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F o l l o w s from (g). Wu, Ws (g) was proved in [ 4 1 ] . A means fWu D Wu, Cr fWs c Ws. (b) l a m i n a t i o n is a f o l i a t i o n F whose leaves F are Cr and U TkF is a continuous bundle, 1 < k < r (where Tk denotes x x x x the k ' t h order tangent). For instance the unstable manifolds of a Cr Anosov diffeomorphism form a invariance means invarianceunder a l l Proof. F i r s t we shall construct This is the hardest p a r t o f ( 4 . 1 ) . f at V Cr lamination. In the case of flows, t i m e - t maps.

Note t h a t such a t r i v i a l i z a b l e ble. Such a m e t r i c on The f o l l o w i n g theorem was s t a t e d i n c o r r e c t l y that F HOLDER SECTION THEOREM. over the compact metric space zable metric on h: X ÷ X. Let E. Let X. E. (The hypothesis 0 < k < I x E X and be a fiber map covering the homeomorphism be such that y, y' E E . x a of f(y')ifx ~ kly - y'l x Proof. h- I f-invariant section Extend f to XxY satisfies and h- I are Lipschitz satisfies 0 < b < 1 . of is b-H~lder. by c o m m u t a t i v i t y of XxY E~E' f Suppose further that ka b < 1 Then the unique be a Finslered Banach bundle Assume that the Finsler is induced by a triviali- f: E ÷ E and the Lipschitz constant f in 6 .

Hood If X h: X 1 ÷ X X0 C Xl In §§4,5,6 we w i l l C1 is a without boundary, and V compact manifold, is a compact is a compact neighborhood of Xl is normally expanding at V, then V V has a compact neighbor- hX0 D X0. such that f o r a whole c l a s s o f X Suppose need a s t r o n g e r v e r s i o n o f ( 3 . 6 ) : h's. For t h i s we need the f o l l o w i n g it must g i v e a u n i f o r m X0 form o f the I n v e r s e Function Theorem. 7) LEMMA. If A: E ÷ E' is a Lipschitz map having is injective.

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