On the $L^q$-spectrum of planar self-affine measures
HTML articles powered by AMS MathViewer
- by Jonathan M. Fraser PDF
- Trans. Amer. Math. Soc. 368 (2016), 5579-5620 Request permission
Abstract:
We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. This class contains the self-affine measures recently considered by Feng and Wang as well as many other measures. In particular, we allow the defining maps to have non-trivial rotational and reflectional components. Assuming the rectangular open set condition, we compute the $L^q$-spectrum by means of a $q$-modified singular value function.
A key application of our results is a closed form expression for the $L^q$-spectrum in the case where there are no mappings that switch the coordinate axes. This is useful for computational purposes and also allows us to prove differentiability of the $L^q$-spectrum at $q=1$ in the more difficult ‘non-multiplicative’ situation. This has applications concerning the Hausdorff, packing and entropy dimension of the measure as well as the Hausdorff and packing dimension of the support. Due to the possible inclusion of axis reversing maps, we are led to extend some results of Peres and Solomyak on the existence of the $L^q$-spectrum of self-similar measures to the graph-directed case.
References
- Krzysztof Barański, Multifractal analysis on the flexed Sierpiński gasket, Ergodic Theory Dynam. Systems 25 (2005), no. 3, 731–757. MR 2142943, DOI 10.1017/S0143385704000859
- Krzysztof Barański, Hausdorff dimension of the limit sets of some planar geometric constructions, Adv. Math. 210 (2007), no. 1, 215–245. MR 2298824, DOI 10.1016/j.aim.2006.06.005
- Julien Barral and De-Jun Feng, Multifractal formalism for almost all self-affine measures, Comm. Math. Phys. 318 (2013), no. 2, 473–504. MR 3020165, DOI 10.1007/s00220-013-1676-3
- T. Bedford, Crinkly curves, Markov partitions and box dimensions in self-similar sets, Ph.D dissertation, University of Warwick, 1984.
- K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339–350. MR 923687, DOI 10.1017/S0305004100064926
- K. J. Falconer, Generalized dimensions of measures on self-affine sets, Nonlinearity 12 (1999), no. 4, 877–891. MR 1709826, DOI 10.1088/0951-7715/12/4/308
- Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
- Kenneth J. Falconer, Generalized dimensions of measures on almost self-affine sets, Nonlinearity 23 (2010), no. 5, 1047–1069. MR 2630090, DOI 10.1088/0951-7715/23/5/002
- Kenneth Falconer and Jun Miao, Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices, Fractals 15 (2007), no. 3, 289–299. MR 2352690, DOI 10.1142/S0218348X07003587
- Ai-Hua Fan, Ka-Sing Lau, and Hui Rao, Relationships between different dimensions of a measure, Monatsh. Math. 135 (2002), no. 3, 191–201. MR 1897575, DOI 10.1007/s006050200016
- De-Jun Feng, Smoothness of the $L^q$-spectrum of self-similar measures with overlaps, J. London Math. Soc. (2) 68 (2003), no. 1, 102–118. MR 1980246, DOI 10.1112/S002461070300437X
- De-Jun Feng and Yang Wang, A class of self-affine sets and self-affine measures, J. Fourier Anal. Appl. 11 (2005), no. 1, 107–124. MR 2128947, DOI 10.1007/s00041-004-4031-4
- Andrew Ferguson, Thomas Jordan, and Pablo Shmerkin, The Hausdorff dimension of the projections of self-affine carpets, Fund. Math. 209 (2010), no. 3, 193–213. MR 2720210, DOI 10.4064/fm209-3-1
- Jonathan M. Fraser, On the packing dimension of box-like self-affine sets in the plane, Nonlinearity 25 (2012), no. 7, 2075–2092. MR 2947936, DOI 10.1088/0951-7715/25/7/2075
- J. M. Fraser and L. Olsen, Multifractal spectra of random self-affine multifractal Sierpinski sponges in $\Bbb R^d$, Indiana Univ. Math. J. 60 (2011), no. 3, 937–983. MR 2985862, DOI 10.1512/iumj.2011.60.4343
- Steven P. Lalley and Dimitrios Gatzouras, Hausdorff and box dimensions of certain self-affine fractals, Indiana Univ. Math. J. 41 (1992), no. 2, 533–568. MR 1183358, DOI 10.1512/iumj.1992.41.41031
