Stability of diffeomorphisms along one parameter (Q5928697)

It is known that for an Axiom A diffeomorphism \(g\) with strong transversality condition\break \((T_xW^s(x)+ T_xW^u(x)= T_xM\) for all \(x\in M\), where \(M\) is a smooth compact manifold), there exists a sufficiently small neighborhood \(V\) of \(g\) in the set of \(C^1\) diffeomorphism such that if \(f\in V\) then there exists a homeomorphism \(h\) near identity map such that \(gh= fh\). NEWLINENEWLINENEWLINEIn this paper the author further investigates the size of the neighborhoods \(V\) and the distance of the homeomorphism \(h\) with the identity map. The author shows that if \(\{g_\varepsilon\}\) is a one-parameter family of \(C^3\) diffeomorphisms, \(g_0\) satisfies Axiom A and the strong transversality condition, and \(g_\varepsilon\) is a \(C^0 O(\varepsilon^3)\)-close and \(C^1 O(\varepsilon^2)\)-close to \(g_0\), then for all small \(|\varepsilon|\), there is a homeomorphism \(h_\varepsilon\) on \(M\), with \(C^0 O(\varepsilon^2)\) near the identity map, such that \(h_\varepsilon g_0= g_\varepsilon h_\varepsilon\).