[212.2.1] Let be a measure preserving system
for a many body system.
[212.2.2] The detailed microscopic time evolution
is frequently not of interest in applications, because it is much
too detailed to be computable.
[212.2.3] Instead one is interested in a reduced, coarse grained
or averaged time evolution of macroscopic states where
the system is locally or globally in equilibrium.
[212.2.4] Examples are isolated systems at phase coexistence or in
metastable states^{f} (This is a footnote:) ^{f}
This differs from
relaxation to equilibrium discussed in [34]..
[212.2.5] This gives rise to the problem of finding the
time evolution
on subsets
of phase space.

[212.3.1] It is not possible to define as the restriction of to , because for fixed initial state the time evolution produces states . [212.3.2] Equivalently, for fixed time the map maps states to states not in . [212.3.3] The restriction is not defined for all . [212.3.4] This seems to preclude a sensible definition of . [212.3.5] The problem of defining an induced continuous time evolution for mixed states on subsets of small measure was introduced and solved in [1, 21]. [212.3.6] It originated from the general classification theory for phase transitions [35, 36, 37, 38, 39]. [212.3.7] The solution involves discretization of , averaging Kakutani’s induced measure preserving transformation [40, 26] and Kac’s theorem for recurrence times [21, 1].