Wild unstable manifolds

Abstract

LET M be a compact smooth manifold, f: M + M a diffeomorphism. An energy function for f is a Morse function M + R which decreases along non-periodic orbits and has critical points only at the periodic points off. An analogous concept for vector fields was introduced by Meyer [8], who proved that Morse-Smale vector fields on compact manifolds always have energy functions. Shub[lZ] and Takens [15] have since asserted, without proof, that the theorem is also true for Morse-Smale diffeomorphisms. In this paper we show, using an example on the three-sphere, that this conjecture is false. The example is given in 02, after a section consisting of definitions and a construction of energy functions for Morse-Smale diffeomorphisms of surfaces. The distinguishing characteristic of the counterexample on S’ is an unstable manifold whose closure is a wildly embedded arc: under certain hypotheses it is shown that such phenomena preclude the existence of energy functions. In investigating further examples of diffeomorphisms with such wild phase portraits we see that no possible extension of the notion of labeled diagram will serve to classify conjugacy classes of Morse-Smale diffeomorphisms in dimension 3. These examples are discussed in 63. All our examples are gradient-like: there is no complicated behavior in the sense of [lo]. Note. The major results of Shub and Takens in the articles referenced above are still true. An alternative proof of Shub’s theorem has appeared in [13] and elsewhere, and Takens’ arguments work just as well using non-degenerate Liapunov functions, as defined in 91.