On weak concepts of stability

Extract

The manifold in this paper is assumed to be connected differentiable of class C∞. Let D^r(M) and Ӿ^r(M) be the set of all diffeomorphisms and vector fields of class C^r on a manifold M with Whitney C^r topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈D^r(M) or X ∈ Ӿ^r(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in D^r(M) or Ӿ^r(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ D^r(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿ^r(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.