2018/07/25 by Huang, Wen, Shao, Song, Ye, Xiangdong
#37B05 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.10155
In this paper we give an answer to Furstenberg's problem on topological disjointness. Namely, we show that a transitive system (X,T) is disjoint from all minimal systems if and only if (X,T) is weakly mixing and there is some countable dense subset D of X such that for any minimal system (Y,S), any point y∈ Y and any open neighbourhood V of y, and for any nonempty open subset U⊂ X, there is x∈ D∩ U satisfying that \n∈ \mathbb Z+: Tnx∈ U, Sny∈ V\ is syndetic. Some characterization for the general case is also described. As applications we show that if a transitive system (X,T) is disjoint from all minimal systems, then so are (Xn,T(n)) and (X, Tn) for any n∈ \mathbb N. It turns out that a transitive system (X,T) is disjoint from all minimal systems if and only if the hyperspace system (K(X),TK) is disjoint from all minimal systems.