Arbitrarily slow approach to limiting behavior

ABSTRACT. Let f(k, t): RN x [0, oo) -- R be jointly continuous in k and t, with lim(t)--(oo) f(k, t) = F(k) discontinuous for a dense set of k's. It is proven that there exists a dense set T of k's such that, for k e T , |f(k, t) - F(k)| approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function g(t) -- 0 . This result is applied to diffusion and conduction in quasiperiodic media and yields arbitrarily slow approaches to limiting behavior as time or volume becomes infinite. Such a slow approach is in marked contrast to the power laws widely found for random media, and, in fact, implies that there is no law whatsoever governing the asymptotics.