Proof. The cocycle $D\Phi ^{t}|_{E^{cu}/E^{0}}$ is $C^{1}$ by the regularity of the splitting mentioned previously and, by hypothesis $D\Phi ^{t}|_{E^{cu}/E^{0}}$ , satisfies the inequalities in the definition of fiber bunching with $\alpha = 1$ . Same for $E^{cs}$ .

Proof. Note that by the compactness of X and continuity of $\sigma ^{t}:X \to \mathbb R$ , the map $\mu \to \int \sigma ^{t} \,d{\kern-1pt}\mu $ is continuous and, hence, by

$$ \begin{align*}\int \rho \, d{\kern-1pt}\mu = \inf_{t> 0} \frac{1}{t} \int \sigma^{t} \,d{\kern-1pt}\mu \end{align*} $$

and the analogous equation for $\tau $ , we obtain upper and lower semicontinuity of $\mu \mapsto \rho _{\mu }$ .

Proof. The proof is nearly identical to that of Theorem 2.9. Namely, one uses continuity of $\mathcal {A}\mapsto \int \sigma ^{t}(\mathcal {A}) \, d{\kern-1pt}\mu $ and the subadditive ergodic theorems.

Proof. First, we would like to consider all cocycles $D\tilde {\Phi }$ constructed on $F_{\Phi }$ as existing on the same bundle over the same base map.

For $x \in \mathcal {O}$ , there exists a unique length-minimizing geodesic segment (from the Riemannian structure on N) from x to $h_{\Phi }(x)$ , as long as $h_{\Phi }$ is close to the identity, which may be ensured by reducing the neighborhood $\mathcal {U}$ to some $\mathcal {V} \subseteq \mathcal {U}$ further if needed. By parallel transport of the bundle $F_{\Phi }$ over $\mathcal {O}_{\Phi }$ along such segments, one then obtains a two-dimensional trivial bundle $F_{\Phi }^{\prime }$ over $\mathcal {O}$ . By reducing $\mathcal {V}$ further if needed, the bundle $F_{\Phi }^{\prime }$ obtained is a given by a graph over $F_{0}$ with respect to the fixed Riemannian metric on N and, hence, by orthogonal projection they may be identified.

As all maps above are continuous, the construction describes a continuous map $Th: \mathcal {U} \times F_{0} \to TN$ , so that $Th_{\Phi }(\cdot ) := Th(\Phi , \cdot )$ are bundle isomorphisms $F_{0} \to F_{\Phi }$ fibering over $h_{\Phi }$ . Hence, conjugating by $Th_{\Phi }$ , we may regard $D\tilde {\Phi }$ on $F_{\Phi }$ as a cocycle on $F_{0}$ over $\Phi _{0}$ . By continuous dependence on $\Phi $ , this defines a continuous map $\Phi \to D\tilde {\Phi }$ , where $D\tilde {\Phi }$ are now regarded as elements of the space of cocycles over $\Phi _{0}$ on $F_{0}$ with the $C^{0}$ -topology.

As all rotation numbers $\rho _{\mathcal {O}_{\tilde {\Phi }}}$ defined previously are preserved by conjugation, it suffices to check continuity of the rotation numbers of the conjugated cocycles, which is given by Proposition 2.11. Thus, the map $\Phi \mapsto \rho _{\mathcal {O}_{\tilde {\Phi }}}$ is continuous and, finally, because

$$ \begin{align*}\rho_{\mathcal{O}_{\tilde{\Phi}}} \ell(\mathcal{O}_{\Phi}) = \rho_{\mathcal{O}_{\Phi}} \ell(\mathcal{O}),\end{align*} $$

and the periods vary continuously, the map $\Phi \mapsto \rho _{\mathcal {O}_{\Phi }}$ is continuous as well.

Proof. The choice of a different section $\Sigma $ or a different point v of the orbit changes $P_{g^{\prime }}$ by conjugation, so the property that $P_{g^{\prime }} \in Q$ needs only to be assured at one fixed point and one fixed section, which is done by Theorem 2.13.

Proof. Openness follows directly from Lemma 3.1, because by the continuity of $\vec {\unicode{x3bb} }^{u}$ , the continuations of $\mathcal {O}$ will satisfy the same condition defining $\mathcal {G}_{d}^{k}$ .

For density, we start by assuming that $g_{0} \in \mathcal {G}^{\infty }_{\mathcal {O}}$ , for some $\mathcal {O}$ , which is possible by the density of $\mathcal {G}^{\infty }$ in $\mathcal {G}^{k}$ . It remains to check that the property defining $\mathcal {G}_{d}^{k}$ is indeed an open dense in $J^{k-1}_{s}$ , so that we may apply Corollary 2.16 to finish the proof. By Remark 2.17, it suffices to check that having eigenvalues distinct with distinct norms, apart from complex conjugate pairs, is an open and dense Sp $(2n)$ .

Openness is clear, because the eigenvalues depend continuously on the matrix entries. For density, we note that the condition of distinct eigenvalues is given by the complement of the equation $\Delta = 0$ , where $\Delta $ is the discriminant of the characteristic polynomial, which is a non-empty Zariski open set in Sp $(2n)$ and, thus, dense in the analytic topology. In particular, the set of diagonalizable matrices is dense. As diagonalizable matrices are symplectically diagonalizable, by the lemma following this proof, by a small perturbation on the norm of the diagonal blocks, we obtain the density of eigenvalues of distinct norms.

Proof. Recall that eigenvalues of $A \in \text {Sp}(2n)$ appear in 4-tuples

$$ \begin{align*} \{\unicode{x3bb}, \overline{\unicode{x3bb}}, \unicode{x3bb}^{-1}, \overline{\unicode{x3bb}}^{-1}\} \end{align*} $$

for $\unicode{x3bb} \notin \mathbb R$ and in pairs $\{\unicode{x3bb} , \unicode{x3bb} ^{-1}\}$ for $\unicode{x3bb} \in \mathbb R$ . For each $\unicode{x3bb} $ , we let $E_{\unicode{x3bb} } = E_{\unicode{x3bb} ^{-1}}$ be the two-dimensional subspace spanned by the eigenspaces of $\unicode{x3bb} $ and $\unicode{x3bb} ^{-1}$ .

