Abstract. It is shown that the class of compacta which can occur as attractors of continuous flows on topological manifolds coincides with the class of finite dimensional compacta having the shape of a finite polyhedron.

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Abstract. A one-dimensional compactum (in fact, a certain solinoid) is constructed, such that there does not exist a self-map on a manifold having this compactum as attractor. The counterexampleIn [2] it is shown that a finite-dimensional compactum can be an attractor of a flow on a manifold if and only if it has the shape of a finite polyhedron. Here a flow is a continuous mapping IxR-iI satisfying the usual functional equation [5, Chapter 4, §7, Theorem 12], and X is a topological manifold. If 1 is replaced by the integers Z we simply get a cyclic group of homeomorphisms, generated by some element / : X » X. We may generalize still further and consider arbitrary self-maps / : X -> X instead of homeomorphisms. A compact subset AC X with f(A) = A is an attractor of / if there is a neighborhood U of A in X, such that f(U) C U and for each neighborhood V of A in X there is n e N with fn(U) C V; this implies fm(U) C V for all m>n. In this situation the classification of attractors seems to be much more complicated than in [2], but at least we are able to produce an example of a one-dimensional compactum that can never be an attractor of a self-map.It is well known that the dyadic solenoid is an attractor of a homeomorphism of a three-dimensional manifold (see [4; 2, Example 3]). The dyadic solenoid is the limit of an inverse sequence of circles Sxn^ = {z e C | |z| = 1}, and the bonding map gn : 5'(1n+1) -► Sxn^ is given by g"(z) = z2. Of course, we might as well choose different bonding maps such as g"(z) = zv", where (^")"eN is an arbitrary sequence of integers. The compacta obtained in this way are generalized solenoids. Theorem 1. The generalized solenoid obtained from a sequence of pairwise relatively prime integers vn cannot be an attractor of a self map on a manifold.