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

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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 ) ] . A l s o , by Lemma 5 . 5 , we have ||gx-hx|| < e | | x | | p . 5). 10. 6) is similar. (Orbit Approximation Lemma). Let A, C, e be positive constants with e < 1 and A > 1. Let s be a1 -1 such that 0 < s < ~-r. Suppose that for x € cpIS . 1 } Lr\. ). 10. ,n(x). }. ,n(x). ,n(x), where K i s a s i n Lemma 6 . 7 . 10) where < e[l+2p-2Ap-2K||x||p-2]' 2-(p-2)K-1A2||x||2 « .

I Also II S x l 1/k. 14, we have eJ 3 l|x|| < «J a ' < 8. n dh,-n ^ R ^ ,-n „ contracts angles to h x g-nx,h x in K ^ by a factor of 1/k. But since £ „ maps J -n -n , -n * g x g x,h x J K -n „ to K -n ^ by a rotation, i vthis implies that dh n g x h x h x contracts angles in K by a factor of 1/k. 2. ,x }, dh contracts angles in X m i=l i s h"nx K by a factor of 1/2. ) 1 S 1 By the results of [Kl], §4, there is a constant b Q with m -1 w 0 < b n < 1 such that if x" €' M - U cp7 (B ) , then dg _ x x s --• g x i=l g" contracts angles in K _, by a factor of b n .

Cp-2) a, \2 + (kXa2)i O) C2l|x| k X +1 C 9 = C (k,p) -(p-2) is a constant. Therefore, in both cases, n(x) where < C(k,p)||x|r(p"2), C = C(k,p) = max(C l5C ) Remark. 17. 4. } n = n(x,k,p) angles in K k, say g~nx n(x,k,p) are k = 2. and let / P r 2 2/p p = I 2 , . ] , p = p(i). 3, + 39 by a factor of then 2 1/k . dg g" x contracts MARLIES GERBER 40 by Proof. 2) hold. Let be represented in the x (a l9 a«). , i 0. , or Then by the estimates in a? is non-zero. 5. Let k > 1 and let p = ["Ykxl » p = p(i).

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