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An R-reversible diffeomorphism on R 2N is studied possessing a hyperbolic fixed point. If the stable manifold of the hyperbolic fixed point and the fixed point set Fix R of R have a nontrivial local topological crossing, then an infinite number of R-symmetric periodic orbits of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of the set Fix R and the stable manifold for the perturbed diffeomorphism.

Swan’s theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define A ( r ) {A^{(r)}} -maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan’s theorem and thus generalize Swan’s theorem to include non-linear differential operators as morphisms. An A ( r ) {A^{(r)}} -manifold structure is introduced on the space of sections of a fiber bundle through charts with A ( r ) {A^{(r)}} -maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of A ( r ) {A^{(r)}} -maps.

The stable/unstable manifold theorem for hyperbolic diffeomorphisms has proven to be of extreme importance in differentiable dynamics. We prove a stable/unstable "manifold" theorem for area preserving homeomorphisms of orientable two manifolds having isolated fixed points of index less than 1. The proof relies upon the concept of free modification which was first developed by Morton Brown for homeomorphisms of two manifolds and later extended by Pelikan and Slaminka for area preserving homeomorphisms of two manifolds.

The relation between local stable manifolds of an ordinary differential equation and its discretization is studied. We show that a local stable manifold of a hyperbolic fixed point of an ordinary differential equation is the limit of local stable manifolds of the same fixed point of its discretizations as the discretization parameter h > 0 h > 0 approaches 0.

Perturbations of a dynamical system are defined and the behavior of compact asymptotically stable sets under these perturbations is determined. The occurrence of critical points in a perturbed planar dynamical system is also investigated. In [1] it is shown that if C is an asymptotically stable cycle of a planar dynamical system 7r and if 7ri is a net of planar dynamical systems which converges to 7r, then there are limit cycles Ci of ri such that Ci-*C. This paper presents a similar result in a more general setting for perturbed asymptotically stable sets. If irj is a net of dynamical systems which converges to a dynamical system rr and if M is a compact asymptotically stable set of vr, then eventually there are asymptotically stable sets Mi of 7r arbitrarily close to M. Moreover, if M is invariant with respect to all ri, then Mi-*M. R, R+, and Rwill denote the reals, nonnegative reals, and nonpositive reals respectively. A dynamical system vr on a topological space X is a mapping of Xx R onto X which satisfies the following three conditions (where xITt= ir(x, t)): (i) 7r is continuous in the product topology. (ii) x7rO=x for each x E X. (iii) (x7rt)vrs=xr(t+s) for each x E X and s, t E R. If A c X and B cR, then ATrB will denote the set {x7rr: x E A, t E B}. L+(x) and L-(x) will denote the positive limit set of x and the negative limit set of x respectively. A subset M of X is called an (negative) attractor if there is a neighborhood U of M such that (L-(x)) L+(x) c M for every x E U. If M is an (negative) attractor, then (A-(M)) A+(M) will denote the largest such neighborhood. A subset S of X is called a section with respect to I7 iff (S7Tt) nS= 0 for all t$O. In a topological space X it is possible to define limits of nets of subsets Received by the editors April 12, 1971 and, in revised form, January 7, 1972. AMS 1970 subject classifications. Primary 34C35, 34D10; Secondary 34C05.

The property of having no C' first integrals other than constants is shown to be generic in Diffr( M) for each r = 1, 2,..., where n is the dimension of M. Let M be a compact Riemannian manifold, and consider the space Diff r(M) of the Cr diffeomorphisms from M to itself with the Cr topology. Peixoto has shown in [2] that the property of having no smooth first integrals other than constants is C' generic. This note describes how a combination of ideas of Arraut [1] and Takens [5] yields a proof that the nonexistence of (sufficiently) smooth, nonconstant first integrals is Cr generic for any r = 1, 2,..., x. If f: M -* M is a diffeomorphism, then a map g: M -R is a first integral of f if (i) g is constant along f-orbits: g(f(x)) = g(x) for all x in M, and (ii) g is not a constant. Peixoto's proof in [2] that the nonexistence of sufficiently smooth first integrals is a C' generic property is based on three observations. First, Sard's theorem implies that if g is sufficiently smooth (Cn where n is the dimension of M), and has no regular values in its image, then g is constant. Second, by Pugh's general density theorem, the property of having the nonwandering set equal to the closure of the hyperbolic periodic points is generic in Diff'(M) [3]. This is why Peixoto's argument works only to establish C' genericity. Third, if g is a C' first integral of f and p is a hyperbolic periodic point of f, then p is necessarily a critical point of g. Peixoto establishes this using a geometric argument based on the stable manifold theorem; a more elementary proof can be found in [4]. For f satisfying Pugh's genericity condition, it follows from continuity that every nonwandering point of f is a critical point of g. Since M is compact, every level set of g contains nonwandering points, so g has no regular values in its image, and thus is constant. By combining a theorem of Takens and an argument due to Arraut we can avoid using Pugh's theorem and so obtain the following result: THEOREM. Suppose M is a compact manifold of dimension n. For each r= 1, 2, ..., x, there is a residual subset Sr of Diff r(M) with the property that if f is in Sr then no Cr first integral off has any regular values in its image. In particular, if r > n, then no f in Sr has any nonconstant Cr first integrals. Received by the editors September 5, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F35.