Corps différentiels et flots géodésiques I: Orthogonalité aux constantes pour les équations différentielles autonomes

Abstract: Orthogonality to the constants is property of an algebraic differential equation that originated from the model-theoretic study of differential fields and that expresses remarkable independence properties for its solutions

In this article, we study the property of orthogonality to the constants in a differential algebraic language for autonomous differential equations and describe some effective methods to establish this property. The main result is a criterion for orthogonality to the constants (and its version for families) for real absolutely irreducible D-varieties \((X,v)\) based on the dynamical properties of the associated real analytic flow \((M,\phi) \). More precisely, we show that if there exists a compact region \(K\) of \(M\), Zariski-dense in \(X\) and such that the restriction of the flow \(\phi \) to \(K \) is topologically weakly mixing then the generic type of \((X,v)\) is orthogonal to the constants.

This criterion will be applied in a second part to study from this model-theoretic point of view the geodesic flow of compact Riemannian varieties (presented algebraically) with negative curvature.