  The time-dependent structure of the Lyapunov vectors corresponding to the
steps of Lyapunov spectra and their basis set representation are discussed for
a quasi-one-dimensional many-hard-disk systems. Time-oscillating behavior is
observed in two types of Lyapunov modes, one associated with the time
translational invariance and another with the spatial translational invariance,
and their phase relation is specified. It is shown that the longest period of
the Lyapunov modes is twice as long as the period of the longitudinal momentum
auto-correlation function. A simple explanation for this relation is proposed.
This result gives the first quantitative connection between the Lyapunov modes
and an experimentally accessible quantity.
