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

Research paper by Tim Austin

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

#### 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