On dimensions of visible parts of self-similar sets with finite rotation groups

Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville; Wu, Meng

doi:10.1090/proc/15843

On dimensions of visible parts of self-similar sets with finite rotation groups
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by Esa Järvenpää, Maarit Järvenpää, Ville Suomala and Meng Wu PDF

Proc. Amer. Math. Soc. 150 (2022), 2983-2995 Request permission

Abstract:

We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the weak separation condition. The bound is valid for all visible parts and it depends on the direction and the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points. Our proof relies on Furstenberg’s dimension conservation principle for self-similar sets.

References

Ida Arhosalo, Esa Järvenpää, Maarit Järvenpää, MichałRams, and Pablo Shmerkin, Visible parts of fractal percolation, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 2, 311–331. MR 2928497, DOI 10.1017/S0013091509001680
Christoph Bandt and Siegfried Graf, Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc. 114 (1992), no. 4, 995–1001. MR 1100644, DOI 10.1090/S0002-9939-1992-1100644-3
Artur Bartoszewicz, Małgorzata Filipczak, Szymon Gła̧b, Franciszek Prus-Wiśniowski, and Jarosław Swaczyna, On generating regular Cantorvals connected with geometric Cantor sets, Chaos Solitons Fractals 114 (2018), 468–473. MR 3856669, DOI 10.1016/j.chaos.2018.07.026
Mario Bonk, Juha Heinonen, and Steffen Rohde, Doubling conformal densities, J. Reine Angew. Math. 541 (2001), 117–141. MR 1876287, DOI 10.1515/crll.2001.089
Roy O. Davies and Henryk Fast, Lebesgue density influences Hausdorff measure; large sets surface-like from many directions, Mathematika 25 (1978), no. 1, 116–119. MR 492190, DOI 10.1112/S0025579300009335
Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
Kenneth J. Falconer and Jonathan M. Fraser, The visible part of plane self-similar sets, Proc. Amer. Math. Soc. 141 (2013), no. 1, 269–278. MR 2988729, DOI 10.1090/S0002-9939-2012-11312-7
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
J. M. Fraser, A. M. Henderson, E. J. Olson, and J. C. Robinson, On the Assouad dimension of self-similar sets with overlaps, Adv. Math. 273 (2015), 188–214. MR 3311761, DOI 10.1016/j.aim.2014.12.026
Jonathan M. Fraser, Douglas C. Howroyd, Antti Käenmäki, and Han Yu, On the Hausdorff dimension of microsets, Proc. Amer. Math. Soc. 147 (2019), no. 11, 4921–4936. MR 4011524, DOI 10.1090/proc/14613
Hillel Furstenberg, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 405–422. MR 2408385, DOI 10.1017/S0143385708000084
Esa Järvenpää, Maarit Järvenpää, Paul MacManus, and Toby C. O’Neil, Visible parts and dimensions, Nonlinearity 16 (2003), no. 3, 803–818. MR 1975783, DOI 10.1088/0951-7715/16/3/302
Esa Järvenpää, Maarit Järvenpää, and Juho Niemelä, Transversal mappings between manifolds and non-trivial measures on visible parts, Real Anal. Exchange 30 (2004/05), no. 2, 675–687. MR 2177426, DOI 10.14321/realanalexch.30.2.0675
Ka-Sing Lau and Sze-Man Ngai, Multifractal measures and a weak separation condition, Adv. Math. 141 (1999), no. 1, 45–96. MR 1667146, DOI 10.1006/aima.1998.1773
J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257–302. MR 63439, DOI 10.1112/plms/s3-4.1.257
J. M. Marstrand, The dimension of Cartesian product sets, Proc. Cambridge Philos. Soc. 50 (1954), 198–202. MR 60571, DOI 10.1017/s0305004100029236
Pertti Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 227–244. MR 0409774, DOI 10.5186/aasfm.1975.0110
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Pertti Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48 (2004), no. 1, 3–48. MR 2044636, DOI 10.5565/PUBLMAT_{4}8104_{0}1
Toby C. O’Neil, The Hausdorff dimension of visible sets of planar continua, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5141–5170. MR 2327025, DOI 10.1090/S0002-9947-07-04460-1
Tuomas Orponen, Slicing sets and measures, and the dimension of exceptional parameters, J. Geom. Anal. 24 (2014), no. 1, 47–80. MR 3145914, DOI 10.1007/s12220-012-9326-0
T. Orponen, On the dimension of visible parts, J. Eur. Math. Soc., to appear, arXiv:1912.10898, 2020.
Eino Rossi, Visible part of dominated self-affine sets in the plane, Ann. Fenn. Math. 46 (2021), no. 2, 1089–1103. MR 4313141, DOI 10.5186/aasfm.2021.4668
Víctor Ruiz, Dimension of homogeneous rational self-similar measures with overlaps, J. Math. Anal. Appl. 353 (2009), no. 1, 350–361. MR 2508873, DOI 10.1016/j.jmaa.2008.12.008
Martin P. W. Zerner, Weak separation properties for self-similar sets, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3529–3539. MR 1343732, DOI 10.1090/S0002-9939-96-03527-7