A Hartman-Grobman theorem for algebraic dichotomies

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 ${\int }_{-\mathrm{\infty }}^t{e}^{-\alpha (t-s)}ds$ and ${\int }_t^{+\mathrm{\infty }}{e}^{-\alpha (s-t)}ds$ which are convergent. However, the algebraic dichotomy will leads us to ${\int }_{-\mathrm{\infty }}^t{\left(\frac{\mu (t)}{\mu (s)}\right)}^{-\alpha }ds$ or ${\int }_t^{+\mathrm{\infty }}{\left(\frac{\mu (s)}{\mu (t)}\right)}^{-\alpha }ds$, whose the convergence is unknown in the sense of Riemann.