5 Proof of Theorems A and B
Summarizing 4, given a solution \((\mathcal{h},\overline{\partial}_L,\varphi)\) to the vortex equations for a complex line bundle \(L \to (M,g)\) and upon fixing an \(n\)-th root \(SM^{1/n}\) of \(SM\), we obtain a vortex thermostat on \(SM^{1/n}\). After possibly applying a (non-unitary) gauge transformation to \((\mathcal{h},\overline{\partial}_L,\varphi)\), we can assume that the thermostat \(\phi\) arises from \(\lambda=a-\mathbb{V}\theta\), where \(a\) encodes a fractional differential on \(M\), that is, a section \(A\) of \(K^{(m+n)/n}\) and \(\theta\) a \(1\)-form on \(M\) so that the following equations hold \[\tag{5.1} K_g-\delta_g\theta=-1+\ell |A|^2_g \qquad \text{and}\qquad \overline{\partial}A=\ell\,\theta^{0,1}\otimes A,\] where for simplicity of notation we write \(\overline{\partial}\) for \(\overline{\partial}_{K^{(m+n)/n}}\) and where \(\ell=m/n\). Thus, by Lemma 4.3 and Lemma 4.4 our setup consists of \((X,H,\mathbb{V})\) on \(SM^{1/n}\) as well as real-valued functions \(a,\theta\) satisfying \(\mathbb{V}\mathbb{V}a=-(1+\ell)^2 a\) and \(\mathbb{V}\mathbb{V}\theta=-\theta\) so that \[\tag{5.2} \begin{aligned} K_g&=-1-X \theta-H\mathbb{V}\theta+\ell|\boldsymbol{A}|^2,\\ \frac{X\mathbb{V}a}{1+\ell}&=Ha+\frac{\ell\theta \mathbb{V}a}{1+\ell}-\ell a\mathbb{V}\theta, \end{aligned}\] where \(\ell\) is a positive rational number and \(\boldsymbol{A}=\frac{\mathbb{V}a}{1+\ell}+\mathrm{i}a\).
5.1 Dominated splitting
Applying Theorem 3.3 we obtain:
Every vortex thermostat admits a dominated splitting. Moreover, if all closed orbits of \(\phi\) are hyperbolic saddles, then \(\phi\) is Anosov.
Proof. Using Theorem 3.3 we need to show that there exists a smooth function \(p : SM^{1/n} \to \mathbb{R}\) so that \[\kappa_{p}=\kappa+Fp+p(p-\mathbb{V}\lambda)<0.\] Recall that \(\lambda=a-\mathbb{V}\theta\). Taking \(p=\theta+\mathbb{V}a/(1+\ell)\) we compute \[\begin{aligned} \kappa_p-\kappa&=F\left(\theta+\frac{\mathbb{V}a}{1+\ell}\right)-\left(\theta+\frac{\mathbb{V}a}{1+\ell}\right)\left(\theta+\frac{\mathbb{V}a}{1+\ell}-\mathbb{V}a+\mathbb{V}\mathbb{V}\theta\right)\\ &=X\theta+Ha+\frac{\ell\theta \mathbb{V}a}{1+\ell}-\ell a\mathbb{V}\theta+\lambda \mathbb{V}p-\ell\left(\theta+\frac{\mathbb{V}a}{1+\ell}\right)\frac{\mathbb{V}a}{1+\ell}\\ &=X\theta+Ha-(1+\ell)a^2-(\mathbb{V}\theta)^2+2a\mathbb{V}\theta-\ell\left(\frac{\mathbb{V}a}{1+\ell}\right)^2\\ &=X\theta+Ha -\ell|\boldsymbol{A}|^2-a^2-(\mathbb{V}\theta)^2+2a\mathbb{V}\theta\\ &=-1-K_g-H\mathbb{V}\theta+H a-a^2-(\mathbb{V}\theta)^2+2a\mathbb{V}\theta\\ &=-1-(K_g-H\lambda+\lambda^2)=-1-\kappa\end{aligned}\] where we have used that \(\mathbb{V}\mathbb{V}\theta=-\theta\) and \(\mathbb{V}\mathbb{V}a=-(1+\ell)^2 a\) as well as (3.2), (5.2). We conclude that \(\kappa_p=-1\) and the existence of a dominated splitting follows.
