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- W2949198784 abstract "Let $lambda$ be a probability measure on $mathbb T^{n-1}$ where $n=2$ or 3. Suppose $lambda$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $mu $ on $SL_n(mathbb Z)backslash SL_n(mathbb R)$ by putting $lambda$ on some unstable horospherical orbit of the right translation of $a_t=mathrm{diag}(e^t,..., e^t, e^{-(n-1)t})$ $(t>0)$. We prove that if the average of $mu$ with respect to the flow $a_t$ has a limit, then it must be a scalar multiple of the probability Haar measure. As an application we show that if the entropy of $lambda$ is large, then Dirichlet's theorem is not improvable $lambda$ almost surely." @default.
- W2949198784 created "2019-06-27" @default.
- W2949198784 creator A5063891918 @default.
- W2949198784 date "2011-03-07" @default.
- W2949198784 modified "2023-09-24" @default.
- W2949198784 title "Convergence of measures under diagonal actions on homogeneous spaces" @default.
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- W2949198784 doi "https://doi.org/10.48550/arxiv.1103.1244" @default.
- W2949198784 hasPublicationYear "2011" @default.
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