Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flow of dimension $3$

TL;DR

This paper introduces a geometric approach to analyze the structure of certain 3-dimensional compact mixing Anosov flows, demonstrating their algebraic disintegration through invariant foliations and rational factors.

Contribution

It develops a new geometric framework for semi-minimality and applies it to show generic disintegration of specific algebraic differential equations related to Anosov flows.

Findings

Certain 3D mixing Anosov flows are generically disintegrated.

Invariant foliations play a key role in understanding the flow structure.

The framework links rational factors and algebraic foliations to flow disintegration.

Abstract

In this article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation $(X,v)$, based on the study of rational factors of $(X,v)$ and of algebraic foliations on $X$, invariant under the Lie-derivative of the vector field $v$. We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disintegrated.