A strange attractor is an infinite-point attractor with non-integer dimension. Although they consist of an infinite number of points, strange attractors do not fill state space. Instead, they are contained within a bounded region and are often highly structured. In fact, Strange attractors are a type of fractal, exhibiting self-similarity on all scales.
If you examined the evolution of a system characterized by a strange attractor, you would notice some interesting things. As with any attractor, the trajectory of the system would migrate towards the strange attractor region of state space and return there if displaced. But a trajectory following a strange attractor would never repeat itself, no matter how long you watched. The system would never take on the exact same state twice.
Furthermore, if you started the system at two similar states and watched the resulting evolution, you would see the two trajectories diverge from each other expoentially. Even if the starting points were almost identical, given a little time, the resulting outcomes would look totally different from each other. This sensitivity to initial conditions is a hallmark of chaotic systems.