1 Introduction

1.1 Background

This paper is concerned with the description and study of a class of dynamical systems determined by the solutions of a pair of partial differential equations naturally arising in Abelian gauge theories on a closed oriented Riemannian \(2\)-manifold \((M,g)\) of negative Euler characteristic. Let \(SM\) denote the unit tangent bundle of \((M,g)\). Given a smooth function \(\lambda\in C^{\infty}(SM)\), we may consider the ODE for \(\gamma:\mathbb{R}\to M\) \[\tag{1.1} \ddot{\gamma}=\lambda(\gamma,\dot{\gamma})J\dot{\gamma},\] where \(J:TM\to TM\) denotes rotation by \(\pi/2\) according to the orientation of the surface, and the acceleration of \(\gamma\) is computed using the Levi-Civita connection of \(g\). Equation (1.1) describes the motion of particle on \(M\) driven by a force orthogonal to its velocity with magnitude determined by \(\lambda\). As such it is easy to see that the speed of \(\gamma\) remains constant and thus \((\gamma,\dot{\gamma})\) defines a flow in \(SM\). If \(\lambda=0\), we obtain the geodesic flow of \(g\), the prototype example of a conservative dynamical system. If \(\lambda\) only depends on position (i.e. it is the pull-back of a function on \(M\)), we still obtain a volume preserving flow (a magnetic flow), but the situation changes if \(\lambda\) is allowed to depend on velocities. For instance we may take \(\lambda\) as the restriction to \(SM\) of a 1-form on \(M\) and in that case we obtain a Gaussian thermostat as studied in [33, 34]. In general, these flows are not volume preserving and here we are concerned with thermostat flows as defined by (1.1) when \(\lambda\) arises from a higher order differential on \(M\).

The dynamical properties that we shall investigate are hyperbolicity and domination. Hyperbolicity has played a prominent role in dynamics [31], but weaker forms of hyperbolicity, like domination have in recent decades come under intense focus [6]. The notion of dominated splitting was introduced by Mañé in the context of the proof of the stability conjecture (cf. [30]), but it has appeared in several other contexts and under different names. It can be regarded as a projective form of hyperbolicity and it can also be characterized in terms of the singular value decomposition of the linear Poincaré flow [4]. The notion is particularly relevant in our setting: for volume preserving flows on 3-manifolds domination is equivalent to hyperbolicity, but for dissipative thermostats this is no longer the case. Thus in the results below some effort will be spent in studying when we can upgrade our flows from having a dominatted splitting to being Anosov.

Let us give the benchmark example that motivates our construction. Let \(A\) be a holomorphic cubic differential on \(M\) so that \(\overline{\partial}A=0\), and suppose the pair \((g,A)\) is linked by the additional equation \[K_{g}=-1+2|A|^2_{g},\] where \(K_{g}\) is the Gauss curvature. By the work of Labourie [22] and Loftin [23], such a pair gives rise to a properly convex projective structure on \(M\) and hence to an associated divisible strictly convex set \(\tilde{M}\subset \mathbb{RP}^2\). The set \(\tilde{M}\) comes equipped with a distance function, the so-called Hilbert metric — see for instance [20] for details — while \(g\) is known as the Blaschke metric. The Hilbert metric is the distance function of a Finsler metric whose geodesic flow is known to be Anosov [3]. If we choose \(\lambda\) to be the imaginary part of \(A\) — regarded as a function on \(SM\) — then the thermostat flow determined by (1.1) is a suitable reparametrization of the geodesic flow of the Hilbert metric. While the work of Labourie interprets the pair of equations \(\overline{\partial}A=0\) and \(K_{g}=-1+2|A|^2_{g}\) as an instance of Hitchin’s Higgs bundle equations [17], they may also be interpreted as an example of the so-called Abelian vortex equations [11]. One can, in fact, consider similar equations for differentials of any order, not just 3, and investigate the dynamical properties of the associated thermostat. This was done in [25], but here we uncover a larger landscape that allows for example the consideration of holomorphic differential of fractional order, i.e. holomorphic sections of \(K^{m/n}\) where \(K\) is the canonical line bundle of \((M,g)\). The natural habitat of our thermostats is not the unit sphere bundle anymore, but rather root bundles covering \(SM\) to accommodate for the fractional degrees.

