[page 634, §1]

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

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connecting and . [634.1.2] Of course . [634.1.3] In practical applications might be a more or less general subset of , e.g., a KMS-state in . [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

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of -almost invariant -indistinguishable states near . [634.1.6] Depending on the invariant states and macroscopic algebra of interest a similar weak*-neighbourhood can be defined for other subsets of .

[634.2.1] The time translations with time scale translate any initial state along its orbit according to

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where denotes the initial instant, and the time scale. [634.2.2] Discretizing time as

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with such that produces discretized orbits , for all as iterates of . [634.2.3] For every initial state define

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as the first return time of into the set . [634.2.4] For all invariant one has . [634.2.5] For states that never return to one sets . [634.2.6] For all let

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denote the subset of states with recurrence time with interpreted as

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[634.2.7] The states generate a one parameter family of resolutions of the identity resulting in a one parameter family of measures on denoted as with . [634.2.8] Their mixture

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

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define a discrete probability density on . [635.0.2] It may be interpreted as a properly weighted probability of recurrence into the neighbourhood of the straight line segement .