2023/09/11 by Shao, Hua
#37B40 #37B55 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2309.05225
Let f0,∞=\fn\n=0∞ be a sequence of continuous self-maps on a compact metric space X. Firstly, we obtain the relations between topological sequence entropy of a nonautonomous dynamical system (X,f0,∞) and that of its finite-to-one extension. We then prove that the topological sequence entropy of (X,f0,∞) is no less than its corresponding measure sequence entropy if X has finite covering dimension. Secondly, we study the supremum topological sequence entropy of (X,f0,∞), and confirm that it equals to that of its n-th compositions system if f0,∞ is equi-continuous; and we prove the supremum topological sequence entropy of (X,fi,∞) is no larger than that of (X,fj,∞) if i≤ j, and they are equal if f0,∞ is equi-continuous and surjective. Thirdly, we investigate the topological sequence entropy relations between (X,f0,∞) and (M(X),f0,∞) induced on the space M(X) of all Borel probability measures, and obtain that given any sequence, the topological sequence entropy of (X,f0,∞) is zero if and only if that of (M(X),f0,∞) is zero; the topological sequence entropy of (X,f0,∞) is positive if and only if that of (M(X),f0,∞) is infinite. By applying this result, we obtain some big differences between entropies of nonautonomous dynamical systems and that of autonomous dynamical systems. Finally, we study whether multi-sensitivity of (X,f0,∞) imply positive or infinite topological sequence entropy.