1.2 Vortices

We now proceed to describe in detail the geometric setting for our pair of PDEs.

Let \(L \to M\) be a complex line bundle of positive degree \(\deg(L)\). For a triple consisting of a Hermitian bundle metric \(\mathcal{h}\) on \(L\), a del-bar operator \(\overline{\partial}_L\) on \(L\) and a \((1,\! 0)\)-form \(\varphi\) on \(M\) with values in \(L\), we consider the following pair of equations \[\tag{1.2} R(\mathrm{D})+\frac{1}{2}\varphi\wedge\varphi^*+\mathrm{i}\ell\Omega_g=0 \qquad \text{and} \qquad \overline{\partial}_L\varphi=0.\] Here we write \(\ell:=\deg(L)/|\chi(M)|\), \(\mathrm{D}\) denotes the Chern connection on \(L\) with respect to \((\mathcal{h},\overline{\partial}_L)\), \(R(\mathrm{D})\) its curvature, \(\Omega_g\) the area form of \(g\) and \(\varphi^*:=\mathcal{h}(\cdot,\varphi)\). We assume \(\mathcal{h}\) to be conjugate linear in the second variable, so that \(\varphi^*\) is a \((0,\! 1)\)-form on \(M\) with values in the dual \(L^{-1}\) of \(L\).

The pair (1.2) of equations are a minor variation of the Abelian vortex equations on a Riemann surface, hence we refer to them as vortex equations as well. The usual Abelian vortex equations concern a triple \((\mathcal{h},\overline{\partial}_{L^{\prime}},\Phi)\), where \(\Phi\) is a section of a complex line bundle \(L^{\prime}\) over an oriented Riemannian \(2\)-manifold \((M,g)\). Besides \(\Phi\) being holomorphic, one requests that the Chern connection \(\mathrm{D}\) determined by \((\mathcal{h},\overline{\partial}_{L^{\prime}})\) satisfies \[\tag{1.3} \mathrm{i}\Lambda R(\mathrm{D})+\frac{1}{2}\Phi\otimes \Phi^*-\frac{c}{2}=0,\] where \(c\) is some real constant and \(\Lambda\) denotes the \(L^2\)-adjoint of wedging with the area form \(\Omega_g\). The Abelian vortex equations are a modification of the Ginzburg–Landau model for superconductors and were first studied by Noguchi [28] and Bradlow [7] (for background, see also [19]). A general framework for the so-called symplectic vortices over closed Riemann surfaces was described in [9].

1.3 Vortex thermostats

Since \(L\) has positive degree and \(\chi(M)<0\), there exist unique positive coprime integers \((m,n)\) so that we have an isomorphism \(L^n\simeq K^m\) of complex line bundles. We fix an \(n\)-th root \(SM^{1/n}\) of the unit tangent bundle \(\pi : SM \to M\) of \((M,g)\). By this we mean a principal \(\mathrm{SO}(2)\)-bundle \(\pi_n : SM^{1/n} \to M\) which is an equivariant \(n\)-fold cover of \(\pi: SM \to M\), see 2.2 below for details.

Following [8], we call three linearly independent vector fields \((X,H,V)\) on a smooth \(3\)-manifold \(N\) a generalised Riemannian structure, if they satisfy the commutator relations \[[V,X]=H,\qquad [V,H]=-X, \qquad [X,H]=K_g V,\] for some smooth function \(K_g\) on \(N\). A (generalised) thermostat is a flow \(\phi\) on \(N\) which is generated by a vector field of the form \(X+\lambda V\), where \(\lambda\) is a smooth function on \(N\). The root \(SM^{1/n}\) is equipped with a generalised Riemannian structure by pulling back the natural Riemannian structure on \(SM\) determined by \(g\) and the orientation (where \(X\) is the geodesic vector field and \(V\) the vertical vector field). In 4 we show how to associate a thermostat to a solution \((\mathcal{h},\overline{\partial}_L,\varphi)\) of the vortex equations on \(L \to (M,g)\). We call such flows vortex thermostats.

