Axiom A - Omega Stability

Omega Stability

An important property of Axiom A systems is their structural stability against small perturbations. That is, trajectories of the perturbed system remain in 1-1 topological correspondence with the unperturbed system. This property is important, in that it shows that Axiom A systems are not exceptional, but are in a sense 'generic'.

More precisely, for every C1-perturbation f_ε of f, its non-wandering set is formed by two compact, f_ε-invariant subsets Ω₁ and Ω₂. The first subset is homeomorphic to Ω(f) via a homeomorphism h which conjugates the restriction of f to Ω(f) with the restriction of f_ε to Ω₁:

If Ω₂ is empty then h is onto Ω(f_ε). If this is the case for every perturbation f_ε then f is called omega stable. A diffeomorphism f is omega stable if and only if it satisfies axiom A and the no-cycle condition (that an orbit, once having left an invariant subset, does not return).