Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem (Q1129922)

The following problem is considered. Given \(f,N\) germs of \(C^r\) diffeomorphisms in \((\mathbb R^n,0)\), hyperbolic at \(0\), \(\| f(x)-N(x)\| =o(\| x\| ^s)\), \(s\geq 1\), determine \(g\), the germ of a diffeomorphism in \((\mathbb R^n,0)\) with \(Dg(0)= \text{Id}\), in such a way that \(g^{-1}\circ f\circ g=N\). Moreover, if \(f\) and \(N\) preserves a geometric structure, determine \(g\) in such a way that it preserves the same structure. The solution of the problem is based on the deformation method: let \( f_{\epsilon}\) be a differentiable family with \(f_{0}=N\), \(f_1=f\). It is required to find a family satisfying the relation \(g^{-1}_{\epsilon}\circ f_{\epsilon}\circ g_{\epsilon}=f_0\). The infinitesimal variant of the last relation is the cohomology equations. An estimate for the cohomology equations allows the authors to receive a solution of the problem. An analogous result is also proved for flows. Such theorems were sketched in [\textit{S. Sternberg}, Am. J. Math. 81, 578-604 (1959; Zbl 0211.56304)]. A proof by other methods appeared in [\textit{M. Chaperon}, `Géométrie différentielle et singularités des systèmes dynamiques', Astérisque 138/139 (1986; Zbl 0601.58002)].