Let $\omega $ be the canonical symplectic form on $\mathbb R^{2n}$ and extend $\omega $ and A to $\omega _{\mathbb {C}}$ and $A_{\mathbb {C}}$ in the complexification $\mathbb {C}^{2n} = \mathbb R^{2n} \otimes \mathbb {C}$ . By definition, $A_{\mathbb {C}}$ and $\omega _{\mathbb {C}}$ agree with A and $\omega $ on $\mathbb R^{2n} \otimes 1$ .

The identity for eigenvectors $v_{\unicode{x3bb} }$ and $v_{\eta }$ :

$$ \begin{align*}\omega_{\mathbb{C}}(v_{\unicode{x3bb}}, v_{\eta}) = \omega_{\mathbb{C}}(A_{\mathbb{C}} v_{\unicode{x3bb}}, A_{\mathbb{C}} v_{\eta}) = \unicode{x3bb}\eta\, \omega_{\mathbb{C}}(v_{\unicode{x3bb}}, v_{\eta}),\end{align*} $$

implies that, unless $\unicode{x3bb} \eta = 1$ , we have $\omega _{\mathbb {C}}(v_{\unicode{x3bb} }, v_{\eta }) = 0$ . Therefore, $E_{\unicode{x3bb} } \otimes \mathbb {C}$ is symplectically orthogonal to $E_{\eta } \otimes \mathbb {C}$ for any $\unicode{x3bb} \neq \eta , \eta ^{-1}$ .

In particular, this implies that the $E_{\unicode{x3bb} } \otimes 1$ are symplectic subspaces with respect to $\omega $ the real form, and symplectically orthogonal to each other. In each $E_{\unicode{x3bb} }$ , A can be put in Jordan real form with respect to a symplectic basis. By orthogonality, we may construct a symplectic basis for $\mathbb R^{2n}$ by taking the union of symplectic bases for the $E_{\unicode{x3bb} }$ . Then, let P be the matrix which sends the standard $\mathbb R^{2n}$ basis to the constructed symplectic basis.

Proof. Fix a $C^{2}$ -open set $\mathcal {U} \subseteq \mathcal {G}^{k}$ . First, because $\mathcal {G}_{d}^{k}$ is $C^{2}$ -open and dense and $\mathcal {G}^{\infty }$ is $C^{k}$ -dense in $\mathcal {G}^{k}$ , we may fix some $g_{0} \in \mathcal {U} \cap \mathcal {G}_{d}^{\infty }$ . Let $\mathcal {O}$ be as in Proposition 3.2.

Suppose that the vector $\vec {\unicode{x3bb} }^{u}(\mathcal {O}, g)$ has $2c$ entries in $\mathbb {C} \setminus \mathbb R$ , for some $c> 0$ . It suffices to show that there exists a metric $g^{\prime }$ in $\mathcal {U}$ that has a periodic orbit $\mathcal {O}^{\prime }$ such that $\vec {\unicode{x3bb} }^{u}(\mathcal {O}^{\prime }, g^{\prime })$ has $2(c-1)$ complex entries and all real entries distinct.

Along $\mathcal {O}$ there is a dominated splitting $E^{u} = E^{-}_{1} \oplus \cdots \oplus E^{-}_{k}$ such that each $E_{i}$ is either one- or two-dimensional. Fix the smallest index $i \in \{1,\ldots , k\}$ such that $E^{\pm }_{i}$ is two-dimensional and let $P_{g_{0}}$ denote the Poincaré return map of the geodesic flow for a fixed section $\Sigma $ transverse to the flow small enough so that $\mathcal {O} \cap \Sigma =:\{v\}$ . By shrinking $\mathcal {U}$ further if needed we may assume that $\mathcal {U} \subseteq \mathcal {G}_{d}^{k}$ , that is, that the dominated splitting for $\varphi _{g_{0}}$ along $\mathcal {O}$ persists for the continuation of $\mathcal {O}$ for all $g \in U$ ; thus, by Lemma 3.1, the map $g \mapsto \theta _{g}:= |\text {arg}(\unicode{x3bb} _{g})|$ is well defined and continuous, where $\unicode{x3bb} _{g}$ is an eigenvalue of $D\varphi _{g}$ on $E^{-}_{i}$ on the continuation of $\mathcal {O}$ .

Let $g_{1}$ be given as in the lemma and, for $0 \leq s \leq 1$ , we let $g_{s} = sg_{1} + (1-s)g_{0}$ , which, if $g_{1}$ is taken sufficiently close to $g_{0}$ , also satisfies $\{g_{s}\} \subseteq \mathcal {U} \cap \mathcal {G}_{\mathcal {O}}^{k}$ . Clearly, the map $[0,1] \to \mathcal {G}^{k}$ given by $s \mapsto g_{s}$ is continuous. In addition, note that, by Proposition 2.15(2), $\mathcal {O}$ is not only a closed orbit of $\varphi _{g_{s}}$ for all $s \in [0,1]$ , but it, in fact, has the same arc-length parametrization with respect to all $g_{s}$ .

For the geodesic flow of $g_{0}$ , fix w a transverse homoclinic point of v, that is, $w \in W^{u}(v) \cap W^{cs}(v)$ . Fix some $\varepsilon> 0$ so that the geodesic flow has local product structure at scale $2\varepsilon $ . Then there exists $t_{1}, t_{2}> 0$ such that $\varphi ^{-t_{2}}_{g_{0}}(w) \in W^{u}_{\varepsilon }(v)$ , $\varphi ^{t_{1}}_{g_{0}}(w) \in W^{s}_{\varepsilon }(v)$ and also a $C> 0$ such that, for all $t> 0$ ,

$$ \begin{align*} d(\varphi^{-(t_{2}+t)}_{g_{0}}(w), \varphi_{g_{0}}^{-t}(v)) < C\varepsilon e^{-t},\end{align*} $$

$$ \begin{align*}d(\varphi^{t_{1}+t}_{g_{0}}(w), \varphi_{g_{0}}^{t}(v)) < C\varepsilon e^{-t}. \end{align*} $$

Hence, for $n \in {\mathbb N}$ the $\gamma _{n}: \mathbb R \to SM$ given by

$$ \begin{align*}\gamma_{n}(t) = \varphi^{\tilde{t}-(t_{2}+n\ell)}_{g_{0}}(w)\quad \text{where }\tilde{t} = t \text{ mod } (t_{2} + t_{1} + 2n \ell)\end{align*} $$

are $\varepsilon _{n}$ -pseudo-orbits where $\varepsilon _{n} < 2C\varepsilon e^{-n\ell }$ , by the fact that the minimal expansion of the geodesic flow is $\tau = 1$ by the assumption on curvature.

