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

Rémi Jaoui

doi:10.5802/cml.68

Abstract

Rational factors, invariant foliations and algebraic disintegration of compact mixing Anosov flows of dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>3</mml:mn></mml:math>

Highlights

Many applications of geometric stability theory to concrete geometric settings originated from Zilber’s trichotomy on minimal types — respectively called disintegrated, locally-modular or non-locally modular — in stable theories
We develop a complementary geometric framework for the notion of the semi-minimality, which provide effective tools to establish this property, based on the study of rational factors and invariant foliations of a given autonomous differential equations (X, v)
We apply these techniques to study certain three dimensional algebraic differential equations — namely, algebraically presented, compact, mixing Anosov flows of dimension three — and establish disintegration of the system of algebraic relations shared by their generic solutions

Summary

Introduction

Many applications of geometric stability theory to concrete geometric settings originated from Zilber’s trichotomy on minimal types — respectively called disintegrated, locally-modular (non-disintegrated) or non-locally modular — in stable theories. Theorem B provides a substential improvement of some of results of [14] which only ensures, for an autonomous differential equation (X, v) satisfying the hypotheses of Theorem B, the existence of a generically disintegrated rational factor π : (X, v) (Y, w) of positive dimension Building on this result, Theorem B reduces to proving that an an algebraically presented, compact, mixing Anosov flow (X, v) of dimension three does not admit any non-trivial rational factor. The decisive argument in our study of rational factors through invariant foliations is a result of Plante in [23], which ensures that the strongly stable and the strongly unstable distributions W ss and W su do not admit any algebraic leaves To conclude this introduction, we illustrate more concretely the content of Theorem B with a separate real-analytic instance of it, when applied to a geodesic motion in negative curvature: Corollary C. The last section is dedicated to the study of rational factors of compact, mixing, Anosov flows of dimension 3 and to the proofs of Theorem B and Corollary C

Lie derivative

Foliations on a smooth algebraic variety

Foliations invariant by a vector field

Rational factors of mixing Anosov flows of dimension 3