H. Narnhofer, W. Thirring C* Dynamical Systems that Asymptotically are Highly Anticommutative (26K, LaTeX) ABSTRACT. It is shown that with probability 1 on $\Theta$ resp. on $g_X$ the irrational rotation algebra ${\cal M}_\Theta$ with respect to the CAT map and the generalized Price--Powers shift ${\cal A}_X$ are asymptotically highly anticommutative.