2.1 Notation

If $H=C^k,H^s,C^{-\infty }$ etc. is a regularity scale and $E\to M$ a smooth bundle over a smooth compact manifold M, we will use the notation $H(M;E)$ for sections of E with regularity H, while if N is a smooth manifold, we use the notation $H(M,N)$ for the space of maps from M to N with regularity H.

2.2 Microlocal calculus

On a closed manifold M, we will denote by $\Psi ^{m}(M;V)$ the space of classical pseudo-differential operators of order $m\in \mathbb {R}$ acting on a vector bundle V over M (see [Reference Grigis and SjöstrandGS94]; the operators could map sections of two distinct vector bundles, but this will not be needed here). We recall that for fixed $m \in \mathbb {R}$ , this is a Fréchet space: indeed, using a fixed smooth cutoff function $\theta $ supported in a small neighborhood of the diagonal, a fixed system of charts, each $A\in \Psi ^{m}(M;V)$ has Schwartz kernel $\kappa _A$ that can be decomposed as $\theta \kappa _A+(1-\theta )\kappa _A$ . For the $(1-\theta )\kappa _A$ part we can use the $C^\infty (M\times M; V\otimes V^*)$ topology, while for $\chi \kappa _A$ we can use the semi-norms of the full symbols of $\chi \kappa _A$ using the local charts and the left quantization in the charts. We also denote by $H^{s}(M)$ the $L^2$ -based Sobolev space of order $s\in \mathbb {R}$ , with norm given by fixing an arbitrary Riemannian metric $g_{0}$ on M. More precisely, denoting by $\Delta $ the non-negative Laplacian associated to this metric, we define

and $H^s(M)$ is the completion of $C^\infty (M)$ with respect to this norm. This definition is naturally extended to sections of vector bundles. What is important is that the spaces and the norm (up to a scaling factor) do not depend on the choice of metric $g_{0}$ . For $k\in \mathbb {N}, \alpha \in (0,1)$ , the spaces $C^{k,\alpha }(M)$ are the usual Hölder spaces and $\mathcal {D}^{\prime }(M)$ will denote the space of distributions dual to $C^\infty (M)$ . We will denote by $\langle \cdot ,\cdot \rangle _{L^2}$ the continuous extension of the pairing

$$ \begin{align*} C^\infty(M) \times C^\infty(M) \ni (f,f^{\prime}) \mapsto \int_{M}f\bar{f}^{\prime} d{\textrm{vol}}_{g_0}, \end{align*} $$

to the pairing $H^{s}(M) \times H^{-s}(M)\to \mathbb {C}$ for each $s\in \mathbb {R}$ (and likewise for sections of bundles).

2.3 Symmetric tensors and X-ray transform

In this subsection, we assume that the metric g is fixed and that its geodesic flow $\varphi ^g_t$ is Anosov on the unit tangent bundle $SM$ of g. We denote by $\mu ^{\textrm{L}}$ the Liouville measure, normalized to be a probability measure on $SM$ . For the sake of simplicity, we drop the index g in the notation. Given an integer $m \in \mathbb {N}$ , we denote by $\otimes ^m T^*M \rightarrow M$ , $S^mT^*M \rightarrow M$ the respective vector bundle of m-tensors and symmetric m-tensors on M. Given $f \in C^\infty (M;S^m T^*M)$ , we denote by $\pi _m^*f \in C^\infty (SM)$ the canonical morphism $\pi _m^*f : (x,v) \mapsto f_x(v,\ldots ,v)$ . We also introduce the trace operator $\operatorname {\textrm{Tr}} : C^\infty (M;S^{m+2} T^*M) \rightarrow C^\infty (M;S^{m} T^*M)$ defined pointwise in $x \in M$ by

$$ \begin{align*} \operatorname{\textrm{Tr}}(f) = \sum_{i=1}^n f(\textbf{e}_i,\textbf{e}_i, \cdot, \ldots, \cdot), \end{align*} $$

where $(\textbf{e}_1, \ldots , \textbf{e}_n)$ denotes an orthonormal basis of $TM$ in a neighborhood of a fixed point $x_0 \in M$ . Observe that, for $f = \sum _{i,j=1}^n f_{ij} \textbf{e}_i^* \otimes \textbf{e}_j^* \in C^\infty (M;S^2T^*M)$ defined around $x_0$ , we have

$$ \begin{align*} \begin{aligned} \int_{SM} \pi_2^*f\, d\mu_{g}^L &= \int_{M} \bigg(\int_{S_x M} \pi_2^*f(x,v)\, dS_x(v) \bigg)d{\textrm{vol}}(x)\\ & = \sum_{i,j=1}^n \int_{M} f_{ij}(x) \bigg( \int_{S_x M} v_i v_j\, dS_x(v) \bigg)d{\textrm{vol}}(x) \\ & = C_n \sum_{i=1}^n \int_{M} f_{ii}(x) d{\textrm{vol}}(x) = C_n \int_{M} \operatorname{\textrm{Tr}}_{g}(f) d{\textrm{vol}}, \end{aligned} \end{align*} $$

for some constant $C_n = \int _{\mathbb {S}^{n-1}} v_1^2 dv$ depending on $n=\dim M$ . This justifies the claim that the trace-free condition (1.2) was equivalent to the fact that the pullback of the symmetric tensor to $SM$ was of average $0$ .

The natural derivation of symmetric tensors is $D := \sigma \circ \nabla $ , where $\nabla $ is the Levi-Civita connection and $\sigma : \otimes ^m T^*M \rightarrow S^m T^*M$ is the operation of symmetrization. This operator satisfies the important identity

(2.1) $$ \begin{align} X \pi_m^* = \pi_{m+1}^* D, \end{align} $$

where X denotes the geodesic vector field on $SM$ . The operator D is elliptic [Reference Goüezel and LefeuvreGL, Lemma 2.4] with trivial kernel when m is odd and one-dimensional kernel when m is even, given by the Killing tensors $c\sigma (g^{\otimes m/2}), c \in \mathbb {R}$ (this is a simple consequence of (2.1) combined with the fact that the geodesic flow is ergodic in the Anosov setting). We denote by $\langle \cdot , \cdot \rangle $ the scalar product on $C^\infty (M;S^mT^*M)$ induced by the metric g (see [Reference Goüezel and LefeuvreGL, §2] for further details). The formal adjoint of D with respect to this scalar product is $D^*=-\operatorname {\textrm{Tr}}\circ \nabla $ . We also denote by the same $\langle \cdot , \cdot \rangle $ the natural $L^2$ scalar product on $C^\infty (SM)$ induced by the Liouville measure $\mu ^{{\textrm{L}}}$ . The formal adjoint of $\pi _m^*$ with respect to these two scalar products is denoted by

$$ \begin{align*} {\pi_m}_*: \mathcal{D}^{\prime}(SM)\to \mathcal{D}^{\prime}(M; S^mT^*M), \end{align*} $$

where $\mathcal {D}^{\prime }$ denotes the space of distributions, dual to $C^\infty $ .

We recall that $\mathcal {C}$ , the set of free homotopy classes in M, is in one-to-one correspondence with the set of conjugacy classes of $\pi _1(M,x_0)$ for some arbitrary choice of $x_0 \in M$ (see [Reference KlingenbergKli74]) and for each $c \in \mathcal {C}$ there exists a unique closed geodesic $\gamma (c) \in c$ . We denote its Riemannian length with respect to g by $L(c)=\ell _g(\gamma (c))$ . The X-ray transform on $SM$ is the operator defined by

$$ \begin{align*} I : C^0(SM) \rightarrow \ell^\infty(\mathcal{C}), \quad If(c) = \dfrac{1}{L(c)} \int_0^{L(c)} f(\varphi_t(z))\,dt, \end{align*} $$

where $z \in \gamma (c)$ is any point. This is a continuous linear operator when $\ell ^\infty (\mathcal {C})$ is endowed with the sup norm on the sequences. Then the X-ray transform $I_m$ of symmetric m-tensors is simply defined by $I_m := I \circ \pi _m^*$ . Using (2.1), we immediately have

(2.2) $$ \begin{align} \{ Dp ~|~ p \in C^\infty(M;S^{m-1}T^*M) \} \subset \ker I_m \cap C^\infty(M;S^{m}T^*M). \end{align} $$

Using the ellipticity of D, any tensor $f \in C^\infty (M; S^m T^*M)$ can be decomposed uniquely as a sum

(2.3) $$ \begin{align} f=Dp+h, \end{align} $$

with $p \in C^\infty (M;S^{m-1}T^*M)$ and $h \in C^\infty (M;S^{m}T^*M)$ is such that $D^*h=0$ . We call $Dp$ the potential part of f and h the solenoidal part. The same decomposition holds in Sobolev regularity $H^s(M)$ , $s \in \mathbb {R}$ , and in $C^{k,\alpha }(M)$ regularity, $k \in \mathbb {N}, \alpha \in (0,1)$ . We will write $h = \pi _{\ker D^*}f$ and the solenoidal projection is a pseudodifferential operator of order $0$ [Reference Goüezel and LefeuvreGL, Lemma 2.6] (here $\Delta _g := D^*_g D_g$ is the Laplacian on $1$ -forms). The X-ray transform is said to be solenoidal injective (or s-injective for short) if (2.2) is an equality. It is conjectured that $I_m$ is s-injective as long as the metric is Anosov, but it is only known in the following cases:

• • for $m=0,1$ [Reference Dairbekov and SharafutdinovDS10];

• • for any $m \in \mathbb {N}$ in dimension $2$ [Reference GuillarmouGui17, Reference Paternain, Salo and UhlmannPSU14];

• • for any $m \in \mathbb {N}$ , in any dimension in non-positive curvature [Reference Croke and SharafutdinovCS98].

It is also known that $\ker I_m/\operatorname {\textrm{ran}} D$ is finite-dimensional for general Anosov geodesic flow (see [Reference Dairbekov and SharafutdinovDS03, Theorem 1.5] or [Reference GuillarmouGui17, Remark 3.7]).

The direct study of the analytic properties of $I_m$ is difficult as this operator involves integrals over the set of closed orbits, which is not a manifold. Nevertheless, in [Reference GuillarmouGui17], the second author introduced an operator $\Pi _m$ that involves a sort of integration of tensors over ‘all orbits’, and this space is essentially the manifold $SM$ . The construction of $\Pi _m: C^\infty (M;S^mT^*M)\to \mathcal {D}^{\prime }(M;S^mT^*M)$ relies on microlocal tools coming from [Reference Dyatlov and ZworskiDZ16, Reference Faure and SjöstrandFS11], but a simpler definition that uses the fast mixing of the flow $\varphi _t$ is given by

(2.4) $$ \begin{align} \begin{aligned} \Pi_m:= {\pi_m}_*(\Pi+\langle \cdot, 1\rangle) \pi_m^* \quad \textrm{with}{\kern6pc} \\ \Pi: C^\infty(SM)\to \mathcal{D}^{\prime}(SM), \quad \langle \Pi f,f^{\prime}\rangle:=\lim_{T\to \infty}\int_{-T}^T \langle e^{tX}f,f^{\prime}\rangle \, dt \end{aligned}\end{align} $$

if $\langle f,1\rangle =\int _{SM} f \, d\mu ^{\textrm{L}}=0$ and $\Pi (1):=0$ . The convergence of the integral as $T\to \infty $ is ensured by the exponential decay of correlations [Reference LiveraniLiv04] (but also follows from the existence of the variance [Reference Katsuda and SunadaKS90]). We can thus write, for $\langle f,1\rangle =0$ ,

$$ \begin{align*}\langle \Pi f,f^{\prime}\rangle =\int_{\mathbb{R}} \langle f\circ \varphi_t,f^{\prime}\rangle_{L^2(SM)}\, dt.\end{align*} $$

We note the following useful properties of $\Pi $ , proved in [Reference GuillarmouGui17, Theorem 1.1]:

• • $\Pi : H^s(SM)\to H^{-s}(SM)$ is bounded for all $s>0$ ;

• • if $f\in H^s(SM)$ with $s>0$ , then $X\Pi f=0$ ;

• • if f and $Xf$ belong to $H^s(SM)$ for $s>0$ , then $\Pi Xf=0$ . (In [Reference GuillarmouGui17], f is assumed to be in $H^{s+1}(SM)$ , but one can reduce to the case $f\in H^s(SM)$ by using a density argument and [Reference Dyatlov and ZworskiDZ19, Lemma E.45].)

As is well known (see; for example; [Reference Katsuda and SunadaKS90, Proof of Proposition 1.2.]), we can make a link between $\Pi $ and the variance in the central limit theorem for Anosov geodesic flows. Let us quickly explain this fact by using the fast mixing of the flow. The variance of $\varphi _t$ with respect to the Liouville measure $\mu ^{\textrm{L}}$ is defined for $u\in C^\alpha (SM), \alpha \in (0,1)$ real-valued, by

(2.5) $$ \begin{align} {\textrm{Var}}_{\mu^{\textrm{L}}}(u):=\lim_{T\to \infty}\frac{1}{T}\int_{SM}\bigg(\int_0^Tu(\varphi_t(z)) \, dt\bigg)^2d\mu^{\textrm{L}}(z), \end{align} $$

under the condition that $\int _{SM}u\, d\mu ^{\textrm{L}}=0$ . We observe, since $\varphi _t$ preserves $\mu ^{\textrm{L}}$ , that

$$ \begin{align*} \begin{aligned} {\textrm{Var}}_{\mu^{\textrm{L}}}(u)=& \lim_{T\to \infty}\frac{1}{T}\int_{SM}\int_0^T\int_{0}^T u(\varphi_{t-s}(z))u(z) \, dtds d\mu^{\textrm{L}}(z)\\ =& \lim_{T\to \infty} \int_0^1\int_{\mathbb{R}}{\textbf{1}}_{[(t-1)T,tT]}(r)\langle u\circ \varphi_r,u\rangle_{L^2} \, dr\, dt, \end{aligned}\end{align*} $$

where the $L^2$ pairing is with respect to $\mu ^{\textrm{L}}$ . By exponential decay of correlations [Reference LiveraniLiv04], we have, for $|r|$ large,

$$ \begin{align*}|\langle u\circ \varphi_r,u\rangle_{L^2} |\leq Ce^{-\nu |r|}\|u\|_{C^\alpha}^2\end{align*} $$

for some $\alpha>0,\nu >0$ , $C>0$ independent of u. Thus, by the Lebesgue theorem,

(2.6) $$ \begin{align} {\textrm{Var}}_{\mu^{\textrm{L}}}(u)=\langle \Pi u,u\rangle, \end{align} $$

if $\langle u,\textbf{1}\rangle =0$ , where $\textbf{1}$ denotes the constant function equal to $1$ , showing that the quadratic form associated to our operator $\Pi $ is nothing more than the variance. For a symmetric $2$ -tensor h satisfying $\langle h,g\rangle _{L^2}=\int _{M}{\textrm{Tr}}_{g}(h)\, d{\textrm{vol}}_{g}=0$ , we have $\int _{SM}\pi _2^*h \, d\mu _{g}^L=0$ and

$$ \begin{align*} \langle\Pi_2h,h\rangle =\langle \Pi \pi_2^*h,\pi_2^*h\rangle={\textrm{Var}}_{\mu^{\textrm{L}}}(\pi_2^*h).\end{align*} $$

We have the following properties for $\Pi _m$ .