For n sufficiently large, there exist unique periodic $w_{n}$ which $L\varepsilon _{n}$ -shadow $\gamma _{n}$ . Let $w_{n,s}$ be continuations of $w_{n}$ for $0 \leq s \leq 1$ (where $w_{n,0} = w_{n}$ , by definition). Let $w_{s}$ be the hyperbolic continuations of w. By uniqueness of shadowing, note that the $w_{n,s}$ can also be constructed by shadowing segments of the orbit of $w_{s}$ . The following lemma shows we can extend the dominated splitting of $\mathcal {O}$ to the new orbits we defined.

Lemma 3.7. There exists N large so that for each $0 < s < 1$ , the compact invariant set

$$ \begin{align*}K_{N,s} = \bigcup_{n \geq N} \mathcal{O}(w_{n,s}) \cup \mathcal{O}(w_{s}) \cup \mathcal{O},\end{align*} $$

for the geodesic flow $\varphi _{g_{s}}$ of $g_{s}$ admits a dominated splitting for the bundle $E^{u} = E^{-}_{s,1} \oplus \cdots \oplus E^{-}_{s,k}$ over $K_{m,s}$ coinciding with the dominated splitting of $E^{u}$ over $\mathcal {O}$ , and similarly for $E^{s} = E^{+}_{s,1} \oplus \cdots \oplus E^{+}_{s,k}$ .

Proof sketch (see [Reference Bonatti and VianaBV04], Lemma 9.2)

We sketch the proof for $s = 0$ , which is almost identical to the result cited. Then, because dominated splittings over compact invariant sets persists under $C^{1}$ -small perturbations by an invariant cone argument, this shows the result for all $s \in [0,1]$ . Consider the case of $E^{u}$ .

As $w\in W^{u}(v) \cap W^{cs}(v)$ , one can extend the dominated splitting of $\mathcal {O}$ to $\mathcal {O}(w)$ as follows. Consider the bundles over $\mathcal {O}$ given by $F^{i} = E^{-}_{1} \oplus \cdots \oplus E^{-}_{j+1}$ , and $G^{i} = E^{-}_{j} \oplus \cdots \oplus E^{-}_{k}$ for $i,j= 1, \ldots , k-1$ . Then we define

$$ \begin{align*}E^{j}(w) : = \phi^{cs}_{v, w} F^{j}(v) \cap \phi^{cu}_{v,w}G^{j}(v)\end{align*} $$

and extend the $E^{j}$ bundles to $\mathcal {O}(w)$ by the derivative of the flow. Proof of continuity and domination of this splitting follows closely that in [Reference Bonatti and VianaBV04].

For N sufficiently large we observe that $K_{N,0}$ is contained in an arbitrarily small neighborhood of $\mathcal {O} \cup \mathcal {O}(w)$ , so the dominated splitting extends by continuity.

For each n, we let $\theta _{n}: [0,1] \to S^{1} = \mathbb R/2\pi \mathbb {Z}$ be defined by setting $\theta _{n}(s)$ to be the argument of the eigenvalue of $D\varphi _{g_{s}}$ along $E_{s,i}^{-}$ on the closed orbit $w_{n,s}$ . By Lemma 3.1, the $\theta _{n}$ are continuous so for each n they may be lifted to some $\tilde {\theta }_{n}: [0,1] \to \mathbb R$ .

The main result about these rotation numbers, whose proof is postponed to the next section due to its length, is:

By continuity, one then finds $n,s$ such that $\tilde {\theta }_{n}(s)$ is an integer multiple of $2\pi $ , that is, such that the eigenvalues in $\vec {\unicode{x3bb} }^{u}(\mathcal {O}(w_{n,s}), g_{s})$ corresponding to the subspace $E^{-}_{i,s}$ are real. By another perturbation using Corollary 2.16, there exists a metric such that these eigenvalues become distinct. Then, by induction on the other eigenspaces with complex eigenvalues, all eigenvalues are real and distinct.

To finish the proof, openness follows again by Lemma 3.1, because the requirements on the products of the eigenvalues is an open condition.

Proof of Lemma 3.8

Fix N large enough so that $K_{N,s}$ satisfies the conclusion of Lemma 3.7. We begin with the following result.

Proof. First, note that it suffices to prove that $E_{s}$ is trivializable for $s = 0$ , because the bundles $E_{s}$ vary continuously in the ambient space $TSM$ as s varies.

We construct a non-vanishing section of the frame bundle F associated with $E_{0}$ over some $K_{N^{\prime },0}$ for $N^{\prime }$ large, which is equivalent to a continuous choice of basis for $E_{0}$ , proving triviality of the bundle.

For $\delta> 0$ , let $B_{\delta }(\mathcal {O})$ a $\delta $ -tubular neighborhood of $\mathcal {O}$ . If $\delta $ is sufficiently small relative to the scale of local product structure of the Anosov flow, for all $n \geq N^{\prime }$ , and $N^{\prime }$ sufficiently large, $\mathcal {O}(w_{n})_{\delta } := B_{\delta }(\mathcal {O}) \cap \mathcal {O}(w_{n})$ consists of a connected segment of the embedded circle $\mathcal {O}(w_{n})$ and, moreover, $\mathcal {O}(w)_{\delta } := B_{\delta }(\mathcal {O}) \cap \mathcal {O}(w)$ consists of the complement of a connected closed interval in $\mathcal {O}(w)$ , that is, two immersed connected components (see Figure 1).

Figure 1 Proof of Proposition 3.9. The closed orbit $\mathcal {O}$ is schematically represented by the black dot.

