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On the $ L^q$-dimensions of measures on Hueter-Lalley type self-affine sets


Authors: Jonathan M. Fraser and Tom Kempton
Journal: Proc. Amer. Math. Soc. 146 (2018), 161-173
MSC (2010): Primary 28A80, 28A78, 37C45
DOI: https://doi.org/10.1090/proc/13672
Published electronically: August 1, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the $ L^q$-dimensions of self-affine measures and the Käenmäki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the $ L^q$-
dimensions are equal to the value predicted by Falconer for a range of $ q$. As a corollary this gives a wider class of self-affine sets for which the Hausdorff dimension can be explicitly calculated. Our proof combines the potential theoretic approach developed by Hunt and Kaloshin with recent advances in the dynamics of self-affine sets.


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  • [B] Krzysztof Barański, Multifractal analysis on the flexed Sierpiński gasket, Ergodic Theory Dynam. Systems 25 (2005), no. 3, 731-757. MR 2142943, https://doi.org/10.1017/S0143385704000859
  • [Ba] Balázs Bárány, On the Ledrappier-Young formula for self-affine measures, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 3, 405-432. MR 3413884, https://doi.org/10.1017/S0305004115000419
  • [BK] B. Bárány and A. Käenmäki, Ledrappier-Young formula and exact dimensionality of self-affine measures, preprint, (2015), available at: arXiv:1511.05792.
  • [BR] B. Bárány and M. Rams Dimension maximising measures for self-affine systems, to appear in Trans. Amer. Math. Soc..
  • [BF] 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, https://doi.org/10.1007/s00220-013-1676-3
  • [F1] K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339-350. MR 923687, https://doi.org/10.1017/S0305004100064926
  • [F2] K. J. Falconer, Generalized dimensions of measures on self-affine sets, Nonlinearity 12 (1999), no. 4, 877-891. MR 1709826, https://doi.org/10.1088/0951-7715/12/4/308
  • [F3] Kenneth J. Falconer, Generalized dimensions of measures on almost self-affine sets, Nonlinearity 23 (2010), no. 5, 1047-1069. MR 2630090, https://doi.org/10.1088/0951-7715/23/5/002
  • [FK1] K. J. Falconer and T. Kempton, Planar self-affine sets with equal Hausdorff, box and affinity dimensions, to appear in Ergod. Th. Dynam. Syst., available at arXiv:1503.01270.
  • [FK2] K. J. Falconer and T. Kempton. The dimension of projections of self-affine sets and measures, preprint (2015), http://arxiv.org/abs/1511.03556.
  • [FW] 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, https://doi.org/10.1007/s00041-004-4031-4
  • [Fr] Jonathan M. Fraser, On the $ L^q$-spectrum of planar self-affine measures, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5579-5620. MR 3458392, https://doi.org/10.1090/tran/6523
  • [HL] Irene Hueter and Steven P. Lalley, Falconer's formula for the Hausdorff dimension of a self-affine set in $ {\bf R}^2$, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 77-97. MR 1314970, https://doi.org/10.1017/S0143385700008257
  • [HK] Brian R. Hunt and Vadim Yu. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity 10 (1997), no. 5, 1031-1046. MR 1473372, https://doi.org/10.1088/0951-7715/10/5/002
  • [K] Antti Käenmäki, On natural invariant measures on generalised iterated function systems, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 2, 419-458. MR 2097242
  • [Ke] T. Kempton, The scenery flow for self-affine measures, preprint, arxiv:1505.01663, 2015.
  • [Ki] James F. King, The singularity spectrum for general Sierpiński carpets, Adv. Math. 116 (1995), no. 1, 1-11. MR 1361476, https://doi.org/10.1006/aima.1995.1061
  • [MS] I. Morris and P. Shmerkin, On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems, preprint, arxiv:1602.08789, 2016.
  • [N] 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, https://doi.org/10.1090/S0002-9939-97-03974-9
  • [O] L. Olsen, Self-affine multifractal Sierpinski sponges in $ \mathbf {R}^d$, Pacific J. Math. 183 (1998), no. 1, 143-199. MR 1616626, https://doi.org/10.2140/pjm.1998.183.143
  • [R] A. Rapaport, On self-affine measures with equal Hausdorff and Lyapunov dimensions, preprint, 2015.

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Additional Information

Jonathan M. Fraser
Affiliation: School of Mathematics, The University of St Andrews, St Andrews, KY16 9SS, United Kingdom
Email: jmf32@st-andrews.ac.uk

Tom Kempton
Affiliation: School of Mathematics, The University of Manchester, Manchester, M13 9PL, United Kingdom
Email: thomas.kempton@manchester.ac.uk

DOI: https://doi.org/10.1090/proc/13672
Received by editor(s): July 8, 2016
Received by editor(s) in revised form: January 19, 2017
Published electronically: August 1, 2017
Additional Notes: The authors were financially supported by an LMS Scheme 4 Research in Pairs grant. The second author also acknowledges financial support from the EPSRC grant EP/K029061/1, and the first author acknowledges financial support from a Leverhulme Trust Research Fellowship (RF-2016-500).
The authors thank the Universities of Manchester and St Andrews for hosting the research visits which led to this work and Kenneth Falconer and Antti Käenmäki for helpful discussions.
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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