• • $\Pi _m$ is a positive self-adjoint pseudodifferential operator of order $-1$ , elliptic on solenoidal tensors; see [Reference GuillarmouGui17, Theorem 3.5] and [Reference Goüezel and LefeuvreGL, Lemma 4.3].

• • $\Pi _mD = 0$ and $D^*\Pi _m=0$ (by [Reference GuillarmouGui17, Theorem 3.5] and $X\pi _{m-1}^*= \pi _m^*D$ ).

• • If $I_m$ is s-injective, then $\Pi _m$ is invertible on solenoidal tensors in the sense that there exists a pseudodifferential operator Q of order $1$ such that $Q\Pi _m=\pi _{\ker D^*}$ ; see [Reference Goüezel and LefeuvreGL, Theorem 4.7].

• • Conversely, if $\Pi _m|_{\ker D^*}$ is injective, then $I_m$ is s-injective. Indeed, by [Reference GuillarmouGui17, Corollary 2.8], if $I_mh=0$ then $\pi _m^*h=Xu$ for some $u\in C^\infty (SM)$ and thus $\Pi _mh={\pi _m}_*\Pi Xu=0$ .

In particular, using the spectral theorem, there is a bounded self-adjoint operator $\sqrt {\Pi _m}$ on $L^2$ such that $\sqrt {\Pi _m}\sqrt {\Pi _m}=\Pi _m$ . We add the following property, the use of which will be crucial in this paper.

Lemma 2.1. If $(M,g)$ has Anosov geodesic flow and $I_2$ is s-injective, there exists a constant $C> 0$ such that, for all tensors $h \in H^{-1/2}(M; S^2T^*M)$ ,

$$ \begin{align*}\langle \Pi_2 h, h \rangle \geq C\|\pi_{\ker D^*}h\|^2_{H^{-1/2}(M)}.\end{align*} $$

Proof In [Reference Goüezel and LefeuvreGL, Theorem 4.4 and Lemma 2.2], the principal symbol of $\Pi _2$ was computed and turned out to be

$$ \begin{align*} \sigma_2 := \sigma(\Pi_2) : (x,\xi) \mapsto |\xi|^{-1} \pi_{\ker i_\xi}A_2^2\pi_{\ker i_\xi}, \end{align*} $$

for some positive definite diagonal endomorphism $A_2$ which is constant on both subspaces $S^2_0T^*M:=\{h\in S^2T^*M |\, {\textrm{Tr}}_g(h)=0\}$ and $\mathbb {R} g=\{\unicode{x3bb} g\in S^2T^*M\, |\, \unicode{x3bb} \in \mathbb {R}\}$ . Here $i_\xi $ is the interior product with the dual vector $\xi ^\sharp \in T_xM$ of $\xi $ with respect to the metric. We introduce the symbol $b\in C^\infty (T^*M)$ of order $-1/2$ defined by $b :(x,\xi ) \mapsto \chi (x,\xi )|\xi |^{-1/2}A_2$ , where $\chi \in C^\infty (T^*M)$ vanishes near the $0$ section in $T^*M$ and is equal to $1$ for $|\xi |>1$ , and define $B := \operatorname {\textrm{Op}}(b)\in \Psi ^{-1/2}(M;S^2T^*M)$ , where $\operatorname {\textrm{Op}}$ is a quantization on M. Using that the principal symbol of $\pi _{\ker D^*}$ is $\pi _{\ker i_\xi }$ (see [Reference Goüezel and LefeuvreGL, Lemma 2.6]), we observe that $\Pi _2 = \pi _{\ker D^*}B^*B\pi _{\ker D^*} + R$ , where $R\in \Psi ^{-2}(M;S^2T^*M)$ . Thus, given $h \in H^{-1/2}(M,S^2 T^*M)$ ,

(2.7) $$ \begin{align} \langle \Pi_2h, h \rangle_{L^2} = \|B\pi_{\ker D^*}h\|^2_{L^2} + \langle Rh, h \rangle_{L^2}. \end{align} $$

By ellipticity of B, there exists a pseudodifferential operator Q of order $1/2$ such that $QB\pi _{\ker D^*} = \pi _{\ker D^*} + R^{\prime }$ , where $R^{\prime }\in \Psi ^{-\infty }(M;S^2T^*M)$ is smoothing. Thus there is $C>0$ such that, for each $h\in C^\infty (M;S^2T^*M)$ ,

$$ \begin{align*} \|\pi_{\ker D^*}h\|^2_{H^{-1/2}} \leq \|QB\pi_{\ker D^*}h\|_{H^{-1/2}}^2 + \|R^{\prime}h\|_{H^{-1/2}}^2 \leq C \|B\pi_{\ker D^*}h\|^2_{L^2} + \|R^{\prime}h\|_{H^{-1/2}}^2. \end{align*} $$

Since Lemma 2.1 is trivial on potential tensors, we can already assume that h is solenoidal, that is, $\pi _{\ker D^*}h=h$ . Recalling (2.7), we obtain that

(2.8) $$ \begin{align} \begin{aligned} \|h\|^2_{H^{-1/2}} & \leq C\langle \Pi_2 h, h \rangle_{L^2} - C\langle Rh, h\rangle_{L^2} + \|R^{\prime}h\|_{H^{-1/2}}^2\\ & \leq C\langle \Pi_2 h, h \rangle_{L^2} + C\|Rh\|_{H^{1/2}}\|h\|_{H^{-1/2}} + \|R^{\prime}h\|_{H^{-1/2}}^2. \end{aligned} \end{align} $$

Now, assume by contradiction that the statement in Lemma 2.1 does not hold, that is, we can find a sequence of tensors $f_n \in C^\infty (M;S^2 T^*M)$ such that $\|f_n\|_{H^{-1/2}}=1$ with $D^*f_n=0$ and

$$ \begin{align*} \|\sqrt{\Pi_2}f_n\|^2_{L^2} =\langle \Pi_2 f_n, f_n \rangle_{L^2} \leq \frac{1}{n}\| f_n\|^2_{H^{-1/2}}=\frac{1}{n} \rightarrow 0. \end{align*} $$

Up to a subsequence, and since R is of order $-2$ , we can assume that $Rf_n \rightarrow v_1$ in $H^{1/2}$ for some $v_1$ , and $R^{\prime }f_n \rightarrow v_2$ in $H^{-1/2}$ . Then, using (2.8), we obtain that $(f_n)_{n \in \mathbb {N}}$ is a Cauchy sequence in $H^{-1/2}$ which thus converges to an element $v_3 \in H^{-1/2}$ such that $\|v_3\|_{H^{-1/2}}=1$ and $D^*v_3=0$ . By continuity, $\Pi _2f_n \rightarrow \Pi _2v_3$ in $H^{1/2}$ and thus $\langle \Pi _2v_3,v_3 \rangle =0$ . Since $v_3$ is solenoidal, we get $\sqrt {\Pi _2}v_3=0$ , thus $\Pi _2v_3=0$ . Since we assumed $I_2$ s-injective, $\Pi _2$ is also injective by [Reference Goüezel and LefeuvreGL, Lemma 4.6]. This implies that $v_3 \equiv 0$ , thus contradicting $\|v_3\|_{H^{-1/2}}=1$ .

We note that the same proof also works for tensors of any order $m \in \mathbb {N}$ . In fact we can even get a uniform estimate.

Lemma 2.2. Let $(M,g_0)$ be a smooth compact Anosov Riemannian manifold with $I_2^{g_0}$ being s-injective. There exist a $C^\infty $ neighborhood $\mathcal {U}_{g_0}$ of $g_0$ and a constant $C> 0$ such that for all $g\in \mathcal {U}_{g_0}$ and all tensors $h \in H^{-1/2}(M; S^2T^*M)$ ,

$$ \begin{align*}\langle \Pi_2^g h, h \rangle_{L^2} \geq C\|\pi_{\ker D_g^*}h\|^2_{H^{-1/2}(M)}.\end{align*} $$

Proof First, let $g_0$ be fixed Anosov metric with $I_2^{g_0}$ s-injective (in particular, it is the case if it has non-positive curvature). Proposition 4.1 (which will be proved later) shows that the operator $\Pi _2=\Pi _2^g$ is a continuous family as a map

$$ \begin{align*} g\in \mathcal{U}_{g_0} \mapsto \Pi_2^g\in \Psi^{-1}(M;S^2T^*M)\end{align*} $$

where $\mathcal {U}_{g_0}\subset C^\infty (M;S^2T^*M)$ is a $C^\infty $ -neighborhood of $g_0$ and $\Psi ^{-1}(M;S^2T^*M)$ is equipped with its Fréchet topology as explained before. Let $h \in \ker D^*_g$ be a solenoidal (with respect to g) symmetric $2$ -tensor, then $h=\pi _{\ker D^*_g}h$ . Let $C_{g_0}> 0$ be the constant provided by Lemma 2.1 applied to the metric $g_0$ . We choose $\mathcal {U}_{g_0}$ small enough so that $\|\Pi ^g_2-\Pi ^{g_0}_2\|_{H^{-1/2}\rightarrow H^{1/2}} \leq C_{g_0}/3$ (this is made possible by the continuity of $g \mapsto \Pi ^g_2 \in \Psi ^{-1}$ ). Then

$$ \begin{align*} \langle \Pi^g_2 h,h \rangle = \langle (\Pi^g_2-\Pi^{g_0}_2)h,h \rangle + \langle \Pi^{g_0}_2 h,h\rangle \geq C_{g_0} \|\pi_{\ker D^*_{g_0}} h\|_{H^{-1/2}}^2 - C_{g_0}/3 \times \|h\|_{H^{-1/2}}^2. \end{align*} $$

But the map is continuous: this follows from the fact that one can construct a full parametrix $Q_g\in \Psi ^{-2}(M)$ of $\Delta _g$ modulo smoothing in a continuous way with respect to g (by standard elliptic microlocal analysis), the fact that $\Delta _g$ is injective since $\ker D_g=0$ for g Anosov (as $D_gu=0$ implies $X\pi _1^*u=0$ , thus $\pi _1^*u$ has to be constant, thus $0$ since $\pi _1^*u(x,-v)=-\pi _1^*u(x,v)$ ) and the continuity of composition of pseudodifferential operators. This implies that for g in a possibly smaller neighborhood $\mathcal {U}_{g_0}$ of $g_0$ , using $h=\pi _{\ker D^*_g}h$ ,

$$ \begin{align*} \langle \Pi^g_2 h,h \rangle \geq C_{g_0} \|\pi_{\ker D^*_{g}} h\|_{H^{-1/2}}^2 - \frac{2C_{g_0}}{3} \times \|h\|_{H^{-1/2}}^2 = \frac{C_{g_0}}{3} \|\pi_{\ker D^*_{g}} h\|_{H^{-1/2}}^2. \end{align*} $$

The proof is complete.

We also observe that the generalization of the previous lemma to tensors of any order is straightforward. As mentioned earlier, an immediate consequence of the previous lemma is the following proposition.

2.4 The space of Riemannian metrics

We fix a smooth metric $g_0 \in \mathcal {M}$ and consider an integer $k \geq 2$ and $\alpha \in (0,1)$ . We recall that the space $\mathcal {M}$ of all smooth metrics is a Fréchet manifold. We denote by $\mathcal {D}_0 := \operatorname {\textrm{Diff}}_0(M)$ the group of smooth diffeomorphisms on M that are isotopic to the identity; this is a Fréchet Lie group in the sense of [Reference HamiltonHam82, Section 4.6]. The right action

$$ \begin{align*} \mathcal{M} \times \mathcal{D}_0\to \mathcal{M},\quad (g,\psi)\mapsto \psi^*g\end{align*} $$

is smooth and proper [Reference EbinEbi68, Reference Ebin, Chern and SmaleEbi70]. Moreover, if g is a metric with Anosov geodesic flow, it is directly seen from ergodicity that there are no Killing vector fields and thus the isotropy subgroup $\{\psi \in \mathcal {D}_0 ~|~ \psi ^*g=g\}$ of g is finite. For negatively curved metrics it is shown in [Reference FrankelFra66] that the action is free, that is, the isotropy group is trivial. One cannot apply the usual quotient theorem [Reference TrombaTro92, pp. 20] in the setting of Banach or Hilbert manifolds but rather smooth Fréchet manifolds instead (using the Nash–Moser theorem). Thus, in the setting of the space of smooth metrics with Anosov geodesic flows (the important fact, to apply Ebin’s slice theorem, is that metrics with Anosov geodesic flows do not have Killing vector fields, that is, infinitesimal isometries; this is due to the fact that $\ker D|_{C^\infty (M,T^*M)} = \{ 0 \}$ as mentioned earlier, which itself follows from the ergodicity of the geodesic flow), which is an open set of a Fréchet vector space, the slice theorem says that there exist a neighborhood $\mathcal {U}$ of $g_0$ , a neighborhood $\mathcal {V}$ of ${\textrm{Id}}$ in $\mathcal {D}_0$ and a Fréchet submanifold $\mathcal {S}$ containing $g_0$ so that

(2.9) $$ \begin{align} \mathcal{S}\times \mathcal{V}\to \mathcal{U}, \quad (g,\psi)\mapsto \psi^*g \end{align} $$

is a diffeomorphism of Fréchet manifolds and $T_{g_0}\mathcal {S}=\{h\in T_{g_0}\mathcal {M} ~|~ D_{g_0}^*h=0\}$ ; see [Reference EbinEbi68, Reference Ebin, Chern and SmaleEbi70]. Moreover, $\mathcal {S}$ parametrizes the set of orbits $g\cdot \mathcal {D}_0$ for g near $g_0$ and $T_g\mathcal {S}\cap T(g \cdot \mathcal {D}_0)=0$ .

On the other hand, if one considers $\mathcal {M}^{k,\alpha }$ , the space of metrics with $C^{k,\alpha }$ regularity and $\mathcal {D}^{k+1,\alpha }_0 := \operatorname {\textrm{Diff}}^{k+1,\alpha }_0(M)$ , the group of diffeomorphisms isotopic to the identity with $C^{k+1,\alpha }$ regularity, then both spaces are smooth Banach manifolds. However, the action of $\mathcal {D}_0^{k+1,\alpha }$ on $\mathcal {M}^{k,\alpha }$ is no longer smooth but only topological, which also prevents us from applying the quotient theorem.