In the special case where \(L\) is the canonical bundle equipped with its standard complex structure and Hermitian metric induced by \(g\) and where \(\varphi\) vanishes identically, the vortex equations (1.2) are equivalent to \(g\) being hyperbolic. The case where \(g\) has non-constant negative Gauss curvature can be dealt with by modifying the complex structure on \(K\). In particular, suitably reparametrised, our family of flows include the geodesic flow of metrics of negative Gauss curvature and more generally the so-called W-flows of Wojtkowski [33, 34] (in the case of negative curvature, c.f. [25]).

1.4 Results

Our goal is to establish hyperbolicity properties for the general class of vortex thermostats. We first show:

The choice of an \(n\)-th root \(SM^{1/n}\) of \(SM\) gives a corresponding \(n\)-th root \(K^{1/n}\) of \(K\) and hence an isomorphism \(\mathcal{Z} : L \to K^{m/n}\) of complex line bundles. While \(\mathcal{Z}\) is in general not an isomorphism of holomorphic line bundles, we can upgrade Theorem A as follows:

We do not know if there is a vortex thermostat which is not Anosov.

As in the case of the usual vortex equations, the equations (1.2) are invariant under a suitable action of the complex gauge group of \(L\), that is, the group \(\mathrm{G}_{\mathbb{C}}\) of automorphisms of \(L\). We show that by possibly applying a complex gauge transformation, we can assume without losing generality that \(\mathcal{h}=\mathcal{h}_0\), where \(\mathcal{h}_0\) denotes the natural Hermitian bundle metric on \(L\simeq K^{m/n}\) determined by \(g\). The \(1\)-form \(\varphi\) is a section of \(K\otimes L\simeq K^{1+\ell}\) and hence we may think of \(\varphi/\ell\) as a differential \(A\) of fractional degree \(1+\ell>1\). Furthermore, since \(K^{m/n}\simeq L\) as complex line bundles, there exists a unique \(1\)-form \(\theta\) on \(M\) so that \(\overline{\partial}_{K^{m/n}}-\overline{\partial}_L=\ell\theta^{0,1}\), where \(\theta^{0,1}\) denotes the \((0,\! 1)\)-part of \(\theta\). By construction, the above isomorphism \(\mathcal{Z}\) of complex line bundles is an isomorphism of holomorphic line bundles if and only if \(\theta\) vanishes identically. In terms of the triple \((g,A,\theta)\) the vortex equations (1.2) are equivalent to \[K_g-\delta_g\theta=-1+\ell |A|^2_g \qquad \text{and}\qquad \overline{\partial}A=\ell\,\theta^{0,1}\otimes A,\] where \(|\cdot|_g\) denotes the pointwise norm induced on \(K^{1+\ell}\) by \(g\) and \(\delta_g\) the co-differential. Thus, we recover the main equations from [25] (see also [24]), but now in the more general setting of fractional differentials. In particular, Theorem A and Theorem B above generalise the results from [25] to the case of differentials of fractional degree. Proving the Anosov property for fractional differential presents new obstacles, particularly those in the range \(0<\ell< 1\).

As in [25], our flows do not preserve a volume form, unless \(\varphi\) vanishes. More precisely, the proof of [25] shows that under the hypotheses of Theorem B the associated vortex thermostat \(\phi\) preserves an absolutely continuous measure if and only if \(\varphi\) vanishes identically. This property implies that vortex thermostats as in Theorem B with \(\varphi\neq 0\) have positive entropy production and thus they provide interesting models in nonequilibrium statistical mechanics [13, 14, 29]. The moduli space of gauge equivalence classes of solutions of the usual vortex equations was described in [7], we expect a similar statement to hold as well in the case considered here; this may be taken up elsewhere.

In 7 we briefly discuss the dominated splitting property for a thermostat that one can associate to the usual vortex equations.

Acknowledgements

We are grateful to Miguel Paternain, Rafael Potrie and the anonymous referee for helpful comments. TM was partially funded by the priority programme SPP 2026 “Geometry at Infinity” of DFG. GPP was partially supported by EPSRC grant EP/R001898/1.

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