On chaotic minimal center of attraction of a Lagrange stable motion for topological semi flows

Let f:R+×X→Xf:R+×X→X be a topological semi flow on a Polish space X. In 1977, Karl Sigmund conjectured that if there is a point x in X   such that the motion f(·,x)f(·,x) has just X as its minimal center of attraction, then all such x form a residual subset of X  . In this paper, we first present a positive solution to this conjecture and then apply it to the study of chaotic dynamics occurring inside or near the minimal center of attraction of a motion f(·,x)f(·,x).