Quantitative multiple recurrence for two and three transformations
Quantitative multiple recurrence for two and three transformations
复制标题
两次和三次转换的定量多重递归
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Wenbo Sun
中科院分区：
文献类型：
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作者：
S. Donoso;Wenbo Sun
AbstractWe provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that:
There exists an ergodic system (X,X, μ,T1, T2) with two commuting transformations such that, for every 0 < ℓ < 4, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA)<μ(A)ℓforeveryn≠0There exists an ergodic system (X,X, μ,T2, T3) with three commuting transformations such that, for every ℓ > 0, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A cap T_3^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA∩T3−nA)<μ(A)ℓforeveryn≠0There exists an ergodic system (X,X, μ,T1, T2) with two transformations generating a 2-step nilpotent group such that, for every ℓ > 0, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA)<μ(A)ℓforeveryn≠0
AbstractWe provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that:
There exists an ergodic system (X,X, μ,T1, T2) with two commuting transformations such that, for every 0 < ℓ < 4, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA)<μ(A)ℓforeveryn≠0There exists an ergodic system (X,X, μ,T2, T3) with three commuting transformations such that, for every ℓ > 0, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A cap T_3^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA∩T3−nA)<μ(A)ℓforeveryn≠0There exists an ergodic system (X,X, μ,T1, T2) with two transformations generating a 2-step nilpotent group such that, for every ℓ > 0, there exists A ∈ X such that $$mu left( {A cap T_1^{ - n}A cap T_2^{ - n}A}
ight) < mu {left( A
ight)^ell }foreveryn
e 0$$μ(A∩T1−nA∩T2−nA)<μ(A)ℓforeveryn≠0