On a differentiable linearization theorem of Philip Hartman

TL;DR

This paper extends Hartman's linearization theorem for bi-circular derivatives to less smooth diffeomorphisms and Banach spaces, simplifying proofs of classical dynamical systems results under weaker regularity conditions.

Contribution

It generalizes Hartman's linearization theorem to $C^{1,eta}$ diffeomorphisms and Banach spaces, broadening the theorem's applicability.

Findings

Extended linearization to $C^{1,eta}$ diffeomorphisms.

Applied results to classical saddle-focus dynamics.

Provided simpler proofs under weaker regularity assumptions.

Abstract

A linear automorphism of Euclidean space is called bi-circular its eigenvalues lie in the disjoint union of two circles $C_1$ and $C_2$ in the complex plane where the radius of $C_1$ is $r_1$, the radius of $C_2$ is $r_2$, and $0 < r_1 < 1 < r_2$. A well-known theorem of Philip Hartman states that a local $C^{1,1}$ diffeomorphism $T$ of Euclidean space with a fixed point $p$ whose derivative $DT_p$ is bi-circular is $C^{1,\beta}$ linearizable near $p$. We generalize this result to $C^{1,\alpha}$ diffeomorphisms $T$ where $0 < \alpha < 1$. We also extend the result to local diffeomorphisms in Banach spaces with $C^{1,\alpha}$ bump functions. The results apply to give simpler proofs under weaker regularity conditions of classical results of L. P. Shilnikov on the existence of horseshoe dynamics near so-called saddle-focus critical points of vector fields in $R^3$.