  We generalize the concept of convective (or velocity-dependent) Lyapunov
exponent $\Lambda(v)$ to an entire spectrum $\Lambda(v,n)$. Our results are
supported by the consistency between the outcome of the chronotopic approach
[{\it S. Lepri et al. J. Stat. Phys., 82 5/6 (1996) 1429}] and a more direct
method. There exists a critical integrated density $n=n_c$, beyond which the
convective exponent exhibits a discontinuous dependence on the velocity, which
originates from the appearance of multiple branches. This phenomenon can be
traced back to a change of concavity of the so-called {\it temporal} Lyapunov
spectrum for $n>n_c$, which is therefore a dynamical invariant.
