| [2] | [2]A.S.Besicovitch and P.A.P.MoranThe measure of product and cylinder sets. J. Lond. Math. Soc.20 (1945), 110-120.0016448 · Zbl 0063.00354 |
| [3] | [3]G.BouligandEnsembles impropres et nombre dimensionnels. Bull. Sci. Math.52 (1928), 320-344. · JFM 54.0644.03 |
| [4] | [4]G.David and S.SemmesFractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure (Oxford University Press, 1997)1616732 · Zbl 0887.54001 |
| [5] | [5]K.J.FalconerFractal Geometry: Mathematical Foundations and Applications (3rd edition) (John Wiley and Sons, England, 2014).3236784 · Zbl 1285.28011 |
| [6] | [6]D.Feng, Z.Wen and J.WuSome dimensional results for homogeneous Moran setsSci. China Math.40 (1997), no. 5, 475-482.10.1007/BF02896955 · Zbl 0881.28003 |
| [7] | [7]J.FraserAssouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc.366 (2014), 6687-6733.10.1090/S0002-9947-2014-06202-83267023 · Zbl 1305.28021 · doi:10.1090/S0002-9947-2014-06202-8 |
| [8] | [8]J.Fraser, J.-J.Miao and S.Troscheit The Assouad dimension of randomly generated fractals. ArXiv Preprint arXiv:1410.6949v2 · Zbl 1392.37024 |
| [9] | [9]A.M.Henderson, E.J.Olson, J. C.Robinson and N.Sharples Equi-homogeneity, Assouad dimension and non-autonomous dynamics. (Submitted). |
| [10] | [10]J.D.HowroydOn dimension and on the existence of sets of finite positive Hausdorff measure. Proc. Lond. Math. Soc.s3-70 (1995), no. 3, 581-604.10.1112/plms/s3-70.3.581 · Zbl 0828.28002 |
| [11] | [11]D.G.LarmanA new theory of dimension. Proc. Lond. Math. Soc.s3-17 (1967), no. 1, 178-192.10.1112/plms/s3-17.1.178 · Zbl 0152.24502 |
| [12] | [12]W.-W.Li, W.-X.Li, J.-J.Miao and L.-F.Xi Assouad dimensions of Moran sets and Cantor-like sets. ArXiv Preprint arXiv:1404.4409v3 · Zbl 1364.28011 |
| [13] | [13]F.Lü, M.-L.Lou, Z.-Y.Wen and L.-F.Xi Bilipschitz embeddings of homogeneous fractals. ArXiv Preprint arXiv:1402.0080 |
| [14] | [14]J.LuukkainenAssouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc.35 (1998), no. 1, 23-76.1608518 · Zbl 0893.54029 |
| [15] | [15]P.A.P.MoranAdditive functions of intervals and Hausdorff measure. Math. Proc. Camb. Phil. Soc.42 (1946), no. 1, 15-23.10.1017/S0305004100022684S0305004100022684 · Zbl 0063.04088 |
| [16] | [16]E.J.OlsonBouligand dimension and almost Lipschitz embeddings. Pacific J. Math.202 (2002), no. 2, 459-474.10.2140/pjm.2002.202.4591888175 · Zbl 1046.37013 |
| [17] | [17]E.J.Olson and J.C.Robinson A simple example concerning the upper box-counting dimension of a Cartesian product. Real Anal. Exchange (2014), (to appear). |
| [18] | [18]J.C.RobinsonInfinite-dimensional dynamical systems: an introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge Texts in Applied Math. (Cambridge University Press, 2001).10.1007/978-94-010-0732-01881888 · Zbl 0980.35001 |
| [19] | [19]J.C.RobinsonDimensions, embeddings, and attractors. Cambridge Tracts in Math., no. 186 (Cambridge University Press, 2011). · Zbl 1222.37004 |
| [20] | [20]J.C.Robinson and N.SharplesStrict inequality in the box-counting dimension product formulas. Real Anal. Exchange38 (2013), no. 1, 95-119. · Zbl 1277.28014 |
| [21] | [21]D.ŽubrinićAnalysis of Minkowski contents of fractal sets and applications. Real Anal. Exchange31 (2005/2006) no. 2, 315-354.2265777 · Zbl 1142.37315 |
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