Nevertheless, recalling that $g_0$ is smooth, if we consider $\mathcal {O}^{k,\alpha }(g_0) := g_0\cdot \mathcal {D}^{k+1,\alpha }_0 \subset \mathcal {M}^{k,\alpha }$ , then this is a smooth submanifold of $\mathcal {M}^{k,\alpha }$ and

$$ \begin{align*} T_{g}\mathcal{O}^{k,\alpha}(g_0) = \{D_g p ~|~ p \in C^{k+1,\alpha}(M;T^*M) \}. \end{align*} $$

Notice that (2.3) in $C^{k,\alpha }$ regularity exactly says that given $g \in \mathcal {O}^{k,\alpha }(g_0)$ , we have the decomposition

(2.10) $$ \begin{align} T_g \mathcal{M} = T_{g}\mathcal{O}^{k,\alpha}(g_0) \oplus \ker D_g^*|_{C^{k,\alpha}(M,S^2T^*M)}. \end{align} $$

Thus, an infinitesimal perturbation of a metric $g \in \mathcal {O}^{k,\alpha }(g_0)$ by a symmetric $2$ -tensor that is solenoidal with respect to g is actually an infinitesimal displacement transversally to the orbit $\mathcal {O}^{k,\alpha }(g_0)$ .

We will need a stronger version of the previous decomposition (2.10) which can be understood as a slice theorem. Knowledge of it goes back to [Reference EbinEbi68, Reference Ebin, Chern and SmaleEbi70]; see also [Reference Guillarmou and LefeuvreGL19, Lemma 4.1] for a short proof in the $C^{k,\alpha }$ category.

Lemma 2.4. Let k be an integer greater than or equal to $2$ and $\alpha \in (0,1)$ , let $g_0$ be a $C^{k+3,\alpha }$ metric with Anosov geodesic flow. There exists a neighborhood $\mathcal {U} \subset \mathcal {M}^{k,\alpha }$ of $g_0$ in the $C^{k,\alpha }$ -topology such that for any $g \in \mathcal {U}$ , there exists a unique $C^{k+1,\alpha }$ -diffeomorphism $\psi $ such that $\psi ^*g$ is solenoidal with respect to $g_0$ . Moreover, the following map is $C^2$ :

$$ \begin{align*}C^{k,\alpha}(M;S^2T^*M)\times C^{k+3,\alpha}(M;S^2T^*M) \to \mathcal{D}_0^{k+1,\alpha}(M), \quad (g,g_0)\mapsto \psi.\end{align*} $$

2.5 Thermodynamic formalism

Let f be a Hölder-continuous function on $S_{g_0}M$ . We recall that its pressure [Reference WaltersWal82, Theorem 9.10] is defined by

(2.11) $$ \begin{align} \textbf{P}(f) := \mathop{\textrm{sup}}\limits_{\mu \in \mathfrak{M}_{\textrm{inv}}} \bigg({\textbf{ h}}_\mu(\varphi_1^{g_0}) + \int_{S_{g_0}M} f\, d\mu\bigg), \end{align} $$

where $\mathfrak {M}_{\textrm{inv}}$ denotes the set of invariant (by the flow $\varphi ^{g_0}$ ) Borel probability measures and ${\textbf {h}}_\mu (\varphi _1^{g_0})$ is the metric entropy of the flow $\varphi _1^{g_0}$ at time $1$ . It is actually sufficient to restrict the $\textrm{sup}$ to ergodic measures $\mathfrak {M}_{\textrm{inv,erg}}$ [Reference WaltersWal82, Corollary 9.10.1]. Since the flow is Anosov, the supremum is always achieved for a unique invariant ergodic measure $\mu _f$ (by [Reference Bowen and RuelleBR75, Theorem 3.3]; see also [Reference Hasselblatt and FisherHF19, Theorem 9.3.4] and the following discussion therein) called the equilibrium state of f, and

(2.12) $$ \begin{align} \mu_f=\mu_{f^{\prime}} \Longrightarrow f-f^{\prime}=X_{g_0}u +c \quad \textrm{for some } u \textrm{ H}\ddot{\rm o}\textrm{lder and } c \textrm{ constant}; \end{align} $$

see [Reference Hasselblatt and FisherHF19, Theorem 9.3.16]. The measure $\mu _f$ is also mixing and positive on open sets, which rules out the possibility of a finite combination of Dirac measures supported on a finite number of closed orbits. Moreover, $\mu _f$ can be written as an infinite weighted sum of Dirac masses $\delta _{g_0}(c_j)$ supported over the geodesics $\gamma _{g_0}(c_j)$ , where $c_j\in \mathcal {C}$ are the primitive classes (see [Reference Parry and AlexanderPar88] for the case ${\textbf {P}}(f)\geq 0$ or [Reference Paulin, Pollicott and SchapiraPPS15, Theorem 9.17] for the general case). For example, when ${\textbf {P}}(f)\geq 0$ ,

(2.13) $$ \begin{align} \int u \, d\mu_f=\lim_{T\to \infty} \frac{1}{N(T,f)}\sum_{\{j | L_{g_0}(c_j)\in [T,T+1]\}}e^{\int_{\gamma_{g_0}(c_j)}f} \int_{\gamma_{g_0}(c_j)}u, \end{align} $$

where $N(T,f):=\sum _{j, L_{g_0}(c_j)\in [T,T+1]}L_{g_0}(c_j)e^{\int _{\gamma _{g_0}(c_j)}f}$ . When $f = 0$ , this is the measure of maximal entropy, also called the Bowen–Margulis measure $\mu _{g_0}^{\textrm{BM}}$ ; in that case $\textbf{P}(0) = {\textbf {h}}_{\textrm{top}}(\varphi _1^{g_0})$ is the topological entropy of the flow. When $f =-J_{g_0}^u$ , where $J_{g_0}^u : x \mapsto \partial _t (|\det d\varphi ^g_t(x)|_{E_u(x)})|_{t=0}$ is the unstable Jacobian, we obtain the Liouville measure $\mu _{g_0}^{\textrm{L}}$ induced by the metric $g_0$ ; in that case, $\textbf{P}(-J_{g_0}^u) = 0$ . If we fix an exponent of Hölder regularity $\nu> 0$ , then the map $C^\nu (S_{g_0}M) \ni f \mapsto \textbf{P}(f)$ is real analytic (see [Reference RuelleRue04, Corollary 7.10] for discrete systems and [Reference Parry and PollicottPP90, Proposition 4.7] for flows).

2.6 Geodesic stretch

We refer to [Reference Croke and FathiCF90, Reference KnieperKni95] for the original definition of this notion.

2.6.1 Structural stability and time reparametrization

We fix a smooth metric $g_0 \in \mathcal {M}$ with Anosov geodesic flow and we view the geodesic flow and vector fields of any metric g close to $g_0$ as living on the unit tangent bundle $S_{g_0}M$ of $g_0$ by simply pulling them back by the diffeomorphism

$$ \begin{align*} (x,v)\in S_{g_0}M\to \bigg(x,\frac{v}{|v|_{g}}\bigg)\in S_gM.\end{align*} $$

We fix some constant $k \geq 2$ and $\alpha \in (0,1)$ . There exist a regularity parameter $\nu>0$ and a neighborhood $\mathcal {U} \subset \mathcal {M}^{k,\alpha }$ of $g_0$ such that, by the structural stability theorem ([Reference de la Llave, Marco and MoriyóndlLMM86, Appendix A] or [Reference Katok, Knieper, Pollicott and WeissKKPW89, Proposition 2.2] for the Hölder regularity case), for any $g \in \mathcal {U}$ , there exists a $C^\nu $ Hölder homeomorphism $\psi _g : S_{g_0}M \rightarrow S_{g_0}M$ , differentiable in the flow direction, which is an orbit conjugacy that is such that

(2.14) $$ \begin{align} d\psi_g(z) X_{g_0}(z) = a_g(z) X_{g}(\psi_g(z)) \quad \text{for all } z \in S_{g_0}M, \end{align} $$

where $a_g$ is in $C^{\nu }(S_{g_0}M)$ . Moreover, the map

$$ \begin{align*}\mathcal{U} \ni g \mapsto (a_g,\psi_g) \in C^\nu(S_{g_0}M) \times C^\nu(S_{g_0}M,S_{g_0}M)\end{align*} $$

is $C^{k-2}$ and $\psi _g$ is homotopic to the identity. For the proof of Theorem 1.3, we will also need the continuity of $a_g=a_{g_0,g}$ and of its g-derivatives of order $\ell \leq k-2$ as a function of the base metric $g_0$ . This continuity follows essentially from the proof of [Reference Katok, Knieper, Pollicott and WeissKKPW89, Proposition 2.2]; we give a proof of this fact in Proposition C.1 in the Appendix.

Note that neither $a_g$ nor $\psi _g$ is unique, but $a_g$ is unique up to a coboundary and in all the following paragraphs; adding a coboundary to $a_g$ will not affect the results. From (2.14), we obtain that for $t \in \mathbb {R}$ , $z \in S_{g_0}M$ , $\varphi ^g_{\kappa _{a_g}(z,t)} (\psi _g(z)) = \psi _g(\varphi ^{g_0}_{t}(z))$ with

(2.15) $$ \begin{align} \kappa_{a_g}(z,t) = \int_0^t a_g(\varphi^{g_0}_s(z))\, ds. \end{align} $$

If $c \in \mathcal {C}$ is a free homotopy class, then

(2.16) $$ \begin{align} L_g(c) = \int_0^{L_{g_0}(c)} a_g(\varphi_s^{g_0}(z))\, ds, \end{align} $$

for any $z \in \gamma _{g_0}(c)$ , the unique $g_0$ -closed geodesic in c.

2.6.2 Definition of the geodesic stretch

We denote by $\widetilde {M}$ the universal cover of M. Given a metric $g \in \mathcal {M}$ on M, we denote by $\widetilde {g}$ its lift to the universal cover. Given two metrics $g_1$ and $g_2$ on M, there exists a constant $c> 0$ such that $c^{-1}g_1 \leq g_2 \leq cg_1$ . This implies that any $\widetilde {g_1}$ -geodesic is a quasi-geodesic for $\widetilde {g_2}$ . We now assume that the two metrics $g_1,g_2$ are Anosov on M. The ideal (or visual) boundary $\partial _\infty \widetilde {M}$ is independent of the choice of g and is naturally endowed with the structure of a topological manifold (see Appendix B) whose regularity inherits that of the foliation (that is, it is at least Hölder continuous and is $C^{2-\varepsilon }$ for any $\varepsilon> 0$ on negatively curved surfaces by [Reference Hurder and KatokHK90]). In negative curvature, we refer to [Reference Bridson and HaefligerBH99, Ch. H.3] and [Reference Knieper, Hasselblat and KatokKni02] for further details. For the general Anosov case, we refer to [Reference KnieperKni12] and Appendix B of the present paper.

We denote by $\mathcal {G}_g := S_{\widetilde {g}}\widetilde {M}/\sim $ (where $z \sim z^{\prime }$ if and only if there exists a time $t \in \mathbb {R}$ such that $\varphi _t(z) =z^{\prime }$ ) the set of g-geodesics on $\widetilde {M}$ : this is a smooth $2n$ -dimensional manifold. Moreover, there exists a Hölder-continuous homeomorphism $\Phi _g : \mathcal {G}_g \rightarrow \partial _\infty \widetilde {M} \times \partial _\infty \widetilde {M} \setminus \Delta $ , where $\Delta $ is the diagonal in $\partial _\infty \widetilde {M} \times \partial _\infty \widetilde {M}$ . Given a point $z \in S_{\widetilde {g}}\widetilde {M}$ , we will denote by $z_+, z_- \in \partial _\infty \widetilde {M}$ the points (in the future and in the past, respectively) on the boundary at infinity of the geodesic generated by z.

We now consider a fixed metric $g_0$ on M and a metric g in a neighborhood of $g_0$ . If $\psi _g$ denotes an orbit equivalence between the two geodesic flows, then $\psi _g$ induces a homeomorphism $\Psi _g : \mathcal {G}_{g_0} \rightarrow \mathcal {G}_g$ . The map

$$ \begin{align*} \Phi_g \circ \Psi_g \circ \Phi_{g_0}^{-1} : \partial_\infty \widetilde{M} \times \partial_\infty \widetilde{M} \setminus \Delta \rightarrow \partial_\infty \widetilde{M} \times \partial_\infty \widetilde{M} \setminus \Delta \end{align*} $$

is nothing more than the identity.

Given $z = (x,v) \in S_{g_0}M$ , we denote by $c_{g_0}(z) : t \mapsto c_{g_0}(z,t)\in M$ the unique geodesic (for the sake of simplicity, we identify the geodesic and its arc-length parametrization) such that $c_{g_0}(z,0)=x, \dot {c}_{g_0}(z,0)=v$ . We consider $\widetilde {c}_{g_0}(z)$ , a lift of $c_{g_0}(z)$ to the universal cover $\widetilde {M}$ , and introduce the function

$$ \begin{align*} b : S_{g_0}M \times \mathbb{R} \rightarrow \mathbb{R}, \quad b(z,t) := d_{\widetilde{g}}(\widetilde{c}_{g_0}(z,0),\widetilde{c}_{g_0}(z,t)), \end{align*} $$

which computes the $\widetilde {g}$ -distance between the endpoints of the $\widetilde {g_0}$ -geodesic joining $\widetilde {c}_{g_0}(z,0)$ to $\widetilde {c}_{g_0}(z,t)$ . It is an immediate consequence of the triangle inequality that $(z,t) \mapsto b(z,t)$ is a subadditive cocycle for the geodesic flow $\varphi ^{g_0}$ , that is,

$$ \begin{align*} b(z,t+s) \leq b(z,t) + b(\varphi_t^{g_0}(z),s) \quad \text{for all } z \in S_{g_0}M, \text{ for all } t,s \in \mathbb{R} \end{align*} $$

As a consequence, by the subadditive ergodic theorem (see [Reference WaltersWal82, Theorem 10.1], for instance), we obtain the following lemma.

Lemma 2.6. Let $\mu $ be an invariant probability measure for the flow $\varphi _t^{g_0}$ . Then the quantity

$$ \begin{align*} I_\mu(g_0,g,z) := \lim_{t \rightarrow +\infty} b(z,t)/t \end{align*} $$

exists for $\mu $ -almost every $z \in S_{g_0}M$ , $I_\mu (g_0,g,\cdot ) \in L^1(S_{g_0}M,d\mu )$ , and this function is invariant by the flow $\varphi _t^{g_0}$ .

