Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure. (English) Zbl 1469.28009
Let \(V\) be a finite-dimensional vector space over the reals with integral dimension \(\geq 2\). Let \(\mathrm{GL}(V)\) be the automorphism group of \(V\) and let \(\mathcal{M}(\mathrm{GL}(V))\) be the collection of Borel probability measures on \(\mathrm{GL}(V)\). For \(\mu\in\mathcal{M}(\mathrm{GL}(V)\), denote by \(S_\mu\) the smallest closed subsemigroup of \(\mathrm{GL}(V)\) such that \(\mu(S_\mu) = 1\). Further, assume that \(S_\mu\) is strongly irreducible and proximal. Under these two assumptions, there exists a unique \(\nu\in\mathcal{M}(\mathrm{GL}(V))\) which is \(\mu\)-stationary. This measure \(\nu\) is called the Furstenberg measure corresponding to \(\mu\).
The authors prove that \(\nu\) is exact dimensional and provide a Ledrappier-Young type formula for its dimension.
The authors prove that \(\nu\) is exact dimensional and provide a Ledrappier-Young type formula for its dimension.
Reviewer: Peter Massopust (München)
References:
| [1] | B\'{a}r\'{a}ny, Bal\'{a}zs; K\"{a}enm\"{a}ki, Antti, Ledrappier-Young formula and exact dimensionality of self-affine measures, Adv. Math., 318, 88-129 (2017) · Zbl 1457.37032 · doi:10.1016/j.aim.2017.07.015 |
| [2] | B\'{a}r\'{a}ny, Bal\'{a}zs; Hochman, Michael; Rapaport, Ariel, Hausdorff dimension of planar self-affine sets and measures, Invent. Math., 216, 3, 601-659 (2019) · Zbl 1414.28014 · doi:10.1007/s00222-018-00849-y |
| [3] | Bougerol, Philippe; Lacroix, Jean, Products of random matrices with applications to Schr\"{o}dinger operators, Progress in Probability and Statistics 8, xii+283 pp. (1985), Birkh\"{a}user Boston, Inc., Boston, MA · Zbl 0572.60001 · doi:10.1007/978-1-4684-9172-2 |
| [4] | Benoist, Yves; Quint, Jean-Fran\c{c}ois, 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] 62, xi+323 pp. (2016), Springer, Cham · Zbl 1366.60002 |
| [5] | Einsiedler, Manfred; Ward, Thomas, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics 259, xviii+481 pp. (2011), Springer-Verlag London, Ltd., London · Zbl 1206.37001 · doi:10.1007/978-0-85729-021-2 |
| [6] | Falconer, Kenneth, Techniques in fractal geometry, xviii+256 pp. (1997), John Wiley & Sons, Ltd., Chichester · Zbl 0869.28003 |
| [7] | Falconer, Kenneth J.; Jin, Xiong, Exact dimensionality and projections of random self-similar measures and sets, J. Lond. Math. Soc. (2), 90, 2, 388-412 (2014) · Zbl 1305.28010 · doi:10.1112/jlms/jdu031 |
| [8] | D. -J. Feng, Dimension of invariant measures for affine iterated function systems. preprint, 2019. 1901.01691. |
| [9] | Feng, De-Jun; Hu, Huyi, Dimension theory of iterated function systems, Comm. Pure Appl. Math., 62, 11, 1435-1500 (2009) · Zbl 1230.37031 · doi:10.1002/cpa.20276 |
| [10] | Furstenberg, Hillel, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems, 28, 2, 405-422 (2008) · Zbl 1154.37322 · doi:10.1017/S0143385708000084 |
| [11] | Guivarc’h, Yves, Produits de matrices al\'{e}atoires et applications aux propri\'{e}t\'{e}s g\'{e}om\'{e}triques des sous-groupes du groupe lin\'{e}aire, Ergodic Theory Dynam. Systems, 10, 3, 483-512 (1990) · Zbl 0715.60008 · doi:10.1017/S0143385700005708 |
| [12] | Hochman, Michael, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180, 2, 773-822 (2014) · Zbl 1337.28015 · doi:10.4007/annals.2014.180.2.7 |
| [13] | M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy in \(\mathbbR^d \). to appear in Memoirs of the American Mathematical Society, 2015. http://arxiv.org/abs/1503.09043. |
| [14] | M. Hochman and A. Rapaport, Hausdorff dimension of planar self-affine sets and measures with overlaps. preprint, 2019. arXiv:1904.09812. · Zbl 1414.28014 |
| [15] | Hochman, Michael; Solomyak, Boris, On the dimension of Furstenberg measure for \(SL_2(\mathbb{R})\) random matrix products, Invent. Math., 210, 3, 815-875 (2017) · Zbl 1398.37012 · doi:10.1007/s00222-017-0740-6 |
| [16] | Jordan, Thomas; Pollicott, Mark; Simon, K\'{a}roly, Hausdorff dimension for randomly perturbed self affine attractors, Comm. Math. Phys., 270, 2, 519-544 (2007) · Zbl 1119.28004 · doi:10.1007/s00220-006-0161-7 |
| [17] | Kac, M., On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc., 53, 1002-1010 (1947) · Zbl 0032.41802 · doi:10.1090/S0002-9904-1947-08927-8 |
| [18] | Ledrappier, F., Quelques propri\'{e}t\'{e}s des exposants caract\'{e}ristiques. \'{E}cole d’\'{e}t\'{e} de probabilit\'{e}s de Saint-Flour, XII-1982, Lecture Notes in Math. 1097, 305-396 (1984), Springer, Berlin · Zbl 0541.00006 · doi:10.1007/BFb0099434 |
| [19] | Ledrappier, F.; Young, L.-S., The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2), 122, 3, 509-539 (1985) · Zbl 0605.58028 · doi:10.2307/1971328 |
| [20] | Lessa, Pablo, Entropy and dimension of disintegrations of stationary measures, Trans. Amer. Math. Soc. Ser. B, 8, 105-129 (2021) · Zbl 1462.37007 · doi:10.1090/btran/60 |
| [21] | Maker, Philip T., The ergodic theorem for a sequence of functions, Duke Math. J., 6, 27-30 (1940) · Zbl 0027.07705 |
| [22] | Mattila, Pertti, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics 44, xii+343 pp. (1995), Cambridge University Press, Cambridge · Zbl 0819.28004 · doi:10.1017/CBO9780511623813 |
| [23] | Oseledec, V. I., A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Ob\v{s}\v{c}., 19, 179-210 (1968) · Zbl 0236.93034 |
| [24] | Parry, William, Topics in ergodic theory, Cambridge Tracts in Mathematics 75, x+110 pp. (1981), Cambridge University Press, Cambridge-New York · Zbl 1096.37001 |
| [25] | Ruelle, David, Ergodic theory of differentiable dynamical systems, Inst. Hautes \'{E}tudes Sci. Publ. Math., 50, 27-58 (1979) · Zbl 0426.58014 |
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