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Carmen Chicone & Richard Swanson. Linearization via the Lie Derivative

Abstract
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 C2 flow is C1 linearizable at a hyperbolic sink, and a C2 flow in the
plane is C1 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.
Contents
1 Introduction 2
2 Continuous Conjugacy 4
3 Smooth Conjugacy 7
3.1 Hyperbolic Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Smooth Linearization on the Line . . . . . . . . . . . . . 32
3.2 Hyperbolic Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Linearization of Special Vector Fields 45
4.1 Special Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Saddles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Infinitesimal Conjugacy and Fiber Contractions . . . . . . . . . . 50
4.4 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . 51


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