[212.4.1] Consider a subset G⊂Γ with small but positive measure
μG>0 of a measure preserving many body system
Γ,G,μ,ΓTt˙.
[page 213, §0]
[213.0.1] Because of μG>0 the subset G becomes a
probability measure space G,S,ν with
induced probability measure ν=μ/μG
and S=G∩G being the trace of G
in G [41].

[213.1.1] The measure preserving continuous
time evolution ΓTt˙ is discretized by
setting

with k∈Z and τ>0
the discretization time step.
[213.1.2] A character x∈G is called recurrent, if
there exists an integer k≥1 such that ΓTkτx∈G.
[213.1.3] If G∈G and μ is invariant under ΓT, then
almost every character in G is recurrent
by virtue of the Poincarè recurrence theorem.
[213.1.4] A subset G is called recurrent, if μ-almost every point
x∈G is recurrent.
[213.1.5] By Poincarè’s recurrence theorem
the recurrence time tGx
of the character x∈G, defined as

tGx=τmink≥1:ΓTkτx∈G, | | (12) |

is positive and finite for almost every x∈G.
[213.1.6] For every k≥1 let

denote the set of characters with recurrence time kτ.
[213.1.7] Then the number

is the probability to find a recurrence time kτ.
[213.1.8] The numbers pk define a discrete probability
density
pkδs˙-kτ on the arithmetic
progression s˙-kτ,k∈N,s˙∈R.
[213.1.9] Every probability measure ϱs˙ on G,S
at time instant s˙ is then defined on the same arithmetic
progression through

for all B∈S and s˙∈R.
[213.1.10] The induced time evolution GT on the subset G
is defined for every B∈S as the average [1, 21]

GTϱB,s˙=∑k=1∞pkϱB,s˙-kτ | | (16) |

where s˙∈R.
[213.1.11] For characters ϱ=x∈G, one recovers
the first step in the discretized microscopic time evolution
GTxs˙=xs˙-tGx as expected.
[213.1.12] For mixed states ϱ this formula allows
the transition from the microscopic to the macroscopic
time evolution.
[213.1.13] It assigns an averaged translation to the first step in
the induced time evolution of mixed states.