We define the geodesic stretch of the metric g, relative to the metric $g_0$ , with respect to the measure $\mu $ by

$$ \begin{align*} I_\mu(g_0,g) := \int_{S_{g_0}M} I_\mu(g_0,g,z)\,d\mu(z). \end{align*} $$

When the measure $\mu $ in the previous definition is ergodic, the function $I_\mu (g_0,g,\cdot )$ is thus ( $\mu $ -almost everywhere) equal to the constant $I_\mu (g_0,g)$ . We recall that $\delta _{g_0}(c)$ is the normalized measure supported on $\gamma _{g_0}(c)$ , that is,

$$ \begin{align*} \delta_{g_0}(c) : u \mapsto \dfrac{1}{L_{g_0}(c)} \int_0^{L_{g_0}(c)} u(\varphi^{g_0}_t(z))\,dt. \end{align*} $$

We can actually describe the stretch using the time reparametrization $a_g$ .

Lemma 2.7. Let $\mu $ be an ergodic invariant measure with respect to the flow $\varphi _t^{g_0}$ . Then,

$$ \begin{align*} I_{\mu}(g_0,g) = \int_{SM_{g_0}} a_g \,d\mu = \lim_{j \rightarrow +\infty} \dfrac{L_{g}(c_j)}{L_{g_0}(c_j)}, \end{align*} $$

where $(c_j)_{j \geq 0} \in \mathcal {C}^{\mathbb {N}}$ is such that (the existence of $c_j$ follows from [Reference SigmundSig72, Theorem 1]) $\delta _{g_0}(c_j) \rightharpoonup _{j \rightarrow +\infty } \mu $ .

Proof We first prove the left-hand equality. Let $\widetilde {M}$ be the universal covering of M and $\Gamma $ the group of deck transformations. Denote as above by $\widetilde {\psi }_{g} : S_{\widetilde {g}_0}\widetilde {M} \rightarrow S_{\widetilde {g}} \widetilde {M}$ the lift of the conjugacy between the geodesic flow of the metrics $ \widetilde g$ and $\widetilde g_0$ . Then, for all $\gamma \in \Gamma $ ,

$$ \begin{align*}\widetilde{\varphi}^g_{\kappa_{a_g}(z,t)} (\widetilde{\psi}_g(z)) = \widetilde{\psi}_g(\widetilde{\varphi}^{g_0}_{t}(z)) \quad \text{and} \quad \widetilde{\psi}_{g}(\gamma_*{z})= \gamma_* \widetilde{\psi}_{g}(z). \end{align*} $$

If $\pi : T\widetilde {M} \to \widetilde {M}$ is the canonical projection the function $d_{\widetilde {g}}(\pi (\widetilde {\psi }_g(z)), \pi (z))$ is $\Gamma $ -invariant. This follows since

$$ \begin{align*} d_{\widetilde{g}}(\pi(\widetilde{\psi}_g(\gamma_*z)), \pi (\gamma_*z)) &= d_{\widetilde{g}}(\pi(\gamma_*\widetilde{\psi}_g(z)), \pi (\gamma_*z))\\ &= d_{\widetilde{g}}(\gamma\pi(\widetilde{\psi}_g(z)), \gamma \pi (z)) =d_{\widetilde{g}}(\pi(\widetilde{\psi}_g(z)), \pi (z)). \end{align*} $$

Hence, by the compactness of M and the continuity of $d_{\widetilde {g}}(\pi (\widetilde {\psi }_g(z)), \pi (z))$ there is a constant $C>0$ such that $d_{\widetilde {g}}(\pi (\widetilde {\psi }_g(z)), \pi (z)) \le C$ for all $z\in S\widetilde {M}$ . Using the triangle inequality, we obtain

$$ \begin{align*} | b(z,t) -\kappa_{a_g}(t,z) | &= | d_{\widetilde{g}}( \pi (\widetilde \varphi^{g_0}_{t}(z)) ,\pi(z)) -d_{\widetilde{g}}( \pi(\widetilde{\varphi}^g_{\kappa_{a_g}(z,t)} (\widetilde{\psi}_g(z))), \pi (\widetilde{\psi}_g(z))) |\\ &= |d_{\widetilde{g}}( \pi (\widetilde{\varphi}^{g_0}_{t}(z)),\pi(z)) - d_{\widetilde{g}}( \pi (\widetilde{\psi}_g( \widetilde{\varphi}^{g_0}_{t}(z))), \pi (\widetilde{\psi}_g(z)) )| \\ &\le d_{\widetilde{g}}( \pi (\widetilde{\varphi}^{g_0}_{t}(z)), \pi (\widetilde{\psi}_g(\widetilde{ \varphi}^{g_0}_{t}(z))) ) + d_{\widetilde{g}}(\pi (\widetilde{\psi}_g(z)), \pi(z)) \le 2C. \end{align*} $$

This implies, using (2.15), that

$$ \begin{align*} \lim_{t \rightarrow +\infty} b(z,t)/t = \lim_{t \rightarrow + \infty} \kappa_{a_g}(z,t)/t = \lim_{t \rightarrow +\infty} \dfrac{1}{t}\int_0^{t} a_g(\varphi_s^{g_0}(z)) \,ds = \int_{S_{g_0}M} a_g\,d\mu, \end{align*} $$

for $\mu $ -almost every $z \in S_{g_0}M$ , by the Birkhoff ergodic theorem [Reference WaltersWal82, Theorem 1.14]. By (2.16) we also have

$$ \begin{align*}\int_{S_{g_0}M}a_g\,d\mu_f=\lim_{j\to \infty}\langle \delta_{g_0}(c_j),a_g\rangle=\lim_{j\to \infty} \frac{L_{g}(c_j)}{L_{g_0}(c_j)},\end{align*} $$

thus the proof is complete.

As a consequence, we immediately obtain the following corollary.

Corollary 2.8. Let g belong to a fixed neighborhood $\mathcal {U}$ of $g_0$ in $\mathcal {M}^{k,\alpha }$ , and assume that for any sequence of primitive free homotopy classes $(c_j)_{j \geq 0} \in \mathcal {C}^{\mathbb {N}}$ such that $L_{g_0}(c_j) \rightarrow \infty $ , we have $\lim _{j\to \infty }L_{g}(c_j)/L_{g_0}(c_j)= 1$ . Then, for any equilibrium state $\mu _f$ with respect to $\varphi _t^{g_0}$ associated to some Hölder function f, we have $I_{\mu _f}(g_0,g)=1$ .

Combining this with the results of [Reference Guillarmou and LefeuvreGL19, Theorem 1], namely the local rigidity of the marked length spectrum, we also easily obtain the following theorem.

Theorem 2.9. Let $(M,g_0)$ be a smooth Riemannian n-dimensional manifold with Anosov geodesic flow, topological entropy ${\textbf {h}}_{\textrm{top}}(g_0) = 1$ , and assume that its curvature is non-positive if $n \geq 3$ . Then there exists $k \in \mathbb {N}$ large enough, depending only on n, $\varepsilon> 0$ small enough such that the following statement holds: there is $C>0$ depending on $g_0$ so that, for each $g \in C^{k}(M;S^2T^*M)$ with $\|g-g_0\|_{C^k} \leq \varepsilon $ , if

$$ \begin{align*} {\textbf{h}}_{\textrm{top}}(g)=1, \quad \lim_{j \rightarrow +\infty} \dfrac{L_g(c_j)}{L_{g_0}(c_j)} = 1, \end{align*} $$

for some sequence $(c_j)_{j \in \mathbb {N}}$ of primitive free homotopy classes such that $\delta _{g_0}(c_j) \rightharpoonup _{j \rightarrow +\infty } \mu ^{\textrm{BM}}_{g_0}$ , then g is isometric to $g_0$ .

Proof Given a metric g, we have by [Reference KnieperKni95, Theorem 1.2] (in [Reference KnieperKni95] the metric is assumed to be negatively curved, but the argument applies also for Anosov flows, as is shown in [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, Proposition 3.8]: it corresponds to Proposition 3.10 below in the case $f:=1$ and $f^{\prime }=a_g$ ) that

(2.17) $$ \begin{align} {\textbf{h}}_{\text{top}}(g) \geq \dfrac{{\textbf{ h}}_{\text{top}}(g_0)}{I_{\mu^{\textrm{BM}}_{g_0}}(g_0,g)}, \end{align} $$

with equality if and only if $\varphi ^{g_0}$ and $\varphi ^g$ are, up to a scaling, time-preserving conjugate, that is, there exists a homeomorphism $\psi $ such that $\psi \circ \varphi _{g_0}^{ct} = \varphi _g^t \circ \psi $ with $c:= {\textbf { h}}_{\text {top}}(g)/{\textbf {h}}_{\text {top}}(g_0)$ .

In particular, restricting to metrics with entropy $1$ , we obtain that $I_{\mu ^{\textrm{BM}}_{g_0}}(g_0,g) \geq 1$ with equality if and only if the geodesic flows are conjugate, that is, if and only if $L_g = L_{g_0}$ (by the Livsic theorem). As a consequence, given $g_0, g$ with entropy $1$ such that $L_{g}(c_j)/L_{g_0}(c_j) \rightarrow _{j \rightarrow +\infty } 1$ for some sequence $\delta _{g_0}(c_j) \rightharpoonup _{j \rightarrow +\infty } \mu ^{\textrm{BM}}_{g_0}$ , we obtain that $I_{\mu ^{\textrm{BM}}_{g_0}}(g_0,g)=1$ , hence $L_g=L_{g_0}$ . If $k \in \mathbb {N}$ was chosen large enough at the beginning, we can then conclude by the local rigidity of the marked length spectrum [Reference Guillarmou and LefeuvreGL19, Theorem 1].

In Theorem 2.9, we assume that $g_0$ has entropy $1$ . This is actually a harmless assumption in so far as the same result holds true on metrics of constant topological entropy ${\textbf {h}}_{{\textrm{top}}}(g)=\unicode{x3bb}>0$ . Recall that by considering $\unicode{x3bb} ^2 g_0$ for some constant $\unicode{x3bb}> 0$ , the entropy scales as ${\textbf { h}}_{\textrm{top}}(\unicode{x3bb} ^2 g_0) = {\textbf {h}}_{\textrm{top}}(g_0)/\unicode{x3bb} $ [Reference PaternainPat99, Lemma 3.23] and we can thus always reduce to the previous case ${\textbf {h}}_{{\textrm{top}}}(g_0) = 1$ . We also observe that the previous theorem implies the local rigidity of the marked length spectrum: if $L_g=L_{g_0}$ , then ${\textbf {h}}_{\text {top}}(g_0)={\textbf {h}}_{\text {top}}(g)$ because the topological entropy ${\textbf {h}}_{\text {top}}(g)$ is the first pole of the Ruelle zeta function [Reference Parry and PollicottPP90, Theorem 9.1]

$$ \begin{align*} \zeta_{g}(s) := \prod_{c \in \mathcal{C}} (1-e^{-s L_g(c)}). \end{align*} $$

We can then apply Theorem 2.9 to deduce that g is isometric to $g_0$ . We will provide an alternative proof of this fact in the next section without using the proof of [Reference Guillarmou and LefeuvreGL19].

3.1 The proof of Theorem 1.2

In what follows, we will compute the derivatives of the map $\Psi ,\Phi ,F$ . As mentioned earlier, they are $C^{k-2}$ by [Reference ContrerasCon92, Theorem C], and explicit computations of their derivatives can be found in [Reference Parry and PollicottPP90, Proposition 4.10] (subshift case) and [Reference Katok, Knieper, Pollicott and WeissKKPW90, Reference Katok, Knieper and WeissKKW91] (topological entropy case). The first step in the proof is the following proposition.

Proposition 3.2. The non-negative functional $\Psi :\mathcal {M}^{k,\alpha }\to \mathbb {R}^+$ defined in (3.2) satisfies the following property: there exist a neighborhood $\mathcal {U}$ of $g_0$ in $C^{5,\alpha }(M,S^2T^*M)$ and a constant $C_{g_0}$ depending on $g_0$ such that, for all $g\in \mathcal {U}$ ,

$$ \begin{align*} \Psi(g)\geq \tfrac{1}{8}(\langle \Pi_2^{g_0}(g-g_0),(g-g_0)\rangle_{L^2} - \langle (g-g_0),g_0 \rangle^2_{L^2})-C_{g_0}\|g-g_0\|^3_{C^{5,\alpha}}. \end{align*} $$

Proof We shall compute the Taylor expansion of $\Psi $ at $g=g_0$ to second order. By [Reference Parry and PollicottPP90, Proposition 4.10], we have, for $h\in T_{g}\mathcal {M}^{k,\alpha }$ ,

$$ \begin{align*} dF_g.h = -\int_{S_{g_0}M} da_g.h \, dm_g \end{align*} $$

where $m_g$ is the equilibrium measure of $-V_g$ . In particular, observe that for $g=g_0$ , we have

(3.6) $$ \begin{align} dF_{g_0}.h = -\int_{S_{g_0}M} da_{g_0}.h \,d \mu^{\textrm{L}}_{g_0}, \end{align} $$

since $m_{g_0} = \mu ^{\textrm{L}}_{g_0}$ . Next, we get, for $h\in T_{g_0}\mathcal {M}^{k,\alpha }$ ,

(3.7) $$ \begin{align} d\Phi_{g_0}.h = \int_{S_{g_0}M} da_{g_0}.h \, d\mu^{\textrm{L}}_{g_0}=-dF_{g_0}.h, \end{align} $$

thus $d\Psi _{g_0}.h=0$ for all $h\in T_{g_0}\mathcal {M}^{k,\alpha }$ .

Let us next compute the second derivative $d^2\Psi _{g_0}(h,h)$ . First, we have

$$ \begin{align*} d^2\Phi_{g_0}=\int_{S_{g_0}M}d^2a_{g_0}(h,h)\, d\mu_{g_0}^L. \end{align*} $$

Then, by [Reference Parry and PollicottPP90, Proposition 4.11] we know that

(3.8) $$ \begin{align} d^2\textbf{P}_{-V_{g_0}}(dV_{g_0}.h, dV_{g_0}.h) &= \operatorname{\textrm{Var}}_{\mu^{\textrm{L}}_{g_0}}(dV_{g_0}.h-\langle dV_{g_0}.h,1\rangle)=\langle \Pi^{g_0}dV_{g_0}.h,dV_{g_0}.h\rangle_{L^2}, \nonumber \\ d\textbf{P}_{-V_{g_0}}(dV_{g_0}.h) &= \int_{S_{g_0}M} dV_{g_0}.h \, d\mu^{\textrm{L}}_{g_0}, \end{align} $$

where $\operatorname {\textrm{Var}}_{\mu ^{\textrm{L}}_{g_0}}(h)$ is the variance defined in (2.5), equal to $\langle \Pi ^{g_0} h,h\rangle _{L^2}$ by (2.6) and ${\Pi ^{g_0}1=0}$ . Therefore,

$$ \begin{align*}\begin{aligned} d^2F_{g_0}(h,h)=& -d{\textbf{P}}_{-V_{g_0}}.d^2V_{g_0}(h,h)+d^2{\textbf{ P}}_{-V_{g_0}}(dV_{g_0}.h,dV_{g_0}.h)\\ =& -d{\textbf{P}}_{-V_{g_0}}.d^2a_{g_0}(h,h)+\langle \Pi^{g_0}da_{g_0}.h,da_{g_0}.h\rangle_{L^2}. \end{aligned} \end{align*} $$

All together, we finally get

$$ \begin{align*}d^2\Psi_{g_0}(h,h)=\langle \Pi^{g_0}da_{g_0}.h,da_{g_0}.h\rangle_{L^2}.\end{align*} $$

To conclude, we claim in Lemma 3.3 below that $da_{g_0}.h-\tfrac {1}{2}\pi _2^*h$ is a coboundary, so that

$$ \begin{align*}d^2\Psi_{g_0}(h,h)=\langle \Pi^{g_0}\pi_2^*h,\pi_2^*h\rangle_{L^2}=\tfrac{1}{4}(\langle \Pi_2^{g_0}h,h\rangle_{L^2}-\langle h,g_0\rangle_{L^2}^2).\end{align*} $$

The statement of the proposition is then simply the Taylor expansion of $\Psi (g)$ at $g=g_0$ , with $h=g-g_0$ . (We need the map to be $C^3$ for the Taylor expansion, hence the need for the $C^{5,\alpha }$ regularity since we lose two derivatives as mentioned at the beginning of §3.)

