3 Dominated splitting and hyperbolicity for thermostats
We are interested in the questions: when is this cocycle hyperbolic? When does it have a dominated splitting? We start with some definitions.
The cocycle \(\Psi_t\) is free of conjugate points if any non-trivial solution of the Jacobi equation \(\ddot{y}-V(\lambda)\dot{y}+\kappa y=0\) with \(y(0)=0\) vanishes only at \(t=0\).
The cocycle \(\Psi_t\) is said to be hyperbolic if there exists a splitting \(E=E^{u}\oplus E^{s}\) where \(E^u,E^s\) are continuous \(\rho\)-invariant line subbundles of \(TSM\), and constants \(C>0\) and \(0<\zeta<1<\eta\) such that for all \(t>0\) we have \[\Vert\Psi_{-t}|_{E^u}\Vert\leqslant C\,\eta^{-t}\quad \text{and}\quad \Vert\Psi_{t}|_{E^s}\Vert\leqslant C\,\zeta^{t}.\]
We also say:
The cocycle \(\Psi_t\) is said to have a dominated splitting if there is a continuous \(\rho\)-invariant splitting \(E=E^{u}\oplus E^{s}\), and constants \(C>0\) and \(0<\tau<1\) such that for all \(t>0\) we have \[\|\Psi_{t}|_{E^{s}(x,v)}\|\|\Psi_{-t}|_{E^{u}(\phi_{t}(x,v))}\|\leq C\,\tau^{t}.\]
Obviously hyperbolicity implies dominated splitting. It also implies that there are no conjugate points [9]. Moreover the cocycle \(\Psi_{t}\) is hyperbolic if and only if the thermostat flow \(\phi\) is Anosov (cf. for instance [40] where it is proved that the subbundles \(E^{s,u}\) of \(E\) lift to subbundles of \(TSM\) to give the usual definition of Anosov flow). We shall say that \(\phi\) has a dominated splitting if \(\Psi_{t}\) has a dominated splitting (this is the adequate notion of dominated splittings for flows, see e.g. [1]). For the case of flows on 3-manifolds, as it is our case, the existence of a dominated splitting can produce hyperbolicity if one has additional information on the closed orbits. Indeed [1] implies that if all closed orbits of \(\phi\) are hyperbolic saddles, then \(SM=\Lambda\cup\mathcal T\) where \(\Lambda\) is a hyperbolic invariant set and \(\mathcal T\) consists of finitely many normally hyperbolic irrational tori.
A very convenient way to establish the aforementioned properties for cocycles is to use quadratic forms as in [28, 41, 42]. In particular, we have [42]:
Let \(Q\) be a continuous non-degenerate quadratic form on \(E\). Suppose furthermore that the derivative \[\dot{Q}\!\left([\xi]\right):=\left.\frac{d}{dt}\right\vert_{t=0}Q\!\left([d\phi_t(\xi)]\right)\] exists for all \([\xi] \in E\). Then \(\Psi_t\) has a dominated splitting if \(\dot{Q}([\xi])>0\) for all \([\xi]\neq 0\) with \(Q([\xi])=0\). If the stronger property \(\dot{Q}([\xi])>0\) for all \([\xi]\neq 0\) holds, then \(\Psi_t\) is hyperbolic.
In what follows it will be helpful to understand how the spaces \(E^{u,s}\) are constructed using \(Q\). This is explained in detail in [42], so here we just give a brief summary adapted to our situation. We let \(\mathcal L_{+}(x,v)\) denote the set of all 1-dimensional subspaces \(W\) such that \(Q_{(x,v)}\) is positive on \(W\). The condition on the quadratic form \(Q\) ensures that \(\Psi_{t}\) acts as a contraction on \(\mathcal L_{+}\) and hence there is a unique point of intersection \[E^{u}(x,v)=\bigcap_{t>0}\Psi_{t}(\phi_{-t}(x,v))\mathcal L_{+}(\phi_{-t}(x,v)). \tag{3.1}\] All our quadratic forms \(Q\) below will have the property that \(Q(0,b)=0\) (using the identification \(E\simeq SM \times \mathbb{R}^2\)) and hence we can construct \(E^{u}\) (and \(E^s\)) simply by applying the procedure (3.1) to the vertical subspace \(\mathbb{R}(0,1)\), that is, \[E^{u}(x,v)=\lim_{t\to\infty}\Psi_{t}(\phi_{-t}(x,v))\mathbb{R}\left( \begin{array}{c} 0 \\ 1 \end{array} \right). \tag{3.2}\]
Let us put these ideas to use. Define \({\mathbb K}=\kappa +FV\lambda\).
