Geometric and combinatorial properties of self-similar multifractal measures. (English) Zbl 07682668

Rutar, Alex

Ergodic Theory Dyn. Syst. 43, No. 6, 2028-2072 (2023).

Summary: For any self-similar measure \(\mu\) in \(\mathbb{R}\), we show that the distribution of \(\mu\) is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of \(\mu\) to certain compact subsets of \(\mathbb{R}\), determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some \(q\in\mathbb{R}\), there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

MSC:

28A80	Fractals
37C45	Dimension theory of smooth dynamical systems

Keywords:

iterated function system; self-similar; multifractal analysis; weak separation condition

References:

[1]	Arbeiter, M. and Patzschke, N.. Random self-similar multifractals. Math. Nachr.181(1) (1996), 5-42. · Zbl 0873.28003
[2]	Bandt, C. and Graf, S.. Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc.114(4) (1992), 995-1001. · Zbl 0823.28003
[3]	Cawley, R. and Mauldin, R. D.. Multifractal decompositions of Moran fractals. Adv. Math.92(2) (1992), 196-236. · Zbl 0763.58018
[4]	Deng, G. and Ngai, S.-M.. Differentiability of \({L}^q\) -spectrum and multifractal decomposition by using infinite graph-directed IFSs. Adv. Math.311 (2017), 190-237. · Zbl 1459.28004
[5]	Deng, Q.-R., Lau, K.-S. and Ngai, S.-M.. Separation Conditions for Iterated Function Systems with Overlaps(Contemporary Mathematics, 600). Eds. Carfì, D., Lapidus, M., Pearse, E. and Van Frankenhuijsen, M.. American Mathematical Society, Providence, RI, 2013, pp. 1-20.
[6]	Falconer, K. J.. Techniques in Fractal Geometry. Wiley, Chichester, NY, 1997. · Zbl 0869.28003
[7]	Feng, D.-J.. Lyapunov exponents for products of matrices and multifractal analysis. Part I: Positive matrices. Israel J. Math.138(1) (2003), 353-376. · Zbl 1037.37023
[8]	Feng, D.-J.. Smoothness of the \({L}^q\) -spectrum of self-similar measures with overlaps. J. Lond. Math. Soc. (2)68(01) (2003), 102-118. · Zbl 1041.28004
[9]	Feng, D.-J.. The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math.195(1) (2005), 24-101. · Zbl 1078.11062
[10]	Feng, D.-J.. Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices. Israel J. Math.170(1) (2009), 355-394. · Zbl 1181.37073
[11]	Feng, D.-J.. On the topology of polynomials with bounded integer coefficients. J. Eur. Math. Soc. (JEMS)18(1) (2016), 181-193. · Zbl 1332.11066
[12]	Feng, D.-J. and Hu, H.. Dimension theory of iterated function systems. Comm. Pure Appl. Math.62(11) (2009), 1435-1500. · Zbl 1230.37031
[13]	Feng, D.-J. and Lau, K.-S.. Multifractal formalism for self-similar measures with weak separation condition. J. Math. Pures Appl. (9)92(4) (2009), 407-428. · Zbl 1184.28009
[14]	Feng, D.-J., Lau, K.-S. and Wang, X.-Y.. Some exceptional phenomena in multifractal formalism: Part II. Asian J. Math.9(4) (2005), 473-488. · Zbl 1134.28008
[15]	Feng, D.-J., Lau, K.-S. and Wu, J.. Ergodic limits on the conformal repellers. Adv. Math.169(1) (2002), 58-91. · Zbl 1033.37017
[16]	Fraser, J. M., Henderson, A. M., Olson, E. J. and Robinson, J. C.. On the Assouad dimension of self-similar sets with overlaps. Adv. Math.273 (2015), 188-214. · Zbl 1317.28014
[17]	Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. and Shraiman, B. I.. Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A33(2) (1986), 1141-1151. · Zbl 1184.37028
