Pointwise Equidistribution and Translates of Measures on Homogeneous\n Spaces

2017/03/21 by Osama Khalil, Khalil, Osama
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1703.07224

openalex publication_date 2017/03/21 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

Let (X, mathfrakB,\μ) be a Borel probability space. Let T_n:\nX\→ X be a sequence of continuous transformations on X. Let \ν\nbe a probability measure on X such that \(1)/(N)\∑_n=1^N (T_n)_\∗\n\ν \→ \μ in the weak-\∗ topology. Under general conditions, we\nshow that for \ν almost every x\∈ X, the measures\n\(1)/(N)\∑_n=1^N \δ_{Tn x} get equidistributed towards \μ if\nN is restricted to a set of full upper density. We present applications of\nthese results to translates of closed orbits of Lie groups on homogeneous\nspaces. As a corollary, we prove equidistribution of exponentially sparse\norbits of the horocycle flow on quotients of SL(2,\ℝ), starting from\nevery point in almost every direction.\n

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