- Thomas Jordan and Michal Rams, Multifractal analysis for Bedford-McMullen carpets, Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 147–156. MR 2739077, DOI 10.1017/S0305004110000472
- R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergodic Theory Dynam. Systems 16 (1996), no. 2, 307–323. MR 1389626, DOI 10.1017/S0143385700008828
- R. Kenyon and Y. Peres, Hausdorff dimensions of sofic affine-invariant sets, Israel J. Math. 94 (1996), 157–178. MR 1394572, DOI 10.1007/BF02762702
- James F. King, The singularity spectrum for general Sierpiński carpets, Adv. Math. 116 (1995), no. 1, 1–11. MR 1361476, DOI 10.1006/aima.1995.1061
- Ka-Sing Lau, Self-similarity, $L^p$-spectrum and multifractal formalism, Fractal geometry and stochastics (Finsterbergen, 1994) Progr. Probab., vol. 37, Birkhäuser, Basel, 1995, pp. 55–90. MR 1391971
- R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
- Curt McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1–9. MR 771063, DOI 10.1017/S0027763000021085
- Sze-Man Ngai, A dimension result arising from the $L^q$-spectrum of a measure, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2943–2951. MR 1402878, DOI 10.1090/S0002-9939-97-03974-9
- Tian-Jia Ni and Zhi-Ying Wen, The $L^q$-spectrum of a class of graph directed self-affine measures, Dyn. Syst. 24 (2009), no. 4, 517–536. MR 2573002, DOI 10.1080/14689360903143722
- Lars Olsen, Random geometrically graph directed self-similar multifractals, Pitman Research Notes in Mathematics Series, vol. 307, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. MR 1297123, DOI 10.2307/4351476
- L. Olsen, A multifractal formalism, Adv. Math. 116 (1995), no. 1, 82–196. MR 1361481, DOI 10.1006/aima.1995.1066
- L. Olsen, Self-affine multifractal Sierpinski sponges in $\mathbf R^d$, Pacific J. Math. 183 (1998), no. 1, 143–199. MR 1616626, DOI 10.2140/pjm.1998.183.143
- L. Olsen, Multifractal geometry, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 3–37. MR 1785619
- Lars Olsen, Random self-affine multifractal Sierpinski sponges in $\Bbb R^d$, Monatsh. Math. 162 (2011), no. 1, 89–117. MR 2747346, DOI 10.1007/s00605-009-0160-9
- Yuval Peres and Boris Solomyak, Existence of $L^q$ dimensions and entropy dimension for self-conformal measures, Indiana Univ. Math. J. 49 (2000), no. 4, 1603–1621. MR 1838304, DOI 10.1512/iumj.2000.49.1851
- Henry W. J. Reeve, The packing spectrum for Birkhoff averages on a self-affine repeller, Ergodic Theory Dynam. Systems 32 (2012), no. 4, 1444–1470. MR 2955322, DOI 10.1017/S0143385711000368
- Alfréd Rényi, Dimension, entropy and information, Trans. 2nd Prague Conf. Information Theory, Publ. House Czech. Acad. Sci., Prague, 1960; Academic Press, New York, 1960, pp. 545–556. MR 0129049
- A. Rényi, Probability theory, North-Holland Series in Applied Mathematics and Mechanics, Vol. 10, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. Translated by László Vekerdi. MR 0315747
- Rolf Riedi, An improved multifractal formalism and self-similar measures, J. Math. Anal. Appl. 189 (1995), no. 2, 462–490. MR 1312056, DOI 10.1006/jmaa.1995.1030
- Robert S. Strichartz, Self-similar measures and their Fourier transforms. III, Indiana Univ. Math. J. 42 (1993), no. 2, 367–411. MR 1237052, DOI 10.1512/iumj.1993.42.42018
- Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI 10.1017/s0143385700009615
Additional Information
- Jonathan M. Fraser
- Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom
- Address at time of publication: School of Mathematics, The University of Manchester, Manchester M13 9PL, United Kingdom
- MR Author ID: 946983
- Email: jon.fraser32@gmail.com
- Received by editor(s): January 21, 2014
- Received by editor(s) in revised form: July 14, 2014
- Published electronically: June 24, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 5579-5620
- MSC (2010): Primary 28A80, 37C45; Secondary 28A78, 15A18, 26A24
- DOI: https://doi.org/10.1090/tran/6523
- MathSciNet review: 3458392