Pablo Lessa
Published 2019-08-05Version 1
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 $\text{GL}(\mathbb{R}^d)$ 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 $\text{GL}(\mathbb{R}^d)$.
Absolute continuity of stationary measures
When is a group action determined by its orbit structure?
On the dimension of limit sets on $\mathbb{P}(\mathbb{R}^3)$ via stationary measures: variational principles and applications
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