One of t h e f irst s tcps in t h e c o n s t r u c t i o n of r igo rous s t a t i s t i c a l d y n a m i c s is to p r o v e t h e ex i s t ence of t h e t i m e e v o l u t i o n . T he p ionee r ing p a p e r in t h i s d i r ec t ion is due to LANFORD (1.2). He s tud ied t h e t i m e evo lu t i on for t he equ i l i b r i um s t a t i s t i c a l s t a t e s of a classical gas of pa r t i c les i n t e r a c t i n g in one d i m e n s i o n v ia b o u n d e d pa i r p o t e n t i a l s of f in i te range . More r ecen t ly , u s i n g a d i f fe rent t e c h n i q u e , SINAI o b t a i n e d s imi la r resu l t s for h a r d c o r e par t ic les i n t e r a c t i n g v ia f in i te range po t en t i a l s . His p r o c e d u r e appl ies b o t h to one -d imens iona l s y s t e m s (3) and to sys tems in more d imens ions in t h e reg ion of sma l l e n o u g h dens i t ies (4). T h e ex t ens ion to t h e l ong r ange case in one d i m e n s i o n has b e e n o b t a i n e d in ref. (5). All these conclus ions d e p e n d s t rong ly e i t h e r on t he d i m e n s i o n a l i t y or on t he smal l densi t ies . On t he o t h e r h a n d m a n y s t a t i s t i ca l p h e n o m e n a , as for i n s t a n c e t h e phase t r a n s i t i o n s , c a n n o t occur in t he above s i tua t ions . I n t h i s no te we p r o v e t he ex is tence of t i m e evo lu t i on in v d imens ions . L e t &r {(qi), (p~)} be t h e p h a s e space of in f in i t e ly m a n y classical pa r t i c les m o v i n g in ~ d imens ions . We assume t h a t t h e y i n t e r a c t v i a a f in i te range pa i r p o t e n t i a l ~ , w h i c h is also b o u n d e d t o g e t h e r w i t h i t s first and second der iva t ives . W e suppose also t h a t ~ is supe r s t ab l e (6,7). T he e q u i l i b r i u m s ta t e s of t h e gas are r ep resen ted b y t h e prob a b i l i t y measures on s w h i c h sa t i s fy t h c e q u i l i b r i u m equa t i ons of LANFORD a n d RUELLE (7,8).