Local dimensions of self-similar measures satisfying the finite neighbour condition. (English) Zbl 1514.28007

Hare, Kathryn E.; Rutar, Alex

Nonlinearity 35, No. 9, 4876-4904 (2022).

The authors investigate sets of local dimensions for self-similar measures in \(\mathbb{R}\) satisfying the so-called finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. Additionally, an explanation of how these results allow computations of the sets of local dimensions in many explicit cases is given. These results contextualize and generalize a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.

Reviewer: Peter Massopust (München)

MSC:

28A80

Fractals

Keywords:

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

References:

[1]	Cawley, R.; Mauldin, R. D., Multifractal decompositions of Moran fractals, Adv. Math., 92, 196-236 (1992)
[2]	Feng, D-J, Lyapunov exponents for products of matrices and multifractal analysis: I. Positive matrices, Isr. J. Math., 138, 353-376 (2003)
[3]	Feng, D-J, Smoothness of the L^q-spectrum of self-similar measures with overlaps, J. London Math. Soc., 68, 102-118 (2003)
[4]	Feng, D-J, The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers, Adv. Math., 195, 24-101 (2005)
[5]	Feng, D-J, Lyapunov exponents for products of matrices and multifractal analysis: II. General matrices, Isr. J. Math., 170, 355-394 (2009)
[6]	Feng, D-J; Lau, K-S, The pressure function for products of non-negative matrices, Math. Res. Lett., 9, 363-378 (2002)
[7]	Feng, D-J; Lau, K-S, Multifractal formalism for self-similar measures with weak separation condition, J. Math. Pures Appl., 92, 407-428 (2009)
[8]	Hare, K. E.; Hare, K. G., Local dimensions of overlapping self-similar measures, Real Anal. Exch., 44, 247 (2019)
[9]	Hare, K. E.; Hare, K. G.; Matthews, K. R., Local dimensions of measures of finite type: appendix (2015)
[10]	Hare, K.; Hare, K.; Matthews, K., Local dimensions of measures of finite type, J. Fractal Geom., 3, 331-376 (2016)
[11]	Hare, K. E.; Hare, K. G.; Shing Ng, M. K., Local dimensions of measures of finite type II: measures without full support and with non-regular probabilities, Can. J. Math., 70, 824-867 (2018)
[12]	Hare, K.; Hare, K.; Rutar, A., When the weak separation condition implies the generalized finite type condition, Proc. Am. Math. Soc., 149, 1555-1568 (2021)
[13]	Hentschel, H. G E.; Procaccia, I., The infinite number of generalized dimensions of fractals and strange attractors, Physica D, 8, 435-444 (1983)
[14]	Hille, E.; Phillips, R. S., Functional Analysis and Semi-groups (1957), Providence, RI: American Mathematical Society, Providence, RI
[15]	Hu, T-Y; Lau, K-S, Multifractal structure of convolution of the Cantor measure, Adv. Appl. Math., 27, 1-16 (2001)
[16]	Lau, K-S; Ngai, S-M, Multifractal measures and a weak separation condition, Adv. Math., 141, 45-96 (1999)
[17]	Lau, K-S; Ngai, S-M, A generalized finite type condition for iterated function systems, Adv. Math., 208, 647-671 (2007)
[18]	Lau, K-S; Wang, X-Y, Iterated function systems with a weak separation condition, Stud. Math., 161, 249-268 (2004)
[19]	Ngai, S-M; Wang, Y., Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc., 63, 655-672 (2001)
[20]	Patzschke, N., Self-conformal multifractal measures, Adv. Appl. Math., 19, 486-513 (1997)
[21]	Rutar, A., Geometric and combinatorial properties of self-similar multifractal measures, Ergod. Theor. Dynam. Syst.
[22]	Shmerkin, P., A modified multifractal formalism for a class of self-similar measures with overlap, Asian J. Math., 9, 323-348 (2005)
[23]	Testud, B., Phase transitions for the multifractal analysis of self-similar measures, Nonlinearity, 19, 1201-1217 (2006)
[24]	Varjú, P. P., Recent progress on Bernoulli convolutions, Proceedings of the 7th European Congress of Mathematics (2018)
[25]	Zerner, M., Weak separation properties for self-similar sets, Proc. Am. Math. Soc., 124, 3529-3539 (1996)