2017/08/03 by Joseph Auslander, Auslander, Joseph, Xiongping Dai +1
Mathematics · #20M20 #37B05 #37B20 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1708.00996
openalex publication_date 2017/08/03 · openalex created_date 2017/08/17 · openalex updated_date 2026/07/28
Let π\colon T× X→ X with phase map (t,x)↦ tx, denoted (π,T,X), be a semiflow on a compact Hausdorff space X with phase semigroup T. If each t∈ T is onto, (π,T,X) is called surjective; and if each t∈ T is 1-1 onto (π,T,X) is called invertible and in latter case it induces π-1\colon X× T→ X by (x,t)↦ xt:=t-1x, denoted (π-1,X,T). In this paper, we show that (π,T,X) is equicontinuous surjective iff it is uniformly distal iff (π-1,X,T) is equicontinuous surjective. As applications of this theorem, we also consider the minimality, distality, and sensitivity of (π-1,X,T) if (π,T,X) is invertible with these dynamics. We also study the pointwise recurrence and Gottschalk's weak almost periodicity of ℤ-flow with compact zero-dimensional phase space.