Finally, the addendum regarding the Anosov property when the closed orbits of \(\phi\) are hyperbolic saddles is a consequence of [1]. Indeed, in our situation the invariant normally hyperbolic irrational tori cannot arise since \(V\) must be transversal to them. If we had one such torus \(T\), then the projection map \(\pi_{n}:SM^{1/n}\to M\) restricted to \(T\) would be a local diffeomorphism which is absurd since \(\chi(M)<0\).
5.2 The Anosov property
While we have an isomorphism \(\mathcal{Z} : L \to K^{m/n}\) of complex line bundles, the two line bundles need not be isomorphic as holomorphic line bundles. We do however obtain:
Suppose \(\mathcal{Z} : L \to K^{m/n}\) is an isomorphism of holomorphic line bundles, then the associated vortex thermostat is Anosov.
Recall from (4.4) that we write \(\overline{\partial}_L=\overline{\partial}_{K^{m/n}}-\ell\,\theta^{0,1}\) for some \(1\)-form \(\theta\) on \(M\). The isomorphism \(\mathcal{Z}\) being an isomorphism of holomorphic line bundles translates to \(\theta\) vanishing identically. We thus henceforth restrict to the case where \(\theta\equiv 0\), so that the equations (5.1) become \[\tag{5.3} K_g=-1+\ell|A|^2_g,\quad \text{and}\quad \overline{\partial}A=0.\]
We start with the following comparison lemma:
Let \(h\) be the positive Hopf solution of \(Fh+h^2+Bh-1=0\). Then \[\frac{-c+\sqrt{c^2+4}}{2}\leqslant h\leqslant \frac{c+\sqrt{c^2+4}}{2}\] where \(c=\max |B|\) and \(B=\left(\frac{1-\ell}{1+\ell}\right)\mathbb{V}a\).
Proof of Lemma 5.1. We fix \((x,v) \in SM^{1/n}\). Recall from 3 that the existence of a dominated splitting implies that the positive Hopf solution \(h\) may be constructed using the limiting procedure \[h(x,v)=\lim_{R \to \infty}\eta_R(0),\] where for \(R>0\) the function \(\eta_R\) denotes the solution to the ODE \[\dot{\eta}(t)+\eta^2(t)+B(\phi_t(x,v))\eta(t)-1=0\] with \(\eta_R(-R)=0\). Since \(B\geqslant -c\) and \(h\) is positive, we have \[\dot{\eta}=-\eta^2-B\eta+1\leqslant -\eta^2+c\eta+1.\] Hence if \(\gamma\) solves the constant coefficients Riccati equation \[\dot{\gamma}+\gamma^2-c\gamma-1=0\] then \(\eta(t)\leqslant \gamma(t)\) for \(t\geqslant t_0\) provided \(\eta(t_0)=\gamma(t_0)\) by ODE comparison. The solution \(\gamma_R\) to \(\dot{\gamma}+\gamma^2-c\gamma-1=0\) with \(\gamma_{R}(-R)=0\) is given by \[\gamma_{R}(t)=\frac{1-e^{(-R-t)/E}}{-C_{-}+C_{+}e^{(-R-t)/E}}\] where \[C_{\pm}=\frac{c\pm\sqrt{c^2+4}}{2}\] and \(E=1/(C_{+}-C_{-})\). Thus \[\eta_{R}(0)\leqslant \gamma_{R}(0)\to -1/C_{-}=C_{+}\] as \(R\to\infty\) and thus \(h(x,v)\leqslant \frac{c+\sqrt{c^2+4}}{2}\).
