3 Dominated splittings and hyperbolicity
In this section we summarize the main dynamical set up that we shall use; in the first three subsections we follow closely the presentation in [25]. For background on the notion of dominated splittings we refer to [10].
3.1 Definitions
Let \(N\) be a smooth closed \(3\)-manifold and \(\phi : N\times \mathbb{R}\to N\) a continuous flow. A cocycle over \(\phi\) with values in \(\mathrm{GL}(2,\mathbb{R})\) is a continuous map \(\Psi : N \times \mathbb{R}\to \mathrm{GL}(2,\mathbb{R})\) such that \[\Psi_{t_1+t_2}(x)=\Psi_{t_1}(\phi_{t_2}(x))\Psi_{t_2}(x)\] for all \(t_1,t_2 \in \mathbb{R}\) and \(x \in N\). Note that the cocycle condition ensures that on the trival vector bundle \(E=N\times \mathbb{R}^{2}\) we obtain a continuous linear flow \(\rho : E \times \mathbb{R}\to E\) by defining \[\rho_t((x,a))=(\phi_t(x),\Psi_t(x)a)\] for all \((x,a) \in E=N\times \mathbb{R}^2\) and \(t \in \mathbb{R}\).
We say \(E\) admits a continuous \(\rho\)-invariant splitting if there exist continuous \(\rho\)-invariant line bundles \(E^{s,u}\) so that \(E=E^u\oplus E^s\). We fix a norm \(|\cdot|\) on \(\mathbb{R}^2\).
The cocycle \(\Psi\) is said to be hyperbolic is there exists a continuous \(\rho\)-invariant splitting \((E^s,E^u)\) and positive constants \(C,\mu>0\) so that \[\Vert\left.\Psi_t(x)\right|_{E^s(x)}\Vert \leqslant Ce^{-\mu t}\quad\text{and}\quad \Vert\left.\Psi_{-t}(x)\right|_{E^u(x)}\Vert \leqslant Ce^{-\mu t}\] for all \(x \in N\) and \(t>0\).
Here \(\Vert\cdot\Vert\) denotes the operator norm induced on \(\mathrm{Hom}(E^{s,u}(x),E^{s,u}(\phi_t(x)))\) by the norm \(|\cdot |\), respectively. A weaker notion than that of hyperbolicity is to ask that for all \(x \in N\), any direction not contained in the suspace \(E^{s}(x)\) converges exponentially fast to \(E^u(\phi_t(x))\) when applying \(\rho_t(x)\). This condition is equivalent to the following notion:
The cocycle \(\Psi\) is said to admit a dominated splitting if there exists a continuous \(\rho\)-invariant splitting \((E^u,E^s)\) and positive constants \(C,\mu>0\) so that \[\tag{3.1} \Vert\left.\Psi_t(x)\right|_{E^s(x)}\Vert \Vert\left.\Psi_{-t}(\phi_t(x))\right|_{E^u(\phi_t(x))}\Vert\leqslant Ce^{-\mu t}\] for all \(x \in N\) and \(t>0\).
