Séminaire Bourbaki vol. 1968/69 Exposés 347-363 by Pierre Gabriel (auth.)

By Pierre Gabriel (auth.)

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Sur l'intersection de l'horicycle oppos4 & V avec n'importe quel horicycle normalis4 par H, et de ramener le calcul au cas du groupe SL(2), oh il reste & sommer des progressions ggom4triques. On trouve alors que 47 (37) Cp. 2 - --[- ,~() ,~p ++S'Pu>+"~) oa Gp(Z) = (I - p-~)-1 si ~ e _c. o<£U Cette formule, valable pour p fini, suppose que G est d4ploy4, et que la composante p-adique de ~- est triviale. Pour p infini, et dans les m~mes hypoth&ses, on trouve (38) cj~-~ = "I [ %( + ~) oa G~ c

Smale introduit enfin, en plus des conditions a) et b) avec les notations du th4or&me 6, la condition suivante (Axiome c) si wS(~i) D wu(~j) ~ ~, y~j tels que les vari~t4s pr4c4dentes, et B il existe deux points p@riodiques WS(x) et WU(y) de [14]) : x(~ i et poss&dent un point d'intersection transversale. (wU(oi) d4signe ici l~analogue de wS(oi ) pour TH~OR~4E 7 ([14]). tions a), b), c) THEORi~E 8 ([14]). a), b), c). ) L'ensemble des diff4omorphismes de est un ouvert non vide de Soit fun f'~EU il existe un hom4omor~hisme U v4rifi~nt les condi- D(M).

On d4finit ensuite la vari4t4 instable t4 stable de de tel Cette sous-vari~t6, d@sign@e par en U fP. x de f de p4riode p est hyperbolique s'il est ~u point On d4finit les vari4t4s stables et instables comme les vari4t4s stables et instables de 2 (I. Kupka [6], S. Smale [11]). Les deux propri4t4s fP en WS(x) et x. suiv~ntessont g4n~- riques : a) t o u s l e s points p4riodiques du diff@omorphisme b) si WS(x) et En x et WU(y) y s0nt hTperboliques ; sont deux ~oints p4riodiques (distincts ou non) d#e f se ooupsnt transversalement.

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