Ergodic adiabatic invariants of chaotic systems

For a slowly time-dependent Hamiltonian system exhibiting motion which ergodically covers the energy surface, the phase-space volume enclosed inside this surface is an adiabatic invariant. In this paper the scaling of the error in the adiabatic approximation is investigated for this situation via numerical experiments on chaotic billiard systems. It is found that the scaling depends on the long-time behavior of correlations in the chaotic system.