Obstructions to finitely smooth linearizations
The following assertion concerning a finitely-smooth linear approximation \(\Lambda\) of a diffeomorphism \(F\) in \(\mathbb R^ n\) are proved: Every \(C^ k\)-germ of a diffeomorphism \(F(x)=\Lambda x+o(\| x\|^ k)\) \((x\in\mathbb R^ n)\) is adjoint to a linear one in the class \(C^ k\) iff \((\max | \lambda |)^ k<\min | \lambda |\) \((\lambda\in\text{spec}\, \Lambda)\). The condition is equivalent to \[ | \lambda_ i| \neq | \lambda |^ a=| \lambda_ 1|^{a_ 1}... | \lambda_ n|^{a_ n} \] \((\lambda_ i\in\text{spec}\, \Lambda\), \(a\in\mathbb R^ n_+\), \(| a| =k)\). For every \(k<\infty\), there are hyperbolic \(C^ k\)-germs in the plane which are not adjoint in \(C^ k\) to any \(C^{k+1}\)-germ.
scientific article; zbMATH DE number 4104022