Note that we may assume that the map $D\varphi _{g_{0}}^{\ell (w_{n})}$ preserves orientation on $E_{0}$ over any periodic orbit $\mathcal {O}(w_{n})$ , because otherwise it would have real eigenvalues (any $A \in \text {GL}(2, \mathbb R)$ with negative determinant has real eigenvalues) and we would obtain a proof of Lemma 3.8. Hence, the bundle $E_{0}$ is trivializable over any $\mathcal {O}(w_{n})$ . It is also clearly so over $\mathcal {O}(w)$ , because it is an immersed real line, and we may assume this is also true for $\mathcal {O}$ , because otherwise, again, we would have real eigenvalues.

By shrinking $\delta $ further if necessary, there exists a well-defined closest point projection $p: B_{\delta }(\mathcal {O}) \to \mathcal {O}$ which is a surjective submersion. Fix a trivialization of $E_{0}$ over $\mathcal {O}$ , that is, a non-vanishing section $S: \mathcal {O} \to F$ , which is possible by the previous paragraph.

For $x \in B_{\delta }(\mathcal {O}) \cap K_{N^{\prime }, 0} =: K_{\delta }$ , again shrinking $\delta $ further if necessary, there exists a unique length-minimizing geodesic segment between x and $p(x)$ , and by parallel transporting $E_{0}(p(x))$ along such segments and then projecting orthogonally onto $E_{0}(x)$ one obtains a continuous bundle map $E_{0}|_{K_{\delta }} \to E_{0}|_{\mathcal {O}}$ which is an isomorphism on fibers. This map induces a map $F|_{K_{\delta }} \to F_{\mathcal {O}}$ and so by pulling back the non-vanishing section $S: \mathcal {O} \to F$ we obtain a non-vanishing section, which we now denote by $S: K_{\delta } \to F$ because its restriction to $\mathcal {O}$ agrees with the previous S, of F over $K_{\delta }$ .

Recall that $\mathcal {O}(w)_{\delta }$ consists of two connected immersed components homeomorphic to $\mathbb R$ . As $\mathcal {O}(w)$ is contractible, it is possible to define a determinant on $F|_{\mathcal {O}(w)}$ ; up to scalar it is unique and, hence, there is a well-defined continuous sign function on each fiber. Then we claim that $S|_{\mathcal {O}(w)_{\delta }}$ has the same determinant sign on both components, so that it may be extended to a continuous section $\mathcal {O}(w) \to F|_{\mathcal {O}(w)}$ . Suppose this is not the case for a contradiction.

Define the line bundle $L:= \bigwedge ^{2} E^{0}$ over K, which restricted to individual orbits is trivial since $E_{0}$ is. At each point $x \in K$ there is a natural map $F(x) \to L(x)$ given by $(e_{1}, e_{2}) \mapsto e_{1} \wedge e_{2}$ , which extends to a continuous global map $W: F \to L$ . Considering the image of $S|_{\mathcal {O}(w)_{\delta }}$ under W, we obtain a section $\mathcal {O}(w)_{\delta } \to L$ , which has opposite signs in the two connected components. Let $B: \mathcal {O}(w) \to L$ be any extension of this section to all of $\mathcal {O}(w)$ ; by the previous remark, B must have an odd number of zeros.

By continuity,

$$ \begin{align*} \mathcal{O}(w_{n}) \setminus B_{\delta/2}(\mathcal{O}) \to \mathcal{O}(w) \setminus B_{\delta/2}(\mathcal{O}) \end{align*} $$

as $n \to \infty $ , so by continuity of the bundle for n sufficiently large, we can parallel transport the section B on $\mathcal {O}(w) \setminus B_{\delta /2}(\mathcal {O})$ to $\mathcal {O}(w_{n}) \setminus B_{\delta /2}(\mathcal {O})$ to obtain a section $B_{n}$ on $\mathcal {O}(w_{n}) \setminus B_{\delta /2}(\mathcal {O})$ that has the same number of zeros as B on $\mathcal {O}(w) \setminus B_{\delta /2}(\mathcal {O})$ , that is, oddly many.

On the other hand, as $n \to \infty $ ,

$$ \begin{align*}B_{n}|_{\mathcal{O}(w_{n})_{\delta} \setminus B_{\delta/2}(\mathcal{O})} \to (W\circ S)|_{\mathcal{O}(w_{n})_{\delta} \setminus B_{\delta/2}(\mathcal{O})}\end{align*} $$

and, hence, for n large enough, $B_{n}$ has constant sign on $\mathcal {O}(w_{n})_{\delta } \setminus B_{\delta /2}(\mathcal {O})$ . Thus, $B_{n}$ extends to $\mathcal {O}(w_{n})_{\delta }$ without any zeros. Hence, we obtain a global section $B_{n}$ on $\mathcal {O}(w_{n})$ with an odd number of zeros, contradicting the triviality of L over $\mathcal {O}(w_{n})$ .

Hence, we may extend $S|_{\mathcal {O}(w)_{\delta }}$ continuously to all of $\mathcal {O}(w)$ . Since in $K_{\delta }$ the section S is continuous, and again $\mathcal {O}(w_{n}) \setminus B_{\delta /2}(\mathcal {O}) \to \mathcal {O}(w) \setminus B_{\delta /2}(\mathcal {O})$ as $n \to \infty $ , we can then continuously extend $S|_{\mathcal {O}(w)\setminus B_{\delta /2}(\mathcal {O})}$ to $\mathcal {O}(w_{n}) \setminus B_{\delta /2}(\mathcal {O})$ while agreeing with S in $K_{\delta }$ . As $S|_{\mathcal {O}(w)}$ is non-vanishing, the section obtained in this way is also globally non-vanishing.

The projectivization $\mathcal {P} E_{s}$ of the bundle $E_{s}$ then defines a trivial circle bundle over $K_{N,s}$ , and we fix a trivializing bundle isomorphism $\phi _{s}: \mathcal {P} E_{s} \to K_{N,s} \times S^{1}$ . By conjugating with $\phi _{s}$ , the derivative of the geodesic flow then defines a continuous cocycle $\mathcal {A}_{s}$ on $K_{N,s} \times S^{1}$ over the geodesic flow, so we may apply the results of §2.3 for $\mathcal {A}_{s}$ .

