×

When the weak separation condition implies the generalized finite type condition. (English) Zbl 1466.28010

In this well-written paper, the authors prove that an iterated function system (IFS) of similarities on \(\mathbb R\) that satisfies the weak separation condition and has an interval \([0,1]\) as its self-similar set is of generalized finite type. But it is unknown if the assumption that the self-similar set is an interval is necessary. In addition, the IFS satisfies the convex generalized finite type condition if and only if it satisfies the finite neighbour condition.

MSC:

28A80 Fractals
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Deng, Qi-Rong; Lau, Ka-Sing; Ngai, Sze-Man, Separation conditions for iterated function systems with overlaps. Fractal geometry and dynamical systems in pure and applied mathematics. I. Fractals in pure mathematics, Contemp. Math. 600, 1-20 (2013), Amer. Math. Soc., Providence, RI · Zbl 1321.28014
[2] Falconer, Kenneth, Techniques in fractal geometry, xviii+256 pp. (1997), John Wiley & Sons, Ltd., Chichester · Zbl 0869.28003
[3] Feng, De-Jun, Smoothness of the \(L^q\)-spectrum of self-similar measures with overlaps, J. London Math. Soc. (2), 68, 1, 102-118 (2003) · Zbl 1041.28004
[4] Feng, De-Jun, The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers, Adv. Math., 195, 1, 24-101 (2005) · Zbl 1078.11062
[5] Feng, De-Jun, Lyapunov exponents for products of matrices and multifractal analysis. II. General matrices, Israel J. Math., 170, 355-394 (2009) · Zbl 1181.37073
[6] Feng, De-Jun, On the topology of polynomials with bounded integer coefficients, J. Eur. Math. Soc. (JEMS), 18, 1, 181-193 (2016) · Zbl 1332.11066
[7] Feng, De-Jun; Lau, Ka-Sing, Multifractal formalism for self-similar measures with weak separation condition, J. Math. Pures Appl. (9), 92, 4, 407-428 (2009) · Zbl 1184.28009
[8] Fraser, J. M.; Henderson, A. M.; Olson, E. J.; Robinson, J. C., On the Assouad dimension of self-similar sets with overlaps, Adv. Math., 273, 188-214 (2015) · Zbl 1317.28014
[9] Hare, Kathryn E.; Hare, Kevin G.; Simms, Grant, Local dimensions of measures of finite type III-measures that are not equicontractive, J. Math. Anal. Appl., 458, 2, 1653-1677 (2018) · Zbl 1376.28008
[10] HR Kathryn E. Hare and Alex Rutar, Local dimensions of self-similar measures satisfying the finite neighbour condition, preprint. 2101.07400.
[11] Hu, Tian-You; Lau, Ka-Sing, Multifractal structure of convolution of the Cantor measure, Adv. in Appl. Math., 27, 1, 1-16 (2001) · Zbl 0991.28008
[12] Lau, Ka-Sing; Ngai, Sze-Man, Multifractal measures and a weak separation condition, Adv. Math., 141, 1, 45-96 (1999) · Zbl 0929.28007
[13] Lau, Ka-Sing; Ngai, Sze-Man, A generalized finite type condition for iterated function systems, Adv. Math., 208, 2, 647-671 (2007) · Zbl 1113.28006
[14] Lau, Ka-Sing; Ngai, Sze-Man; Rao, Hui, Iterated function systems with overlaps and self-similar measures, J. London Math. Soc. (2), 63, 1, 99-116 (2001) · Zbl 1019.28005
[15] Ngai, Sze-Man; Wang, Yang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63, 3, 655-672 (2001) · Zbl 1013.28008
[16] Nhu Nguyen, Iterated function systems of finite type and the weak separation property, Proc. Amer. Math. Soc., 130, 2, 483-487 (2002) · Zbl 0986.28010
[17] Ru Alex Rutar, Geometric and combinatorial properties of self-similar multifractal measures, preprint.2008.00197.
[18] Shmerkin, Pablo, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal., 24, 3, 946-958 (2014) · Zbl 1305.28012
[19] Va P\'eter P. Varj\'u, Recent progress on Bernoulli convolutions, Apollo-University of Cambridge Repository (Jan. 2018), https://www.repository.cam.ac.uk/handle/1810/270472. 1810.08905.
[20] Zerner, Martin P. W., Weak separation properties for self-similar sets, Proc. Amer. Math. Soc., 124, 11, 3529-3539 (1996) · Zbl 0874.54025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.