  We demonstrate the preservation of the Lyapunov modes by the underlying
tangent space dynamics of hard disks.
  This result is exact for the zero modes and correct to order $\epsilon$ for
the transverse and LP modes where $\epsilon$ is linear in the mode number.
  For sufficiently large mode numbers the dynamics no longer preserves the mode
structure.
  We propose a Gram-Schmidt procedure based on orthogonality with respect to
the centre space that determines the values of the Lyapunov exponents for the
modes.
  This assumes a detailed knowledge of the modes, but from that predicts the
values of the exponents from the modes.
  Thus the modes and the exponents contain the same information.