Lemma 3.3. Consider a smooth deformation $(g_\unicode{x3bb} )_{\unicode{x3bb} \in (-1,1)}$ of $g_0$ inside $\mathcal {M}^{k,\alpha }$ . Then there exists a Hölder-continuous function $f : S_{g_0}M \rightarrow \mathbb {R}$ such that

$$ \begin{align*} \pi_2^*(\partial_\unicode{x3bb} g_\unicode{x3bb}|_{\unicode{x3bb}=0}) - 2 \partial_\unicode{x3bb} a_\unicode{x3bb} |_{\unicode{x3bb}=0} = X_{g_0}f.\end{align*} $$

Proof Let c be a fixed free homotopy class, and let $\gamma _0 \in c$ be the unique closed $g_0$ -geodesic in the class c, which we parametrize by unit-speed $z_0 : [0,\ell _{g_0}(\gamma _0)]\rightarrow S_{g_0}M$ . We define $z_\unicode{x3bb} (s)=\psi _\unicode{x3bb} (z_0(s)) = (\alpha _\unicode{x3bb} (s),\dot {\alpha }_\unicode{x3bb} (s))$ (the dot is the derivative with respect to s), where $\psi _\unicode{x3bb} $ is the conjugacy between $g_\unicode{x3bb} $ and $g_0$ : this gives a non-unit-speed parametrization of $\gamma _\unicode{x3bb} $ , the unique closed $g_\unicode{x3bb} $ -geodesic in c. We recall that $\pi : TM \rightarrow M$ is the projection. Using (2.14), we obtain

$$ \begin{align*} &\int_0^{\ell_{g_0}(\gamma_0)} g_\unicode{x3bb}(\dot{\alpha}_\unicode{x3bb}(s),\dot{\alpha}_\unicode{x3bb}(s))\,ds\\ &\quad = \int_0^{\ell_{g_0}(\gamma_0)} g_\unicode{x3bb}(\partial_s(\pi \circ z_\unicode{x3bb}(s)),\partial_s(\pi \circ z_\unicode{x3bb}(s)))\,ds \\ &\quad = \int_0^{\ell_{g_0}(\gamma_0)} g_\unicode{x3bb}(\partial_s(\pi \circ \psi_\unicode{x3bb} \circ z_0(s)),\partial_s(\pi \circ \psi_\unicode{x3bb} \circ z_0(s)))\,ds \\ &\quad = \int_0^{\ell_{g_0}(\gamma_0)} a_\unicode{x3bb}^{2}(z_0(s))\underbrace{g_\unicode{x3bb}(d\pi(X_{g_\unicode{x3bb}}(z_\unicode{x3bb}(s))),d\pi(X_{g_\unicode{x3bb}}(z_\unicode{x3bb}(s))))}_{=1}\, ds\\ &\quad = \int_0^{\ell_{g_0}(\gamma_0)} a_\unicode{x3bb}^{2}(z_0(s))\,ds. \end{align*} $$

Since $s \mapsto \alpha _0(s)$ is a unit-speed geodesic for $g_0$ , it is a critical point of the energy functional (with respect to $g_0$ ). Thus, by differentiating the previous identity with respect to $\unicode{x3bb} $ and evaluating at $\unicode{x3bb} =0$ , we obtain

$$ \begin{align*} \int_0^{\ell_{g_0}(\gamma_0)} \partial_\unicode{x3bb} g_\unicode{x3bb}|_{\unicode{x3bb}=0}(\dot{\alpha}_0(s),\dot{\alpha}_0(s))\,ds = 2 \int_0^{\ell_{g_0}(\gamma_0)} \partial_\unicode{x3bb} a_\unicode{x3bb}|_{\unicode{x3bb}=0}(z_0(s))\,ds. \end{align*} $$

As a consequence, $\pi _2^*(\partial _\unicode{x3bb} g_\unicode{x3bb} |_{\unicode{x3bb} =0}) - 2 \partial _\unicode{x3bb} a_\unicode{x3bb} |_{\unicode{x3bb} =0}$ is a Hölder-continuous function in the kernel of the X-ray transform: by the usual Livsic theorem, there exists a function f (with the same Hölder regularity), differentiable in the flow direction, such that $\pi _2^*(\partial _\unicode{x3bb} g_\unicode{x3bb} |_{\unicode{x3bb} =0}) - 2 \partial _\unicode{x3bb} a_\unicode{x3bb} |_{\unicode{x3bb} =0} = X_{g_0}f$ .

As a corollary, we obtain the following result.

Corollary 3.4. For $k\geq 5$ , $\alpha \in (0,1)$ , there exist a neighborhood $\mathcal {U}$ of $g_0$ in $C^{k,\alpha }(M;S^2T^*M)$ and constants $C_{g_0},C_{g_0}^{\prime }>0$ depending on $g_0$ such that, for all $g\in \mathcal {U}$ ,

$$ \begin{align*}C_{g_0}\|\pi_{\ker D_{g_0}^*}(g-g_0)\|^2_{H^{-1/2}(M)}\leq \Psi(g)+\tfrac{1}{4}\langle (g-g_0),g_0\rangle_{L^2}^2 +C_{g_0}^{\prime}\|g-g_0\|_{C^{5,\alpha}}^3.\end{align*} $$

There exist a neighborhood $\mathcal {U}^{\prime }$ of $g_0$ in $C^{k,\alpha }(M;S^2T^*M)$ and a constant $C_{g_0}^{\prime \prime }>0$ depending on $g_0$ such that, for all $g\in \mathcal {U}^{\prime }$ , there is a diffeomorphism $\psi \in C^{k+1,\alpha }(M)$ such that

$$ \begin{align*}C_{g_0}\|\psi^*g-g_0\|^2_{H^{-1/2}(M)}\leq \Psi(g)+\tfrac{1}{4}\langle (\psi^*g-g_0),g_0\rangle_{L^2}^2 +C_{g_0}^{\prime\prime}\|\psi^*g-g_0\|_{C^{5,\alpha}}^3\end{align*} $$

The next step is to control the term $\langle (\psi ^*g-g_0),g_0\rangle _{L^2}$ by the geodesic stretch. We will show the following proposition.

Proposition 3.5. There is $k \in \mathbb {N}$ large enough, depending only on $n=\dim M$ , such that if $g_0$ is smooth with Anosov geodesic flow, and non-positive curvature in the case $n>2$ , there exist $C_{g_0}>0$ and $C_n>0$ , an open neighborhood $\mathcal {U}$ in $C^{k,\alpha }(M;S^2T^*M)$ of $g_0$ , such that. for each $g\in \mathcal {U}$ , there is a diffeomorphism $\psi $ satisfying

$$ \begin{align*} & C_{g_0}\|\psi^*g-g_0\|^2_{H^{-1/2}(M)}\\&\quad\leq {\textbf{ P}}\bigg(\!-J_{g_0}^u-a_g+\int_{S_{g_0}M}a_g\, d\mu_{g_0}^L\bigg)+C_n(I_{\mu_{g_0}^L}(g_0,g)-1)^2 \\ &\quad\leq {\textbf{ P}}\bigg(\!-J_{g_0}^u-a_g+\int_{S_{g_0}M}a_g\, d\mu_{g_0}^L\bigg)+C_n({\textbf{ P}}(-J_{g_0}^u-a_g+1))^2\\ &\quad\leq {\textbf{ P}}\bigg(\!-J_{g_0}^u-a_g+\int_{S_{g_0}M}a_g\, d\mu_{g_0}^L\bigg)+({\textrm{Vol}}_g(M)-{\textrm{Vol}}_{g_0}(M))^2. \end{align*} $$

Here $C_{g_0}$ depends on $g_0$ and $C_n$ on n only.

Proof We write the Taylor expansion of $\Phi (\psi ^*g)=\Phi (g)=I_{\mu _{g_0}^L}(g_0,g)-1$ at $g=g_0$ : by Lemma 3.6 and Lemma 3.3,

$$ \begin{align*}d\Phi_{g_0}.h=\int_{S_{g_0}M}da_{g_0}.h\, d\mu_{g_0}^L=\frac{1}{2}\int_{S_{g_0}M}\pi_2^*h\, d\mu_{g_0}^L=C_n\langle h,g_0\rangle_{L^2}\end{align*} $$

for some $C_n>0$ depending only on $n=\dim M$ . Then

$$ \begin{align*} \Phi(g)=\Phi(\psi^*g)=C_n\langle \psi^*g-g_0,g_0\rangle_{L^2}+\mathcal{O}(\|\psi^*g-g_0\|^2_{C^{5,\alpha}}).\end{align*} $$

Combining with Corollary 3.4, we obtain

(3.9) $$ \begin{align} C_{g_0}\|\psi^*g-g_0\|^2_{H^{-1/2}(M)}\leq \Psi(g)+2C_n^{-2}\Phi(g)^2+ C_{g_0}^{\prime\prime}\|\psi^*g-g_0\|_{C^{5,\alpha}}^3 \end{align} $$

if $\|\psi ^*g-g_0\|_{C^{5,\alpha }}$ is small enough, which is the case if $\|g-g_0\|_{C^{5,\alpha }}$ is small enough by Lemma 2.4. To obtain the first inequality of Proposition 3.5, we apply Sobolev embedding and interpolation estimates (the interpolation estimate $\|u\|_{H^c}\leq \|u\|_{H^a}^{t}\|u\|_{H^b}^{1-t}$ for $c=ta+(1-t)b$ is obtained by applying the Hadamard three-line theorem to the holomorphic function $s\mapsto \sum _{j}(1+\unicode{x3bb} _j)^{s}\langle u,e_j\rangle _{L^2}^2$ on ${\textrm{Re}}(s)\in [a,b]$ , where $e_j$ is an orthonormal basis of eigenfunctions of any positive elliptic self-adjoint differential operator of order $2$ on symmetric tensors and $\unicode{x3bb} _j$ gives the corresponding eigenvalues) [Reference TaylorTay96, Ch. 4] and get, for some constants $c_{g_0}>0,c^{\prime }_{g_0}>0$ depending on $g_0$ only,

$$ \begin{align*} \|\psi^*g-g_0\|^3_{C^{5,\alpha}} \leq c_{g_0} \|\psi^*g-g_0\|^3_{H^{({n}/{2})+5+\alpha^{\prime}}} \leq c^{\prime}_{g_0} \|\psi^*g-g_0\|^2_{H^{-1/2}} \|\psi^*g-g_0\|_{H^{k}}, \end{align*} $$

if $k>({3}/{2})n+16+3\alpha $ and $\alpha ^{\prime }>\alpha $ . This means that if $\|\psi ^*g-g_0\|_{H^k}$ is small enough, depending on the constants $C_{g_0}, C_{g_0}^{\prime \prime },c_{g_0},c^{\prime }_{g_0}$ , one can absorb the $\|\psi ^*g-g_0\|^3_{C^{5,\alpha }}$ term of (3.9) into the left-hand side and get the first inequality of Proposition 3.5. The smallness of $\|\psi ^*g-g_0\|_{H^k}$ is implied by the smallness of $\|g-g_0\|_{C^{k,\alpha }}$ by Lemma 2.4. The same exact argument applies by replacing $\Phi (g)$ by $F(g)$ using that $dF_{g_0}=-d\phi _{g_0}$ ; this proves the second inequality of Proposition 3.5. The last inequality is similar since

$$ \begin{align*}{\textrm{Vol}}_g(M)-{\textrm{Vol}}_{g_0}(M)=\frac{1}{2}\int_{M}{\textrm{Tr}}_{g_0}(h)\, d{\textrm{vol}}_{g_0}+\mathcal{O}(\|h\|^2_{C^{5,\alpha}})=\frac{1}{2}\langle h,g_0\rangle_{L^2}+\mathcal{O}(\|h\|^2_{C^{5,\alpha}})\end{align*} $$

for $h:=g-g_0$ . The proof is complete.

To conclude the proof of Theorem 1.2, we need to estimate $\Phi (g)$ and $F(g)$ in terms of $\mathcal {L}_\pm (g)$ . Recall that (see [Reference Paulin, Pollicott and SchapiraPPS15, Corollary 9.17])

$$ \begin{align*}\begin{aligned} {\textbf{P}}(-V_g)=&\lim_{T\to \infty}\frac{1}{T}\log \sum_{c\in\mathcal{C}, L_{g_0}(c)\in [T,T+1]}e^{-\int_{\gamma_{g_0}(c)}V_g}\\ =&\lim_{T\to \infty}\frac{1}{T}\log \sum_{c\in\mathcal{C}, L_{g_0}(c)\in [T,T+1]}e^{-\int_{\gamma_{g_0}(c)}J^u_{g_0}}e^{L_{g_0}(c)-L_{g}(c)}. \end{aligned}\end{align*} $$

Thus, if we order $\mathcal {C}=(c_j)_{j\in \mathbb {N}}$ by the lengths (that is, $L_{g_0}(c_j)\geq L_{g_0}(c_{j-1})$ ), and we define

$$ \begin{align*} \mathcal{L}_+(g):=\limsup_{j\to \infty}\frac{L_g(c_j)}{L_{g_0}(c_j)}-1, \quad \mathcal{L}_-(g):=\liminf_{j\to \infty}\frac{L_{g}(c_j)}{L_{g_0}(c_j)}-1,\end{align*} $$

we see that for all $\delta>0$ small, there is $T_0>0$ large so that, for all j with $L_{g_0}(c_j)\in [T,T+1]$ with $T\geq T_0$ ,

$$ \begin{align*} e^{\min((T+1)(-\mathcal{L}_+(g)-\delta),T(-\mathcal{L}_+(g)-\delta))}\leq e^{L_{g_0}(c_j)-L_{g}(c_j)} \leq e^{\max((T+1)(-\mathcal{L}_-(g)+\delta),T(-\mathcal{L}_-(g)+\delta))}.\end{align*} $$

We deduce, using ${\textbf {P}}(-V_{g_0})=0$ , that

$$ \begin{align*} -\mathcal{L}_+(g)-\delta\leq {\textbf{P}}(-V_g)\leq -\mathcal{L}_-(g)+\delta.\end{align*} $$

Since $\delta>0$ is arbitrarily small, we obtain $|F(g)|\leq \max (|\mathcal {L}_+(g)|,|\mathcal {L}_-(g)|)$ . Similarly, Lemma 2.7 shows that $|\Phi (g)|\leq \max (|\mathcal {L}_+(g)|,|\mathcal {L}_-(g)|)$ . So the proof of Theorem 1.2 is complete by combining these bounds with Proposition 3.5 (the right-hand side in the first and second inequalities of Proposition 3.5 being $F(g)+\Phi (g)+C_n \Phi (g)^2$ and $F(g)+\Psi (g)+C_nF(g)^2$ ).

