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By Klaus Thomsen

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Extra info for C*-Algebras of Homoclinic and Heteroclinic Structure in Expensive Dynamics

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Let Q ∈ Mn (Z)∩Gln (Q) be hyperbolic. 19) Zn Q Zn Q Zn Q ... , 40 3. THE HOMOCLINIC ALGEBRA OF EXPANSIVE ACTIONS where Qt is the transpose of Q. It can happen that these inductive limit groups are not isomorphic. This is for example the case when Q= 65 7 , 24 67 cf. 6 of [BJKR]. In this case the homoclinic group of σQ is not isomorphic to SQ (as it is in the case where Q ∈ Gln (Z)). e. as dimension groups) the arguments for the stronger statement have been included in Appendix C. Presumably the phenomenon is not exceptional at all; the methods explored in Appendix C can be used to find other examples, albeit in a rather unsystematic way.

As observed above −→ C(G) has torsion-free K-theory and finite covering dimension. Since ψ induces a positively expansive surjective group endomorphism of G/Fk it follows that also C (G/Fk ) has torsion-free K-theory and that dim G/Fk = dim G < ∞ for all k. Since C (G/Fk ) and pk Mlk (C (G/Fk )) pk have the same K-theory we conclude that K∗ (Aψ (G)) is torsion-free. 7 of [G] that Aψ (G) is in fact an AT-algebra. It is then approximately divisible in the sense of [BKR] by a result of Elliott, [Ell1].

Let K be a compact subset of X and p ∈ Per X such that • p ∈ K, • ϕ−|p| (K) ⊆ K, and • there is an > 0 such that x ∈ K, d (ϕn (x), ϕn (p)) ≤ ∀n ∈ Z ⇒ x = p. 2) lim sup d ϕ−n (x), ϕ−n (p) = 0, n→∞ x∈K δ where K δ = x ∈ K : d ϕj (x), ϕj (p) ≤ δ, j ≤ 0 . 41 42 4. THE HETEROCLINIC ALGEBRA −j −1 Proof. Set L = |p|−1 -invariant and conj=0 ϕ (K). Then L is compact, ϕ tains the orbit of p. 3) κ < min ϕi (p) − ϕj (p) : i, j ∈ {0, 1, 2, . . 4) x, y ∈ L, d(x, y) ≤ κ ⇒ 1 d (ϕ(x), ϕ(y)) ≤ min ϕi (p) − ϕj (p) : i, j ∈ {0, 1, 2, .

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