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By Marlies Gerber

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Extra info for Conditional Stability and Real Analytic Pseudo-Anosov Maps

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L et A, C be p o s i t i v e c o n s t a n t s . L e t s be af -1 0 < s ^ -r-. Suppose t h a t f o r x € (p. (9 ) - { x . 7. ,n(x). }, J 0 0 ^ ^ i s [|dg;n(x)|| < JQ Proof• and lldg^^li i S Jfl. i) •'• ^ . i ^ g < [I 5 V + x g x KAP-2||X||P-2]C||X|! -Cp-2) JQ = exp[CKAp~ ]. where The proof of the second inequality is similar. 9. 5) and | | g " 1 x - h " 1 y | | < [ 1 + K m a x ( | | x | | P " 2 , | | y | | p - 2 ) ] | | x - y | | + e||x||P. 6) Proof. By Lemma 6 . 7 , we have ||hx-hy|| < ||x-y||sup{||dh z | | : ||z|| < max( ||x|| ,||y||) } < l|x-y||[l+K m a x ( | | x | | p " 2 5 | | y | | p " 2 ) ] .

1. Let the stable and unstable curves for h be s u defined in terms of the E , and E , line fields, as in x, n x, n the third paragraph of §4. , then E , is the l x, n ~+ ~7 tangent line to 0 x € S? }, i,3 as defined in §2.

3) |s 2 (t)| > |a2|exp[(«n \)|a 2| Since a„ ^ 0, T is equal to the absolute value of the time t such that s„(t) = \ka„. T where C < |t|], t < 0. 3), -2(p-2) C'la I P = C'(k) = £S2£. We now c o n v e r t t h i s t o an e s t i m a t e i n t e r m s of ||x||. map #.. ( z ) = — z p 1,3 p $. t a k i n g x t o ( a . , a 0 ) i s g i v e n by 1)3 1 2 ° ^ we have a 2 + a 2 = - ^ | | x | | p . P Case 1. |a2| > ( ^ ) 2 a±. Since t h e , CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS 9 Then 9 U U 9 a^ + a^ < ( k X + l ) a * T where C <.

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