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

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.

Generic planar algebraic vector fields are disintegrated

“…Extension of a rational factor into an invariant foliation. We refer to [Har80] for the theory of saturation of coherent algebraic sheaves on a smooth algebraic variety and to [Jao18] for an exposition of the theory of (possibly singular) algebraic foliations in this language.…”

“…More recently, the disintegration property has been studied for its own sake in specific families of differential equations in order to classify the possible algebraic relations shared by their solutions: Painlevé equations have been extensively studied from this point of view in [NP14], [NP17]; Schwarzian differential equations in [FS18], [CFN18], as well as, geodesics of two dimensional pseudo-Riemanian manifolds with negative curvature in [Jao19], [Jao18].…”

“…Trichotomy in DCF 0 . The strategy for the proof of Theorem A and its geometric formulation(Theorem B) follows the strategy of [Jao19] and [Jao18] to study geodesic flows of pseudo-Riemmanian varieties with negative curvature in dimension two.…”

“…Let's conclude by a few words about the proof of Theorem C, which uses the technology developed in [Jao18] to study rational factors of a differential equation (X, v) from information on the invariant foliations of (X, v).…”