Symposium on Algebraic Topology in Honor of Jose Adem by Gitler S. (ed.)

By Gitler S. (ed.)

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Sn , fn , Sn+1 associated (in the obvious way) to s. Then x = fn fn−1 · · · f0 (x)Efn fn−1 · · · f0 (y) = y , so find k with gk (x ) = y . Fix α0 ⊇ s with α0 (n+ 1) = k. Then clearly α0 ∈ A α0 (as (x, y) ∈ En+1 ), so A ∩ Ns = ∅. 13 Generic Compressibility Let E be a countable Borel equivalence relation on X, and let D : E → R+ be a Borel cocycle. E is D-aperiodic if |[x]E |x is infinite, for all x ∈ X, where | · |x is defined as in Section 8. The following extends a result of Wright [Wr]. 1 (Kechris-Miller).

Now set A = {φm (Sij ) : m < r, Si ⊆ φ−1 m (A ∩ Am ), j < pi }. Clearly A ⊆ A and  µ(A \ A ) ≤ µ   Sij ) : m < r, i < n} {φm (Si \ j 0. Then for all sufficiently large k, there is a refinement B of rank 2k , such that µ({x : T (x) ∈ OrbitAB (x)}) < δ. Proof. Let T : A0 → A0 be a Borel automorphism which induces E|A0 . We can assume that T is aperiodic by throwing away a null set.

Given an array A, we use A to denote A0 ∪ · · · ∪ An−1 , and φi,j to denote φj ◦ φ−1 : Ai → Aj . It is useful to think of φi,j as a link between Ai , i Aj . The rank of A is given by rank(A) = n. A refinement of A of rank k is a system B = B0 , . . , Bk−1 , ψ0 , . . , ψk−1 , where {Bi } is a partition of a conull subset of A0 . This gives rise to a subarray AB of A defined by AB = B i,j ,φ i,j i,j

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