  In this paper we prove that if $f$ is a self-mapping of a nonempty subset $K$
of a normed space $X$ that satisfies some mild conditions, then the minimal
displacement of large iterations $f^n$ always dominates that of $f$ along
certain $f^n$-invariant regions. As a consequence, we deduce that when $X$ is a
Banach space, $K$ is closed convex and $f$ is continuous with $f^n$ being
compact for some $n\geq 1$, then $f$ has at least one fixed point. This offers
a new approach resulting in a streamlined proof of the long-standing asymptotic
conjecture.
