  We construct both normal and anomalous deterministic biased diffusions to
obtain the Einstein relation for their time-averaged transport coefficients. We
find that the difference of the generalized Lyapunov exponent between biased
and unbiased deterministic diffusions is related to the normalized velocity
based on the ensemble average. By Hopf's ergodic theorem, the ratios between
the time-averaged velocity and the Lyapunov exponent for single trajectories
converge to a universal constant, which is proportional to the strength of the
bias. We confirm this theory using numerical simulations.
