Invariant measures ofC 2-diffeomorphisms of Markov type inR d

A class of uniformly expanding, piecewiseC²-diffeomorphisms from domainsIR^d(bounded or not) into themselves is considered. It is shown that the number of the extreme points of Fix (P)={gG:Pg=g} whereP is the Frobenius-Perron operator associated with andG={gL¹: g0 g=1}, can be determined in an effective way. Moreover, it is shown that the sequence {P^jg} is convergent inL¹ for anygG, and in the topology of uniform convergence for anygG(1). The limit is a linear projectionR inL¹ (defined by (3.1)) which mapsG onto Fix (P) (see Th. 3.1).Dedicated to professor A. Lasota on the occasion of his 60th birthday