Dimension maximizing measures for self-affine systems

Bárány, Balázs; Rams, Michał

doi:10.1090/tran/7103

Dimension maximizing measures for self-affine systems
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by Balázs Bárány and Michał Rams PDF

Trans. Amer. Math. Soc. 370 (2018), 553-576 Request permission

Abstract:

In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and the strong separation condition. The main result of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space.

References

Artur Avila, Jairo Bochi, and Jean-Christophe Yoccoz, Uniformly hyperbolic finite-valued $\textrm {SL}(2,\Bbb R)$-cocycles, Comment. Math. Helv. 85 (2010), no. 4, 813–884. MR 2718140, DOI 10.4171/CMH/212
Balázs Bárány, On the Ledrappier-Young formula for self-affine measures, Math. Proc. Cambridge Philos. Soc. 159 (2015), no. 3, 405–432. MR 3413884, DOI 10.1017/S0305004115000419
Jairo Bochi and Nicolas Gourmelon, Some characterizations of domination, Math. Z. 263 (2009), no. 1, 221–231. MR 2529495, DOI 10.1007/s00209-009-0494-y
Jairo Bochi and MichałRams, The entropy of Lyapunov-optimizing measures of some matrix cocycles, J. Mod. Dyn. 10 (2016), 255–286. MR 3538864, DOI 10.3934/jmd.2016.10.255
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Second revised edition, Lecture Notes in Mathematics, vol. 470, Springer-Verlag, Berlin, 2008. With a preface by David Ruelle; Edited by Jean-René Chazottes. MR 2423393, DOI 10.1007/978-3-540-77695-6
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
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
K. J. Falconer, The dimension of self-affine fractals. II, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 1, 169–179. MR 1131488, DOI 10.1017/S0305004100075253
K. Falconer and T. Kempton, Planar self-affine sets with equal Hausdorff, box and affinity dimensions, Ergodic Theory Dynam. Systems (2016), FirstView:1-20, available at arXiv:1503.01270.
Irene Hueter and Steven P. Lalley, Falconer’s formula for the Hausdorff dimension of a self-affine set in $\textbf {R}^2$, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 77–97. MR 1314970, DOI 10.1017/S0143385700008257
Antti Käenmäki, Measures of full dimension on self-affine sets, Acta Univ. Carolin. Math. Phys. 45 (2004), no. 2, 45–53. 32nd Winter School on Abstract Analysis. MR 2138274
Philip T. Maker, The ergodic theorem for a sequence of functions, Duke Math. J. 6 (1940), 27–30. MR 2028
I. D. Morris and P. Shmerkin, On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems, preprint, 2016, available at arXiv:1602.08789.
A. Rapaport, On self-affine measures with equal Hausdorff and Lyapunov dimensions, Trans. Amer. Math. Soc., to appear, available at arXiv:1511.06893.
V. A. Rohlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation 1952 (1952), no. 71, 55. MR 0047744
K. Simon, B. Solomyak, and M. Urbański, Invariant measures for parabolic IFS with overlaps and random continued fractions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 5145–5164. MR 1852098, DOI 10.1090/S0002-9947-01-02873-2
Boris Solomyak, Measure and dimension for some fractal families, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 531–546. MR 1636589, DOI 10.1017/S0305004198002680
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
Jean-Christophe Yoccoz, Some questions and remarks about $\textrm {SL}(2,\mathbf R)$ cocycles, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 447–458. MR 2093316