Cohomology equations near hyperbolic points and geometric versions of sternberg linearization theorem

We prove that if two germs of diffeomorphisms preserving a volume, symplectic, or contact structure are tangent to a high enough order and the linearization is hyperbolic, it is possible to find a smooth change of variables that sends one into the other and which, moreover, preserves the same geometric structure. This result is a geometric version of Sternberg's linearization theorem, which we recover as a particular case. An analogous result is also proved for flows.