PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 88, Number 2. June 1983 AN ERGODIC THEOREM FOR SEMIGROUPS OF CONTRACTIONS S. GUTMAN AND A. PAZY Abstract. An ergodic theorem for semigroups of nonlinear contractions having precompact trajectories in a Banach space is proved. • 1. The main result. Throughout this note X will be a real Banach space, C C X a closed subset of X and S(t), t > 0, a semigroup of contractions on C, that is a family of mappings S(t): C — C, t > 0, satisfying: (i) lim,_,0S(r);e = S(t0)x for ;0 s* 0, x G C. (ii) S(t + s)x = S(t)S(s)x for t, s > 0, x G C. (iii)||S(0*-S(i).yll < \\x-y\\foit>0,x,yGC. For x G C we denote by a(x) = {S(t)x : t 3=0} the trajectory starting at x and by u(x) = \y : y = lim S(tn)x, for some sequence tn -> oo r„^oo the possibly empty w-limit set of X. If oi(x) ¥= 0 then it follows from its definition that u(x) is invariant under S(t), t > 0, i.e. S(t): u>(x) -» oi(x) for t 3=0 and (1) limdist(5(Ojc,co(x)) = 0, f-00 where dist(z, B) is the distance between the point z and the set B. Assuming, as we will do below, that for some x G C the trajectory a(x) is precompact, it follows easily that u(x) is nonempty and compact. In this case, u(x) can be given the structure of a compact commutative group and the following much stronger asser- tion, which is our main result, holds. Theorem 1 (the ergodic theorem). Let X, Y be real Banach spaces, C C X be closed and let S(t), t > 0, be a semigroup of contractions on C. If for some x G C the trajectory y(x) is precompact, then u(x) is a compact commutative group, and for every f: C -» Y which is uniformly on bounded subsets of C we have (2) Urn ±(Tf(S(t)x)dt=f f(.)d., where d_ is the unique normalized Haar's measure on ic(x). 2. The proof of Theorem 1. Let C C X be a closed subset of the Banach space X and let S(i)> t > 0, be a semigroup of contractions on C. A subset ñ of C is called minimal under S(t), t > 0, if it is the closure of the trajectory y(y) = {S(t)y: t > 0} Received by the editors June 25, 1982. 1980 Mathematics Subject Classification. Primary 47H20, 47A35. Key words and phrases. Semigroups of contractions, ergodic theorem, Haar's measure, u-limit set. ©1983 American Mathematical Society 0002-9939/82/0000-1096/S01.50 254 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use