# Non-conventional ergodic averages for several commuting actions of an amenable group

Research paper by **Tim Austin**

Indexed on: **14 Dec '16**Published on: **01 Nov '16**Published in: **Journal d'Analyse Mathématique**

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#### Abstract

Abstract
Let (X, μ) be a probability space, G a countable amenable group, and (F
n
)
n
a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages
$$\frac{1}{{\left {{F_n}} \right }}\sum\limits_{g \in {F_n}} {\mathop \Pi \limits_{i = 1}^d } \left( {{f_i} \circ T_1^g \cdots T_i^g} \right)$$
associated to a commuting tuple of μ-preserving actions
\({T_1}, \ldots {T_d}:G \curvearrowright X\)
and f
1,..., f
d ∈ L
∞(μ). We prove that these averages always converge in
\({\left\ \cdot \right\ _2}\)
, and that they witness a multiple recurrence phenomenon when f
1 =... = f
d
= 1
A
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.AbstractLet (X, μ) be a probability space, G a countable amenable group, and (F
n
)
n
a left Følner sequence in G. This paper analyzes the non-conventional ergodic averages
$$\frac{1}{{\left {{F_n}} \right }}\sum\limits_{g \in {F_n}} {\mathop \Pi \limits_{i = 1}^d } \left( {{f_i} \circ T_1^g \cdots T_i^g} \right)$$
associated to a commuting tuple of μ-preserving actions
\({T_1}, \ldots {T_d}:G \curvearrowright X\)
and f
1,..., f
d ∈ L
∞(μ). We prove that these averages always converge in
\({\left\ \cdot \right\ _2}\)
, and that they witness a multiple recurrence phenomenon when f
1 =... = f
d
= 1
A
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.XμGF
n
n
n
nG
$$\frac{1}{{\left {{F_n}} \right }}\sum\limits_{g \in {F_n}} {\mathop \Pi \limits_{i = 1}^d } \left( {{f_i} \circ T_1^g \cdots T_i^g} \right)$$
$$\frac{1}{{\left {{F_n}} \right }}\sum\limits_{g \in {F_n}} {\mathop \Pi \limits_{i = 1}^d } \left( {{f_i} \circ T_1^g \cdots T_i^g} \right)$$μ
\({T_1}, \ldots {T_d}:G \curvearrowright X\)
\({T_1}, \ldots {T_d}:G \curvearrowright X\)f1fdL∞μ
\({\left\ \cdot \right\ _2}\)
\({\left\ \cdot \right\ _2}\)f1f
d
d
A
AAX