Pointwise multiple averages for systems with two commuting transformations

被引:10
作者
Donoso, Sebastian [1 ]
Sun, Wenbo [2 ]
机构
[1] Univ Chile, Ctr Math Modeling, Beauchef 851, Santiago, Chile
[2] Northwestern Univ, Dept Math, 2033 Sheridan Rd, Evanston, IL 60208 USA
基金
美国国家科学基金会;
关键词
NONCONVENTIONAL ERGODIC AVERAGES; NORM CONVERGENCE; RECURRENCE;
D O I
10.1017/etds.2016.127
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We show that for every ergodic measure-preserving system (X, X, mu, S, T) with commuting transformations S and T, the average i/N-3 Sigma(N-1)(i,j,k=0) f(0)(S(j)T(k)x)f(1)(S(i+j)T(k)x)f(2)(S(j)T(i+k)x) converges for mu-almost every x is an element of X as N -> infinity for all f(0), f(1), f(2) is an element of L-infinity(mu). We also show that if (X, X, mu, S, T) is an ergodic measurable distal system, then the average 1/N Sigma(N-1)(i=0) f(1)(S(i)x) f(2)(T(i)x) converges for mu-almost every x is an element of X as N -> infinity for all f(1), f(2) is an element of L-infinity (mu).
引用
收藏
页码:2132 / 2157
页数:26
相关论文
共 15 条
[1]   On the norm convergence of non-conventional ergodic averages [J].
Austin, Tim .
ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2010, 30 :321-338
[2]  
Becker Howard, 1996, LONDON MATH SOC LECT, V232
[3]  
BOURGAIN J, 1990, J REINE ANGEW MATH, V404, P140
[4]   Multiple recurrence for two commuting transformations [J].
Chu, Qing .
ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2011, 31 :771-792
[5]   A pointwise cubic average for two commuting transformations [J].
Donoso, Sebastian ;
Sun, Wenbo .
ISRAEL JOURNAL OF MATHEMATICS, 2016, 216 (02) :657-678
[6]  
Furstenberg H., 1981, Recurrence in Ergodic Theory and Combinatorial Number Theory
[7]  
GLASNER E, 2003, Mathematical Surveys and Monographs, V101
[8]   Nonconventional ergodic averages and nilmanifolds [J].
Host, B ;
Kra, B .
ANNALS OF MATHEMATICS, 2005, 161 (01) :397-488
[9]   Ergodic seminorms for commuting transformations and applications [J].
Host, Bernard .
STUDIA MATHEMATICA, 2009, 195 (01) :31-49
[10]  
JEWETT RI, 1970, J MATH MECH, V19, P717