2 Preliminaries on general thermostats

Let \(M\) be a closed oriented surface equipped with a Riemannian metric \(g\), \(SM\) its unit circle bundle and \(\pi:SM\to M\) the canonical projection. The latter is in fact a principal \(\mathrm{SO}(2)\)-bundle and we let \(V\) be the infinitesimal generator of the action of \(\mathrm{SO}(2)\).

Given a unit vector \(v\in T_{x}M\), we will denote by \(Jv\) the unique unit vector orthogonal to \(v\) such that \(\{v,Jv\}\) is an oriented basis of \(T_{x}M\). There are two semibasic 1-forms \(\omega_1\) and \(\omega_2\) on \(SM\), which are defined by the formulas: \[(\omega_1)_{(x,v)}(\xi):=g\left(d_{(x,v)}\pi(\xi),v\right);\] \[(\omega_2)_{(x,v)}(\xi):=g\left(d_{(x,v)}\pi(\xi),Jv\right).\] The form \(\omega_1\) is the canonical contact form of \(SM\) whose Reeb vector field is the geodesic vector field \(X\).

A basic theorem in 2-dimensional Riemannian geometry asserts that there exists a unique 1-form \(\psi\) on \(SM\) – the Levi-Civita connection form of \(g\) – such that \(\psi(V)=1\) and \[\tag{2.1} \begin{aligned} d\omega_1&=-\omega_2\wedge\psi,\\ d\omega_2&=-\psi\wedge\omega_1,\\ d\psi&=-(K_g\circ\pi)\,\omega_1\wedge\omega_2, \end{aligned}\] where \(K_g\) denotes the Gaussian curvature of \(g\). In fact, the form \(\psi\) is given by \[\psi_{(x,v)}(\xi)=g\left( \frac{DZ}{dt}(0),Jv\right),\] where \(Z:(-\varepsilon,\varepsilon)\to SM\) is any curve with \(Z(0)=(x,v)\), \(\dot{Z}(0)=\xi\) and \(\frac{DZ}{dt}\) is the covariant derivative of \(Z\) along the curve \(\pi\circ Z\).

For later use it is convenient to introduce the vector field \(H\) uniquely defined by the conditions \(\omega_2(H)=1\) and \(\omega_1(H)=\psi(H)=0\). The vector fields \(X,H,V\) are dual to \(\omega_1,\omega_2,\psi\) and as a consequence of (2.1) they satisfy the commutation relations \[=H,\quad [V,H]=-X,\quad [X,H]=K_gV.\] The equations (2.1) also imply that the vector fields \(X,H\) and \(V\) preserve the volume form \(\omega_1\wedge d\omega_1\) and hence the Liouville measure. Note that the flow of \(H\) is given by \(R^{-1}\circ \phi^0_t\circ R\), where \(R(x,v)=(x,Jv)\) and \(\phi^0_t\) is the geodesic flow of \(g\).

Let \(\lambda\) be an arbitrary smooth function on \(SM\). For several of the results that we will describe below, we will not need \(\lambda\) to be a special polynomial in the velocities. We consider a (generalised) thermostat flow on \((M,g)\), that is, a flow \(\phi\) defined by \[\frac{D\dot{\gamma}}{dt}=\lambda(\gamma,\dot{\gamma})\,J\dot{\gamma}.\] It is easy to check that \[F:=X+\lambda V\] is the generating vector field of \(\phi\).

Now let \(\Theta:=-\omega_1\wedge d\omega_1=\omega_1\wedge\omega_2\wedge\psi\). This volume form generates the Liouville measure \(d\mu\) of \(SM\).

2.1 Jacobi equations

It is easy to derive the ODEs governing the behaviour of \(d\phi_{t}\) using the bracket relations above. Given \(\xi\in T_{(x,v)}SM\) (the initial conditions), if we write \[d\phi_{t}(\xi)=xF+yH+uV\] then \[\begin{aligned} \dot{x} &=\lambda\,y;\\ \dot{y} &=u;\\ \dot{u} &=V(\lambda)\dot{y}-\kappa y, \end{aligned}\] where \(\kappa:=K_g-H\lambda+\lambda^2\).

