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Multiple ergodic averages for tempered functions
Discrete and Continuous Dynamical Systems ( IF 1.1 ) Pub Date : 2020-08-11 , DOI: 10.3934/dcds.2020314
Andreas Koutsogiannis ,

Following Frantzikinakis' approach on averages for Hardy field functions of different growth, we add to the topic by studying the corresponding averages for tempered functions, a class which also contains functions that oscillate and is in general more restrictive to deal with. Our main result is the existence and the explicit expression of the $ L^2 $-norm limit of the aforementioned averages, which turns out, as in the Hardy field case, to be the "expected" one. The main ingredients are the use of, the now classical, PET induction (introduced by Bergelson), covering a more general case, namely a "nice" class of tempered functions (developed by Chu-Frantzikinakis-Host for polynomials and Frantzikinakis for Hardy field functions) and some equidistribution results on nilmanifolds (analogous to the ones of Frantzikinakis' for the Hardy field case).

中文翻译:

多项遍历平均值可调节功能

遵循Frantzikinakis对不同增长的Hardy场函数的平均值的方法,我们通过研究调质函数的相应平均值来增加话题,该类也包含振荡的函数,并且通常更难以处理。我们的主要结果是上述平均值的$ L ^ 2 $-范数极限的存在和明确的表达,与在Hardy字段中的情况一样,这证明是“预期的”。主要成分是现在使用的经典的PET归纳法(由Bergelson引入)的使用,涵盖了更一般的情况,即“好”类的调和函数(由Chu-Frantzikinakis-Host为多项式开发,由Frantzikinakis为Hardy领域开发)函数)和一些关于尼尔曼流形的均等分布结果(类似于Frantzikinakis的那些)
更新日期:2020-08-11
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