By A. H. Assadi, P. Vogel (auth.), Andrew Ranicki, Norman Levitt, Frank Quinn (eds.)
Read or Download Algebraic and Geometric Topology: Proceedings of a Conference held at Rutgers University, New Brunswick, USA July 6–13, 1983 PDF
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Extra resources for Algebraic and Geometric Topology: Proceedings of a Conference held at Rutgers University, New Brunswick, USA July 6–13, 1983
First, pass to a subsequence such that ψ n (ui ) → vi ∈ X. Next, choose a chain of overlapping rectangles connecting v1 to v2 . In case v1 or v2 is a zero of q, choose the initial or terminal rectangle so it also contains ψ n (ui ) for inﬁnitely many n. Let U1 , . . , Um be open sets in the overlap of adjacent rectangles. Then by the lemma above, we can ﬁnd yj ∈ Uj and xj ∈ E such that ψ n (xj ) → yj along a further subsequence. 61, v1 is generic for the same measure as y1 . Similarly yi and yi+1 are generic for the same measure, as are ym and v2 .
Note that F : X → X, while locally aﬃne, is not Anosov — it preserves a splitting of the tangent space, but the splitting is not along the stable and unstable manifolds of F . 12 Bers embedding In this section we discuss the Teichm¨ uller space of a quite general Riemann surface X, and its embedding as a domain in a complex Banach space. Ideal boundary. Let X = H/ΓX be a hyperbolic Riemann surface, presented as the quotient of the upper halfplane by a Fuchsian group. We do not assume that X has ﬁnite volume.
62 Given a sequence N ⊂ N and a nonempty open set U ⊂ X, there is an x ∈ Ei and a further subsequence N ′ ⊂ N along which ψ n (x) → y ∈ U . Proof. Let K ⊂ U be a compact set of positive measure, and let Kn = ψ −n (K). Then since ψ is measure preserving, the sum over n ∈ N of the |q|-measure of Kn diverges, and thus (by Borel-Cantelli) there is a set of A ⊂ E of positive Lebesgue measure consisting of points with ψ n (x) ∈ K for inﬁnitely many n ∈ A. Passing to a convergent subsequence yields the lemma.
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