Then the rotation numbers have the following characterization over periodic orbits.

Lemma 3.10. For a closed orbit $\mathcal {O}(u)$ of a point $u \in K_{N,s}$ , the argument $\theta (u)$ of the eigenvalue of the return map of the geodesic flow on $E^{-}_{s,i}$ satisfies

$$ \begin{align*}\theta(u) = \ell(u) \cdot \rho_{\mathcal{O}(u)} \, (\mathrm{mod} \, \, 2\pi),\end{align*} $$

where $\ell (u)$ is the period of u, and $\rho _{\mathcal {O}(u)}$ is as in Remark 2.10 for the cocycle $\mathcal {A}_{s}$ defined above.

Proof. On the one hand, it follows from the definition of $\rho $ that $ \ell (u) \cdot \rho _{\mathcal {O}(u)}$ agrees mod $2\pi $ with the Poincaré rotation number for the map

$$ \begin{align*}(A_{s})_{u}^{\ell(u)}: S^{1} \to S^{1}.\end{align*} $$

On the other hand, the projectivization of the derivative of the flow also defines on the fiber a homeomorphism $S^{1} \to S^{1}$ with Poincaré rotation number equal to the argument of the eigenvalue of the derivative.

As these two statements differ by a conjugation given by $\pi _{2}\circ \phi _{s}(u, \cdot ): S^{1} \to S^{1}$ , where $\pi _{2}: K_{N,s} \times S^{1}$ is the natural projection, by invariance we obtain the result.

Applying Lemma 3.10 to the $\theta _{n}(s)$ , we obtain, for $0 \leq s \leq 1$ ,

$$ \begin{align*} \theta_{n}(s) = \ell(w_{n,s})\rho_{\mathcal{O}(w_{n,s})}\, (\text{mod}\, 2\pi). \end{align*} $$

By continuity of the functions $\theta _{n}$ , we may lift them to $\tilde {\theta }_{n}: [0,1] \to \mathbb R$ satisfying $\tilde {\theta }_{n}(0) = \ell (w_{n,0})\rho _{\mathcal {O}(w_{n,0})}.$ By the continuity of $\rho _{\mathcal {O}(w_{n,s})}$ in s, given by Proposition 2.12, our choice of lift then implies

$$ \begin{align*}\tilde{\theta}_{n}(s) = \ell(w_{n,s})\rho_{\mathcal{O}(w_{n,s})} \quad\textrm{for}\ 0 \leq s \leq 1. \end{align*} $$

Let $\theta (s)$ be the argument of the eigenvalue of the $D\varphi _{g_{s}}$ on $E^{-}_{i,s}$ on the periodic orbit $\mathcal {O}$ (recall $\mathcal {O}$ is a closed geodesic for all $g_{s}$ with $\ell (\mathcal {O})$ fixed), and repeat the constructions above to obtain $\tilde {\theta }(s)$ as well satisfying

(3.1) $$ \begin{align} \tilde{\theta}(s) = \ell(\mathcal{O}) \rho_{\mathcal{O}}(s), \end{align} $$

where $\rho _{\mathcal {O}}(s)$ is $\rho _{\mathcal {O}}$ of the geodesic flow of $g_{s}$ .

As $\mu _{\mathcal {O}(w_{n,s})} \to \mu _{\mathcal {O}}$ (where $\mu _{\mathcal {O}}$ is the invariant probability measure supported on the closed orbit $\mathcal {O}$ ) we have $\rho _{\mathcal {O}(w_{n,s})} \to \rho _{\mathcal {O}}(s)$ as $ n \to \infty $ by Theorem 2.9. By hypothesis, $\theta (1) \neq \theta (0)$ and because $\ell (\mathcal {O})$ is constant as s varies, equation (3.1) gives that $\rho _{\mathcal {O}}(1) - \rho _{\mathcal {O}}(0) \neq 0$ . Hence, for n large enough, there exists some $\delta> 0$ such that $|\rho _{\mathcal {O}(w_{n,1})}- \rho _{\mathcal {O}(w_{n,0})}| \geq \delta $ .

Finally, let $\delta _{n} = |\ell (w_{n,1}) - \ell (w_{n,0})|$ . Again, we defer the proof of the following final proposition we need.

With Lemma 3.11, we complete the proof of Lemma 3.8:

$$ \begin{align*} \begin{aligned} |\tilde{\theta}_{n}(1) - \tilde{\theta}_{n}(0)| &= |\ell(w_{n,1})\tilde{\rho}_{\mathcal{O}(w_{n,1})} - \ell(w_{n,0})\tilde{\rho}_{\mathcal{O}(w_{n,0})}| \\[3pt] &\geq |\ell(w_{n,1})(\tilde{\rho}_{\mathcal{O}(w_{n,1})} - \tilde{\rho}_{\mathcal{O}(w_{n,0})}) | \\[3pt] &\quad- |(\ell(w_{n,1})-\ell(w_{n,0}))\tilde{\rho}_{\mathcal{O}(w_{n,0})} |\\[3pt] &\geq \delta \ell(w_{n,1}) - \delta_{n} |\tilde{\rho}_{\mathcal{O}(w_{n,0})} | \\[3pt] &> \delta \ell(w_{n,1}) - M {}_{1}M_{2}> 2\pi \end{aligned} \end{align*} $$

for all n sufficiently large, because $\ell (w_{n,1}) \to \infty $ .

Proof of Proposition 3.11

To bound the variations $\delta _{n}$ , we use exponential shadowing and Hölder continuity of the geodesic stretch, defined in the following. As the geodesic flow is unperturbed on $\mathcal {O}$ and the orbits $\mathcal {O}(w_{n,s})$ approximate $\mathcal {O}$ , the two mentioned properties give us the bound on $\delta _{n}$ .

