# Periodic orbits, symbolic dynamics and topological entropy by Arioli G.

By Arioli G.

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N ; ζ1 , . . , ζn ) + nε. Explicitly, ﬁx ε < ε. Then take r to be the number of balls of diameter δ which are needed to cover the state space of a C∗ -algebra B with dim B ≤ d, where δ is such that |S(ψ1 ) − S(ψ2 )| < ε for states ψ1 and ψ2 on B as soon as ψ1 − ψ2 < δ. Take a coupling Λ and partitions ξ1 , . . , ξn such that HΛ (γ1 , . . , γn ; ξ1 , . . , ξn ) is close to Hϕ (γ1 , . . , γn ) up to ε−ε . Then applying the procedure described above we get λ and ζ1 , . . , ζn . For channels γ, γ : B → A, put γ−γ where a ϕ ϕ = sup x∈B, x ≤1 γ(x) − γ (x) ϕ, = ϕ(a∗ a)1/2 .

Cn ) ≤ Hϕ (A1 , . . , An ) ≤ Hϕ (B) ≤ S(ϕ|B ). Hence the above inequalities are in fact equalities. 2. Further properties of mutual entropy are given in the following proposition. 7. We have: (i) if πϕ : A → B(Hϕ ) is the GNS-representation and ϕ¯ is the normal extension of ϕ to πϕ (A) , then Hϕ (γ1 , . . , γn ) = Hϕ¯ (πϕ ◦ γ1 , . . , πϕ ◦ γn ); (ii) if ψ is another state on A and 0 ≤ λ ≤ 1, then Hλϕ+(1−λ)ψ (γ1 , . . , γn ) ≥ λHϕ (γ1 , . . , γn ) + (1 − λ)Hψ (γ1 , . . , γn ) − (n − 1)(η(λ) + η(1 − λ)); (iii) if ψ is a state on a C∗ -algebra B, 0 ≤ λ ≤ 1 and θk : Bk → B, 1 ≤ k ≤ n, are channels, then on A ⊕ B Hλϕ⊕(1−λ)ψ (γ1 ⊕ θ1 , .

In other words, any abelian model (C, µ, {Ck }nk=1 , P ) for n−1 (A, ϕ, {γk }nk=1 ) deﬁnes an abelian model (C, µ, {Ck }n−1 k=1 , P ) for (A, ϕ, {γk }k=1 ) with Ck = Ck for k < n − 1 and Cn−1 = Cn−1 ∨ Cn . Comparing the entropies of these abelian models, we see that the classical terms as well as the ﬁrst n−2 correction terms coincide. 8 applied to I = In−1 and J = In shows that the entropy of (C, µ, {Ck }n−1 k=1 , P ) is not smaller than the entropy of (C, µ, {Ck }nk=1 , P ). It follows that Hϕ (γ1 , .