3.2 A submanifold of the space of metrics

It is quite natural to describe the stretch functional $\Phi $ on the space

(3.10) $$ \begin{align} \mathcal{N}^{k,\alpha} := \{g \in \mathcal{M}^{k,\alpha} ~|~ \textbf{P}(-V_g) = 0 \} \end{align} $$

and on $\mathcal {N}^{k,\alpha }_{\textrm{sol}} := \mathcal {N}^{k,\alpha } \cap \ker D^*_{g_0}$ . Indeed, as we shall see, this becomes a strictly convex functional near $g_0\in \mathcal {N}_{\textrm{sol}}^{k,\alpha }$ when restricted to $\mathcal {N}_{\textrm{sol}}^{k,\alpha }$ . It is possible that the map is strictly convex globally on $\mathcal {N}_{\textrm{sol}}^{k,\alpha }$ , in which case that would prove the global rigidity of the marked length spectrum.

Given $g \in \mathcal {N}^{k,\alpha }$ , we denote by $m_g$ the unique equilibrium state for the potential $V_g$ . We will also write $\mathcal {N}$ for the case where $k=\infty $ . First we check that these are (infinite-dimensional) manifolds.

Proof To prove this lemma, we will use the notion of differential calculus on Banach manifolds as it is stated in [Reference ZeidlerZei88, Ch. 73]. Note that $\mathcal {M}^{k,\alpha }$ is a smooth Banach manifold and $\mathcal {N}^{k,\alpha } \subset \mathcal {M}^{k,\alpha }$ is defined by the implicit equation $F(g)=0$ for

(3.11) $$ \begin{align} F: g \mapsto \textbf{P}(-V_g) \in \mathbb{R}. \end{align} $$

The map F being $C^{k-2}$ , we need only prove that $dF_{g_0}$ does not vanish by [Reference ZeidlerZei88, Theorem 73.C]. This will immediately give that $T_{g_0}\mathcal {N}^{k,\alpha }=\ker dF_{g_0}$ . In order to do so, we need a deformation lemma. For the sake of simplicity, we write the objects $\cdot _{\unicode{x3bb} }$ instead of $\cdot _{g_\unicode{x3bb} }$ .

We can now complete the proof of Lemma 3.6. We first prove the first part concerning $\mathcal {N}^{k,\alpha }$ . Recall formula (3.6) for $dF_{g_0}$ . Using Lemma 3.3, we obtain

(3.12) $$ \begin{align} d F_{g_0}.h= -\int_{S_{g_0}M} da_{g_0}.h\, d\mu^{\textrm{L}}_{g_0} = - \dfrac{1}{2} \int_{S_{g_0}M} \pi_2^* h\, d\mu^{\textrm{L}}_{g_0} = - C_n \langle h,g_0 \rangle_{L^2}, \end{align} $$

for some constant $C_n> 0$ depending on n. This is obviously surjective and we also obtain

$$ \begin{align*} T_{g_0} \mathcal{N}^{k,\alpha} = \ker dF_{g_0} = \{h \in C^{k,\alpha}(M;S^2T^*M) ~|~ \langle h,g_0 \rangle_{L^2} = 0 \} = (\mathbb{R} g_0)^\bot, \end{align*} $$

where the orthogonal is understood with respect to the $L^2$ -scalar product.

We now deal with $\mathcal {N}^{k,\alpha }_{\textrm{sol}}$ . First observe that $\ker D^*_{g_0}$ is a closed linear subspace of $\mathcal {M}^{k,\alpha }$ and thus a smooth submanifold of $\mathcal {M}^{k,\alpha }$ . By [Reference ZeidlerZei88, Corollary 73.50], it is sufficient to prove that $\ker D^*_{g_0}$ and $\mathcal {N}^{k,\alpha }$ are transverse at $g_0$ . But observe that $g_0 \in \ker D^*_{g_0} \simeq T_{g_0} \ker D^*_{g_0}$ and thus

$$ \begin{align*} T_{g_0} \ker D^*_{g_0} + T_{g_0}\mathcal{N}^{k,\alpha} = T_{g_0} \mathcal{M}^{k,\alpha}, \end{align*} $$

showing transversality.

The case of $\mathcal {N}$ follows directly from the Nash–Moser theorem: F is obviously a smooth tame map from $C^\infty (M;S^2T^*M)$ to $\mathbb {R}$ ; moreover, $dF_g$ has a right inverse $H_g$ since

$$ \begin{align*}dF_g.g= -\int_{S_{g_0}M}da_g.g\, dm_g=-\frac{1}{2}\int_{S_{g_0}M} a_g\, dm_g= -\frac{1}{2}I_{m_g}(g_0,g) \end{align*} $$

where we use the fact that $2da_g.g$ is cohomologous to $a_g$ . This can be seen by differentiating $L_g(c)=\int _{\gamma _{g_0}(c)}a_g$ and applying the Livsic theorem. In particular,the right inverse is given by $H_g.1:=-2g/I_{m_g}(g_0,g)$ . The family of right inverses $g\mapsto H_g$ is smooth since $g\mapsto a_g$ and $g\mapsto m_g$ are smooth by [Reference ContrerasCon92, Theorem C], and it is clearly also tame, thus we can apply directly [Reference HamiltonHam82, Theorem 1.1.3, pp. 172] to deduce that F has a smooth tame right inverse, which shows that $\mathcal {N}$ is a Fréchet submanifold.

We remark that if $L_g=L_{g_0}$ , then $a_g$ is cohomologous to $1$ , so ${\textbf { P}}(-V_{g})={\textbf {P}}(-V_{g_0})=0$ in that case, which means that $g\in \mathcal {N}^{k,\alpha }$ . From the second inequality in Proposition 3.5, we obtain the following corollary.

Corollary 3.7. Let $g_0$ be a smooth metric with Anosov geodesic flow, with non-positive curvature if $n>2$ . There exist $C_{g_0}>0$ and a neighborhood $\mathcal {U}\subset \mathcal {N}^{k,\alpha }$ such that, for all $g\in \mathcal {U}$ , there is a diffeomorphism $\psi \in \mathcal {D}_0^{k+1,\alpha }$ so that

$$ \begin{align*} C_{g_0}\|\psi^*g-g_0\|^2_{H^{-1/2}(M)}\leq I_{\mu_{g_0}^L}(g_0,g)-1.\end{align*} $$

As suggested by this estimate, the functional $\Phi $ turns out to be strictly convex near $g_0$ when restricted on $\mathcal {N}_{\textrm{sol}}^{k,\alpha }$ . First, we have, for $h\in T_{g_0}\mathcal {N}^{k,\alpha }$ ,

$$ \begin{align*} d\Phi_{g_0}.h=-dF_{g_0}.h=0,\end{align*} $$

so that $\Phi :\mathcal {N}^{k,\alpha }\to \mathbb {R}$ has a critical point at $g_0$ . For the second derivative at $g_0$ , the same computation as in the previous section easily gives the following result.

Lemma 3.8. The map $\Phi :\mathcal {N}^{k,\alpha }_{\textrm{sol}} \to \mathbb {R}$ is strictly convex at $g_0$ , and there is $C>0$ such that

$$ \begin{align*}d^2\Phi_{g_0}(h,h) = \tfrac{1}{4} \langle \Pi_2^{g_0} h, h \rangle \geq C \|h\|^2_{H^{-1/2}(M)}\end{align*} $$

for all $h \in T_{g_0}\mathcal {N}^{k,\alpha }_{\textrm{sol}}$ .

3.3 The pressure metric on the space of negatively curved metrics

The results of this paragraph are stated in negative curvature, but it is very likely that one could relax the assumption to the Anosov case. Again, the only obstruction for the moment is that it is still not known whether the X-ray transform $I_2$ (hence the operator $\Pi _2$ ) is injective on solenoidal tensors in the Anosov case when $\textrm{dim}(M) \geq 3$ .

3.3.1 Definition of the pressure metric using the variance

On $\mathcal {M}^-$ , the cone of smooth negatively curved metrics, we introduce the non-negative symmetric bilinear form

(3.13) $$ \begin{align} G_g(h_1,h_2) := \langle \Pi^{g}_2 h_1, h_2 \rangle_{L^2(M,d\operatorname{\textrm{vol}}_g)}, \end{align} $$

defined for $g \in \mathcal {M}$ , $h_j\in T_g \mathcal {M} \simeq C^{\infty }(M; S^2T^*M)$ . It is non-degenerate on $T_g \mathcal {M} \cap \ker D^*_g$ , namely, $G_g(h,h) \geq C_g \|h\|^2_{H^{-1/2}}$ by Lemma 2.2, and the constant $C_g$ turns out to be locally uniform for g near a given metric $g_0$ . Combining these facts, we obtain the following proposition.

Proposition 3.9. Let $g_0\in \mathcal {M}^-$ . Then the bilinear form G defined in (3.13) produces a Riemannian metric on the quotient space $\mathcal {M}^-/\mathcal {D}_0$ near the class $[g_0]$ , where $\mathcal {M}^-/\mathcal {D}_0$ is identified with the slice $\mathcal {S}$ passing through $g_0$ as in (2.9).

Proof It suffices to show that G is non-degenerate on $T\mathcal {S}$ . Let $h\in T_{g}\mathcal {S}$ and assume that $G_g(h,h)=0$ . We can write $h=\mathcal {L}_{V}g+h^{\prime }$ where $D_g^*h^{\prime }=0$ and V is a smooth vector field and $\mathcal {L}_V$ the Lie derivative with respect to V. By Lemma 2.1 we obtain $0=G_g(h,h)\geq C\|h^{\prime }\|_{H^{-1/2}}$ . Thus $h=\mathcal {L}_Vg$ , but we also know that $T_{g}\mathcal {S}\cap \{\mathcal {L}_Vg ~|~ V\in C^\infty (M;T^*M)\}=\{0\}$ since $\mathcal {S}$ is a slice. Therefore $h=0$ .

3.3.2 Definition using the intersection number

We now want to relate the pressure metric previously introduced to some renormalized intersection numbers involving some well-chosen potentials. This will be needed to show that the pressure metric coincides with (a multiple of) the Weil–Petersson metric in the case where M is a surface and one restricts to hyperbolic metrics. This also makes a relation with recent work of [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15].

Let us assume that g is in a fixed $C^2$ -neighborhood of $g_0$ . Since $J^u_{g_0}> 0$ , we obtain that $V_g = J^u_{g_0} + a_g-1> 0$ if g is close enough to $g_0$ . By [Reference SambarinoSam14, Lemma 2.4], there exists a unique constant ${\textbf {h}}_{V_g} \in \mathbb {R}$ such that $\textbf{P}(-{\textbf {h}}_{V_g} V_g)= 0$ . In particular, $\mathcal {N}$ coincides in a neighborhood of $g_0$ with the set $\{ g \in \mathcal {M} ~|~ {\textbf {h}}_{V_g} = 1\}$ . One can express the constant ${\textbf {h}}_{V_g}$ as ${\textbf {h}}_{V_g} = {\textbf {h}}_{\textrm{top}}(\varphi _t^{g_0,V_g})$ , where $\varphi _t^{g_0,V_g}$ is a time reparametrization of the geodesic flow of $g_0$ (see [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, §3.1.1]). More precisely, given a Hölder-continuous positive function $f \in C^\nu (S_{g_0}M)$ on $S_{g_0}M$ , we introduce the unique real number ${\textbf {h}}_f$ such that $\textbf{P}(-{\textbf {h}}_f f)=0$ and we set

$$ \begin{align*} S_{g_0}M \times \mathbb{R} \ni (z,t) \mapsto \kappa_f(z,t) := \int_0^t f(\varphi^{g_0}_s(z)) \,ds. \end{align*} $$

For a fixed $z\in S_{g_0}M$ , this is a homeomorphism on $\mathbb {R}$ and thus allows us to define

(3.14) $$ \begin{align} \varphi_{\kappa_f(z,t)}^{g_0,f}(z) := \varphi^{g_0}_t(z). \end{align} $$

We now follow the approach of [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, §3.4.1]. Given two Hölder-continuous functions $f,f^{\prime } \in C^\nu (S_{g_0}M)$ such that $f> 0$ , one can define an intersection number [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, Eq. (13)]

$$ \begin{align*} \textbf{I}_{g_0}(f,f^{\prime}) := \dfrac{\int_{S_{g_0}M} f^{\prime} \, d\mu_{- {\textbf{h}}_f f}}{\int_{S_{g_0}M} f \,d\mu_{- {\textbf{h}}_f f}} \end{align*} $$

where $d\mu _{- {\textbf {h}}_f f}$ is the equilibrium measure for the potential $-{\textbf {h}}_f f$ . We have the following result, which follows from [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, Proposition 3.8] stated for Anosov flows on compact metric spaces:

Proposition 3.10. (Bridgeman, Canary, Labourie and Sambarino [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15])

Let $f, f^{\prime } : S_{g_0}M \rightarrow \mathbb {R}_+$ be two Hölder-continuous positive functions. Then

$$ \begin{align*} \textbf{J}_{g_0}(f,f^{\prime}) := \dfrac{{\textbf{h}}_{f^{\prime}}}{{\textbf{h}}_f} \textbf{I}_{g_0}(f,f^{\prime}) \geq 1, \end{align*} $$

with equality if and only if ${\textbf {h}_{\textbf {f}}} f$ and $\textbf{h}_{\textbf{f}^{\prime }} f^{\prime }$ are cohomologous for the geodesic flow $\varphi _t^{g_0}$ of $g_0$ . The quantity $\textbf{J}_{g_0}(f,f^{\prime })$ is called the renormalized intersection number.

