The standard proof of the Grobman--Hartman linearization theorem for a flow at a hyperbolic rest point proceeds by first establishing the analogous result for hyperbolic fixed points of local diffeomorphisms. In this exposition we present a simple direct proof that avoids the discrete case altogether. We give new proofs for Hartman's smoothness results: A
flow is
linearizable
at a hyperbolic sink, and a
flow in the plane is
linearizable at a hyperbolic rest point.
Also, we formulate and prove some new results on smooth linearization
for special classes of quasi-linear vector fields where either
the nonlinear part is restricted or additional conditions
on the spectrum of the linear part (not related to resonance conditions)
are imposed.
