Abstract
Let λ be a probability measure on T n − 1 where n = 2 or 3. Suppose λ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure μ on SL n ( Z ) \\ SL n ( R ) by putting λ on some unstable horospherical orbit of the right translation of a t = diag ( e t , … , e t , e − ( n − 1 ) t ) ( t > 0 ) . We prove that if the average of μ 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 λ is large, then Dirichletʼs theorem is not improvable λ almost surely.
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