TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 280, Number 2. December 1983 MULTI-INVARIANT SETS ON TORI BY DANIEL BEREND Abstract. Given a compact metric group G, we are interested in those semigroups 2 of continuous endomorphisms of G, possessing the following property: The only infinite, closed, 2-invariant subset of G is G itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization—for the case of finite- dimensional tori—of those commutative semigroups with the aforementioned prop- erty. 1. Introduction. Let S be a multiplicative semigroup of integers. 2 is lacunary if all the members of (a E 2|o > 0} are powers of a single integer a. Otherwise, 2 is nonlacunary. With this terminology, Furstenberg proves in [1, p. 48] the following Theorem. // 2 is a nonlacunary semigroup of integers and a is an irrational, then 2 a is dense modulo 1. This theorem is a generalization of a theorem of Hardy and Littlewood, which asserts that if r is a fixed positive integer and a is an irrational, then the set [nra | n E N} is dense modulo 1. From the point of view of the theory of diophantine approximations, we are in a special case of the following general situation: Suppose G is a metric group and H is a closed subgroup, which is invariant under a given commutative semigroup 5 of continuous endomorphisms of G. We can form the subgroup H/S of G, consisting of all the elements of G which are carried to H by some endomorphism in S. Now it can be asked how closely, in a properly defined sense, can the elements of G be approximated by elements of H/S. The theorem relates to the case G = R, H = Z, and implies that if 5 is a nonlacunary semigroup, then for every irrational a E R and for every e > 0 there is some m/s E Z/S such that (1.1) \a- m/s\<e/s. Restricting ourselves to the problem of approximating irrational numbers, we may ask which sets S of positive integers have the property that every irrational a can be approximated as in (1.1) by rationals with denominators in S. The theorem supplies a complete answer in the case that 5 is a multiplicative semigroup. Such approxima- tions are possible iff 5 is nonlacunary. (Actually, for lacunary 5 it is easy to find irrationals a for which (1.1) is impossible for sufficiently small e > 0.) Received by the editors May 24, 1982. 1980 Mathematics Subject Classification. Primary 10F10, 54H20;Secondary 28A65, 54H15. Key words and phrases. Invariant set, finite-dimensional torus, semigroup of endomorphisms, ergodic endomorphism, minimal set. ©1983 American Mathematical Society 0002-9947/83 $1.00 + $.25 per page 509 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use