Recall that the $w_{n}$ are constructed by shadowing $\gamma _{n}: \mathbb R \to SM$ given by

$$ \begin{align*}\gamma_{n}(t) = \varphi^{\tilde{t}-(t_{2}+n\ell)}_{g_{0}}(w), \quad\text{where }\tilde{t} = t \text{ mod } (t_{2} + t_{1} + 2n \ell), \end{align*} $$

which is a $\varepsilon _{n}$ -pseudo-orbit, where $t_{1}$ (respectively, $t_{2}$ ) is such that $\varphi ^{t_{1}}_{g_{0}}(w)$ (respectively, $\phi ^{-t_{2}}_{g_{0}}(w)$ ) is in $W^{s}_{\varepsilon }(v)$ (respectively, $W^{u}_{\varepsilon }(v)$ ) and $\varepsilon _{n} < 2C\varepsilon e^{-n\ell }$ .

The following well-known theorem is an adaptation for flows of the usual ‘exponential’ shadowing theorem, which uses the Bowen bracket in its proof. The statement gives a sharper estimate on how well shadowing orbits approximate pseudo-orbits.

In the context of the current proof, we apply the above theorem as follows.

Let $T_{n} = \ell (w_{n})$ , $x = \varphi ^{T_{n}/2}_{g_{0}} (w_{n})$ , and $y = \varphi ^{\tau _{n}}_{g_{0}} (w)$ where $\tau _{n} := T_{n}/2 - (t_{2} + n\ell )$ . For n sufficiently large, $d(\varphi ^{t}_{g_{0}}(x),\varphi ^{s(t)}_{g_{0}}(y)) < \delta $ is satisfied, by the statement of shadowing, for $|t| < T_{n}/2$ and $\delta $ given by the theorem for the $\varepsilon> 0$ fixed before. Then the theorem gives a $t_{n} \in \mathbb R$ such that

$$ \begin{align*} d(\varphi^{t_{n}+t}_{g_{0}}(w_{n}), \varphi^{\tau_{n} + t}_{g_{0}} (w) ) <c e^{\eta(T_{n}/2-|t|)} \quad\text{for } |t| < T_{n}/2.\end{align*} $$

Now we turn to computing the period of $w_{n,1}$ using the facts established previously. By structural stability, there exists $h: SM \to SM$ which conjugates the orbits of $\varphi _{g_{0}}$ to those of $\varphi _{g_{1}}$ . This conjugacy can be taken to be Hölder continuous and $C^{1}$ along the flow direction. Thus, there exists some $a: SM \to \mathbb R$ that is Hölder continuous with some exponent $1 \geq \beta> 0$ , such that for $u \in SM$ :

$$ \begin{align*}dh (u) X_{g_{0}}(u) = a(u) X_{g}(h(u)),\end{align*} $$

where $X_{g}$ (respectively, $X_{g_{0}}$ ) is the vector field generating the geodesic flow for g (respectively, $g_{0}$ ). The function a is referred to as the geodesic stretch, and the proof of the facts above can be found, for instance, in [Reference Guillarmou, Knieper and LefeuvreGKL19, pp. 12–13]

The period of $w_{n,1}$ is given by the formula

$$ \begin{align*}\ell(w_{n,1}) = \int_{0}^{T_{n}} a(\varphi^{t}_{g_{0}}(w_{n})) \,dt\end{align*} $$

By Proposition 2.15(2), because $\mathcal {O}$ is a closed geodesic, with same arclength parametrization for $g_{0}$ and $g_{1}$ , it is clear that $a|_{\mathcal {O}} \equiv 1$ . Therefore, we may compute the difference $\delta _{n} = |\ell (w_{n,1}) - \ell (w_{n,0})|$ as follows:

$$ \begin{align*} \begin{aligned} |\ell(w_{n,1}) - \ell(w_{n,0})| &\leq \int_{-T_{n}/2}^{T_{n}/2} |a(\varphi^{t}_{g_{0}}(w_{n})) - 1| \, dt \\[6pt] &\leq M \int_{-T_{n}/2}^{T_{n}/2} d(\varphi^{t_{n}+t}_{g_{0}}(w_{n}), \mathcal{O})^{\beta} \, dt, \end{aligned}\end{align*} $$

because a is $\beta $ -Hölder continuous and the distance between a point and a compact set is well defined. To estimate the distance, note

$$ \begin{align*}\begin{aligned} d(\varphi^{t_{n}+t}_{g_{0}}(w_{n}), \mathcal{O}) &\leq d(\varphi^{t_{n}+t}_{g_{0}}(w_{n}), \varphi^{\tau_{n}+ t}_{g_{0}} (w) ) + d(\varphi^{\tau_{n}+ t}_{g_{0}} (w) , \mathcal{O}) \\[3pt] &\leq c (e^{\eta(T_{n}/2-|t|)} + e^{-|t|}) \quad\text{for } |t| < T_{n}/2, \end{aligned}\end{align*} $$

because w is a homoclinic point of $\mathcal {O}$ so $d(\varphi ^{\tau _{n}+ t}_{g_{0}} (w) , \mathcal {O}) \leq ce^{-|t|}$ for some $c> 0$ which we assume, by taking the maximum if necessary, is the same as the previous c. Substituting this inequality into the previous integral, we obtain

$$ \begin{align*}|\ell(w_{n,1}) - \ell(w_{n,0})| \leq M\int_{-T_{n}/2}^{T_{n}/2} (e^{\eta(T_{n}/2-|t|)} + e^{-|t|})^{\beta} \, dt < M_{2} < \infty, \end{align*} $$

for $M_{2}$ independent of n, as an easy calculus exercise shows.

Proof. Again, by the density of $\mathcal {G}^{\infty } \subseteq \mathcal {G}^{k}$ and openness of $\mathcal {G}_{p}^{k}$ we may assume that $g_{0} \in \mathcal {G}^{\infty }$ so we can apply Theorem 2.13. For some small $\varepsilon> 0$ , consider the geodesic segment $\gamma = \varphi ^{g_{0}}_{[0, \varepsilon ]}(w)$ . Note that because $\mathcal {O}(w)$ accumulates as $|t| \to \infty $ on the compact set $\mathcal {O}$ , if we take $\varepsilon> 0$ small enough we may take $\pi (\gamma )$ to be disjoint from $\pi (\mathcal {O}(w) \setminus \gamma ) \cup \pi (\mathcal {O})$ , where $\pi : SM \to M$ is the projection map.

