Differential fields and geodesic flows II: Geodesic flows of pseudo-Riemannian algebraic varieties

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

“…Apart from suspensions of Anosov diffeomorphisms, other classical examples of compact Anosov flow come from geodesic motion on compact Riemannian manifolds with negative curvature and are known to be mixing (see [7]). We proved in [14] that these classical examples ensure the existence of unlimited families of algebraic autonomous differential equations satisfying the hypotheses of Theorem B.…”

“…Suppose that (X, v) admits a non-trivial rational factor π : (X, v) (Y, w). Since we already now that (X, v) has no non-trivial rational integral (see [14]), we may assume that w = 0. By Proposition 3.1.5, the tangent foliation F π is v-invariant and does not contain the foliation generated by v, since w = 0.…”

“…Disintegration of 3-dimensional compact mixing Anosov flow. The second aim of this article is to exploit the relationship between invariant foliations and rational factors described above, conjointly with the results of [14], to study disintegration properties of, algebraically presented, compact, mixing Anosov flows of dimension 3.…”

“…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.…”

“…Moreover, by replacing X by the Zariskiclosure of M in X, we can assume that M = X(R) is Zariski-dense in X. Together with the irreducibility property, this implies that X is absolutely irreducible (see for example Lemma 4.1.1 of [14]).…”