Smooth orbit equivalence rigidity for dissipative geodesic flows

As previously, let $X$ be the geodesic vector field on $SM$, and let $V$ be the vertical vector field generating the circle action on the fibers. We define $H:=[V,X]$. The vector fields $\{X,H,V\}$ form an orthonormal basis on $SM$ for the Sasaki metric (the natural lift of $g$ to $SM$). We set $\mathbb{H}:=\mathbb{R}H$ and $\mathbb{V}:=\mathbb{R}V$. We also note that the geodesic vector field splits into $X=\eta_{+}+\eta_{-}$ where $\eta_{\pm}$ are the raising and lowering Guillemin-Kazhdan operators given by

$$\eta_{\pm}:=\dfrac{1}{2}(X\mp iH).$$

(2.1)

The Liouville $1$-form $\alpha\in\mathcal{C}^{\infty}(SM,T^{*}(SM))$ is defined by $\alpha(X)=1$ and $\alpha(H)=\alpha(V)=0$. It is invariant by the geodesic flow in the sense that $\mathcal{L}_{X}\alpha=0$. The $2$-form $d\alpha$ is non-degenerate on the contact plane $\mathbb{H}\oplus\mathbb{V}$, and it satisfies $\iota_{X}d\alpha=0$. Hence

$$\mu:=-\alpha\wedge d\alpha$$

is a volume form invariant by the geodesic flow. We call it the Liouville volume form. It corresponds to the Riemannian volume form induced by the Sasaki metric on $SM$. From now on, the $L^{2}$ space on $SM$ is defined as $L^{2}(SM):=L^{2}(SM,\mu)$.

We also define the $1$-forms $\beta,\psi$ on $SM$ by $\beta(H)=1=\psi(V)$ and $\beta(X)=\beta(V)=0=\psi(X)=\psi(H)$. It is easy to check that $d\alpha=\psi\wedge\beta$ so that $\mu=\alpha\wedge\beta\wedge\psi$. We set $(\mathbb{R}X)^{\ast}:=\mathbb{R}\alpha$, $(\mathbb{R}F)^{\ast}:=\mathbb{R}\lambda$, $\mathbb{H}^{\ast}:=\mathbb{R}\beta$, and $\mathbb{V}^{\ast}:=\mathbb{R}\psi$. We refer to [PSU23, Chapter 3] for further details on the geometric structure on $SM$.

The key difference between generalized thermostats and geodesic or magnetic flows is that the generating vector field $F$ might not preserve the Liouville volume form $\mu$. Recall that the divergence of the vector field $F$ with respect to the volume form $\mu$ is the function $\mathrm{div}_{\mu}F\in\mathcal{C}^{\infty}(SM,\mathbb{R})$ uniquely defined by

$$\mathcal{L}_{F}\mu=(\mathrm{div}_{\mu}F)\mu.$$

The following result is proved in [DP07, Lemma 3.2].

In the geodesic and magnetic cases, we have $V(\lambda)=0$, so the Liouville volume form is preserved. Another way in which the divergence manifests itself is when calculating the adjoint operators with respect to the $L^{2}$ inner product on $SM$:

$$F^{*}=-(F+V(\lambda)),\qquad H^{*}=-H,\qquad V^{*}=-V.$$

This is relevant when extending differential operators to act on the space of distributions. Recall that any differential operator $P$ with smooth real-valued coefficients acts on a distribution $u\in\mathcal{D}^{\prime}(SM)$ by duality, that is $\langle Pu,\varphi\rangle_{\mathcal{D}^{\prime}(SM)}:=\langle u,P^{*}\varphi\rangle_{\mathcal{D}^{\prime}(SM)}$ for any $\varphi\in\mathcal{C}^{\infty}(SM)$. The subspace of distributional solutions to the transport equation

$$\mathcal{D}^{\prime}_{\text{tr}}(SM):=\{u\in\mathcal{D}^{\prime}(SM)\mid(F+V(\lambda))u=0\}$$

thus corresponds to the distributions $u\in\mathcal{D}^{\prime}(SM)$ such that $\langle u,F\varphi\rangle_{\mathcal{D}^{\prime}(SM)}=0$ for all $\varphi\in\mathcal{C}^{\infty}(SM)$. If $V(\lambda)=0$, these are simply the distributions invariant by the flow.

It will prove important to understand how the divergence of a system interacts with smooth orbit equivalences.

The next result, which we have stated in a broader setting than the one we are studying in this paper to highlight its generality, relates the divergences of two flows associated by a smooth orbit equivalence.

