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$L^q$-spectra of box-like graph-directed self-affine measures: closed forms, with rotation

Part of: Classical measure theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  26 January 2026

HUA QIU  and
QI WANG
HUA QIU
Affiliation:
Mathematics, Nanjing University , Nanjing, China (e-mail: huaqiu@nju.edu.cn)
QI WANG*
Affiliation:
Mathematics, Nanjing University , Nanjing, China (e-mail: huaqiu@nju.edu.cn)
*
e-mail: 602023210013@smail.nju.edu.cn
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Abstract

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.

Keywords

L q-spectrabox dimensionself-affine setsgraph-directed constructionclosed form expression

MSC classification

Primary: 28A80: Fractals
Secondary: 37C45: Dimension theory of dynamical systems

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 58
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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References

Barański, K.. Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210 (2007), 215–245.10.1016/j.aim.2006.06.005CrossRefGoogle Scholar
Bárány, B., Hochman, M. and Rapaport, A.. Hausdorff dimension of planar self-affine sets and measures. Invent. Math. 216(3) (2019), 601–659.10.1007/s00222-018-00849-yCrossRefGoogle Scholar
Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
Bochi, J. and Morris, I. D.. Equilibrium states of generalised singular value potentials and applications to affine iterated function systems. Geom. Funct. Anal. 28(4) (2018), 995–1028.10.1007/s00039-018-0447-xCrossRefGoogle Scholar
Cawley, R. and Mauldin, R. D.. Multifractal decompositions of Moran fractals. Adv. Math. 92(2) (1992), 196–236.10.1016/0001-8708(92)90064-RCrossRefGoogle Scholar
Cogburn, R.. The central limit theorem for Markov processes. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory. Ed. L. M. Le Cam, J. Neyman and E. L. Scott. University of California Press, Berkeley, CA, 1972, pp. 485–512.10.1525/9780520423671-031CrossRefGoogle Scholar
Das, T. and Simmons, D.. The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result. Invent. Math. 210 (2017), 85–134.10.1007/s00222-017-0725-5CrossRefGoogle ScholarPubMed
Edgar, G. A.. Fractal dimension of self-affine sets: some examples. Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 341–358.Google Scholar
Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103(2) (1988), 339–350.10.1017/S0305004100064926CrossRefGoogle Scholar
Falconer, K. J.. Generalized dimensions of measures on self-affine sets. Nonlinearity 12(4) (1999), 877–891.10.1088/0951-7715/12/4/308CrossRefGoogle Scholar
Falconer, K. J., Fraser, J. M. and Lee, L. D.. ${L}^q$ -spectra of measures on planar non-conformal attractors. Ergod. Th. & Dynam. Sys. 41(11) (2021), 3288–3306.10.1017/etds.2020.106CrossRefGoogle Scholar
Falconer, K. J. and Kempton, T.. Planar self-affine sets with equal Hausdorff, box and affinity dimensions. Ergod. Th. & Dynam. Sys. 38(4) (2018), 1369–1388.10.1017/etds.2016.74CrossRefGoogle Scholar
Fan, A.-H., Lau, K.-S. and Ngai, S.-M.. Iterated function systems with overlaps. Asian J. Math. 4(3) (2000), 527–552.10.4310/AJM.2000.v4.n3.a3CrossRefGoogle Scholar
Feng, D.-J.. Smoothness of the ${L}^q$ -spectrum of self-similar measures with overlaps. J. Lond. Math. Soc. 68(2, 1) (2003), 102–118.10.1112/S002461070300437XCrossRefGoogle Scholar
Feng, D.-J.. The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195(1) (2005), 24–101.10.1016/j.aim.2004.06.011CrossRefGoogle Scholar
Feng, D.-J.. Lyapunov exponents for products of matrices and multifractal analysis. II. General matrices. Israel J. Math. 170 (2009), 355–394.10.1007/s11856-009-0033-xCrossRefGoogle Scholar
Feng, D.-J. and Käenmäki, A.. Equilibrium states of the pressure function for products of matrices. Discrete Contin. Dyn. Syst. 30(3) (2011), 699–708.10.3934/dcds.2011.30.699CrossRefGoogle Scholar
Feng, D.-J. and Shmerkin, P.. Non-conformal repellers and the continuity of pressure for matrix cocycles. Geom. Funct. Anal. 24(4) (2014), 1101–1128.10.1007/s00039-014-0274-7CrossRefGoogle Scholar
Feng, D.-J. and Wang, Y.. A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11(1) (2005), 107–124.10.1007/s00041-004-4031-4CrossRefGoogle Scholar
Fraser, J. M.. On the packing dimension of box-like self-affine sets in the plane. Nonlinearity 25(7) (2012), 2075–2092.10.1088/0951-7715/25/7/2075CrossRefGoogle Scholar
Fraser, J. M.. On the ${L}^q$ -spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620.10.1090/tran/6523CrossRefGoogle Scholar
Fraser, J. M. and Jurga, N.. The box dimensions of exceptional self-affine sets in ${\mathbb{R}}^3$ . Adv. Math. 385 (2021), 107734.10.1016/j.aim.2021.107734CrossRefGoogle Scholar
