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and

(33.70)

Clearly

(33.71)

33. NONCOMPACT HYPERBOLIC LIMITS

FIGURE 33 .4. The annulus AA, p and the loop SA, p (p).

since SA,p C SA,p·

Represent 'D(t) by an embedding ft : D^2 ---+ M - MA (t). Regarding the loop

S A,p, we may assume that among the disjoint union of embedded loops comprising

ft-^1 (SA,p) we have that ft-^1 (SA,p) is an innermost loop in D. In other words, the

disk D^2 in D bounded by the loop ft-^1 ( S A,p) does not intersect any of the other

loops in ft-l ( S A,p). Define the immersed comparison disk

(33.72)

comparison

disk V

FIGURE 33.5. The original disk 'D(t) and the comparison disk D(t).