In the geodesic and magnetic cases, the pullback $\phi^{*}$ of the smooth orbit equivalence $\phi:\tilde{S}M\to SM$ sends the space $\mathcal{D}^{\prime}_{\text{tr}}(SM)$ to $\mathcal{D}^{\prime}_{\text{tr}}(\tilde{S}M)$. More generally, however, the divergence term $V(\lambda)$ appearing in the transport equation breaks this down.

Instead, a more useful perspective is to look at the following subspace of $2$-currents (or distributional $2$-forms) on $SM$ invariant by $F$:

$$\mathcal{F}(SM):=\{\sigma\in\mathcal{D}^{\prime}(SM,\Lambda^{2}T^{*}(SM))\mid\iota_{F}\sigma=d\sigma=0\}.$$

This set only depends on the foliation corresponding to $F$, i.e., it is invariant under time-changes, so we get a $\mathbb{C}$-linear isomorphism $\phi^{*}:\mathcal{F}(SM)\to\mathcal{F}(\tilde{S}M)$. The 2-form

$$\omega:=\iota_{F}\mu$$

(2.2)

then allows us to establish a relationship with solutions to the transport equation.

Thanks to this identification, we can now define a map $\Phi:\mathcal{D}^{\prime}_{\text{tr}}(SM)\to\mathcal{D}^{\prime}_{\text{tr}}(\tilde{S}M)$ associated to the smooth orbit equivalence $\phi:\tilde{S}M\to SM$ via the following diagram:

(2.3)

This point of view does not affect the wavefront set analysis.

By the properties of wavefront sets under pullback operators (see [Hö03, Theorem 8.2.4] for instance), we thus obtain

$$\textup{WF}(\Phi u)=d\phi^{T}(\textup{WF}(u))$$

for all $u\in\mathcal{D}^{\prime}_{\textup{tr}}(SM)$, where $d\phi^{T}:T^{*}(SM)\to T^{*}(\tilde{S}M)$ is the symplectic lift of $\phi^{-1}$ to the cotangent bundles given by

$$d\phi^{T}(y,\eta):=\left(\phi^{-1}(y),d\phi^{T}_{\phi^{-1}(y)}\eta\right),\quad(y,\eta)\in T^{*}(SM).$$

The space $\mathcal{C}^{\infty}(SM)$ breaks up as

$$\mathcal{C}^{\infty}(SM)=\oplus_{k\in\mathbb{Z}}\Omega_{k},\qquad\Omega_{k}:=\{u\in\mathcal{C}^{\infty}(SM)\mid\,Vu=iku\}.$$

This decomposition is orthogonal with respect to the $L^{2}$ inner product on $SM$ and with $\mathcal{C}^{\infty}$ being replaced by $L^{2}$. For any $u\in\mathcal{C}^{\infty}(SM)$, we shall write $u=\sum_{k\in\mathbb{Z}}u_{k}$, where each $u_{k}\in\Omega_{k}$ is given by

$$u_{k}(x,v):=\dfrac{1}{2\pi}\int_{0}^{2\pi}u(\rho_{t}(x,v))e^{-ikt}\,dt,$$

(2.4)

with $\rho_{t}$ being the flow generated by $V$. More generally, any distribution $u\in\mathcal{D}^{\prime}(SM)$ can be decomposed as $u=\sum_{k\in\mathbb{Z}}u_{k}$, where each $u_{k}\in\mathcal{D}^{\prime}(SM)$ is defined by

$$\langle u_{k},\varphi\rangle:=\langle u,\varphi_{-k}\rangle_{\mathcal{D}^{\prime}(SM)},\quad\varphi\in\mathcal{C}^{\infty}(SM),$$

and satisfies $Vu_{k}=iku_{k}$.

If a distribution on $SM$ only has finitely many non-trivial Fourier modes, we say that it has finite Fourier degree. The smallest $m\in\mathbb{N}$ such that $u_{k}=0$ for all $|k|>m$ is then called the Fourier degree of $u$.

It also worth noting that the ladder operators $\eta_{\pm}$ in (2.1) take their name from the fact that they act as raising/lowering operators on the Fourier decomposition, that is,

$$\eta_{\pm}:\Omega_{k}\to\Omega_{k\pm 1}$$

for all $k\in\mathbb{Z}$. In particular, we have $(Xu)_{k}=\eta_{-}u_{k+1}+\eta_{+}u_{k-1}$ for any $u\in\mathcal{D}^{\prime}(SM)$.

