  The relation between the Lyapunov modes (delocalized Lyapunov vectors) and
the momentum autocorrelation function is discussed in two-dimensional hard-disk
systems. We show numerical evidence that the smallest time-oscillating period
of the Lyapunov modes is twice as long as the time-oscillating period of
momentum autocorrelation function for both square and rectangular
two-dimensional systems with hard-wall boundary conditions.
