Dynamical systems generating large sets of probability distribution functions

For a topological dynamical system (X,f)(X,f) we consider the structure of the set F(f)F(f) of asymptotic distributions of the distances between pairs of trajectories. If f   has the weak specification property then F(f)F(f) is closed and convex, and it can contain all nondecreasing functions [0,diamX]→[0,1]. We discuss relations to distributional chaos.