2019/03/29 by Zheng, Liqi, Zheng, Zuohuan
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.12640
Let (X,T) be a topological dynamical system with metric d. We define a new function F(x,y)=\limsupn → +∞ infσ∈ Sn \frac 1n ∑k=1n d(Tk x,Tσ(k) y) by using permutation group Sn. It's shown F(x,y)=limn → +∞ infσ∈ Sn \frac 1n ∑k=1n d(Tk x,Tσ(k) y) exists when x,y ∈ X are generic points. Applying this function, we prove (X,T) is uniquely ergodic if and only if F(x,y)=0 for any x,y ∈ X. The characterizations of ergodic measures and physical measures by F(x,y) are given. We introduce the notion of weak mean equicontinuity and prove that (X,T) is weak mean equicontinuous if and only if the time averages f*(x)=limn → +∞\frac 1n ∑k=1n f(Tk x) exist and are continuous for all f ∈ C(X).