7 Regularity of weak foliations
As we previously mentioned, the case of \(m=2\) has the distinctive feature of having weak bundles of class \(C^{\infty}\). It is natural to ask what happens for \(m\geq 3\). One approach to this question would be to compute the Godbillon–Vey invariant following [33]. Unfortunately for \(m\geq 3\) this calculation does not yield information conducive to an answer. However, for the case \(m\) odd, we can use reversibility of the flow combined with Theorem 5.5 to derive:
Suppose an Anosov thermostat given by the coupled vortex equations has a weak foliation of class \(C^{2}\) and \(m\) is odd. Then \(A\) vanishes identically.
Proof. When \(m\) is odd there is an important additional symmetry in the flow: the flip \(\sigma\) given by \((x,v)\mapsto (x,-v)\). We note that this map is isotopic to the identity. If \(\phi\) denotes the thermostat flow then, \(\sigma\circ\phi_{t}=\phi_{-t}\circ\sigma\). This relation easily implies that \(\sigma\) maps the weak stable foliation to the unstable one. Hence, if one of them is of class \(C^{2}\), the other one is also of class \(C^2\).
As we have already mentioned, Theorem 4.6 in [15] asserts that a smooth Anosov flow on a closed 3-manifold with weak stable and unstable foliations of class \(C^{2}\), is smoothly orbit equivalent to a quasi-fuchsian flow \(\psi\) that depends on a pair of points \(([g_1],[g_2])\) in Teichmüller space. The flow \(\psi\) has smooth weak stable foliation \(C^{\infty}\)-conjugate to the weak stable foliation of the constant curvature metric \(g_1\) and smooth weak unstable foliation \(C^{\infty}\)-conjugate to the weak unstable foliation of the constant curvature metric \(g_2\). But since \(\sigma\) is isotopic to the identity we must have \([g_{1}]=[g_{2}]\) and \(\psi\) is an ordinary geodesic flow preserving a volume form. Thus our thermostat flow preserves a volume form and by Theorem 5.5 we must have \(A=0\).
It is instructive to discuss Theorem 7.1 in the light of the remarks in 6 for \(m=3\). As pointed out, in this case, the thermostat flow is a \(C^{\infty}\) parametrisation of the geodesic foliation of a Hilbert metric. Benoist observes in [3] that the regularity of the weak foliations of the Hilbert geodesic flow coincides with the regularity of the boundary. Hence if the boundary of the strictly convex domain defining the Hilbert metric is \(C^2\), then the associated thermostat flow also has \(C^2\) weak foliations and therefore \(A=0\). This implies that the convex domain is an ellipsoid, thus recovering a result of Benzécri [5] for the case of 2-dimensional domains (note however, that the proof in [5] is more direct and straightforward).