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By Anatole Katok, Jean-Marie Strelcyn, Francois Ledrappier, Feliks Przytycki

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Extra resources for Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities

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For some s e q u e n c e Xip(0) + 0 P Let u 6 B k. q L e t us d e f i n e Zlp • = eXPx(T~)-l(u,xi p (u))~ ! z = eXPx(Tx)-l(u,~(u)). Let us fix e > 0, for p n > O. J Because b i g enough, x1 ÷ x one has and z. + z, for e v e r y i P and Let 38 p(z i ,z) ~ s, P P(x i ,x) <_ a, P p(¢n(z i ),¢n(z)) P ~ s p ( ¢ n ( x i ),%n(x)) P <_ s But z V ( x i ), so t h a t P i P o ( ~ n (z i ) , ¢ n ( x i )) P P and consequently from p(~n(z),~n(x)) As ~ > 0 . 5 + 2~). 7 z ~ U ( x , ~ r,s, ~ ~,7) because u 6 Bk q 0 < q < 6Z r,s,e,y and Thus, to f i n i s h ~n(z) E Un(X).

If Let continuous. probability ~p measure is a b s o l u t e l y - Nikodym = d~p du p. e. one is a c o m p a c t X and is b i m e a s u r a b l e , is c a l l e d 1 j (p-l) op that mapping, ~(A) when 1 = +--j restriction A Radon that J(p) and spaces - algebras p a new mapping see suppose ~m. the to then us A Thus the two measure ~ measurable. that measure function ~1 = +~ Let on ~. be p are mapping such The of that is a b s o l u t e l y define is e a s y on a measurable and 6 A that Jacobian is defined us p the measure of THEORY (Y,~,~) Let continuous, on and ~ = ~(p(A)).

This verified together that with < ~ 4b. 1) will assure that y (V(x) which is a contradiction. We want %ny to c h e c k E U that for all n t 0 (x). 2 it f o l l o w s that U n(x) 9 {w 6 M ; p ( w , ~ n ( x ) ) < Q2(x) _ - - 4bne-n~r,s}. 3) By d e f i n i t i o n E l x ' ÷ Elx. Thus, 4bn -ng e r,s L. 1). to the reader. 1, (cf. Sec. 4), 6Z 0 < q < r,s,e,7' q, 1 and for i big enough, the m a n i f o l d V. (x) n U(x,q) 1 is of the f o r m { e X P x ( T i ) - l ( u , x i ( u ) ) ;Ilull ~ q}, Xi : Bkq ÷ ~ m - k , where k = k(x) = k(x i) for all i under considera- tion.

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