We apply the previous proposition with $f := J^u_{g_0}$ (then ${\textbf {h}}_{J^u_{g_0}} = 1$ ) and $f^{\prime } := V_g$ . Without assuming that $g \in \mathcal {N}$ (that is, we do not necessarily assume that ${\textbf {h}}_{V_g}=1$ ), we have

$$ \begin{align*} \begin{aligned} \textbf{J}_{g_0}(J^u_{g_0},V_g) & = {\textbf{h}}_{V_g} \textbf{I}_{g_0}(J^u_{g_0},V_g) = {\textbf{h}}_{V_g} \dfrac{\int_{S_{g_0}M} (J^u_{g_0} + a_g - \textbf{1}) \, d\mu^{\textrm{L}}_{g_0}}{\int_{S_{g_0}M} J^u_{g_0} d\mu^{\textrm{L}}_{g_0}} \\ & = {\textbf{h}}_{V_g} \dfrac{{\textbf{h}}_L(g_0) + I_{\mu^{\textrm{L}}_{g_0}}(g_0,g) -1}{{\textbf{h}}_L(g_0)} \geq 1, \end{aligned} \end{align*} $$

where ${\textbf {h}}_L(g_0)$ is the entropy of the Liouville measure for $g_0$ . In the specific case where $g \in \mathcal {N}$ , ${\textbf {h}}_{V_g}=1$ and we find that $I_{\mu ^{\textrm{L}}_{g_0}}(g_0,g) \geq 1$ with equality if and only if $a_g$ is cohomologous to $1$ , that is, if and only if $L_g=L_{g_0}$ , or alternatively if and only if $\varphi ^g$ and $\varphi ^{g_0}$ are time-preserving conjugate. This computation holds as long as $J^u_{g_0} + a_g-1> 0$ (which is true in a $C^2$ -neighborhood of $g_0$ ).

In particular, on $\mathcal {N}$ , we have the linear relation

$$ \begin{align*} \textbf{J}_{g_0}(J^u_{g_0},V_g) = 1+ \dfrac{I_{\mu^{\textrm{L}}_{g_0}}(g_0,g) -1}{{\textbf{h}}_L(g_0)}. \end{align*} $$

In the notations of [Reference Bridgeman, Canary, Labourie and SambarinoBCLS15, Proposition 3.11], the second derivative computed for the family $(g_\unicode{x3bb} )_{\unicode{x3bb} \in (-1,1)} \in \mathcal {N}$ is

(3.15) $$ \begin{align} \partial_{\unicode{x3bb}}^2 \textbf{J}_{g_0}(J^u_{g_0},V_{g_\unicode{x3bb}}) |_{\unicode{x3bb} = 0} = \dfrac{1}{{\textbf{h}}_L(g_0)} \partial_{\unicode{x3bb}}^2 I_{\mu^{\textrm{L}}_{g_0}}(g_0,g_\unicode{x3bb}) |_{\unicode{x3bb} = 0} = \dfrac{ \langle \Pi_2^{g_0} \dot{g}_0, \dot{g}_0 \rangle}{4 {\textbf{ h}}_L(g_0)} \end{align} $$

and is called the pressure form. When considering a slice transverse to the $\mathcal {D}_0$ action on $\mathcal {N}$ , it induces a metric called the pressure metric by Lemma 2.1. To summarize, we have the following lemma.

Lemma 3.11. Given a smooth metric $g_0$ , the metric $G_{g_0}$ restricted to $\mathcal {N}$ can be obtained from the renormalized intersection number by

$$ \begin{align*} G_{g_0}(h,h) = 4 {\textbf{h}}_L(g_0) \partial_\unicode{x3bb}^2 \textbf{J}_{g_0}(J^u_{g_0},V_{g_\unicode{x3bb}})|_{\unicode{x3bb} = 0} \end{align*} $$

where $(g_\unicode{x3bb} )_{\unicode{x3bb} \in (-1,1)}$ is any family of metrics such that $g_\unicode{x3bb} \in \mathcal {N}$ and $\dot {g}_0 = h\in T_{g_0}\mathcal {N}$ .

3.3.3 Link with the Weil–Petersson metric

We now assume that $M=S$ is an orientable surface of genus at least $2$ . Let $\mathcal {T}(S)$ be the Teichmüller space of S. We show that the pressure metric coincides with (a multiple of) the Weil–Petersson metric in restriction to $\mathcal {T}(S)$ . We fix a hyperbolic metric $g_0$ . Given $\eta , \rho \in \mathcal {T}(S)$ and $g_\eta ,g_\rho $ the associated hyperbolic metrics, since $\mathcal {T}(S)$ is connected (indeed, a ball in $\mathbb {C}^{3({\textrm{genus}}(M)-1)}$ ) there is topological conjugacy between $g_\eta ,g_\rho $ and $g_0$ and one can defined the time rescaling $a_{g_\eta }$ and $a_{g_{\rho }}$ by using a path of hyperbolic metrics relating $g_0$ to $g_\eta $ or to $g_\rho $ . The intersection number is defined as

$$ \begin{align*} \textbf{I}(\eta,\rho) := \textbf{I}_{g_0}(a_{g_\eta},a_{g_\rho})= \frac{\int_{S_{g_0}M}a_{g_\rho}d\mu_\eta}{\int_{S_{g_0}M}a_{g_\eta}d\mu_\eta},\end{align*} $$

where $[g_\eta ]=\eta , [g_\rho ]=\rho $ and $\mu _\eta $ is the equilibrium state of $-{\textbf {h}}_{a_{g_\eta }}a_{g_{\eta }}$ . Note that ${\textbf {h}}_{a_{g_\eta }}={\textbf { h}}_{\textrm{top}}(\varphi ^{g_0,a_\eta }_t)=1$ since $\varphi ^{g_0,a_\eta }$ is conjugate to the geodesic flow of $g_\eta $ , which in turn has constant curvature, and by [Reference SambarinoSam14, Lemma 2.4], $a_{g_\eta }d\mu _{\eta }/\int _{S_{g_0}M}a_{g_\eta }d\mu _{\eta }$ is the measure of maximal entropy of the flow $\varphi ^{g_0,a_\eta }_t$ , thus also the normalized Liouville measure of $g_\eta $ (viewed on $S_{g_0}M$ ). This number $\textbf{I}(\eta ,\rho )$ is in fact independent of $g_0$ as it can alternatively be written

$$ \begin{align*}\textbf{I}(\eta,\rho)=\lim_{T\to \infty}\frac{1}{N_T(\eta)}\sum_{c\in \mathcal{C}, L_{g_\eta}(c)\leq T}\frac{L_{g_\rho}(c)}{L_{g_\eta}(c)},\end{align*} $$

where $N_T=\sharp \{c\in \mathcal {C} \, | L_{g_\eta }(c)\leq T\}$ (see [Reference Bridgeman, Canary and SambarinoBCS18, Proof of Theorem 4.3]). In particular, taking $g_0 = g_\eta $ , we have

$$ \begin{align*} \textbf{I}(\eta,\rho) = I_{\mu^{\textrm{L}}_{g_\eta}}(g_\eta,g_\rho). \end{align*} $$

As explained in [Reference Bridgeman, Canary and SambarinoBCS18, Theorem 4.3], up to a normalization constant $c_0$ depending on the genus only, the Weil–Petersson metric on $\mathcal {T}(S)$ is equal to

(3.16) $$ \begin{align} \|h\|_{\textrm{WP}}^2= c_0 \partial_{\unicode{x3bb}}^2 \textbf{I}(\eta,\eta_\unicode{x3bb})|_{\unicode{x3bb}=0} = c_0 \partial_{\unicode{x3bb}}^2 I_{\mu^{\textrm{L}}_{g_\eta}}(g_\eta,g_{\eta_\unicode{x3bb}})|_{\unicode{x3bb}=0}, \end{align} $$

where $\dot {\eta }_0=h$ and $(g_{\eta _\unicode{x3bb} })_{\unicode{x3bb} \in (-1,1)}$ is a family of hyperbolic metrics such that $[g_{\eta _\unicode{x3bb} }] = \eta _\unicode{x3bb} $ , $\eta =\eta _0 = [g_0]$ . This fact follows from combined works of Thurston, Wolpert [Reference WolpertWol86] and McMullen [Reference McMullenMM08]: the length of a random geodesic $\gamma $ on $(S,g_0)$ with respect to $g_{\eta _\unicode{x3bb} }$ has a local minimum at $\unicode{x3bb} =0$ and the Hessian is positive definite (Thurston), is equal to the Weil–Petersson norm squared of $\dot {g}$ (Wolpert [Reference Fathi and FlaminioFF93, Reference WolpertWol86]) and is given by a variance (McMullen [Reference McMullenMM08]); here random means equidistributed with respect to the Liouville measure of $g_0$ . We can check that the metric G also corresponds to this metric.

Proposition 3.12. The metric G on $\mathcal {T}(S)$ is a multiple of the Weil–Petersson metric.

Proof This follows directly from (3.15), (3.16) and the fact that ${\textbf {h}}_L(g_\eta )=1$ if $g_\eta $ has curvature $-1$ .

Remark 3.13. We notice that the positivity of the metric in the case of Teichmüller space follows only from some convexity argument in finite dimension. In the case of general metrics with negative curvature, the elliptic estimate of Lemma 2.1 on the variance is much less obvious due to the infinite dimensionality of the space. As it turns out, this is the key for the local rigidity in our results.

4.1 The resolvents of $X_g$ and anisotropic spaces

We first recall the construction of resolvents of $X_g$ from Faure and Sjöstrand [Reference Faure and SjöstrandFS11] (see also [Reference Dyatlov and ZworskiDZ16]) and, in particular, the version used in Dang et al [Reference Dang, Guillarmou, Rivière and ShenDGRS20] that deals with the continuity with respect to the flow $X_g$ . Let $E_{u/s}^*(g)\subset T^*(SM)$ be the annihilators of $E_{u/s}(g)\oplus E_0(g)$ , that is,

$$ \begin{align*} E_u^*(g)(E_u(g) \oplus E_0(g)) = 0, \quad E_s^*(g)(E_s(g) \oplus E_0(g)) = 0. \end{align*} $$

There are two resolvents bounded on $L^2$ for $X_{g}$ defined for ${\textrm{Re}}(\unicode{x3bb} )>0$ by

$$ \begin{align*} R_g^\pm (\unicode{x3bb}):= \pm \int_{0}^\infty e^{-\unicode{x3bb} t}e^{\pm tX_g}f\, dt \end{align*} $$

for $f\in L^2(SM,d\mu ^{\textrm{L}}_g)$ . They solve $(-X_g\pm \unicode{x3bb} )R_g^\pm (\unicode{x3bb} )={\textrm{Id}}$ on $L^2$ . The following results are proved in [Reference Faure and SjöstrandFS11], and we use here the presentation of [Reference Dang, Guillarmou, Rivière and ShenDGRS20, Sections 3.2 and 3.3] due to the need for uniformity with respect to g: there is $c_0>0$ depending only on g, locally uniform with respect to g ( $c_0$ depends only on the Anosov exponents of contraction/dilation of $d\varphi _1^g$ ), such that for each $N_0>0,N_1>16N_0$ , $R_g^\pm (\unicode{x3bb} )$ admits a meromorphic extension in ${\textrm{Re}}(\unicode{x3bb} )>-c_0N_0$ as a bounded operator

(4.1) $$ \begin{align} R_g^-(\unicode{x3bb}): \mathcal{H}^{m_g^{N_0,N_1}}\to \mathcal{H}^{m_g^{N_0,N_1}}, \quad R_g^+(\unicode{x3bb}): \mathcal{H}^{-m_g^{N_0,N_1}}\to \mathcal{H}^{-m_g^{N_0,N_1}} \end{align} $$

where $\mathcal {H}^{\pm m_g^{N_0,N_1}}$ are Hilbert spaces depending on $N_0>0,N_1>0$ satisfying the properties

$$ \begin{align*}H^{2N_1}(SM)\subset \mathcal{H}^{m_g^{N_0,N_1}}\subset H^{-2N_0}(SM), \quad H^{2N_0}(SM)\subset \mathcal{H}^{-m_g^{N_0,N_1}}\subset H^{-2N_1}(SM)\end{align*} $$

and defined by

$$ \begin{align*} \mathcal{H}^{\pm m_g^{N_0,N_1}}=(A_{m_g^{N_0,N_1}})^{\mp 1}L^2(SM), \quad A_{m_g^{N_0,N_1}}:={\textrm{Op}}({e^{m_g^{N_0,N_1}\log f}}), \end{align*} $$

and $A_{m_g^{N_0,N_1}}$ is an invertible pseudo-differential operator with inverse having principal symbol $e^{-m_g^{N_0,N_1}\log f}$ . Here ${\textrm{Op}}$ denotes a quantization (with a fixed small semi-classical parameter to ensure that ${\textrm{Op}}({e^{m_g^{N_0,N_1}\log f}})$ is invertible), while $m_g^{N_0,N_1}\in S^{0}(T^*(SM))$ , $f\in S^1(T^*(SM), [1,\infty ))$ (the usual classes of symbols) are homogeneous of respective degree $0$ and $1$ in $|\xi |>R$ , for some $R>1$ independent of g, and constructed from the lifted flow $\Phi _t^g=((d\varphi _t^{g})^{-1})^T$ acting on $T^*(SM)$ . The function f can be taken depending only on $g_0$ for g in a small enough $C^\infty $ neighborhood $\mathcal {U}$ of $g_0$ . Moreover, there are small conic neighborhoods $C_{u}(g_0)$ and $C_s(g_0)$ of $E^*_u(g_0)$ and $E^*_s(g_0)$ such that, for any smaller open conic neighborhood $C^{\prime }_u(g_0)\subset C_u(g_0)$ of $E_u^*(g_0)$ and $C^{\prime }_s(g_0)\subset C_s(g_0)$ of $E_s^*(g_0)$ , $m_g^{N_1,N_1}$ satisfies

(4.2) $$ \begin{align} \left\{\begin{array}{ll} m_g^{N_0,N_1}(z,\xi)\geq N_1 , & (z,\xi)\in C^{\prime}_s(g_0), \\ m_g^{N_0,N_1}(z,\xi)\geq N_1/8, & (z,\xi)\notin C_u(g_0),\\ m_g^{N_0,N_1}(z,\xi)\leq -N_0, & (z,\xi)\in C^{\prime}_u(g_0), \end{array}\right.\end{align} $$

and $m_g(x,\xi )\in [-2N_0,2N_1]$ for all $(z,\xi )\in T^*(SM)$ . We note that [Reference Dang, Guillarmou, Rivière and ShenDGRS20, Lemma 3.3] shows that $m_g^{N_0,N_1}$ is smooth with respect to the metric g and that f can be taken to be independent of g for g close enough to $g_0$ . The spaces $\mathcal {H}^{m_g^{N_0,N_1}}$ are called anisotropic Sobolev spaces. The pseudodifferential operators $A_{m_g^{N_0,N_1}}$ belong to the class $\Psi ^{2N_1}(SM)$ but also to some anisotropic subclass denoted $\Psi ^{m_g^{N_0,N_1}}(SM)$ admitting composition formulas; we refer to [Reference Faure, Roy and SjöstrandFRS08, Reference Faure and SjöstrandFS11] for details.