2.2 Quotient cocycle

We consider the rank two quotient vector bundle \(E=TSM/\mathbb{R}F\). We use the notation \([\xi]\) with \(\xi \in TSM\) for the elements of \(E\). Note that \(d\phi_t\) descends to the quotient to define a mapping \[\rho : E\times \mathbb{R}\to E, \quad ([\xi],t)\mapsto \rho([\xi],t)=[d\phi_t(\xi)]\] satisfying \(\rho_{t}\circ\rho_{s}=\rho_{t+s}\) for all \(t,s \in \mathbb{R}\). The basis of vector fields \((F,H,V)\) on \(SM\) defines a vector bundle isomorphism \(TSM \simeq SM\times\mathbb{R}^3\) and consequently an identification \(E\simeq SM \times \mathbb{R}^2\). Therefore, for each \(t \in \mathbb{R}\), we obtain a unique map \(\Psi_t : SM \to GL(2,\mathbb{R})\) defined by the rule \[\rho_t((x,v),w)=\left(\phi_t(x,v),\Psi_t(x,v)w\right)\] for all \(((x,v),w) \in E\simeq SM\times \mathbb{R}^2\). The map \(\Psi : SM \times \mathbb{R}\to GL(2,\mathbb{R})\) satisfies \[\Psi_{t+s}(x,v)=\Psi_{s}(\phi_{t}(x,v))\Psi_{t}(x,v)\] for all \((x,v) \in SM\) and \(t,s \in \mathbb{R}\), and hence defines an \(GL(2,\mathbb{R})\)-valued cocycle on \(SM\) with respect to the \(\mathbb{R}\)-action defined by \(\phi\). Explicitly, \(\Psi_t\) is the matrix whose action on \(\mathbb{R}^2\) is given by \[\Psi_t(x,v): \left( \begin{array}{c} y(0) \\ \dot{y}(0) \end{array} \right) \mapsto \left( \begin{array}{c} y(t) \\ \dot{y}(t) \end{array} \right)\] where \(\ddot{y}(t)-V(\lambda)(\phi_{t}(x,v))\dot{y}(t) + \kappa(\phi_t(x,v)) y(t) = 0\).

Note that for thermostats the 2-plane bundle spanned by \(H\) and \(V\) is in general not invariant under \(d\phi_{t}\).

2.3 Infinitesimal generators and conjugate cocycles

Given a cocycle \(\Psi_{t}:SM\times\mathbb{R}\to GL(2,\mathbb{R})\) we define its infinitesimal generator \(\mathbb{B}:SM\to \mathfrak{gl}(2,\mathbb{R})\) as \[\mathbb{B}(x,v):=-\left.\frac{d}{dt}\right|_{t=0}\Psi_{t}(x,v).\] The cocycle \(\Psi_{t}\) can be recovered from \(\mathbb{B}\) as the unique solution to \[\frac{d}{dt}\Psi_{t}(x,v)+\mathbb{B}(\phi_{t}(x,v))\Psi_{t}(x,v)=0,\;\;\;\Psi_{0}(x,v)=\mbox{\rm Id}.\] For the case of thermostats, it is immediate to check that \[\mathbb{B}=\begin{pmatrix} 0 & -1 \\ \kappa & -V\lambda\end{pmatrix}.\] Given a smooth map \(\mathcal P:SM\to GL(2,\mathbb{R})\) (a gauge) we can define a new cocycle by conjugation as \[\tilde{\Psi}_{t}(x,v)=\mathcal{P}^{-1}(\phi_{t}(x,v))\Psi_{t}(x,v)\mathcal{P}(x,v).\] It is easy to check that the infinitesimal generator \(\tilde{\mathbb{B}}\) of \(\tilde{\Psi}_{t}\) is related to \(\mathbb{B}\) by \[\tilde{\mathbb{B}}=\mathcal{P}^{-1}\mathbb{B}\mathcal{P}+\mathcal{P}^{-1}F\mathcal{P}. \tag{2.3}\]