  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 bounded distorsion property
assumption. We express the whole Hausdorff spectrum in terms of a conditional
variational principle, as well as a new large deviations principle. Our
approach provides a new description of the structure of the spectrum in terms
of {\it weak} concavity. Another new point is that we consider 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 is
naturally related to Birkhoff's ergodic theorem and 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.
