5 Conservative Systems

[211.2.1] In classical mechanics the commutative algebra of observables A=C0⁢Γ is the algebra of continuous functions on phase space Γ, that vanish at infinity^e (This is a footnote:) ^e This means that for each a∈C0⁢Γ and ε>0 there is a compact subset K⊆Γ such that a⁢x<ε for all x∈Γ∖K.. [211.2.2] The characters (pure states) are point measures on phase space Γ, and one has the isomorphism Γ≡X⁢A. [211.2.3] By the Riesz representation theorem the states μ∈A*≡C0⁢X⁢A*≡C0⁢Γ* in classical mechanics are probability measures on phase space Γ≡X⁢A. [211.2.4] Every state μ∈C0⁢Γ* 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:Γ→Γ be an invertible map such that S and S-1 are both measurable, i.e. such that S-1⁢G=S⁢G=G where S⁢G:=S⁢x:x∈G for G∈G. [211.3.2] The map S is called a measure preserving transformation and the measure μ on Γ is called invariant under S, if μ⁢G=μ⁢S⁢G=μ⁢S-1⁢G for all G∈G. [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 Γ⁢Tt˙=A*⁢Tt˙. [212.1.2] Pure states (characters) are not invariant under Γ⁢Tt˙. [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,Γ⁢Tt˙ is a measure preserving system.