Give Hit - Or - Miss Topology Of Hyperspace Dynamical System Topological Entropy

The relationship of topological dynamical systems whose underlying space is a Hausdorff locally compact second countable space and the hyperspace dy-namical systems induced by it which is equipped with the hit-or-miss topology on Bowen entropy is systematically explored. The paper is organized as follows:In chapter1, the development and current research status of dynamical systems, hyperspace dynamical systems and entropy are recommended. Then, the main content of this paper is presented.A variety of symbols and relative definitions in this paper are introduced in chapter2, which provided basic knowledge tools for chapter3and4.In chapter3, let (X, d,f) be the topological dynamical systems whose un-derlying space is a Hausdorff locally compact second countable space, d is a compact-type metric,f is a perfect mapping, and (2x,p.2f) and (F, p.2f) arc denoted as the hyperspace dynamical systems induced by (X, d,d) which arc equipped with the hit-or miss topology. h(X,d,f), h(2x, p,2f) and h(F, p,2f) are Bowen topological entropies of the corresponding topological dynamical sys-tems. Firstly, there will be h(X, d, f)≤h(2x,p,2f) and h(X, d, f)≤h(F, p,2f) in the presence of (F, p,2f). Secondly, h(2x, p,2f)=oo and h(F, p,2f)=∞in the presence of (F. p,2f) are proved, when ent*(X,d,f)>0.In chapter4, let (X, d, f) be the topological dynamical systems whose un-derlying space is a Hausdorff locally compact second countable space, d is a compact-type metric,f is a perfect mapping, and (2x,p,2f) and (F, p.2f) are denoted as the hyperspace dynamical systems induced by (X,d,f) which are equipped with the hit-or miss topology. h(X,d,f), h(2x,p,2f) and h(F,p,2f) are Bowen topological entropies of the corresponding topological dynamical sys- tems. Three examples are given to explain that h(2x,p,2f) may be zero, a positive number or infinite when h(X,d,f)=0. Another example is also given to illustrate h(2x,p.2f) may be infinite when h(X, d, f)>0.