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

From Lyapunov modes to the exponents for hard disk systems