[page 634, §1]

[634.1.1] To discuss the question how invariant states can
evolve in time (Problem 2) consider two invariant states
u,v∈B0 and the straight line
segment

S={z=λu+(1-λ)v,0≤λ≤1,u∈B0,v∈B0} | | (31) |

connecting u and v.
[634.1.2] Of course S⊂B0.
[634.1.3] In practical applications
S might be a more or less general subset of
B0, e.g., a KMS-state in ⋃βKβ.
[634.1.4] Straight line segments of invariant states are expected
to be physically important for phase transformations at
thermodynamic coexistence.
[634.1.5] Define a weak*-neighbourhood

of ε-almost invariant
η-indistinguishable states near S.
[634.1.6] Depending on the invariant states S
and macroscopic algebra M of interest
a similar weak*-neighbourhood
G=GS,M,ε,η
can be defined
for other subsets of B0.

[634.2.1] The time translations T-t/τ
with time scale τ translate any initial state z∈G
along its orbit according to

T-t/τKzt0τ=Kzt0-tτ | | (33) |

where t0 denotes the initial instant,
Kzt0/τ=z and τ>0
the time scale.
[634.2.2] Discretizing time as

with k∈Z such that t0=0 produces
discretized orbits Kz-k, k∈N
for all z∈G as iterates of T1.
[634.2.3] For every initial state z∈G define

wGz=mink≥1:T-kKz0∈G | | (35) |

as the first return time of z into the set G.
[634.2.4] For all invariant z∈B0 one has wGz=1.
[634.2.5] For states z that never return to G one
sets wGz=∞.
[634.2.6] For all k≥1 let

denote the subset of states with recurrence time 1≤k≤∞
with k=∞ interpreted as

[634.2.7] The states z∈S generate a one parameter
family of resolutions of the identity resulting
in a one parameter family of measures on
B,B denoted as
Pλu+1-λv with λ∈0,1.
[634.2.8] Their mixture

[page 635, §0]
is again an invariant measure on B,B.
[635.0.1] The numbers

define a discrete probability density on N∪∞.
[635.0.2] It may be interpreted as a properly weighted probability
of recurrence into the neighbourhood G of the straight
line segement S⊂B0.