In this paper,a polynomial version of the Furstenberg joining is introduced and its structure is investigated.Particularly,it is shown that if all polynomials are non-linear,then almost every ergodic component of the ...In this paper,a polynomial version of the Furstenberg joining is introduced and its structure is investigated.Particularly,it is shown that if all polynomials are non-linear,then almost every ergodic component of the joining is a direct product of an infinite-step pro-nilsystem and a Bernoulli system.As applications,some new convergence theorems are obtained.Particularly,it is proved that if T and S are ergodic measure-preserving transformations on a probability space(X,X,μ)and T has zero entropy,then for all c_i∈Z{0},all integral polynomials pjwith deg p_(j)≥2,and all fi,gj∈L^(∞)(X,μ),1≤i≤m and 1≤j≤d,lim N→∞1/N^(N-1)∑n=0f_(1)(T^(cmn)x)·g(1)(S^(p1(n)x)…gd(S^(pd(n)))x)exists in L^(2)(X,μ),which extends a recent result by Frantzikinakis and Host(2023).Moreover,it is shown that for an ergodic measure-preserving system(X,X,μ,T),a non-linear integral polynomial p and f∈L^(∞)(X,μ),the Furstenberg systems of(f^(T^(p(n))x))n∈Zare ergodic and isomorphic to direct products of infinite-step pronilsystems and Bernoulli systems for almost every x∈X,which answers a problem by Frantzikinakis(2022).展开更多
The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic...The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.展开更多
基金supported by National Key R&D Program of China(Grant Nos.2024YFA1013601 and 2024YFA1013600)National Natural Science Foundation of China(Grant Nos.12426201,12371196,12031019 and 12090012)。
文摘In this paper,a polynomial version of the Furstenberg joining is introduced and its structure is investigated.Particularly,it is shown that if all polynomials are non-linear,then almost every ergodic component of the joining is a direct product of an infinite-step pro-nilsystem and a Bernoulli system.As applications,some new convergence theorems are obtained.Particularly,it is proved that if T and S are ergodic measure-preserving transformations on a probability space(X,X,μ)and T has zero entropy,then for all c_i∈Z{0},all integral polynomials pjwith deg p_(j)≥2,and all fi,gj∈L^(∞)(X,μ),1≤i≤m and 1≤j≤d,lim N→∞1/N^(N-1)∑n=0f_(1)(T^(cmn)x)·g(1)(S^(p1(n)x)…gd(S^(pd(n)))x)exists in L^(2)(X,μ),which extends a recent result by Frantzikinakis and Host(2023).Moreover,it is shown that for an ergodic measure-preserving system(X,X,μ,T),a non-linear integral polynomial p and f∈L^(∞)(X,μ),the Furstenberg systems of(f^(T^(p(n))x))n∈Zare ergodic and isomorphic to direct products of infinite-step pronilsystems and Bernoulli systems for almost every x∈X,which answers a problem by Frantzikinakis(2022).
基金This work was supported by National Natural Science Foundation of China(Grant Nos.11431012,11971455,11571335 and 11371339).
文摘The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d 2 N,is introduced and investigated.It turns out that if(X;T)is a topological dynamical system with AP[d]=Δ,then each ergodic measure of(X;T)is isomorphic to a d-step pro-nilsystem,and thus(X;T)has zero entropy.Moreover,it is shown that if(X;T)is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP[d]=RP[d]for each d 2 N.It follows that for a minimal 1-pro-nilsystem,AP[d]=RP[d]for each d 2 N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.
