  The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended
chaotic systems is studied by means of coupled-map lattices. We determine
intrinsic length scales and spatiotemporal correlations of LVs corresponding to
the leading unstable directions by translating the problem to the language of
scale-invariant growing surfaces. We find that the so-called 'characteristic'
LVs exhibit spatial localization, strong clustering around given spatiotemporal
loci, and remarkable dynamic scaling properties of the corresponding surfaces.
In contrast, the commonly used backward LVs (obtained through Gram-Schmidt
orthogonalization) spread all over the system and do not exhibit dynamic
scaling due to artifacts in the dynamical correlations by construction.
