TOPOLOGICAL INVARIANTS AND DETECTION OF PERIODIC-ORBITS

Let f be a smooth flow on a manifold M and C subset-or-equal-to m x (0 , infinity) be an isolated compact set of periodic orbits of f Here we consider the following topological invariants of the pair (f, C): the homology index I(f, C) is-an-element-of H-1(M), the Fuller index I(F) (f, C) is-an-element-of Q, and the p-detection number D(p)(f, C) is-an -element-of Z(p). The latter invariant is defined for a positive integ er p which is relatively prime with the multiplicities of periodic orb its in C. Motivated by problems concerning numerical determination of periodic points, we introduce the notion of p-detectability. We prove that I(f, C) not-equal 0 implies that (f, C) is 1-detectable, but in g eneral this is not the case ff I(F)(f, C) is nontrivial. The condition D(p)(f C) not-equal 0 implies that (f, C) is p-detectable. As a conse quence we prove that if I(F)(f, C) not-equal 0 then (f, C) is p-detect able, provided p is a sufficiently large prime number. We present some applications of these results. (C) 1994 Academic Press, Inc.