Then we apply Theorem 2.13 to $\gamma ^{\prime } \subseteq \gamma $ , where $\gamma ^{\prime } = \varphi ^{g_{0}}_{[\delta , \varepsilon -\delta ]}(w)$ for $\delta> 0$ small, to perturb $D_{w} \varphi ^{\varepsilon }_{g_{0}}$ by perturbing the metric only on a tubular neighborhood $V_{\gamma ^{\prime }}$ of $\gamma ^{\prime }$ small enough (possible by Proposition 2.15(1)) so that

$$ \begin{align*}V_{\gamma^{\prime}} \cap \text{Cl}(\pi(\mathcal{O}) \cup \pi(\mathcal{O}(w) \setminus \gamma)) = \varnothing,\end{align*} $$

where Cl denotes closure.

By equivariance of holonomies, the map $\psi ^{g_{0}}_{v,w}$ can be rewritten as

$$ \begin{align*}\psi^{g_{0}}_{v,w} = h^{s}_{\varphi^{\varepsilon}_{g_{0}}(w),v} \circ D_{w}{\varphi}_{g_{0}}^{\varepsilon}|_{E_{u}} \circ h^{cu}_{v,w}. \end{align*} $$

Then observe that perturbations to the metric of the form described in the previous paragraph affect only the $D_{w}{\varphi }_{g_{0}}^{\varepsilon }|_{E_{u}}$ term in the composition above. Indeed, we recall that $h^{cu}_{w,v}$ depends only on the values of the cocycle on a neighborhood of the $(-\infty , 0]$ part of the orbit $\varphi ^{t}_{g_{0}}(w)$ , and $h^{s}_{\varphi ^{\varepsilon }_{g_{0}}(w),v}$ on a neighborhood of the $[\varepsilon , \infty )$ part of the orbit $\varphi ^{t}_{g_{0}}(w)$ and on the cocycle along $\mathcal {O}$ . By construction of $V_{\gamma ^{\prime }}$ , the cocycle is not perturbed in any of these sets.

It remains to check that for an open and dense set of $1$ -jets of symplectic maps P from a small transversal to the flow at w to a small transversal section to the flow at $\varphi _{g_{0}}^{\varepsilon }(w)$ , the map $\psi ^{g_{0}}_{v,w}$ has the twisting property (we assume both transversals to be tangent to $E^{u}$ at w and at $\varphi _{g_{0}}^{\varepsilon }(w)$ , respectively), if we replace $D_{w}{\varphi }_{g_{0}}^{\varepsilon }|_{E_{u}}$ by $DP|_{E^{u}}$ . This implies, by Theorem 2.13, that we can construct such a small perturbation in the space of metrics, completing the proof.

As both holonomy maps in the composition defining $\psi ^{g_{0}}_{v,w}$ as above are symplectic isomorphisms, an open and dense subset of $\text {Sp}(E^{u}(v) \oplus E^{s}(v))$ is mapped under composition with the holonomies to an open dense set of the $1$ -jets of symplectic maps P as above, so it suffices to check that twisting holds when the map $\psi _{v,w}^{g_{0}}$ takes value in an open and dense subset of $\text {Sp}(E^{u}(v) \oplus E^{s}(v))$ .

Again, observe that the condition defining twisting is given by a Zariski open subset of the matrices $\text {Sp}(E^{u}(v) \oplus E^{s}(v))$ . Hence, as long this set is non-empty the twisting set must also be open and dense in the analytic topology. Then, by the paragraph above, this translates to an open and dense condition in $1$ -jets of symplectic maps P, and as there is no condition imposed on higher jets, we obtain the desired result by Remark 2.17.

To finish the proof, it thus suffices to check that the Zariski open set defining twisting is non-empty in the symplectic group, which is done in the following.

Proof. Note that for fixed $e^{k}_{I}, e^{j}_{I^{\prime }}$ , the property that $(\wedge ^{j} A)(e^{j}_{I}) \wedge e^{l}_{I^{\prime }} \neq 0$ is open in Sp $(2n)$ . Thus, by induction, it suffices to show that for any $e^{j}_{I}, e^{l}_{I^{\prime }}$ one can arrange so that $(\wedge ^{j} A)(e_{I}^{j}) \wedge e^{l}_{I^{\prime }}\neq 0$ and, moreover, A still preserves $E^{u}$ , by an arbitrarily small perturbation of $A \in \text {Sp}(2n)$ . Then, by induction, the proof is completed by performing successively small perturbations over all pairs $I, I^{\prime }$ .

To prove the claim, suppose $(\wedge ^{j} A)(e^{j}_{I}) \wedge e^{l}_{I^{\prime }} = 0$ , and write $ (\wedge ^{j} A)(e^{j}_{I}) = \sum _{J} a_{J} e^{j}_{J}$ . As A is invertible, there exists $J_{0}$ such that $a_{J_{0}} \neq 0$ and such that $|J_{0} \cap I^{\prime }|$ is minimal. As $|J_{0}| + |I^{\prime }| = n$ , we have $|J_{0} \cap I^{\prime }| = \{1,\ldots , n\}\setminus (J_{0} \cup I^{\prime })$ , so we take an arbitrarily chosen bijection $r \mapsto s_{r}$ from $J_{0} \cap I$ to $\{1,\ldots , n\}\setminus (J_{0} \cup I^{\prime })$ .

For $\theta> 0$ , and $r,s \in \{1, \ldots , n\}$ let $R_{\theta }^{r,s}$ be given by rotating the (oriented) planes $\text {span}(e_{r}, e_{s})$ and $\text {span}(f_{r}, f_{s})$ by $\theta $ and preserving the other basis elements. Let $A^{\prime }$ be obtained by composing A with each of $R_{\theta }^{r, s_{r}}$ for $r \in J_{0} \cap I$ (in any order, because the rotation matrices commute). One checks directly that $R_{\theta }^{r,s} \Omega (R_{\theta }^{r,s})^{T} = \Omega $ , where $\Omega $ is the standard symplectic form, so $R^{r,s}_{\theta }$ preserves $E^{u}$ so $A^{\prime }$ is symplectic and preserves $E^{u}$ . Writing

$$ \begin{align*}\prod_{r \in J_{0} \cap I^{\prime}}(\wedge^{k} R_{\theta}^{r, s_{r}})e_{J}^{k} = \sum_{L} b_{L} e^{j}_{L},\end{align*} $$

by a direct computation, one can check that $b_{\{1,\ldots , n\}\setminus I^{\prime }}\neq 0$ if and only if $J = J_{0}$ , which implies that $(\wedge ^{j} A^{\prime })(e^{j}_{I}) \wedge e^{l}_{I^{\prime }} \neq 0$ .

