Abstract
This chapter discusses that every diffeomorphism of a compact manifold is differentiably isotopic to one that is Ω-stable as a discrete dynamical system. It discusses a theorem that every diffeomorphism of a compact manifold is differentiably isotopic to one satisfying Axiom A and the no-cycle condition and corollary that every diffeomorphism is differentiably isotopic to an instable one. The theorem is reminiscent of the theorem that every manifold supports a structurally stable vector field, that is, continuous dynamical system. In the case discussed in the chapter, however, the dynamical system obtained must be more complicated, as the Lefschetz trace formula implies, even for many isotopy classes on 2-manifolds, that there are an infinite number of periodic orbits.
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