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7 Induced Measure Preserving Transformations

[212.4.1] Consider a subset GΓ 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=GG being the trace of G in G [41].

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


with kZ and τ>0 the discretization time step. [213.1.2] A character xG is called recurrent, if there exists an integer k1 such that ΓTkτxG. [213.1.3] If GG 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 xG is recurrent. [213.1.5] By Poincarè’s recurrence theorem the recurrence time tGx of the character xG, defined as


is positive and finite for almost every xG. [213.1.6] For every k1 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,kN,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 BS and s˙R. [213.1.10] The induced time evolution GT on the subset G is defined for every BS as the average [1, 21]


where s˙R. [213.1.11] For characters ϱ=xG, 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.