We consider a compact manifold M" with a volume or a symplectic structure. We remind that a volume structure is given by a nowherezero n-form o9 on M" and that a symplectic structure is given by a closed 2-form o9 on M" which has the property that o9 ^ ... ^ o9 (n/2 times) defines a volume structure. If a volume, resp. symplectic, structure is defined by the differential form o9, we say that o9 is the volume, resp. symplectic, form. Note that on a 2-dimensional manifold, a symplectic form is the same as a volume form. Let Difl~(M) denote the space of C'-automorphisms of M preserving o9, (i.e. if ~o e Difffo,(M) then ~0" (o9)= co) with the Cr-topology. We always assume that M is a C~-manifold and that o9 is a C~ A discrete conservative system on M of class C r is a group morphism �9 : Z~Difff~(M). We shall make in general no distinction between a discrete conservative system �9 and its generating diffeomorphism �9 (1)eDiff~(M). A continuous conservative system on M of class C' is a group morphism 4~: ~,~ DitV~,(M) which is the integral of a C'vectorfield. On the set of discrete, resp. continuous, conservative systems we take the topology induced by Difff~(M), resp. the C'-topology on vectorfields. If q~ e Diff~,(M), r > 1, and p is a periodic point of q~, say with period k, then p is called a hyperbolic periodic point of q~ if d (qgk) lTp(M): Tp(M) Tp(M) has no eigenvalues with norm 1. The stable and the unstable manifold of p with respect to q~ are denoted by W~(p) and W~(p) (for the definition see [4]). A homoclinic point of p is an intersection point of W~(p) and W~(p), which differs from p, in other words the set of homoclinic points of p is (W~(p)c~ W~(p))\ {p}. We shall state our results here only for the discrete case; for the continuous (Hamiltonian) case see w