Topological Invariants and Detection of Periodic Orbits

Let f be a smooth flow on a manifold M and C M × (0, ∞) 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) H1(M), the Fuller index IF(f, C) , and the p-detection number Dp(f, C) p. The latter invariant is defined for a positive integer p which is relatively prime with the multiplicities of periodic orbits in C. Motivated by problems concerning numerical determination of periodic points, we introduce the notion of p-detectability. We prove that I(f, C) ≠ 0 implies that (f, C) is 1-detectable, but in general this is not the case if IF(f, C) is nontrivial. The condition Dp(f, C) ≠ 0 implies that (f, C) is p-detectable. As a consequence we prove that if IF(f, C) ≠ 0 then (f, C) is p-detectable, provided p is a sufficiently large prime number. We present some applications of these results.