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C*-Algebras of Homoclinic and Heteroclinic Structure in by Klaus Thomsen

By Klaus Thomsen

Quantity 206, quantity 970 (end of volume).

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

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1. 9) that there is an N ∈ N such that −N ≤ k and d ϕl (z), ϕl (p) ≤ 2p for all z ∈ W u (p, k, ηp ) and all l ≤ −N . Set Vx = z ∈ X : d ϕl (z), ϕl (x) < and δx = p 2 p 2 , −N ≤ l ≤ 0 ∩ W

2 If necessary, see page 5 of [K-JT] for the definition of Θf,g . 20 2. 11. Assume that both π and π satisfy conditions 1 and 4, and that E, E and E are second countable. It follows that [π ◦ π] = [π ] • [π] in KK (Af (X, E), Af (X , E )), where • is the Kasparov-product. Proof. ) [K-JT]. The present case is greatly simplified by the fact that the degree 1 operator of our Kasparov triples are both zero. Thus the product [π ] • [π] is represented by the triple Eπ ⊗π• Eπ , π• ⊗π• idEπ , 0 , in the notation from [K-JT].

Xi+|w| for some i ∈ Z and some x ∈ X. e. the number of letters in w. The empty word is then the word of length 0. The set of words in X of length k is denoted by Wk (X), so that the set W(X) of all words in X is W(X) = ∞ k=0 Wk (X). Let w ∈ W(X). The context E(w) of w consists of the pair (a, b) ∈ W(X)2 with the property that awb ∈ W(X). A simple but crucial observation is the following: When w, v ∈ W(X), E(w) = E(v) and (a, b) ∈ E(w), then E(awb) = E(avb). This observation is used tacitly (and often) in the following.

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