[18]	Hare, K. E., Hare, K. G. and Matthews, K. R.. Local dimensions of measures of finite type. J. Fractal Geom.3(4) (2016), 331-376. · Zbl 1396.28011
[19]	Hare, K. E., Hare, K. G. and Ng, M. K. S.. Local dimensions of measures of finite type II: measures without full support and with non-regular probabilities. Canad. J. Math.70(4) (2018), 824-867. · Zbl 1457.28008
[20]	Hare, K. E., Hare, K. G. and Rutar, A.. When the weak separation condition implies the generalized finite type condition. Proc. Amer. Math. Soc.149(4) (2021), 1555-1568. · Zbl 1466.28010
[21]	Hare, K. E., Hare, K. G. and Shen, W.. The \({L}^q\) -spectrum for a class of self-similar measures with overlap. Asian J. Math.25(2) (2021), 195-228. · Zbl 1482.28011
[22]	Hare, K. E., Hare, K. G. and Simms, G.. Local dimensions of measures of finite type III—Measures that are not equicontractive. J. Math. Anal. Appl.458(2) (2018), 1653-1677. · Zbl 1376.28008
[23]	Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2)180(2) (2014), 773-822. · Zbl 1337.28015
[24]	Hu, T.-Y. and Lau, K.-S.. Multifractal structure of convolution of the Cantor measure. Adv. Appl. Math.27(1) (2001), 1-16. · Zbl 0991.28008
[25]	Hutchinson, J. E.. Fractals and self similarity. Indiana Univ. Math. J.30(5) (1981), 713-747. · Zbl 0598.28011
[26]	Jordan, T. and Rapaport, A.. Dimension of ergodic measures projected onto self-similar sets with overlaps. Proc. London Math. Soc.122(2) (2021), 191-206. · Zbl 1465.28007
[27]	Lau, K.-S.. Self-similarity, \({L}^p\) -spectrum and multifractal formalism. Fractal Geometry and Stochastics. Eds. Bandt, C., Graf, S. and Zähle, M.. Birkhäuser Basel, Basel, 1995, pp. 55-90. · Zbl 0837.28008
[28]	Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition. Adv. Math.141(1) (1999), 45-96. · Zbl 0929.28007
[29]	Lau, K.-S. and Ngai, S.-M.. A generalized finite type condition for iterated function systems. Adv. Math.208(2) (2007), 647-671. · Zbl 1113.28006
[30]	Lau, K.-S. and Wang, X.-Y.. Iterated function systems with a weak separation condition. Studia Math.161(3) (2004), 249-268. · Zbl 1062.28009
[31]	Lau, K.-S. and Wang, X.-Y.. Some exceptional phenomena in multifractal formalism: Part I. Asian J. Math.9(2) (2005), 275-294. · Zbl 1116.28010
[32]	Ngai, S.-M. and Wang, Y.. Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. (2)63(3) (2001), 655-672. · Zbl 1013.28008
[33]	Patzschke, N.. Self-conformal multifractal measures. Adv. Appl. Math.19(4) (1997), 486-513. · Zbl 0912.28007
[34]	Pesin, Y. and Weiss, H.. A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys.86(1-2) (1997), 233-275. · Zbl 0985.37040
[35]	Rutar, A.. A multifractal decomposition for self-similar measures with exact overlaps. Preprint, 2021, arXiv:2104.06997 [math].
[36]	Schief, A.. Separation properties for self-similar sets. Proc. Amer. Math. Soc.122(1) (1994), 111-115. · Zbl 0807.28005
[37]	Seneta, E.. Non-Negative Matrices and Markov Chains(Springer Series in Statistics), 2nd edn. Springer, New York, 1981. · Zbl 0471.60001
[38]	Shmerkin, P.. A modified multifractal formalism for a class of self-similar measures with overlap. Asian J. Math.9(3) (2005), 323-348. · Zbl 1091.28007
[39]	Testud, B.. Mesures quasi-Bernoulli au sens faible: Résultats et exemples. Ann. Inst. Henri Poincaré B42(1) (2006), 1-35. · Zbl 1134.28303
[40]	Zerner, M. P. W.. Weak separation properties for self-similar sets. Proc. Amer. Math. Soc.124(11) (1996), 3529-3539. · Zbl 0874.54025