Infinitesimal Hartman-Grobman Theorem in Dimension Three

Area of Science:

• Differential Geometry
• Topology
• Singularities Theory

Background:

• Real analytic vector fields in R3 possess a singular point at the origin.
• Understanding the local topological behavior of such fields is crucial in dynamical systems and geometric analysis.
• The principal part of a vector field, often determined by its lowest order terms, dictates local behavior.

Purpose of the Study:

• To establish a local topological equivalence between a real analytic vector field and its principal part.
• To introduce and utilize Newton polyhedra as a tool for defining the principal part.
• To specify the non-degeneracy conditions under which this equivalence holds.

Main Methods:

• Utilizing the concept of topological equivalence in the context of vector fields.
• Defining the principal part of a vector field using Newton polyhedra.
• Applying non-degeneracy conditions to ensure the validity of the equivalence.

Main Results:

• A real analytic vector field in R3 with a singular point at the origin is shown to be locally topologically equivalent to its principal part.
• The principal part is rigorously defined via Newton polyhedra.
• The established equivalence is contingent upon specific non-degeneracy conditions.

Conclusions:

• The principal part, determined by Newton polyhedra, accurately represents the local topological structure of real analytic vector fields.
• This finding simplifies the study of singularities in vector fields by reducing complexity.
• The non-degeneracy conditions are essential for the theoretical framework and practical application.