Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure
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- by Ariel Rapaport PDF
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Abstract:
Assuming strong irreducibility and proximality, we prove that the Furstenberg measure, corresponding to a finitely supported measure on the general linear group of a finite dimensional real vector space, is exact dimensional. We also establish a Ledrappier-Young type formula for its dimension. The general strategy of the proof is based on the argument given by Feng for the exact dimensionality of self-affine measures.References
- Balázs Bárány and Antti Käenmäki, Ledrappier-Young formula and exact dimensionality of self-affine measures, Adv. Math. 318 (2017), 88–129. MR 3689737, DOI 10.1016/j.aim.2017.07.015
- Balázs Bárány, Michael Hochman, and Ariel Rapaport, Hausdorff dimension of planar self-affine sets and measures, Invent. Math. 216 (2019), no. 3, 601–659. MR 3955707, DOI 10.1007/s00222-018-00849-y
- 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, DOI 10.1007/978-1-4684-9172-2
- 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
- Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- Kenneth J. Falconer and Xiong Jin, Exact dimensionality and projections of random self-similar measures and sets, J. Lond. Math. Soc. (2) 90 (2014), no. 2, 388–412. MR 3263957, DOI 10.1112/jlms/jdu031
- D. -J. Feng, Dimension of invariant measures for affine iterated function systems. preprint, 2019. arXiv:1901.01691.
- De-Jun Feng and Huyi Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500. MR 2560042, DOI 10.1002/cpa.20276
- Hillel Furstenberg, Ergodic fractal measures and dimension conservation, Ergodic Theory Dynam. Systems 28 (2008), no. 2, 405–422. MR 2408385, DOI 10.1017/S0143385708000084
- Yves Guivarc’h, Produits de matrices aléatoires et applications aux propriétés géométriques des sous-groupes du groupe linéaire, Ergodic Theory Dynam. Systems 10 (1990), no. 3, 483–512 (French, with English summary). MR 1074315, DOI 10.1017/S0143385700005708
- Michael Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), no. 2, 773–822. MR 3224722, DOI 10.4007/annals.2014.180.2.7
- M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb {R}^{d}$. to appear in Memoirs of the American Mathematical Society, 2015. http://arxiv.org/abs/1503.09043.
- M. Hochman and A. Rapaport, Hausdorff dimension of planar self-affine sets and measures with overlaps. preprint, 2019. arXiv:1904.09812.
- Michael Hochman and Boris Solomyak, On the dimension of Furstenberg measure for $SL_2(\Bbb R)$ random matrix products, Invent. Math. 210 (2017), no. 3, 815–875. MR 3735630, DOI 10.1007/s00222-017-0740-6
- 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, DOI 10.1007/s00220-006-0161-7
- M. Kac, On the notion of recurrence in discrete stochastic processes, Bull. Amer. Math. Soc. 53 (1947), 1002–1010. MR 22323, DOI 10.1090/S0002-9904-1947-08927-8
- F. Ledrappier, Quelques propriétés des exposants caractéristiques, École d’été de probabilités de Saint-Flour, XII—1982, Lecture Notes in Math., vol. 1097, Springer, Berlin, 1984, pp. 305–396 (French). MR 876081, DOI 10.1007/BFb0099434
- F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
- Pablo Lessa, Entropy and dimension of disintegrations of stationary measures, Trans. Amer. Math. Soc. Ser. B 8 (2021), 105–129. MR 4216247, DOI 10.1090/btran/60
- Philip T. Maker, The ergodic theorem for a sequence of functions, Duke Math. J. 6 (1940), 27–30. MR 2028
- 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, DOI 10.1017/CBO9780511623813
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
- William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. MR 614142
- David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581
Additional Information
- Ariel Rapaport
- Affiliation: Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
- MR Author ID: 1088705
- Email: ariel.rapaport@mail.huji.ac.il
- Received by editor(s): August 2, 2020
- Received by editor(s) in revised form: February 10, 2021
- Published electronically: April 28, 2021
- Additional Notes: This research was supported by the Herchel Smith Fund at the University of Cambridge.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 5225-5268
- MSC (2020): Primary 28A80, 37C45
- DOI: https://doi.org/10.1090/tran/8405
- MathSciNet review: 4273191