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# 6 Almost Invariance and Recurrence

[page 634, §1]
[634.1.1] To discuss the question how invariant states can evolve in time (Problem 2) consider two invariant states u,vB0 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 SB0. [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

 G=Bε∩⋃z∈SN⁢z,A1m;η (32)

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 zG along its orbit according to

 T-t/τ⁢Kz⁢t0τ=Kz⁢t0-tτ (33)

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

 t=k⁢τ (34)

with kZ such that t0=0 produces discretized orbits Kz-k, kN for all zG as iterates of T1. [634.2.3] For every initial state zG define

 wG⁢z=min⁡k≥1:T-k⁢Kz⁢0∈G (35)

as the first return time of z into the set G. [634.2.4] For all invariant zB0 one has wGz=1. [634.2.5] For states z that never return to G one sets wGz=. [634.2.6] For all k1 let

 Gk=z∈G:wG⁢z=k (36)

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

 G∞=G∖⋃k∈NGk. (37)

[634.2.7] The states zS 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 λ0,1. [634.2.8] Their mixture

 Q=∫01Pλ⁢u+1-λ⁢v⁢d⁢λ (38)

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

 p⁢k=Q⁢GkQ⁢G (39)

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 SB0.