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By Tolman R. C.

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14) remain valid. 1, and (μ( )ξL\ ) be a system of conditional probabilities for (L , ( x )x∈L , ( ¯ ) ∈F ). We now introduce a system (F ∗ μ( )ξL \ ) on (L , ( x )x ∈L , ( ¯ ) ∈F ). For finite ⊂ L and ξ L \ ∈ ∗L \ , let = {x ∈ L : M(x) ∩ = Ø}. Then, by the properties of a morphism, F(η ∨ ξ L \ ) = F (η ∨ ξ L \ ) ∨ ξ L\ 30 Gibbs states: complements ∗ L\ for some fixed ξ L\ ∈ F ∗ μ( )ξ L \ , whenever η μ( {ξ } = μ( η ∈ The family (F ∗ μ( (L , ( x )x ∈L , ( ¯ ) )ξ L ∈F \ )ξ L\ ∨ ξL \ ∈ {F (ξ ∨ ξ L \ )} {F (η )ξ L\ .

19) lim | (a)|−1 log Z (a) | (b)|−1 log Z˜ (b) + a→∞ | (b)| where Z˜ (b) is now computed with ˜ (b) = {ξ˜ | (b) : ξ˜ ∈ (b)+ }, and is any finite set containing 0. 9). 3) gives U (ξ ∗ | ) + A (τ x ξ ∗ ) = X⊂ x∈ Therefore, if ∗ (ξ ∗ |X ) − x∈ (ξ ∗ |X + x). X ∈ A0 , U (ξ ∗ | ) + A (τ x ξ ∗ ) N ( )| | x∈ and lim | (a)|−1 log Z ∗ a→∞ (a) − log Z ∗ (a) (A ) = 0. 10) and P = P(A ) for all ∈ A. 11) follow from the corresponding properties of P ∗ . 12) follows. 5 Invariant states For each a ∈ Zν a linear map τ a in the space C ∗ of real measures on by (τ a σ )(A) = σ (A ◦ τ a ), σ ∈ C∗, is defined A ∈ C.

We let π(A) be the class of A in L ∞ ( , σ ). 24) Mfinite⊂L\ where the closure is taken with respect to the topology of L ∞ ( , σ ) as weak dual of L 1 ( , σ ). Then Bσ will be called the algebra at infinity associated with σ . e. Bσ consists of the almost everywhere constant functions) by the following cluster property. (C) For all A ∈ C there exists (B ∈ C M , M∩ finite ⊂ L such that = Ø) ⇒ |σ (AB) − σ (A)σ (B)| B . Suppose first that (C) holds. Then for all A ∈ C and B∞ ∈ Bσ , |σ (AB∞ ) − σ (A)σ (B∞ )| B∞ .

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