3.2 The derivative cocycle of a thermostat
Suppose the closed \(3\)-manifold \(N\) is equipped with a generalised Riemannian structure and a thermostat \(\phi\) generated by the vector field \(F=X+\lambda V\) as above. Using the bracket relations (2.5), it is straightforward to derive the ODEs dictating the behavior of \(d\phi_{t}\). Given an initial condition \(\xi\in T_xN\) and if we write \[d\phi_{t}(\xi)=w(t)F(\phi_t(x))+y(t)H(\phi_t(x))+u(t)V(\phi_t(x))\] for real-valued functions \(w,y,u\) on \(\mathbb{R}\), then \[\begin{aligned} \dot{w} &=\lambda\,y;\\ \dot{y} &=u;\\ \dot{u} &=V(\lambda)\dot{y}-\kappa y, \end{aligned}\] where \[\tag{3.2} \kappa:=K_g-H\lambda+\lambda^2.\]
In order to associate a cocycle to a thermostat we consider the rank two quotient vector bundle \(E=TN/\mathbb{R}F\simeq \mathbb{R}H\oplus \mathbb{R}V\). Elements in \(E\) will be denoted by \([\xi]\), where \(\xi \in TN\). The mapping \(d\phi_t\) descends to define a mapping \[\rho : \mathbb{R}\times E \to E, \quad (t,[\xi])\mapsto \rho(t,[\xi])=[d\phi_t(\xi)]\] which satisfies \(\rho_{t_1}\circ\rho_{t_2}=\rho_{t_1+t_2}\) for all \(t_1,t_2 \in \mathbb{R}\). This is sometimes called the linear Poincaré flow. The basis of vector fields \((F,H,V)\) on \(N\) defines a vector bundle isomorphism \(TN \simeq N\times\mathbb{R}^3\) and consequently an identification \(E\simeq N \times \mathbb{R}^2\). Therefore, we obtain a cocycle \(\Psi : N \times \mathbb{R}\to \mathrm{GL}(2,\mathbb{R})\) over \(\phi\) by requiring that for each \(t \in \mathbb{R}\) and all \((x,a) \in E\), we have \[\rho_t((x,a))=(\phi_t(x),\Psi_t(x)a).\] Explicitly, \(\Psi_t\) is the linear map whose action on \(\mathbb{R}^2\) is \[\Psi_t(x): \left( \begin{array}{c} y(0) \\ \dot{y}(0) \end{array} \right) \mapsto \left( \begin{array}{c} y(t) \\ \dot{y}(t) \end{array} \right)\] with \[\ddot{y}(t)-(V\lambda)(\phi_{t}(x))\dot{y}(t) + \kappa(\phi_t(x)) y(t) = 0.\] Observe that for thermostats the \(2\)-plane bundle spanned by \(H\) and \(V\) is in general not invariant under \(d\phi_{t}\).
The cocycle \(\Psi_{t}\) is hyperbolic if and only if the thermostat flow \(\phi_t\) is Anosov (cf. for instance [33]). We will say that \(\phi_t\) admits a dominated splitting if \(\Psi_{t}\) admits a dominated splitting. This is the natural notion for flows, see [1]. For the case of flows on 3-manifolds, as it is our case, the existence of a dominated splitting can produce hyperbolicity if additional information on the closed orbits is available. Indeed [1] implies that if all closed orbits of \(\phi\) are hyperbolic saddles, then \(N=\Lambda\cup\mathcal T\) where \(\Lambda\) is a hyperbolic invariant set and \(\mathcal T\) consists of finitely many normally hyperbolic irrational tori.
Flows with dominated splitting are also called projectively Anosov flows. We note that when the flow \(\phi\) admits a dominated splitting we may write \(TN=\tilde{E}^{s}+\tilde{E}^{u}\), where \(\tilde{E}^{s,u}\) are continuous plane bundles invariant under \(d\phi_t\) and whose intersection is \(\mathbb{R}F\). In general they are integrable but unlike the Anosov case, they may not be uniquely integrable. Also note that the irrational tori in \(\mathcal T\) must be tangent to \(\tilde{E}^{s}\) or \(\tilde{E}^u\) due to the domination condition. We refer to [2] and references therein for a classification of these flows when the bundles \(\tilde{E}^{s,u}\) are of class \(C^2\) (in which case they do determine codimension one foliations of class \(C^2\)).
