A Hartman-Grobman theorem for algebraic dichotomies

TL;DR

This paper extends the Hartman-Grobman linearization theorem to systems with algebraic dichotomies, generalizing previous results for exponential dichotomies and establishing the properties of the conjugating homeomorphism.

Contribution

It introduces a version of the Hartman-Grobman theorem for algebraic dichotomies, broadening the scope of linearization results beyond exponential dichotomies.

Findings

Established a linearization theorem for systems with algebraic dichotomies.

Proved the homeomorphism in the linearization is Hölder continuous with a continuous inverse.

Generalized Palmer's linearization theorem to a broader class of systems.

Abstract

Algebraic dichotomy is a generalization of an exponential dichotomy (Lin, JDE2009). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem. Besides, we prove that the homeomorphism in the linearization theorem (and has a H\"{o}lder continuous inverse). Comparing with exponential dichotomy, algebraic dichotomy is more complicate. The exponential dichotomy leads to the estimates $\int_{-\infty}^{t}e^{-\alpha(t-s)}ds$ and $\int_{t}^{+\infty}e^{-\alpha(s-t)}ds$ which are convergent. However, the algebraic dichotomy will leads us to $\int_{-\infty}^{t}\left(\frac{\mu(t)}{\mu(s)}\right)^{-\alpha}ds$ or $\int_{t}^{+\infty}\left(\frac{\mu(s)}{\mu(t)}\right)^{-\alpha}ds$, whose the convergence is unknown in the sense of Riemann.