Pointwise recurrence for commuting measure preserving transformations

2013/12/18 by Assani, Idris
#37A05 #37A30 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1312.5270

Abstract

Let (X,A, μ) be a probability measure space and let T_i, 1≤ i≤ H, be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages (1)/(N)∑_n=1^N f₁(T₁ⁿx)f₂(T₂ⁿx)⋯ f_H(T_Hⁿx) converge a.e. for every function f_i ∈ L^∞(μ) . As a consequence if T_i = Tⁱ for 1≤ i ≤ H where T is an invertible measure preserving transformation on (X, A, μ) then the averages (1)/(N)∑_n=1^N f₁(Tⁿx)f₂(T²ⁿx)...f_H(T^Hnx) converge a.e. This solves a long open question on the pointwise convergence of nonconventional ergodic averages. For H=2 it provides another proof of J. Bourgain's a.e. double recurrence theorem.

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