3.3 Infinitesimal generators and conjugate cocycles
For a smooth cocycle \(\Psi:N\times\mathbb{R}\to \mathrm{GL}(2,\mathbb{R})\), we define its infinitesimal generator \(\mathbb{B}:N\to \mathfrak{gl}(2,\mathbb{R})\) as follows \[\mathbb{B}(x):=-\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\Psi_{t}(x).\] The cocycle \(\Psi\) can be obtained from \(\mathbb{B}\) as the unique solution to \[\frac{\mathrm{d}}{\mathrm{d}t}\Psi_{t}(x)+\mathbb{B}(\phi_{t}(x))\Psi_{t}(x)=0,\;\;\;\Psi_{0}(x)=\mbox{\rm Id}.\] In the case of thermostats, it is easy to check that we have \[\mathbb{B}=\begin{pmatrix} 0 & -1 \\ \kappa & -V\lambda\end{pmatrix}\] where \(\kappa=K_g-H\lambda+\lambda^2\). Given a gauge, that is, a smooth map \(\mathcal{P}:N\to \mathrm{GL}(2,\mathbb{R})\), we obtain a new cocycle by conjugation \[\tilde{\Psi}_{t}(x)=\mathcal{P}^{-1}(\phi_{t}(x))\Psi_{t}(x)\mathcal{P}(x).\] It is straightforward to check that the infinitesimal generator \(\tilde{\mathbb{B}}\) of \(\tilde{\Psi}_{t}\) is related to \(\mathbb{B}\) by \[\tilde{\mathbb{B}}=\mathcal{P}^{-1}\mathbb{B}\mathcal{P}+\mathcal{P}^{-1}F\mathcal{P}. \tag{3.3}\]
Below we shall use gauges of a particular type. Consider a gauge transformation \(\mathcal{P}:N\to \mathrm{GL}(2,\mathbb{R})\) given by \[\mathcal{P}=\begin{pmatrix} 1 & 0 \\ p & 1\end{pmatrix},\] where \(p\) is a smooth real-valued function on \(N\). A computation using (3.3) shows that the conjugate cocyle \(\tilde{\Psi}_{t}\) via \(P\) has infinitesimal generator given by \[\tilde{\mathbb{B}}=\begin{pmatrix} -p & -1 \\ \kappa_{p} & -V\lambda+p\end{pmatrix},\] where \(\kappa_{p}:=\kappa+Fp+p(p-V\lambda)\). Since the cocycles \(\Psi_{t}\) and \(\tilde{\Psi}_{t}\) are conjugate, they have the same dominated splitting/hyperbolicity properties, but the form of \(\tilde{\mathbb B}\) will expose the origins of these properties when \(\kappa_{p}<0\) (cf. [35]). In both cases, the trace of the matrix is \(-V\lambda\) (minus divergence of \(F\)), giving an indication that \(F\) may not preserve volume.
3.4 Conditions ensuring domination and hyperbolicity
We have [25]:
Let \(N\) be a closed \(3\)-manifold that is equipped with a generalised Riemannian structure \((X,H,V)\) and a thermostat flow \(\phi\) generated by \(F=X+\lambda V\). Suppose there exists a smooth function \(p:N\to\mathbb{R}\) such that \[\kappa_{p}=\kappa+Fp+p(p-V\lambda)<0.\] Then \(\phi\) admits a dominated splitting with \(V\notin E^{s,u}\).
More precisely, in [25], only the case of a thermostat on the unit tangent bundle of an oriented Riemannian \(2\)-manifold \((M,g)\) is considered. However, it is easy to check that the arguments in [25] also prove Theorem 3.3. In [25] we employed quadratic forms to establish this result; we could have used instead a cone-field criterion as described for instance in [10].
The fact that \(V\notin E^{s,u}\) implies that there are uniquely defined continuous (Hölder in fact) functions \(r^{s,u}:N\to \mathbb{R}\) such that \(H+r^{s,u} V\in E^{s,u}\). The invariance of the bundles \(E^{s,u}\) translates into Riccati equations for \(r^{s,u}\) of the form: \[Fr+r^{2}-rV\lambda+\kappa=0.\] Observe that \(h:=r-p\) satisfies the Riccati equation \[\tag{3.4} Fh+h^{2}+h(2p-V\lambda)+\kappa_{p}=0.\] Moreover, the functions \(r^{u,s}\) can be constructed using a limiting procedure as follows. Fix \(x\in N\) and consider for each \(R>0\), the unique solution \(u_{R}\) to the Riccati equation along \(\phi_{t}(x)\) \[\dot{u}+u^2-uV\lambda+\kappa=0\] satisfying \(u_{R}(-R)=\infty\). Then \[\tag{3.5} r^{u}(x)=\lim_{R\to \infty}u_{R}(0).\] Note that \(r^{u}(\phi_{t}(x))=\lim_{R\to\infty}u_{R}(t)\).