The Teichmüller space of $M$, denoted by $\mathcal{T}(M)$, is the space of complex structures on $M$ modulo the equivalence relation that $J\sim\tilde{J}$ if and only if there exists a diffeomorphism $\psi:M\to M$, isotopic to the identity, such that $\psi^{*}\tilde{J}=J$. We will denote such an equivalence class of complex structures by $[J]$.

The mapping class group $\text{MCG}(M)$ is defined as the quotient of orientation preserving diffeomorphisms on $M$ modulo isotopy. They act on $\mathcal{T}(M)$ by pullback, and the quotient space $\mathcal{M}(M):=\mathcal{T}(M)/\text{MCG}(M)$ is the moduli space of complex structures on $M$. See [FM11] for a thorough introduction.

Each complex structure $J$ determines a canonical line bundle $\kappa:=T^{\ast}_{1,0}M$ on $M$. We will denote by $H^{0}_{J}(M,\kappa^{\otimes k})$ the space of $J$-holomorphic sections of the $k$-th tensor power of the canonical line bundle $\kappa$. Locally, its elements have the form $w(z)dz^{k}$ for $k\geq 0$ and $w(z)d\bar{z}^{-k}$ for $k<0$.

Each subspace $\Omega_{k}$ of Fourier modes can be identified with $\mathcal{C}^{\infty}(M,\kappa^{\otimes k})$, the set of smooth sections of the bundle $\kappa^{\otimes k}$. Indeed, we have a $\mathbb{C}$-linear isomorphism

$$\pi_{k}^{\ast}:\mathcal{C}^{\infty}(M,\kappa^{\otimes k})\to\Omega_{k}$$

given by restriction to $SM$, i.e., in local coordinates (for $k\geq 0$),

$$\pi^{\ast}_{k}(wdz^{k})(x,v)=w(x)(dz(v))^{k}.$$

This is a generalization of the map $\pi_{1}^{*}$ which we have already encountered in (1.2) to identify smooth $1$-forms with $\Omega_{-1}\oplus\Omega_{1}$. Note that the definition of $\pi_{k}^{\ast}$ depends on the choice of the metric $g$. We denote by $\pi_{k\ast}$ its $L^{2}$-adjoint. In local coordinates, we have

$$(\pi_{k\ast}u)(x)=\left(\int_{\pi^{-1}(x)}u(x,\cdot)\right)dz^{k}.$$

Once we extend the operators to distributions by duality, the projection onto the $k$-th Fourier mode is simply given by $(2\pi)^{-1}\pi^{\ast}_{k}\pi_{k\ast}$ acting on $\mathcal{D}^{\prime}(SM)$.

Under this identification, we can essentially think of the raising/lowering operators $\eta_{\pm}$ as $\bar{\partial}$ and $\partial$ operators thanks to the following result (see [PSU14, Lemma 2.1] and the ensuing discussion).

As a result, for $k\geq 0$, the operator $\pi_{k}^{*}$ gives us an identification

$$H^{0}_{J}(M,\kappa^{\otimes k})\cong\Omega_{k}\cap\ker\eta_{-}.$$

We also introduce the following terminology:

Equivalently, if we define the Szegö projectors $S_{\pm}:\mathcal{D}^{\prime}(SM)\to\mathcal{D}^{\prime}(SM)$ by

$$S_{+}u=\sum_{k\geq 0}u_{k},\qquad S_{-}u=\sum_{k\leq 0}u_{k},$$

then a distribution $u$ is fiberwise holomorphic if and only if $S_{+}u=u$. The projectors satisfy the commutation relations

$$[S_{+},X]u=\eta_{+}u_{-1}-\eta_{-}u_{0},\qquad[S_{-},X]u=\eta_{-}u_{1}-\eta_{+}u_{0}.$$

(2.5)

We will be interested in the family of fiberwise holomorphic distributions that satisfy the transport equation:

$$\mathcal{D}^{\prime}_{\text{tr},+}(SM):=\{u\in\mathcal{D}^{\prime}(SM)\mid(F+V(\lambda))u=0,\,S_{+}u=u\}.$$

(2.6)

The complex vector space $H^{0}_{J}(M,\kappa)$ of $J$-holomorphic $1$-forms has the same dimension as the genus of $M$ (see [FK92, Proposition III.2.7]). Given a canonical basis $\{a_{j},b_{j}\}$ of the homology $H_{1}(M;\mathbb{Z})$ on $M$, the following result gives us the existence of a useful basis (see [FK92, Proposition, p. 63]).

