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5 Conservative Systems

[211.2.1] In classical mechanics the commutative algebra of observables A=C0Γ is the algebra of continuous functions on phase space Γ, that vanish at infinitye (This is a footnote:) e This means that for each aC0Γ and ε>0 there is a compact subset KΓ such that ax<ε for all xΓK.. [211.2.2] The characters (pure states) are point measures on phase space Γ, and one has the isomorphism ΓXA. [211.2.3] By the Riesz representation theorem the states μA*C0XA*C0Γ* in classical mechanics are probability measures on phase space ΓXA. [211.2.4] Every state μC0Γ* gives rise to a probability measure space Γ,G,μ where G is the σ-algebra of measurable subsets of phase space Γ.

[211.3.1] Let S:ΓΓ be an invertible map such that S and S-1 are both measurable, i.e. such that S-1G=SG=G where SG:=Sx:xG for GG. [211.3.2] The map S is called a measure preserving transformation and the measure μ on Γ is called invariant under S, if μG=μSG=μS-1G for all GG. [page 212, §0]    [212.0.1] An invariant measure is called ergodic with respect to S, if it cannot be decomposed into a convex combination of S-invariant measures.

[212.1.1] Here and in the following the measure preserving transformation is the adjoint time evolution A*Tt˙ which is denoted more briefly as ΓTt˙=A*Tt˙. [212.1.2] Pure states (characters) are not invariant under ΓTt˙. [212.1.3] Examples of invariant probability measures are furnished by the set of equilibrium states of a conservative system with Hamiltonian dynamics. [212.1.4] If μ is an equilibrium state of a conservative system then Γ,G,μ,ΓTt˙ is a measure preserving system.