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.
