Infinitesimal Hartman-Grobman Theorem in Dimension Three

Infinitesimal Hartman-Grobman Theorem in Dimension Three.

In this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.

The Hartman-Grobman Theorem

The Hartman–Grobman Theorem (see [3, page 353]) was proved by Philip Hartman in 1960 [5]. It had been announced byGrobman in 1959 [1], likely unbeknownst to Hartman, and Grobmanpublished his proof in 1962 [2], likely without knowing of Hartman’s work. (Grobman attributes the question to Nemycki and an earlier partial result to R.M. Minc (citing Nauč. Dokl. Vysš. Školy. Fiz.-Mat. Nauki 1 (1958))...

The Hartman-grobman Theorem and the Equivalence of Linear Systems

The connection between computability of a nonlinear problem and its linearization: The Hartman-Grobman theorem revisited

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics [21], Pour-El and Richards asked, “What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?” Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this pa...

On the Grobman-hartman Theorem in Α-hölder Class for Banach Spaces

We consider a hyperbolic diffeomorphism in a Banach space with a hyperbolic fixed point 0 and a linear part Λ. We define σ(Λ) ∈ (0, 1], and prove that for any α < σ(Λ) the diffeomorphism admits local α-Hölder linearization.