Assume \(\mathbb{K}<0\). Then \(\phi\) is Anosov.
Proof. We let \((a,b)\) denote the standard coordinates on \(\mathbb{R}^2\). Using the identification \(E\simeq SM\times \mathbb{R}^2\) we define a quadratic form on \(E\) by the rule \[Q_{(x,v)}(a,b)=(b-V(\lambda)a)a.\] Then \[Q_{\phi_{t}(x,v)}(\Psi_{t}(a,b))=(\dot{y}-V(\lambda)y)y,\] where \(y\) is the unique solution of \[\ddot{y}-V(\lambda)\dot{y}+\kappa y=0,\] with \(y(0)=a\) and \(\dot{y}(0)=b\). A simple calculation shows that \[\dot{Q}=\frac{d}{dt}Q_{\phi_{t}(x,v)}(\Psi_{t}(a,b))=-\mathbb{K}y^2+(\dot{y}-V(\lambda)y)\dot{y}.\] Since \(\mathbb{K}<0\) we see that \[\left.\frac{d}{dt}\right|_{t=0}Q_{\phi_{t}(x,v)}(\Psi_{t}(a,b))>0\] for \((a,b)\neq 0\) and such that \(Q_{(x,v)}(a,b)=0\). Then Proposition 3.4 immediately implies that \(\Psi_{t}\) has a dominated splitting. We can upgrade that to hyperbolic as follows. If we let \(z:=\dot{y}-V(\lambda)y\), then the quadratic form is just \(zy\). By the construction of the subspaces \(E^{s,u}\) (cf. (3.1)) we see that \(E^{s,u}\) do not contain neither \(z=0\), nor \(y=0\). Hence there exist continuous functions \(r^{s,u}:SM\to \mathbb{R}\) such that \(H+r^{s,u} V\in E^{s,u}\). Moreover, we see that \(r^{u}-V\lambda>0\) and \(r^{s}-V\lambda<0\). Consider now a solution with initial conditions \((y(0),\dot{y}(0))\in E^{u}\). Then \(z=(r^{u}-V\lambda)y\) and \(\dot{z}=-\mathbb{K}y=-\mathbb{K}(r^{u}-V\lambda)^{-1}z\). This gives exponential growth for \(z\) and hence the desired exponential growth for \(\Psi_{t}\) on \(E^{u}\). Arguing in a similar way with \(E^{s}\), we deduce that \(\Psi_{t}\) is hyperbolic.
By considering the quadratic form \(Q=y\dot{y}\) we can deduce with a similar proof that if \(\kappa<0\) the thermostat flow \(\phi\) is Anosov. This is because \(\dot{Q}=\dot{y}^{2}-\kappa y^{2}+V(\lambda)y\dot{y}\). We have \(r^{u}>0\) and hyperbolicity follows from \(\dot{y}=r^{u}y\) when \((y(0),\dot{y}(0))\in E^{u}\).
In fact we can generalise this further as follows.
Let \(p:SM\to\mathbb{R}\) be a smooth function such that \[\kappa_{p}:=\kappa+Fp+p(p-V\lambda)<0.\] Then \(\phi\) has a dominated splitting. If in addition \(\kappa_{p}+\frac{(V\lambda)^{2}}{4}<0\), then the flow is Anosov.