Finally, under the assumption in Theorem 3.3 that \(\kappa_p<0\) we get the important additional information that \(h^{u}:=r^{u}-p>0\) and \(h^{s}:=r^{s}-p<0\). We call these the positive and negative Hopf solutions given that they play a similar role as the solutions introduced by E. Hopf in [18] for the geodesic flow.
The property \(V\notin E^{s,u}\) allows a convenient visualization of the domination condition in terms of the behaviour of solutions to the Riccati equation as depicted in Figure [fig:riccati].
domination (11.5,32.5)\(E^s\) (11.5,2.25)\(E^u\) (58,28)\(r^u\) (95,8)\(r^s\)
The reader might find this figure useful when following some of the arguments below, particularly the proof of Lemma 5.1. To prove that our flows are Anosov we shall use the following lemma that “upgrades" the domination condition to hyperbolicity under additional information on the solutions \(r^{s,u}\).
Under the same assumptions as in Theorem 3.3, suppose in addition that either
\(r^{u}>0\) and \(r^{s}<0\); or
\(V\lambda-p-\frac{\kappa_{p}}{r^{u}-p}>0\) and \(V\lambda-p-\frac{\kappa_{p}}{r^{s}-p}<0\).
Then \(\phi_t\) is Anosov.
Proof. We first consider (1). For a given initial condition \((y(0),\dot{y}(0))\in E^{u}\) we know that under the coycle \(\Phi_t\), we have \(\dot{y}=r^{u}y\). If \(r_u>0\) we can find a uniform constant \(\mu>0\) such that \(|y(-t)|\leqslant e^{-\mu t}|y(0)|\) for \(t>0\). This gives uniform exponential growth for \(\Psi_t\) on \(E^u\). Arguing with \(r^{s}<0\) we get uniform exponential contraction for \(\Psi_t\) on \(E^{s}\) thus showing that \(\Psi_t\) is hyperbolic.
Assume now condition (2) and consider a solution with initial conditions \((y(0),\dot{y}(0))\in E^{u}\). Then \(\dot{y}=r^{u}y\) and let \(z:=(r^{u}-p)y\) (recall that \(r^{u}-p>0\)). Then a calculation shows that \(\dot{z}=(V\lambda-p)z-\kappa_{p}y=(V\lambda-p-\frac{\kappa_{p}}{r^{u}-p})z\). 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.
In [25] we used condition (1) to prove that thermostat flows with \(\theta=0\) are Anosov when \(\ell\) is an integer \(\geq 1\). Remarkably, for the case of fractional differentials in the range \(0<\ell<1\), we will crucially need alternative (2).
While we shall not use the next proposition, it complements Theorem 3.3 quite nicely and it gives an indication of the importance of the property \(V\notin E^{s,u}\).
Suppose the thermostat determined by \(\lambda\) is such that \(\Psi_t\) admits a continuous invariant splitting \(E=E^u\oplus E^s\) with \(V\notin E^{u,s}\). Then the splitting is dominated and there exists a hyperbolic \(\mathrm{SL}(2,\mathbb{R})\)-cocycle \(\Psi^{hyp}_{t}\) such that \[\Psi_{t}=e^{\frac{1}{2}\int_{0}^{t}V\lambda}\;\Psi_{t}^{hyp}.\]
Proof. We know that the existence of a splitting with \(V\notin E^{u,s}\) gives rise to two continuous functions \(r^{u,s}:N\to\mathbb{R}\) satisfying the Riccati equation \[Fr+r^2-rV\lambda+\kappa=0.\] Moreover, \(r^u-r^s\neq 0\).
