I work in probability and I am looking for an important example of action of an amenable countable group in other areas of math for which the (pointwise) ergodic theorem is actually quite important. For example, I would like to know if in your area of expertise there is a group action for which probabilistic or statistical statements are of interest: e.g. asymptotics of some empirical mean ect.. Thank you :)
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$\begingroup$ One rather direct application which you might already know is the Hardy-Littlewood type maximal inequities for certain classes of amenable locally compact groups. You can look at E. Linderstrass' paper in Invent. Math. 146 (2001) and Greenleaf-Emerson's paper in Advances in Math (1974). I am hoping this relevant to what you are after. $\endgroup$– BigMCommented Apr 21, 2017 at 4:32
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$\begingroup$ In general, mixing properties (of flows over homogeneous spaces) are intimately related to the spectral gap of the action (and through that, to decay of matrix coefficients and representation theory), this has led to many counting results (the Eskin-McMullen theorem comes into mind). $\endgroup$– AsafCommented Apr 21, 2017 at 7:47
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$\begingroup$ Asaf this is very interesting. Thank you. But those examples are going to be for action of non-discrete (countable) groups right? Do you know anything of that flavor but with coutable groups? $\endgroup$– lettaCommented Apr 21, 2017 at 16:02
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