Let M denote a smooth (Cm) n-dimensional manifold and let $: M x R -+ M denote a Cr flow (or dynamical system) which is generated by a CT vector field 4 (= (d/dt) +(t, x) j t = 0) on M (0 max(1, r}. In this case, we can only obtain Ck smoothness for functions on M. Let K denote a compact invariant set for ‘p, i.e., K is a compact subset of M and if x E K, then 9(x, t) E K for all t E R. Two topological tools which have been used to describe the behavior of p near K are “Lyapunov functions” and “isolating blocks.” A Lyapunov function is a real-valued function (V) which is defined on a neighborhood of K and whose derivatives r and v in the direction of the vector field have special properties. We shall be especially interested in the cases of a monotone Lyu$unov function (77 is strictly decreasing on every trajectory in the complement of K) and of a hy~erbolt’c