2020/06/04 by Moragues, Andreu Ferré
#37A15 #37A30 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2006.03170
We prove that given a measure preserving system (X,B,μ,T1,…,Td) with commuting, ergodic transformations Ti such that TiTj-1 are ergodic for all i ≠ j, the multicorrelation sequence a(n)=∫X f0 ⋅ T1nf1 ⋅ \dotso ⋅ Tdn fd dμ can be decomposed as a(n)=a_\textrmst(n)+a_\textrmer(n), where a_\textrmst is a uniform limit of d-step nilsequences and a_\textrmer is a nullsequence (that is, limN-M → ∞ (1)/(N-M) ∑n=MN-1 |a_\textrmer|2=0). Under some additional ergodicity conditions on T1,…,Td we also establish a similar decomposition for polynomial multicorrelation sequences of the form a(n)=∫X f0 ⋅ ∏i=1dTi^pi,1(n)f1⋅\dotso ⋅ ∏i=1dTi^pi,k(n)fk dμ, where each pi,k: ℤ → ℤ is a polynomial map. We also show, for d=2, that if T1, T2, T1T2-1 are invertible and ergodic, we have large triple intersections: for all ε>0 and all A ∈ B, the set \n ∈ ℤ : μ(A ∩ T1-nA ∩ T2-nA)>μ(A)3-ε\ is syndetic. Moreover, we show that if T1, T2, T1T2-1 are totally ergodic, and we denote by pn the n-th prime, the set \n ∈ ℕ : μ(A ∩ T1-(pn-1)A ∩ T2-(pn-1)A)>μ(A)3-ε\ has positive lower density.