Recall that the infinitesimal generator for the cocycle \(\Psi_t\) is: \[\mathbb{B}=\begin{pmatrix} 0 & -1 \\ \kappa & -V\lambda\end{pmatrix}.\] Consider a gauge transformation \(\mathcal{P}:N\to \mathrm{GL}(2,\mathbb{R})\) given by \[\mathcal{P}=\begin{pmatrix} 1 & 0 \\ p & 1\end{pmatrix}\] with \(p=\frac{V\lambda}{2}\). Then the conjugate cocyle \(\tilde{\Psi}_{t}\) via \(\mathcal{P}\) has infinitesimal generator given by \[\tilde{\mathbb{B}}=-\frac{1}{2}V\lambda\;\text{\rm Id}+\begin{pmatrix} 0 & -1 \\ \kappa_{p} & 0\end{pmatrix}.\] To complete the proof we need to prove that the cocycle generated by \[\begin{pmatrix} 0 & -1 \\ \kappa_{p} & 0\end{pmatrix}\] is hyperbolic. Note that \(h^{u,s}:=r^{u,s}-p\) satisfies the Riccati equation \[Fh+h^{2}+\kappa_{p}=0.\] The quadratic form \[Q(a,b)=2ab-([h^{u}]^{2}+[h^{s}]^{2})a^2\] has the property that \[\dot{Q}=(b-h^{u}a)^{2}+(b-h^{s}a)^{2}>0\] unless \(a=b=0\). (Note that \(\dot{b}+\kappa_{p}a=0\) and \(\dot{a}=b\).) Now the hyperbolicity follows for instance from [35].
We do not know of any example of a thermostat as in Proposition 3.7 that is not Anosov.
3.5 Bi-contact structures
It is possible to recast the discussion of Subsection 3.4 in terms of the notion of bi-contact structure introduced by Eliashberg and Thurston [12] and further studied by Mitsumatsu [27] in the context of projective Anosov flows.
If \(N\) is a closed 3-manifold, we shall say that a bi-contact pair is a pair of contact forms \((\tau_{+},\tau_{-})\) such that \(\tau_{+}\wedge d\tau_{+}\) and \(\tau_{-}\wedge d\tau_{-}\) give rise to opposite orientations and \(\text{ker}\,\tau_{+}\cap \text{ker}\,\tau_{-}\) is 1-dimensional at every point. It turns out (cf. [12, 27]) that the flow of a non-zero vector field \(F\) has a dominated splitting (or is a projective Anosov flow) iff there is a bi-contact pair \((\tau_{+},\tau_{-})\) such that \(F\in \text{ker}\,\tau_{+}\cap \text{ker}\,\tau_{-}\).
Suppose now that \(N\) is endowed with a generalized Riemannian structure \((X,H,V)\) and \(\lambda,p\in C^{\infty}(N)\) are given functions. We consider a new frame \((F,H_{p},V)\), where \(F:=X+\lambda V\) and \(H_{p}=H+pV\). If we denote by \((\alpha,\beta,\psi)\) the co-frame dual to \((X,H,V)\), then a simple computation shows that \((\alpha,\beta,\tilde{\psi})\) is the co-frame dual to \((F,H_{p},V)\), where \[\tilde{\psi}=-\lambda\alpha-p\beta+\psi.\] Then we have:
The pair \((\beta,\tilde{\psi})\) is a bi-contact pair iff \(\kappa_{p}<0\).
We omit the proof of the lemma (which is a fairly straightforward computation), since we will not use it in subsequent sections. Since \(F\in \text{ker}\,\beta\cap \text{ker}\,\tilde{\psi}\) we see that with this lemma we essentially recover Theorem 3.3. The conditions appearing in Lemma 3.5 can now be rephrased in a more pleasing way in terms of the bi-contact pair \((\beta,\tilde{\psi})\). Indeed condition (1) is equivalent to \[d\beta(F,H+r^{u}V)>0\;\;\text{and}\;\;d\beta(F,H+r^{s}V)<0\] while condition (2) is equivalent to \[d\tilde{\psi}(F,H+r^{u}V)>0\;\;\text{and}\;\;d\tilde{\psi}(F,H+r^{s}V)<0.\] Again, we omit the verification of these equivalences as they will not be used in the sequel.
...