Stability, and Almost Sensitivity of Induced Maps

Abstract

Suppose that X is a compact Hausdorff space and f : X → X is continuous. We consider the space K(X), the space of all compact subsets of X with Hausdorff metric H. Let f˜: K(X) → K(X) defined by  f ̃(K)=f(K) . We discuss some interconnections between the orbit of f ̃ and the orbit of  f. By assuming the transitivity of f ̃, we conclude that X contains a cantor set C with (orb(f ̃,C)) ̅ = K(X). That is, orbit of a nowhere dense set is dense in the hyperspace.  Along with this we introduce ”almost sensitivity” and ”stability” in K(X). We prove that f is stable in X if and only if f ̃  is stable in K(X). Again we prove that transitive maps are always ’almost sensitive’ in K(X) and hence the base map is ’almost sensitive’ in X.