TY - JOUR
T1 - Non-conventional ergodic averages for several commuting actions of an amenable group
AU - Austin, Tim
N1 - Publisher Copyright:
© 2016, Hebrew University Magnes Press.
PY - 2016/11/1
Y1 - 2016/11/1
N2 - Let (X, μ) be a probability space, G a countable amenable group, and (Fn)n a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages (Formula Presented) associated to a commuting tuple of μ-preserving actions T1, … Td: G↷ X and f1,.., fd ∈ L∞(μ). We prove that these averages always converge in ‖ ⋅ ‖ 2, and that they witness a multiple recurrence phenomenon when f1 =.. = fd = 1A for a non-negligible set A ⊆ X. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.
AB - Let (X, μ) be a probability space, G a countable amenable group, and (Fn)n a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages (Formula Presented) associated to a commuting tuple of μ-preserving actions T1, … Td: G↷ X and f1,.., fd ∈ L∞(μ). We prove that these averages always converge in ‖ ⋅ ‖ 2, and that they witness a multiple recurrence phenomenon when f1 =.. = fd = 1A for a non-negligible set A ⊆ X. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof relies on an adaptation from earlier works of the machinery of sated extensions.
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U2 - 10.1007/s11854-016-0036-6
DO - 10.1007/s11854-016-0036-6
M3 - Article
AN - SCOPUS:84996508905
SN - 0021-7670
VL - 130
SP - 243
EP - 274
JO - Journal d'Analyse Mathematique
JF - Journal d'Analyse Mathematique
IS - 1
ER -