Proof. The quadratic form to consider is \(Q=zy\), where \(z:=\dot{y}-py\). A calculation shows that \[\dot{Q}=z^{2}-\kappa_{p}y^{2}+zyV\lambda.\] We see that \(\dot{Q}>0\) whenever \(zy=0\), but \((y,z)\neq 0\). The claim in the theorem again follows from Proposition 3.4. Also note that \[\dot{Q}=\left(z-\frac{yV\lambda}{2}\right)^{2}-\left(\kappa_{p}+\frac{(V\lambda)^{2}}{4}\right)y^{2}>0,\] unless \((z,y)=0\). Hence the flow is Anosov by Proposition 3.4.
Let us see the main issue with upgrading the last theorem to “hyperbolic” as in the proof of Proposition 3.5. Certainly we get continuous (Hölder in fact) functions \(r^{s,u}\). To be definite consider the case of \(E^u\) and initial conditions \((y(0),\dot{y}(0))\in E^{u}\). Then \(\dot{y}=r^{u}y\) and \(z=(r^{u}-p)y\) with \(r^{u}-p>0\) as before. But now \(\dot{z}=(V\lambda-p)z-\kappa_{p}y=(V\lambda-p-\frac{\kappa_{p}}{r^{u}-p})z\). To get exponential growth we either need: \[\tag{3.3} r^{u}>0,\;\;\text{or}\;\;\;V\lambda-p-\frac{\kappa_{p}}{r^{u}-p}>0\] and it is not clear how to get any of these conditions in this generality. In the special cases above \(p=0\) or \(p=V\lambda\), we do get one of these conditions. In all these cases the function \(r=r^{u,s}\) satisfies the Riccati equation \[Fr+r^{2}-rV\lambda+\kappa=0,\] which is easily derived using the invariance of \(E^{s,u}\) and the Jacobi equation \(\ddot{y}-V(\lambda)\dot{y}+\kappa y=0\). Observe that \(h:=r-p\) satisfies the Riccati equation \[Fh+h^{2}+h(2p-V\lambda)+\kappa_{p}=0. \tag{3.4}\] Using (3.2) we can also give a construction of functions \(r^{u,s}\) at the level of the Riccati equation as follows. Fix \((x,v)\) and consider for each \(R>0\), the unique solution \(u_{R}\) to the Riccati equation along \(\phi_{t}(x,v)\) \[\dot{u}+u^2-uV\lambda+\kappa=0\] satisfying \(u_{R}(-R)=\infty\). Then (3.2) translates easily into \[r^{u}(x,v)=\lim_{R\to \infty}u_{R}(0). \tag{3.5}\] Note that \(r^{u}(\phi_{t}(x,v))=\lim_{R\to\infty}u_{R}(t)\). These limiting solutions exist whenever the cocycle \(\Psi_{t}\) has no conjugate points [2]. It is easy to check that in all the cases we consider below, the cocycle \(\Psi_t\) is free of conjugate points.
This remark attempts to clarify the role of the function \(p\) in terms of conjugate cocycles and infinitesimal generators as in Subsection 2.3. As we have already pointed out, the infinitesimal generator \(\mathbb B\) for a thermostat is given by \[\mathbb B=\begin{pmatrix} 0 & -1 \\ \kappa & -V\lambda\end{pmatrix}.\] Consider a gauge transformation \(\mathcal P:SM\to GL(2,\mathbb{R})\) given by \[\mathcal P=\begin{pmatrix} 1 & 0 \\ p & 1\end{pmatrix}.\] A calculation using (2.3) shows that the conjugate cocyle \(\tilde{\Psi}_{t}\) via \(\mathcal{P}\) has infinitesimal generator given by \[\tilde{\mathbb{B}}=\begin{pmatrix} -p & -1 \\ \kappa_{p} & -V\lambda+p\end{pmatrix}.\] The cocycles \(\Psi_{t}\) and \(\tilde{\Psi}_{t}\) share the same dominated splitting/hyperbolicity properties by virtue of being conjugate, but the form of \(\tilde{\mathbb B}\) exposes clearly the origins of these properties via \(\kappa_{p}<0\) (cf. [42]). The trace of both matrices, which is \(-V\lambda\) (minus divergence of \(F\)), indicates the dissipative nature of thermostats.