By Klaus Thomsen

Quantity 206, quantity 970 (end of volume).

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

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Using the description of the ideals in a C ∗ -algebra of an ´etale equivalence relation given in [Re1] it follows that Af (X, U ) is full in Af (X, E) in the sense of [Br]. 14 follows from [Re1] and [Br], at least in the separable case. 8 of [MRW] by showing that conditions 1,4,5 and 6 are suﬃcient to make E ×π E into a (Rf (X, E), Rf (X , E ))-equivalence in the sense of [MRW]. CHAPTER 3 The homoclinic algebra of expansive actions Let S be a countable set and (X, d) a locally compact metric space.

Let A be a ﬁnite set, sometimes called the alphabet, and X ⊆ AZ a shift space. Thus X is a closed subset of AZ which is shift-invariant in the sense that σ(X) = X, 23 24 3. THE HOMOCLINIC ALGEBRA OF EXPANSIVE ACTIONS where σ is the shift on AZ , viz. σ (xi )i∈Z j = xj+1 for all j ∈ Z. The words in X n are the elements w ∈ ∞ n=0 A which occur in some element of X in the sense that w = xi+1 xi+2 . . 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.

If i=1 , y = m i=1 Q 2N +1 Φ x, (wi )i=1 , y = 0 it follows that Q2N +1 x = y and hence that x = y = 0 since U ∩ S = {0}. This shows that Φ is injective and the lemma follows. 21. e. there is a sequence ϕk : Tn → Tn , k = 1, 2, . . , of continuous surjective group endomorphisms such that SQ is isomorphic, as a topological group, to the projective limit of the sequence Tn ϕ1 Tn ϕ2 Tn ϕ3 Tn ϕ4 . . In particular, SQ is connected and divisible. Proof. For each k, equip the subgroup Ωk = Qt −k (Zn ) + Qt −k+1 (Zn ) + · · · + Qt k−1 (Zn ) + Qt k (Zn ) of Qn with the discrete topology.