Fraser, J. M., Lee, L., Morris, I. D. and Yu, H.. ${L}^q$ -spectra of self-affine measures: closed forms, counterexamples, and split binomial sums. Nonlinearity 34(9) (2021), 6331–6357.10.1088/1361-6544/ac14a2CrossRefGoogle Scholar
Hochman, M.. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2) 180(2) (2014), 773–822.10.4007/annals.2014.180.2.7CrossRefGoogle Scholar
Käenmäki, A.. On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2) (2004), 419–458.Google Scholar
Käenmäki, A. and Reeve, H. W. J.. Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets. J. Fractal Geom. 1(1) (2014), 83–152.10.4171/jfg/3CrossRefGoogle Scholar
Kenyon, R. and Peres, Y.. Hausdorff dimensions of sofic affine-invariant sets. Israel J. Math. 94 (1996), 157–178.10.1007/BF02762702CrossRefGoogle Scholar
Kenyon, R. and Peres, Y.. Measures of full dimension on affine-invariant sets. Ergod. Th. & Dynam. Sys. 16(2) (1996), 307–323.10.1017/S0143385700008828CrossRefGoogle Scholar
King, J.. The singularity spectrum for general Sierpiński carpets. Adv. Math. 116(1) (1995), 1–11.10.1006/aima.1995.1061CrossRefGoogle Scholar
Kolossváry, I.. The ${L}^q$ spectrum of self-affine measures on sponges. J. Lond. Math. Soc. 108 (2023), 666–701.10.1112/jlms.12767CrossRefGoogle Scholar
Lalley, S. P. and Gatzouras, D.. Hausdorff and box dimensions of certain self–affine fractals. Indiana Univ. Math. J. 41(2) (1992), 533–568.10.1512/iumj.1992.41.41031CrossRefGoogle Scholar
Lau, K.-S. and Ngai, S.-M.. ${L}^q$ -spectrum of Bernoulli convolutions associated with P. V. Numbers. Osaka J. Math. 36(4) (1999), 993–1010.Google Scholar
Lau, K.-S. and Ngai, S.-M.. A generalized finite type condition for iterated function systems. Adv. Math. 208(2) (2007), 647–671.10.1016/j.aim.2006.03.007CrossRefGoogle Scholar
Lau, K.-S. and Wang, X.-Y.. Some exceptional phenomena in multifractal formalism. I. Asian J. Math. 9(2) (2005), 275–294.10.4310/AJM.2005.v9.n2.a13CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C.. Hausdorff dimension in graph-directed constructions. Trans. Amer. Math. Soc. 309(2) (1988), 811–829.10.1090/S0002-9947-1988-0961615-4CrossRefGoogle Scholar
McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.10.1017/S0027763000021085CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L.. Markov Chains and Stochastic Stability (Communications and Control Engineering). Springer-Verlag London, Ltd., London, 1993.10.1007/978-1-4471-3267-7CrossRefGoogle Scholar
Morris, I. D.. An inequality for the matrix pressure function and applications. Adv. Math. 302 (2016), 280–308.10.1016/j.aim.2016.07.025CrossRefGoogle Scholar
Morris, I. D.. An explicit formula for the pressure of box-like affine iterated function systems. J. Fractal Geom. 6(2) (2019), 127–141.10.4171/jfg/72CrossRefGoogle Scholar
Morris, I. D. and Shmerkin, P.. On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems. Trans. Amer. Math. Soc. 371(3) (2019), 1547–1582.10.1090/tran/7334CrossRefGoogle Scholar
Ngai, S.-M.. A dimension result arising from the ${L}^q$ -spectrum of a measure. Proc. Amer. Math. Soc. 125(10) (1997), 2943–2951.10.1090/S0002-9939-97-03974-9CrossRefGoogle Scholar
Ngai, S.-M. and Wang, Y.. Hausdorff dimension of self-similar sets with overlaps. J. Lond. Math. Soc. (2) 63(3) (2001), 655–672.10.1017/S0024610701001946CrossRefGoogle Scholar
Ni, T.-J. and Wen, Z.-Y.. The ${L}^q$ -spectrum of a class of graph directed self-affine measures. Dyn. Syst. 24(4) (2009), 517–536.10.1080/14689360903143722CrossRefGoogle Scholar
Olsen, L.. A multifractal formalism. Adv. Math. 116(1) (1995), 82–196.10.1006/aima.1995.1066CrossRefGoogle Scholar
Olsen, L.. Self-affine multifractal Sierpinski sponges in ${\mathbb{R}}^d$ . Pacific J. Math. 183(1) (1998), 143–199.10.2140/pjm.1998.183.143CrossRefGoogle Scholar
Olsen, L.. Bounds for the ${L}^q$ -spectra of a self-similar multifractal not satisfying the open set condition. J. Math. Anal. Appl. 355(1) (2009), 12–21.10.1016/j.jmaa.2009.01.037CrossRefGoogle Scholar
Qiu, H., Wang, Q. and Wang, S.-F.. ${L}^q$ -spectra of non-conformal planar graph-directed measures. Nonlinearity 37(9) (2024), 095003.10.1088/1361-6544/ad6055CrossRefGoogle Scholar
Roberts, G. O. and Rosenthal, J. S.. General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004), 20–71.10.1214/154957804100000024CrossRefGoogle Scholar
Shmerkin, P.. On Furstenberg’s intersection conjecture, self-similar measures, and the ${L}^q$ norms of convolutions. Ann. of Math. (2) 189(2) (2019), 319–391.10.4007/annals.2019.189.2.1CrossRefGoogle Scholar
Solomyak, B.. Measure and dimension for some fractal families. Math. Proc. Cambridge Philos. Soc. 124(3) (1998), 531–546.10.1017/S0305004198002680CrossRefGoogle Scholar
Strichartz, R. S.. Self-similar measures and their Fourier transforms III. Indiana Univ. Math. J. 42(2) (1993), 367–411.10.1512/iumj.1993.42.42018CrossRefGoogle Scholar