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On self-affine measures with equal Hausdorff and Lyapunov dimensions

Author: Ariel Rapaport
Journal: Trans. Amer. Math. Soc. 370 (2018), 4759-4783
MSC (2010): Primary 37C45; Secondary 28A80
Published electronically: January 18, 2018
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Abstract: Let $ \mu $ be a self-affine measure on $ \mathbb{R}^{d}$ associated to a self-affine IFS $ \{\varphi _{\lambda }(x)=A_{\lambda }x+v_{\lambda }\}_{\lambda \in \Lambda }$ and a probability vector $ p=(p_{\lambda })_{\lambda }>0$. Assume the strong separation condition holds. Let $ \gamma _{1}\ge \cdots \ge \gamma _{d}$ and $ D$ be the Lyapunov exponents and dimension corresponding to $ \{A_{\lambda }\}_{\lambda \in \Lambda }$ and $ p^{\mathbb{N}}$, and let $ \mathbf {G}$ be the group generated by $ \{A_{\lambda }\}_{\lambda \in \Lambda }$. We show that if $ \gamma _{m+1}>\gamma _{m}=\cdots =\gamma _{d}$, if $ \mathbf {G}$ acts irreducibly on the vector space of alternating $ m$-forms, and if the Furstenberg measure $ \mu _{F}$ satisfies $ \dim _{H}\mu _{F}+D>(m+1)(d-m)$, then $ \mu $ is exact dimensional with $ \dim \mu =D$.

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  • [B1] Balázs Bárány, On the Ledrappier-Young formula for self-affine measures, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 3, 405-432. MR 3413884,
  • [B2] J. Bourgain, On the Furstenberg measure and density of states for the Anderson-Bernoulli model at small disorder, J. Anal. Math. 117 (2012), 273-295. MR 2944098,
  • [B3] Jean Bourgain, Finitely supported measures on $ SL_2(\mathbb{R})$ which are absolutely continuous at infinity, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2050, Springer, Heidelberg, 2012, pp. 133-141. MR 2985129,
  • [BK] B. Bárány and A. Käenmäki, Ledrappier-Young formula and exact dimensionality of self-affine measures, arXiv:1511.05792, 2015.
  • [BL1] J. Bliedtner and P. Loeb, A reduction technique for limit theorems in analysis and probability theory, Ark. Mat. 30 (1992), no. 1, 25-43. MR 1171093,
  • [BL2] Philippe Bougerol and Jean Lacroix, Products of random matrices with applications to Schrödinger operators, Progress in Probability and Statistics, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 886674
  • [BQ1] Yves Benoist and Jean-François Quint, Random walks on reductive groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 62, Springer, Cham, 2016. MR 3560700
  • [BQ2] Y. Benoist and J.-F. Quint, On the regularity of stationary measures, preprint (2015).
  • [F1] Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
  • [F2] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • [F3] K. J. Falconer, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 2, 339-350. MR 923687,
  • [F4] Kenneth Falconer, Dimensions of self-affine sets: a survey, Further developments in fractals and related fields, Trends Math., Birkhäuser/Springer, New York, 2013, pp. 115-134. MR 3184190,
  • [FH] De-Jun Feng and Huyi Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math. 62 (2009), no. 11, 1435-1500. MR 2560042,
  • [FK] K. Falconer and T. Kempton, Planar self-affine sets with equal Hausdorff, box and affinity dimensions, arXiv:1503.01270, 2015.
  • [HS] M. Hochman and B. Solomyak, On the dimension of the Furstenberg measure for $ Sl(2,\mathbb{R})$-random matrix products, in preparation.
  • [JM] Maarit Järvenpää and Pertti Mattila, Hausdorff and packing dimensions and sections of measures, Mathematika 45 (1998), no. 1, 55-77. MR 1644341,
  • [JPS] Thomas Jordan, Mark Pollicott, and Károly Simon, Hausdorff dimension for randomly perturbed self affine attractors, Comm. Math. Phys. 270 (2007), no. 2, 519-544. MR 2276454,
  • [M1] Curt McMullen, The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J. 96 (1984), 1-9. MR 771063
  • [M2] P. Mattila, Fourier analysis and Hausdorff dimension, Cambridge University Press, Cambridge, 2015.
  • [M3] Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
  • [P] William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. MR 614142
  • [S] Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531-546. MR 1636589,

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Additional Information

Ariel Rapaport
Affiliation: The Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem

Keywords: Self-affine measures, Furstenberg measure, random matrices.
Received by editor(s): November 27, 2015
Received by editor(s) in revised form: September 26, 2016
Published electronically: January 18, 2018
Additional Notes: The author was supported by ERC grant 306494
Article copyright: © Copyright 2018 American Mathematical Society

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