LASOTA-YORKE MAPS WITH HOLES: CONDITIONALLY INVARIANT PROBABILITY
MEASURES AND INVARIANT PROBABILITY MEASURES ON THE SURVIVOR SET 
   
                               By
       Carlangelo Liverani and Ve'ronique Maume-Deschamps    


			   ABSTRACT
   

Let $T~:~I~\longrightarrow~I$ be a Lasota-Yorke map on the interval $I$,   
let $Y$ be a non trivial sub-interval of $I$ and  
$g^0~:~I~\longrightarrow~\R^+$, be a strictly positive potential which  
belongs to BV and admits a conformal    
measure $m$. We give  constructive conditions on $Y$ ensuring the existence   
of absolutely continuous (w.r.t. $m$) conditionally invariant probability   
measures to non absorption in $Y$. These conditions imply also existence   
of an invariant probability measure on the set $X_\infty$ of points which    
never fall into $Y$. Our conditions allow rather ``large'' holes.