[212.2.1] Let Γ,G,μ,ΓTt˙ be a measure preserving system
for a many body system.
[212.2.2] The detailed microscopic time evolution ΓTt˙:Γ→Γ
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.
[212.2.5] This gives rise to the problem of finding the
time evolution
GTt˙:G→G on subsets G⊂Γ
of phase space.

[212.3.1] It is not possible to define GTt˙=ΓTt˙G
as the restriction of ΓTt˙ to G, because
for fixed initial state x0∈G⊂Γ
the time evolution ΓTt˙ produces states
ΓTt˙x0=xt˙∉G.
[212.3.2] Equivalently, for fixed time t˙
the map ΓTt˙ maps states x∈G to
states not in G.
[212.3.3] The restriction ΓTt˙G is not defined for all t˙∈R.
[212.3.4] This seems to preclude a sensible definition of GTt˙.
[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 ΓTt˙,
averaging Kakutani’s induced measure preserving transformation
[40, 26] and Kac’s theorem for recurrence times
[21, 1].