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Conditional Stability and Real Analytic Pseudo-Anosov Maps by Marlies Gerber

By Marlies Gerber

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T M - K is greater than 5 gx gx ! = C (J,C) is a constant. Thus, CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS if 33 + JLhx ,gxdhxv t , * Kgx' lldSx-*hx,gxdhx" - C'"X||P-2. 6) Now choose e > 0 e(l+C,J^ 2 / p ) < C f . 7. 6) would contradict Thus H, dh K c K , and consequently, nx ^gx x x §^ dh K + c (»_ )"1K+ = K* . The proof that dCh""1) K~ c K~ . xx hx,gx gx hx ^ xx h x is similar. 8. p < a? and C with and h € 3P~ ( g ^ ) (i) There e x i s t p o s i t i v e c o n s t a n t s dh K and and C2pP~ < 1/4, x € IK - {x i > e, p, such t h a t i f with ||x|| < p, l

Thus -1 — = k V2k Vk Va{ + a 2 -33 . ^g „ „ x^ Hence > B, B = ( V 2 k ) " 2 / p < 1. where T h e r e f o r e we have |[g-n(x)x . h-n(x)x|| ^ ilg-n(x)xll provided If g~ x (t.. , t ? ) ^ ^ B " eL||x|| < 1. e(x) is the angle from the line through to the line through x. and h~ n x x. and in the smooth coordinate system, then |sine(x)| ,-n(x) „ M -n(x) IIJS *-JlJ2 *IL < II g . 8) if (s,,s 2 ) eL||x|| < 1 coordinate system. and ($. )~ {(s,,s 2 ): |e(x)| < a. | s 2 | > 2s,} h ~ n ( x ) x { (* i 9 J cp i )" 1 {(s 1 ,s 2 ): eL||x|| < B | s i n aj .

X h > = are one-dimensional subspaces of Clearly, such line fields Jy invariant. T M. 1, stable and unstable curves are defined as in §4. 2). ,x } to be slopes with respect to the 1 m (£,,£9> coordinate systems defined in §5. If we do not restrict to a single S? , i > m, or a single these coordinates are only defined up to changes of sign in each of the coordinates. Consequently the slopes are also defined only up to sign. ,x }, then we can still discuss the change in the absolute value of the difference between the slopes *See footnote at end of Section 6.

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