Under- and over-independence in measure preserving systems

2018/07/09 by Adams, Terry, Bergelson, Vitaly, Sun, Wenbo
#37A05 #37A15 #37A25 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.02966

Abstract

We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of results obtained in this paper: ⋅ (Existence of density-1 UI and OI set) Let (X,B,μ,T) be an invertible probability measure preserving weakly mixing system. Then for any d∈ℕ, any non-constant integer-valued polynomials p₁,p₂,…,p_d such that p_i-p_j are also non-constant for all i≠ j, (i) there is A\inB such that the set \n∈ℕ\colonμ(A∩ T^p₁(n)A∩…∩ T^p_d(n)A)μ(A)^d+1\ is of density 1. ⋅ (Existence of Cesàro OI set) Let (X,B,μ,T) be a free, invertible, ergodic probability measure preserving system and M∈ℕ. %Suppose that X contains an ergodic component which is aperiodic. Then there is A\inB such that (1)/(N)∑_n=M^N+M-1μ(A∩ TⁿA)gt;μ(A)² for all N∈ℕ. ⋅ (Nonexistence of Cesàro UI set) Let (X,B,μ,T) be an invertible probability measure preserving system. For any measurable set A satisfying μ(A) ∈ (0,1), there exist infinitely many N ∈ ℕ such that (1)/(N) ∑_n=0^N-1 μ( A ∩ TⁿA) gt; μ(A)².

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