Geometric proof of the Grobman-Hartman Theorem

TL;DR

This paper provides geometric proofs of the Grobman-Hartman theorem for diffeomorphisms and ODEs, utilizing covering relations, cone conditions, and isolating segments, and establishes the Hölder continuity of the conjugating homeomorphisms.

Contribution

It introduces geometric proof techniques for the Grobman-Hartman theorem and proves Hölder continuity of the conjugacy, expanding the theoretical understanding of the theorem.

Findings

Geometric proofs for the Grobman-Hartman theorem for diffeomorphisms and ODEs.

Establishment of Hölder continuity for the conjugating homeomorphisms.

Application of covering relations, cone conditions, and isolating segments in proofs.

Abstract

We give geometric proofs for Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone condition for ODEs. We prove also the H\"older condition for the conjugating homeomorphims.