# Invariant Manifolds, Entropy and Billiards. Smooth Maps with by Anatole Katok, Jean-Marie Strelcyn, Francois Ledrappier,

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

Publication by means of Katok, Anatole, Strelcyn, Jean-Marie

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Additional info for Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities

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Inequality [] that the n u m b e r < _ <(x) b~-- the i e q u a l i t i e s I < < < min(~,~). 1. 1 gives ~ = #x" of > i, then C = C(x) a new > i. C In- by C = K( 1 + ~ 2 ( I , ~ , ~ , < ) ) . 1). follows [Pes] I. exactly a l o n g the lines of the p r o o f S i n c e the p r o o f for the sake of c o m p l e t e n e s s , in [Pes] 1 is s o m e w h a t give the p r o o f in our in full detail. 1) defined. o f0) (z) is Then (fn .... o f0 ) (z) = ~ - 4 b ( n + l ) F n x ( Z ) = -4b(n+l) = 1 (T,n+lxOeXp-~n+l o ~ n + l o e x P x o(Tx)-l) (z) x 31 Let now y (V(x) .

Let B(HI) c H2 dimensional L e t us d e n o t e H2 and of t w o c H2' L e t us e x t e n d to E1 the o p e r a t o r s by taking and B I H1 by by A I UB = where [ H1 U: H 2 ÷ HI when H 1 = H 2 = E, because of A is an i s o m e t r y , otherwise and P tensor that will E P ~ E E, We we can consider the can also situation assume A 1 = a-iA and to t h e that case a = i, B 1 = a-iB instead B. L e t us p a s s Let we can reduce d i m E = 2p. to some introductory Euclidean P A E denote and power now be a r e a l and P P A E c ® E.

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).