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Exact dimension of Furstenberg measures

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Abstract

For a probability measure \( \mu \) on \( SL_d({\mathbb {R}}) \), we consider the Furstenberg stationary measure \( \nu \) on the space of flags. Under general non-degeneracy conditions, if \( \mu \) is discrete and if \( \sum _g \log \Vert g\Vert \, \mu (g) < + \infty \), then the measure \( \nu \) is exact-dimensional.

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Notes

  1. We thank Yves Coudène for this observation.

  2. Comparing with Theorem 1.1, the content of (8) for \( Q_1\) is that the numbers \( \gamma ^{Q_1}_{i,j} \) are the dimensions of certain conditional measures on specific 1-dimensional leaves in \( {\mathcal {F}}\).

  3. Observe that relation (7) depends on Theorem 1.6, which will be proven in Section 5.

  4. cf. notations of the Introduction.

  5. By convention, \( \log \left( \frac{{\mathbb {P}}_{(X,Y)}(A)}{({\mathbb {P}}_{X}\times {\mathbb {P}}_{Y})(A)}\right) {\mathbb {P}}_{(X,Y)}(A) =0 \) if \( {\mathbb {P}}_{X,Y} (A) = 0,\) the sum is \( +\infty \) if there is one \( A \in {\mathcal {A}}\) such that \( {\mathbb {P}}_{X,Y} (A) \not = 0\) and \( ({\mathbb {P}}_{X}\times {\mathbb {P}}_{Y})(A) =0\).

  6. We will not discuss how, when restricted to \( T_Q\), our arguments would still hold under the hypothesis that the stationary measure is unique on \({\mathbb {R}}{\mathbb {P}}^{d-1}\) and that \( \chi _1 > \chi _2\). This would contain (and be very close to) [Rap21].

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Authors and Affiliations

  1. Université de Paris et Sorbonne Université, CNRS, LPSM, Boîte Courrier 158, 4, Place Jussieu, 75252, Paris Cedex 05, France

    François Ledrappier

  2. IMERL, Facultad de Ingeniería, Julio Herrera y Reissig 565, 11300, Montevideo, Uruguay

    Pablo Lessa

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  1. François Ledrappier
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Ledrappier, F., Lessa, P. Exact dimension of Furstenberg measures. Geom. Funct. Anal. 33, 245–298 (2023). https://doi.org/10.1007/s00039-023-00631-0

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