Entropy and dimension of disintegrations of stationary measures
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- by Pablo Lessa;
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 105-129
- DOI: https://doi.org/10.1090/btran/60
- Published electronically: February 17, 2021
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Abstract:
We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for $\text {SL}_2(\mathbb {R})$ to disintegrations of stationary measures for $\operatorname {GL}(\mathbb {R}^d)$ onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace.
The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simplicity of the Lyapunov spectrum for the given measure on $\operatorname {GL}(\mathbb {R}^d)$.
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Bibliographic Information
- Pablo Lessa
- Affiliation: Facultad de Ingeniería, IMERL, Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay
- MR Author ID: 956036
- Email: plessa@fing.edu.uy
- Received by editor(s): August 5, 2019
- Received by editor(s) in revised form: July 14, 2020, and November 13, 2020
- Published electronically: February 17, 2021
- © Copyright 2021 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 105-129
- MSC (2020): Primary 37F35
- DOI: https://doi.org/10.1090/btran/60
- MathSciNet review: 4216247