By James R. Milgram

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1i ; i1 /. To prove (iii), consider f k W A i ! Ai as f in sublemma 1 and take a parametrization W Œ0; 1 S 1 ! Ai 38 2 Standard Form there. We identify Ai with Œ0; 1 S 1 through . 1 (Fig. 2). 0; Let us pass to the universal covering Œ0; 1 Let IQ be the lift of the arc x S 1 the same a/: R. 1; a C n/ for some n 2 Z. Thus there is a lift fQk W Œ0; 1 R ! Œ0; 1 R S 1 ! 1; 2aCn/. Ai / C 2 i which proves (iii). mod 1/; i /, assertion (iv) follows from (iii). 4 Amphidrome Annuli 39 Now we can fix the ambiguity of the number a unsettled in Sublemma 1.

A. @ isotopic to a homeomorphism f 0jA . / W A. / ! A . / . / such that, for each annulus Aj in A. f 0 j A . / /r W Aj ! Aj . / . / is a linear twist or a special twist, where r denotes the number of the annuli in A . / . Applying this isotopy for each D 1; 2; : : : ; s, we get a pseudo-periodic map f 0 W ˙g ! ˙g in standard form, which is isotopic to the original f W ˙g ! ˙g : t u This completes the proof of (i). Proof (of (ii)). Suppose we are given two pseudo-periodic maps f; f 0 W ˙g ! ˙g in standard form.

Of course if fAi griD1 is an invariant system of annular neighborhoods of a precise cut system fCi griD1 subordinate to a pseudo-periodic map f W ˙g ! Ai / D Ai and f interchanges the boundary components. m; ; / of a boundary curve C of A oriented by the orientation ! induced from A is defined as the valency of C with respect to the periodic map @A ! 1. A a homeomorphism such f j @A ! @A is periodic. Ai / the screw number. mod i / and 0 Ä ıi < i ; D 0; 1: ! Proof. The oriented Ai will be denoted by Ai ; the induced orientation in @Ai by !