The lower bound can also be proved in the same way. Since \(B\leqslant c\), we have \[\dot{\eta}=-\eta^2-B\eta+1\geqslant -\eta^2-c\eta+1.\] And now we compare with solutions of \[\dot{\gamma}+\gamma^2+c\gamma-1=0,\] in particular those \(\gamma_{R}\) with \(\gamma_{R}(-R)=\infty\). One gets \[\eta_{R}(0)\geqslant \gamma_{R}(0)\to \frac{-c+\sqrt{c^2+4}}{2}\] as \(R\to\infty\) and thus \(h(x,v)\geqslant \frac{-c+\sqrt{c^2+4}}{2}\).
For what follows we need a bound on \(|A|_g^2\).
Suppose \((g,A)\) satisfies \(K_{g}=-1+\ell|A|^{2}_{g}\) and \(\overline{\partial}A=0\). Then \(K_{g}<0\).
In the case where \(A\) is a differential of integral degree \(d\geqslant 2\), the lemma was proved in [25]. It is easy to check that the proof also holds in the case of a differential of fractional degree \(d>1\). We refer the reader to [25] for details.
We are now ready to prove Theorem B.
Proof of Theorem B. We already know that the flow admits a dominated splitting. To prove the Anosov property we shall use Lemma 3.5. We will prove that in the range \(\ell\geqslant 1\) our flows fit alternative (1) and for \(0< \ell \leqslant 1\), they fit alternative (2). We shall prove the claims for the unstable bundle. The proofs for the stable bundle are quite analogous.
We note that Lemma 5.2 gives \[-1< \frac{\sqrt{\ell}}{1+\ell}\mathbb{V}a < 1. \tag{5.4}\] Also note that for our thermostat \(p=\mathbb{V}a/(1+\ell)\), \(\kappa_{p}=-1\) and \(h=r^u-p\).
Assume first that \(\ell\geqslant 1\). We shall prove that \(r^u>0\). This is equivalent to \[h+\frac{\mathbb{V}a}{1+\ell}>0. \tag{5.5}\] In view of (5.4) and (5.5) it is enough to prove that \[h\geqslant 1/\sqrt{\ell}.\] From the definition of \(c\) in Lemma 5.1 and the bound \(\frac{\sqrt{\ell}}{1+\ell}\mathbb{V}a < 1\) we derive \(c\leqslant (1-\ell)/\sqrt{\ell}\). Hence \[\frac{-c+\sqrt{c^2+4}}{2}\geqslant 1/\sqrt{\ell}\] and the desired bound follows from Lemma 5.1.
Assume now that \(0<\ell\leqslant 1\). Condition (2) in Lemma 3.5 for \(r^u\) becomes \[\left(\frac{\ell}{1+\ell}\right)\mathbb{V}a+1/h>0. \tag{5.6}\] In view of (5.4) and (5.6) it is enough to prove that \[h\leqslant 1/\sqrt{\ell}.\] From the definition of \(c\) in Lemma 5.1 and the bound \(\frac{\sqrt{\ell}}{1+\ell}\mathbb{V}a < 1\) we derive \(c\leqslant (1-\ell)/\sqrt{\ell}\). Hence \[\frac{c+\sqrt{c^2+4}}{2}\leqslant 1/\sqrt{\ell}\] and the desired bound follows from Lemma 5.1.
As we have mentioned in the introduction, in the special case where \(A\) is a cubic holomorphic differential, a solution \((g,A)\) to (5.3) gives rise to a properly convex projective structure on \(M\). The monodromy representation of such a properly convex projective structure is an example of an Anosov representation as introduced by Labourie [21]. In recent work [5] Bochi, Potrie & Sambarino show how Anosov representations can be used to construct certain cocycles admitting a dominated splitting. At the time of writing, it is however quite unclear if there is any relation between [5] and our construction which goes beyond the special case of cubic holomorphic differentials.