An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C1 diffeomorphisms to C1 Hamiltonian vector fields.

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ABSTRACT. In this note, we give an example of a diffeomorphism / on a three dimensional manifold M such that /has a property called quasi-Anosov but such that / does not have a hyperbolic structure (is not Anosov). Mané has given a method of extending / to a diffeomorphism g on a larger dimensional manifold V such that g has a hyperbolic structure on M as a subset of V. This gives a counterexample to a question of M. Hirsch.M. Hirsch asks in [2], if a diffeomorphism g: V -► V has a compact invariant submanifold M C V with a hyperbolic structure as a subset of V, does it folfollow that g restricted to M is Anosov (has a hyperbolic structure). He proves this is true in certain cases if g has a dense orbit in V. Ricardo MarTé notes that g restricted to M has a property he calls quasi-Anosov [5]. He asks if a quasiAnosov diffeomorphism is always Anosov. C. Robinson [6] gives an example of a quasi-Anosov flow (not a diffeomorphism) that is not Anosov on an eleven dimensional manifold. In this note, we give an example of a quasi-Anosov diffeomorphism / on a three dimensional manifold. (This is the minimal dimension.) Mane gives a method in [5] of embedding our result in a diffeomorphism g of a manifold V such that g has a hyperbolic structure on M. This gives a counterexample to the question of Hirsch as stated above. However, we do not know if g can be constructed so M is contained in the nonwandering set of g. Also, the results of Hirsch [2] show that if our f:M3 -*■ M3 is embedded in g: V-► V so that g has a hyperbolic structure on M and the dimension of Kis four or five, then g cannot have a point with a dense orbit in all of V.1. Definitions and Theorem. A diffeomorphism is quasi-Anosov if the fact that | Tf"v\ is bounded for all n £ Z implies that v = 0. Here Tf is the induced map on tangent vectors of M, v E TM. If A C M is invariant by a diffeomorphism /, we say that /has a hyperbolic structure on A if there are 0 < X < 1, C> 0, and a splitting TM IA = F" 0 Es such that for n > 0,