The space of symmetric matrices with positive definite imaginary part and size given by the genus of $M$ is called the Siegel upper half-space $\mathcal{H}(M)$. We thus get a well-defined period matrix map

$$\Pi:\mathcal{T}(M)\to\mathcal{H}(M).$$

The following form of Torelli’s theorem tells us that period matrices capture a lot of the information about the complex structure.

We refer to [FK92, Theorem III.12.3] for a proof.

Recall that the Anosov property means that there exists a flow-invariant continuous splitting

$$T(SM)=\mathbb{R}F\oplus E^{s}\oplus E^{u}$$

and uniform constants $C\geq 1$ and $0<\rho<1$ such that for all $t\geq 0$ we have

$$\|d\varphi_{t}|_{E^{s}}\|\leq C\rho^{t},\qquad\|d\varphi_{-t}|_{E^{u}}\|\leq C\rho^{t}.$$

(2.8)

In the geodesic case, the contact form $\alpha$ is preserved, so $\ker\alpha=\mathbb{H}\oplus\mathbb{V}=E^{s}\oplus E^{u}$. It is then known that $E^{s}\cap\mathbb{V}=\{0\}=E^{u}\cap\mathbb{V}$. For a generalized thermostat, we instead know by [DP07, Lemma 4.1] that

$$(\mathbb{R}F\oplus E^{s})\cap\mathbb{V}=\{0\}=(\mathbb{R}F\oplus E^{u})\cap\mathbb{V}.$$

(2.9)

Here $\mathbb{R}F\oplus E^{s/u}$ are the weak stable and unstable bundles. This implies that there exist $r^{s/u}\in\mathcal{C}^{0}(SM,\mathbb{R})$ such that

$$Y^{s}:=H+r^{s}V\in\mathbb{R}F\oplus E^{s},\qquad Y^{u}:=H+r^{u}V\in\mathbb{R}F\oplus E^{u}.$$

(2.10)

In fact, the weak stable and unstable bundles are $\mathcal{C}^{1}$ (see [Has94, Corollary 1.8]), so the functions $r^{s/u}$ are also $\mathcal{C}^{1}$ (and smooth along the flow since each bundle $\mathbb{R}F\oplus E^{s/u}$ is $\varphi_{t}$-invariant). The Anosov property implies that $r^{s}\neq r^{u}$ everywhere. One may in fact show that $r^{s}<r^{u}$, so the basis $\{F,Y^{s},Y^{u}\}$ is positively-oriented.

The dual bundles are defined by

$$(\mathbb{R}F)^{*}(E^{s}\oplus E^{u})=0=(E^{s})^{\ast}(\mathbb{R}F\oplus E^{s})=(E^{u})^{\ast}(\mathbb{R}F\oplus E^{u}).$$

One can check we have similar estimates to (2.8) for $(E^{s})^{*}$ and $(E^{u})^{*}$, with $d\varphi_{t}$ replaced by $d\varphi_{t}^{-T}$ (inverse transpose). Translated to the setting of the cotangent bundle, property (2.9) then becomes

$$(E^{s})^{\ast}\cap\mathbb{H}^{*}=\{0\}=(E^{u})^{\ast}\cap\mathbb{H}^{\ast}.$$

(2.11)

Further note that

$$(E^{s})^{*}\oplus(E^{u})^{*}=\Sigma,$$

where

$$\Sigma:=\{(y,\eta)\in T^{*}(SM)\mid\eta(F(y))=0\}$$

is the characteristic set of the operator $F$ (usually defined without the zero section).

The tensor tomography problem is interesting in its own right, particularly as it pertains to the injectivity of the X-ray transform for thermostats. We will need the following property in the case $n=1$.

The term ‘attenuated’ refers to the presence of the divergence $V(\lambda)$ in the transport equation. Note that such a term appears for Gaussian thermostats and generalized thermostats of higher Fourier degree, but not for magnetic or geodesic flows.

The fact that geodesic flows satisfy the (attenuated) tensor tomography problem was first proved in negative curvature in [GK80] for $n\geq 0$ and then generalized to the Anosov case in [DS03] for $n\leq 1$, [PSU14] for $n\leq 2$, and [Gui17] for $n\geq 2$. It was also shown in [DP05] that Anosov magnetic flows satisfy the (attenuated) tensor tomography problem of order $n\leq 1$.