Linearization via the Lie Derivative

oleh: Carmen Chicone, Richard Swanson

Format: Article
Diterbitkan: Texas State University 2000-11-01

Deskripsi

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 ${cal C}^2$ flow is ${cal C}^1$ linearizable at a hyperbolic sink, and a ${cal C}^2$ flow in the plane is ${cal C}^1$ 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.