Arbitrarily slow approach to limiting behavior

Abstract:

Let $f({\mathbf {k}},t):{\mathbb {R}^N} \times [0,\infty ) \to \mathbb {R}$ be jointly continuous in ${\mathbf {k}}$ and $t$, with ${\lim _{t \to \infty }}f({\mathbf {k}},t) = F({\mathbf {k}})$ discontinuous for a dense set of ${\mathbf {k}}$’s. It is proven that there exists a dense set $\Gamma$ of ${\mathbf {k}}$’s such that, for ${\mathbf {k}} \in \Gamma$, $|f({\mathbf {k}},t) - F({\mathbf {k}})|$ approaches 0 arbitrarily slowly, i.e., roughly speaking, more slowly than any expressible function $g(t) \to 0$. 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.