By D. Mumford

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X 2b/ 36 2 Standard Form for some b 2 Q. Take a parametrization W S 1 ! x/ D . x C a/ holds, where a 2 Q is arbitrary at this point. f 2 '/. x C a/ f D '. x/ D . x/ D '. mod 1/: 2 This ambiguity cannot be settled now. Since ' W S 1 ! @0 A and W S 1 ! 1; x/ D Then S1 ! x/. 4 Amphidrome Annuli 37 as required. mod 1/. t u This sublemma has the following corollary. 3. A a homeomorphism such that f j @A W @A ! @A is periodic. Suppose Ai be an amphidrome annulus in A with respect to f . Ai / the screw number of f in Ai .

1 Proof (of (i)). We must show that a given pseudo-periodic map f W ˙g ! ˙g is isotopic to a pseudo-periodic map in standard form. Let fCi griD1 be the precise system of cut curves subordinate to f (Chap. 1). C / D C ; S C being riD1 Ci . ˙g C / W ˙g C ! ˙g C 48 2 Standard Form is isotopic to a periodic map. Then f jB W B ! B is also isotopic to a periodic map (Cf. A /; so we may assume that f j B is already periodic. s/ accordingly. Since f cyclically permutes the connected components of A . / , all the annuli contained in A .

2. A be a linear twist with respect to a parametrization S 1 ! 1. 2 (U NIQUENESS OF LINEARIZATION). Let f; f 0 W A ! A be linear twists of an annulus A. Suppose that f j @A D f 0 j @A; and that the screw number of f in A is equal to the screw number of f 0 in A. Then there is an isotopy 26 2 Standard Form h W A ! D f j @A/ on @A, and (iii) f D h1 1 f 0 h1 . 2. This shows that the quality of being linear is independent of the parametrization up to a sort of isotopy given by conditions (i), (ii) and (iii) above.