Eventually, [Reference Dang, Guillarmou, Rivière and ShenDGRS20, Proposition 6.1] shows that there is a small open neighborhood $W_\delta $ of the circle $\{\unicode{x3bb} \in \mathbb {C} ~|~ |\unicode{x3bb} |=\delta \}$ for some small $\delta>0$ so that

(4.3) $$ \begin{align} \mathcal{U}\times W_\delta \ni (g,\unicode{x3bb}) \mapsto A_{m_g^{N_0,N_1}}R_g^-(\unicode{x3bb})(A_{m_g^{N_0,N_1}})^{-1} \in\mathcal{L}(H^1(SM),L^2(SM)) \end{align} $$

is continuous. (In [Reference Dang, Guillarmou, Rivière and ShenDGRS20, Proposition 6.1], a small semi-classical parameter $h>0$ appears: we can just fix this parameter small enough. It does not play any role here except in the quantization procedure ${\textrm{Op}}$ . We also add that in [Reference Dang, Guillarmou, Rivière and ShenDGRS20, Proposition 6.1], $N_1$ is chosen to be equal to $20N_0$ for notational convenience, but the proof does not use that fact.)

4.2 The operator $\Pi _2^g$ in terms of resolvents

Following [Reference GuillarmouGui17], the link between $\Pi ^g$ and the resolvent is given by the Laurent expansion

$$ \begin{align*} \Pi^g= R_g^{+}(0)-R_g^{-}(0),\end{align*} $$

where $R_g^{+}(\unicode{x3bb} )$ has a pole of order $1$ , $R_g^\pm (0)$ is defined by

$$ \begin{align*} R_g^{\pm}(\unicode{x3bb})=\pm\unicode{x3bb}^{-1}\langle\cdot,\textbf{1}\rangle +R_g^\pm(0)+\mathcal{O}(\unicode{x3bb}),\end{align*} $$

and $R_g^-(0)=-(R_g^+(0))^*$ where the adjoint is with respect to the Liouville measure.

Lemma 4.2. Let $\chi \in C_c^\infty (\mathbb {R})$ be even and equal to $1$ in $[-T,T]$ and supported in the interval $(-T-1,T+1)$ . Then

(4.4) $$ \begin{align} \Pi^g &= \int_{\mathbb{R}} \chi(t) e^{tX_g}\, dt-R_g^+(0) \int_0^{+\infty} \chi^{\prime}(t) e^{tX_g}\, dt \nonumber \\ &\quad +R_g^-(0) \int_{0}^{+\infty} \chi^{\prime}(t) e^{-tX_g}\, dt -\langle \cdot,\textbf{1}\rangle \int_{\mathbb{R}} \chi. \end{align} $$

Proof For ${\textrm{Re}}(\unicode{x3bb} )>0$ , we can write, by integration by parts,

$$ \begin{align*}\begin{aligned} R_g^{\pm}(\unicode{x3bb})=& \pm \int_0^\infty \chi(t)e^{-t(\unicode{x3bb} \mp X_g)}dt\pm \int_0^\infty (1-\chi(t))e^{-t(\unicode{x3bb} \mp X_g)}dt\\ =& \pm \int_0^\infty \chi(t)e^{-t(\unicode{x3bb} \mp X_g)}dt- R_g^\pm(\unicode{x3bb})\int_0^\infty \chi^{\prime}(t) e^{t(\pm X_g-\unicode{x3bb})}dt. \end{aligned} \end{align*} $$

Then taking the limit as $\unicode{x3bb} \to 0$ , we obtain

$$ \begin{align*}R_g^{\pm}(0)=\pm \int_0^\infty \chi(t)e^{\pm tX_g}dt-R_g^{\pm}(0)\int_0^\infty \chi^{\prime}(t) e^{\pm tX_g}dt\mp\int_0^\infty \chi(t)dt \langle \cdot,\textbf{1}\rangle,\end{align*} $$

and summing gives the result.

Next, we remark that, using that $\varphi _t^g(x,-v)=-\varphi _{-t}^g(x,v)$ (where multiplication by $-1$ is the symmetry in the fibers of $SM$ ), it is straightforward to check that, for all $t\in \mathbb {R}$ ,

$$ \begin{align*} {\pi_2}_*e^{tX_g}\pi_2^*={\pi_2}_*e^{-tX_g}\pi_2^*,\end{align*} $$

which also implies that ${\pi _2}_*R_g^+(0)e^{tX_g}\pi _2^*=-{\pi _2}_*R_g^-(0)e^{-tX_g}\pi _2^*$ and thus

(4.5) $$ \begin{align} \Pi_2^g &= 2{\pi_2}_*\int_{0}^\infty \chi(t) e^{-tX_g}\, dt \pi_2^* \nonumber \\ &\quad + 2{\pi_2}_* R_g^-(0) \int_0^{+\infty} \chi^{\prime}(t) e^{-tX_g} \, dt\pi_2^* + \bigg(1-\int_{\mathbb{R}} \chi\bigg) \langle \cdot , \textbf{1} \rangle. \end{align} $$

We will prove that these three terms depend continuously on g. Note that

$$ \begin{align*} \bigg(1-\int_{\mathbb{R}} \chi\bigg) \langle f,\textbf{1}\rangle = \bigg(1-\int_{\mathbb{R}} \chi\bigg) \int_{SM}f(z)\,d\mu_g^{\textrm{L}}(z) \end{align*} $$

and thus the g-continuity of this term is immediate. Now, we claim the following result.

Lemma 4.3. There exist $T>0$ large enough and a neighborhood $\mathcal {U}^{\prime }\subset \mathcal {U}$ of $g_0$ in $\mathcal {M}^{\textrm{An}}$ so that for all $x\in M$ and all $g\in \mathcal {U}^{\prime }$ the exponential map of g in the universal cover $\widetilde {M}$ ,

$$ \begin{align*}\exp^{\tilde{g}}_x: \{v\in T_{x}\widetilde{M}; |v|_g\leq T\}\to \widetilde{M}, \end{align*} $$

is a diffeomorphism onto its image and $\Phi _{t}^g(V^*\cap \ker \iota _{X_g})\subset C^{\prime }_u(g_0)$ for all $t\geq T$ , if $\Phi _t^g:=((d\varphi _t^g)^{-1})^T$ is the symplectic lift of $\varphi _t^g$ , $V^*\subset T^*(SM)$ is the annihilator of the vertical bundle $V=\ker d\pi _0\subset T(SM)$ and $\iota _X:T^*(SM)\to \mathbb {R}$ is the contraction $\iota _{X_g}(\xi )=\xi (X_g)$ .

We also mention here as it is used in the following proof that, as a consequence of hyperbolicity,

$$ \begin{align*} V^* \cap E_s^* = V^* \cap E_u^* = \{0 \}. \end{align*} $$

This can be found in [Reference PaternainPat99, Theorem 2.50], for instance (formulated for the tangent bundle $T(SM)$ but the adaptation to $T^*(SM)$ is straightforward). The T in Lemma 4.2 will be chosen accordingly so that Lemma 4.3 is satisfied.

4.3 Proof or Proposition 4.1

Let us define

$$ \begin{align*} \Omega_1^g:= {\pi_2}_*\int_{0}^\infty \chi(t) e^{-tX_g}\, dt \pi_2^*,\quad \Omega_2^g:={\pi_2}_* R_g^-(0) \int_0^{+\infty} \chi^{\prime}(t) e^{-tX_g} \, dt\pi_2^*.\end{align*} $$

Proposition 4.1 is a consequence of the following two lemmas.

Lemma 4.4. For each $g\in \mathcal {U}^{\prime }$ , $\Omega _1^g\in \Psi ^{-1}(M;S^2T^*M)$ with principal symbol

$$ \begin{align*} \sigma(\Omega_1^g)(x,\xi)=c_n|\xi|^{-1}\pi_{\ker i_\xi}A_2^2\pi_{\ker i_\xi}\end{align*} $$

for some $c_n>0$ depending only on $n=\dim M$ and $A_2$ some positive definite endomorphism defined in Lemma 2.1, and the map $g\mapsto \Omega _1^g$ is continuous with respect to the smooth topology on $\mathcal {U}^{\prime }$ and the usual Fréchet topology on $\Psi ^{-1}(M;S^2T^*M)$ .

Proof The fact that, for each $g\in \mathcal {M}^{\textrm{An}}$ , the operator $\Omega _1^g\in \Psi ^{-1}(M;S^2T^*M)$ is proved in [Reference GuillarmouGui17, Theorem 3.5]. The computation of the principal symbol follows from the computation [Reference Sharafutdinov, Skokan and UhlmannSSU05, Reference Stefanov and UhlmannSU04] and is done in detail in our setting in [Reference Goüezel and LefeuvreGL, Theorem 4.4]. We need to check the continuity with respect to g in the $\Psi ^{-1}(M;S^2T^*M)$ topology and we can proceed as in [Reference Stefanov and UhlmannSU04, Propositions 1 and 2]. For $h\in C^\infty (M;S^2T^*M)$ , we can write explicitly in $(x_i)_i$ coordinates in the universal cover $\widetilde {M}$ near a point $p\in \widetilde {M}$ ,

$$ \begin{align*} (\Omega_1^gh(x))_{ij}=\int_{S_x\widetilde{M}}\int_0^\infty \chi(t) \widetilde{h}_{\exp^{\widetilde{g}}_x(tv)}(\partial_t\exp^{\widetilde{g}}_x(tv),\partial_t\exp^{\widetilde{g}}_x(tv)) p_{ij}(x,v) \, dtdS_x(v),\end{align*} $$

where $p_{ij}(x,v)$ are homogeneous polynomials of order $2$ in the v variable, $\widetilde {h}\in C^\infty (\widetilde {M};S^2T^*M)$ is the lift of h to the universal cover $\widetilde {M}$ , and $dS_x$ is the natural measure on the sphere $S_x\widetilde {M}$ . Using Lemma 4.3, we can perform the change of coordinates $(t,v)\in (0,T)\times S_x\widetilde {M}\mapsto y:=\exp ^{\widetilde {g}}_{x}(tv)\in \widetilde {M}$ , and we get the distance $t=d_{\widetilde {g}}(x,y)$ in $\widetilde {M}$ , and

$$ \begin{align*}dtdv=\frac{J^g_x(y)}{(d_{\widetilde{g}}(x,y))^{n-1}}d{\textrm{vol}}_g(y), \quad v=-\nabla^{\widetilde{g}}_yd_{\widetilde{g}}(x,y), \quad \partial_t\exp^{\widetilde{g}}_x(tv)=(\nabla^{\widetilde{g}}_xd_{\widetilde{g}}(x,y)), \end{align*} $$

for some $J^g_{x}(y)$ smooth in $x,y,g$ . We claim that this implies that

$$ \begin{align*} \Omega_1^gh(x)=\int_{M} K_g(x,y)h(y)\, d{\textrm{vol}}_g(y)\end{align*} $$

for some $K_g(x,y)$ which is smooth in $(g,x,y)$ outside the diagonal $x=y$ and, near the diagonal, has the form (for some $L<\infty )$

$$ \begin{align*} K_g(x,y)= \sum_{\ell=1}^L c_\ell(g,x,y)\omega_{\ell, g,x}(x-y)\end{align*} $$

with $c_\ell $ a matrix valued function, smooth in all its variables and $\omega _{\ell , g,x}(v)$ a vector-valued function smooth in $g,x$ , homogeneous of degree $-(n-1)$ in $v\in \mathbb {R}^n$ . Indeed, one can work in the universal cover $\widetilde {M}$ where $x_i$ are globally defined coordinates, so that, writing $h(x)=\sum _{i,j}h_{ij}(x)dx_idx_j$ and $p=\sum _{ij}p_{ij}(x)dx_idx_j$ , we get that $K_g(x,y)$ is a matrix with coefficients

$$ \begin{align*} (K_g(x,y))_{iji^{\prime}j^{\prime}}=\chi(d_{\widetilde{g}}(x,y)) p_{ij}(x)F_i^g(x,y)F_{j}^g(x,y)G_{i^{\prime}}^g(x,y)G_{j^{\prime}}^g(x,y)\frac{J_{x}^g(y)}{d_{\widetilde{g}}(x,y)^{n-1}} \end{align*} $$

where $F^g_i(x,y)=-dx_i(\nabla ^{\widetilde {g}}_yd_{\widetilde {g}}(x,y))$ and $G_i^g(x,y)=dx_i(\nabla ^{\widetilde {g}}_x d_{\widetilde {g}}(x,y))$ . Now we can use the standard fact (see, for example, [Reference Stefanov and UhlmannSU04, Lemma 1]) that

$$ \begin{align*}d_{\tilde{g}}^2(x,y)&=\sum_{ij}H^{1}_{ij}(g,x,y)(x-y)_i(x-y)_j, \\ dx_i(\nabla^g_x d_{\tilde{g}}(x,y))&=\frac{\sum_{ij}H^{2}_{ij}(g,x,y)(x-y)_j}{d_{\tilde{g}}(x,y)}\end{align*} $$

(and likewise for $dx_i(\nabla ^g_y d_{\tilde {g}}(x,y))$ by symmetry) where $H_{ij}^{k}(g,x,y)$ are smooth in all variables and positive definite for $x=y$ . The kernel $K_g$ is thus smooth outside the diagonal (as a function of $g,x,y$ ), and can be written near the diagonal as a sum of terms of the form $c(g,x,y)\omega _{g,x}(x-y)$ where c is smooth in all its variables and $\omega _{g,x}(v)$ is a homogeneous distribution of degree $-(n-1)$ in the variable v, smooth in $g,x$ . The off-diagonal term for the Fréchet topology is then clearly smooth in g, while the near-diagonal term has full local symbols that are Fourier transforms of $c(g,x,x-v)\omega _{g,x}(v)$ :

$$ \begin{align*} \sigma(g;x,\xi)=\int_{\mathbb{R}}e^{iv\xi}c(g,x,x-v)\omega_{g,x}(v)\,dv.\end{align*} $$

It is then a standard and easy exercise to check that this provides uniform bounds on semi-norms of the symbol. (Alternatively, the semi-norms on the full symbol are equivalent to semi-norms in the space of distributions on $M\times M$ that are conormal to the diagonal, defined through differentiations of $K_g(x,y)$ with respect to smooth fields tangent to ${\textrm{diag}}(M\times M)$ ; see [Reference MelroseMel, Ch. 5, Proposition 6.1.1 and its proof]. Such norms for $K_g$ are clearly uniformly bounded in terms of g.) We deduce the continuity (and indeed, smoothness) of $\Omega _1^g$ as an element of $\Psi ^{-1}(M;S^2T^*M)$ with respect to the metric g.