PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume «7. Number 3. Mardi IW3 APPROXIMATING THE ABSOLUTELY CONTINUOUS MEASURES INVARIANT UNDER GENERAL MAPS OF THE INTERVAL ABRAHAM BOYARSKY1 Abstract. Let r: / — / be a nonsingular. piecewise continuous transformation which admits a unique absolutely continuous invariant measure u with density function/*. The main result establishes the fact that/* can be approximated weakly by the densitv functions of a sequence of measures invariant under piecewise linear Markov maps {t„} which approach t uniformly. 1. Introduction. Let t be a nonsingular, measurable transformation from / = [0,1] into itself and let ® denote the Lebesgue measurable subsets of /. A measure ju defined on (/. VÄ)is absolutely continuous if there exists a function /: / — [0, oo), which is integrable with respect to Lebesgue measure m, i.e../G £,(/, Vt>, m) = £,, and for which p(S)= ff(x)m(dx) VSevÖ. The measure ju is said to be invariant (under t) if p(r'[S) = p(S) for all sets S E li>. The Frobenius-Perron operator PT: £, -» £, has proven to be a useful tool in the study of absolutely continuous invariant measures [1,2]. It is defined by f/V/)(*)-=-¿/ f(s)m(ds). The importance of PT lies in the fact that each of its fixed points is the density of a measure invariant under t, i.e., if PTf* = /*, then p( )=fr(x)m(dx) is invariant under t [1]. In [2] a sequence of matrices {P„}, depending on t, is constructed and the following result obtained: Theorem 1. Let r: I — I be a piecewise C2 map with inf |.t' | > 2. // FT has a unique fixed point /*, then the sequence {/„} of fixed points (regarded as functions on I) of {P„} converges to f* in the tx-norm. Received by the editors March 30, 1982. 1980 Mathematics Subject Classification. Primary 28D05; Secondary 41A30. 'This research was supported by NSERC Grant No. A-9072 and an FCAC Grant from the Education Department of Quebec. ©1983 American Mathematical Socictv OOO2-9939/82/OOO0-0854/SO2.25 475 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use