  We conduct the multifractal analysis of the level sets of the asymptotic
behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a
topologically mixing subshift of finite type $X$ endowed itself with a metric
associated with such a potential. We work without additional regularity
assumption other than continuity. Our approach differs from those used
previously to deal with this question under stronger assumptions on the
potentials. As a consequence, it provides a new description of the structure of
the spectrum in terms of {\it weak} concavity. Also, the lower bound for the
spectrum is obtained as a consequence of the study sets of points at which the
asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$
rather than being equal to a constant. Specifically, we compute the Hausdorff
dimension of sets of the form $\{x\in X: \lim_{n\to\infty}
\phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has
interesting geometric applications to fixed points in the asymptotic average
for dynamical systems in $\R^d$, as well as the fine local behavior of the
harmonic measure on conformal planar Cantor sets.