Proof. If B is sufficiently small, we may assume that it is foliated by the transversals $\Sigma (t)$ and, moreover, by passing to a further neighborhood we may assume that the transverse sections are mapped diffeomorphically onto each other by the flow X, that is, we may construct the perturbation in a flow box with transversals given by the $\Sigma (t)$ .

In the flowbox, a classic application of Moser’s trick allows us to assume that the flow is in normal coordinates $\varphi ^{t}_{X} (x_{1}, \ldots , x_{n-1}, s) \mapsto (x_{1}, \ldots , x_{n-1}, s +t)$ , where the image of l is contained in $\{x_{1} = \cdots = x_{n} = 0\}$ and $m = dx^{1} \wedge \cdots \wedge dx^{n}$ . In these coordinates, we may regard $B \cong U \times [0,\varepsilon ]$ , where $U\subseteq \mathbb R^{n-1}$ is a domain and so $Q \subseteq J^{k}_{m}(n-1, \mathbb R)$ , where $J^{k}_{m}(n-1, \mathbb R)$ is the Lie group of k-jets of volume-preserving maps fixing the origin.

Using the flow to identify the fibers of $U \times \mathbb R \to \mathbb R$ , the problem is thus reduced to the construction, for each $\delta> 0$ , of a time-dependent vector field $\{Y_{t}\}_{t \in [0,\varepsilon ]}$ on $\mathbb R^{n-1}$ with the following properties:

1. (a) $Y_{t}(0) = 0$ and $\text {supp}(Y_{t}) \subseteq U$ for $t \in [0,\varepsilon ]$ ;

2. (b) $Y_{t} \equiv 0$ on $[0, \delta ]$ and $Y_{t} \equiv 0$ on $[\varepsilon -\delta , \varepsilon ]$ ;

3. (c) $Y_{t}$ is divergence free for all $t \in [0,\varepsilon ]$ ;

4. (d) the time- $\varepsilon $ map $f: (\mathbb R^{n-1},0) \to (\mathbb R^{n-1},0)$ of $Y_{t}$ has derivative at $0$ in Q;

5. (e) $\|Y_{t}\|_{C^{\infty }} < \delta $ for all $t \in [0,\varepsilon ];$

The construction is given by first specifying the time- $\varepsilon $ map f and then finding an appropriate isotopy within the volume-preserving category to the identity.

Fix some $\theta \in Q$ sufficiently close to the k-jets of I (the identity map) and a map $F: (\mathbb R^{n-1},0) \to (\mathbb R^{n-1},0)$ whose k-jet at the origin is given by $\theta $ . Take some $C^{\infty }$ bump function $\rho : \mathbb R^{n-1} \to \mathbb R$ which interpolates between the constant function $1$ in $B(0, \eta /2)$ to the constant function $0$ outside of $B(0,\eta )$ for some $\eta $ small. Let $F^{\prime } = \rho F$ ; if $\theta $ is sufficiently close to $0$ , then $ \| F^{\prime }-I \|_{C^{k}}$ is small so in particular $F^{\prime }\in \text {Diff}(\mathbb R^{n-1})$ . Applying Moser’s trick, we can find an $f \in \text {Diff}_{m}(\mathbb R^{n-1})$ , that is preserving m, which is $C^{k}$ -close to the identity and which agrees with $F^{\prime }$ where it is conservative, namely, everywhere except $B(0,\eta )\setminus B(0, \eta /2)$ . In particular, the k-jet of f at the origin equals $\theta \in Q$ .

To obtain such an f, we construct a family $s \mapsto h_{s} \in \text {Diff}(\mathbb R^{n-1})$ such that $h_{1} = F$ and $h_{0} =:f$ is conservative. Let $r: B \to \mathbb R$ be the smooth function $C^{k}$ close to $1$ satisfying $F_{*}\mu = r\mu $ . Then, for $s \in [0,1]$ , we solve $(h_{s})_{*} \mu = r^{s} \mu $ , namely div $Z_{s} = r^{s}\log r^{s}$ , where $Z_{s} = \partial _{s} h_{s}$ . Moreover, the proof of the Poincaré lemma shows that we can take $Z_{s}$ to be constant equal to $0$ outside of $B(0,\eta )$ . By the conservative pasting lemma [Reference TeixeiraTe20], there exists $W_{s}$ which agrees with $Z_{s}$ on a neighborhood of $\mathbb R^{n-1} \setminus (B(0,\eta )\setminus B(0, \eta /2))$ and is divergence-free. Then $Z^{\prime }_{s} = Z_{s} - W_{s}$ also satisfies div $Z^{\prime }_{s} = r^{s}\log r^{s}$ and it is identically $0$ where $r = 1$ , so that $h_{s}(x) = F(x)$ on $B(0,\eta )\setminus B(0, \eta /2)$ , where now $\partial _{s} h_{s} = Z^{\prime }_{s}$ . In particular, $f:= h_{0}$ is the identity outside of $B(0,\eta )$ and its k-jet at the origin is given by $\theta $ . This constructs the desired f.

Now let $\alpha :[0,\varepsilon ]\to [0,\varepsilon ]$ be a $C^{\infty }$ function such that $\alpha \equiv 0$ on $[0, \delta ]$ and $\alpha \equiv \varepsilon $ on $[\varepsilon -\delta , \varepsilon ]$ . If $\|f-I\|_{C^{k}}$ is sufficiently small (which is ensured by taking $\theta $ closer to the jets of the identity), the maps $g_{t} := \alpha (t) f + (1-\alpha (t))I$ are all diffeomorphisms and $t \mapsto \partial _{t} g_{t}$ is a time-dependent vector field that satisfies all the desired properties except for being divergence-free.

To repair that, again Moser’s trick constructs a family $s \mapsto g_{t,s}$ such that $g_{t,0} = g_{t}$ and $g_{t,1}$ is conservative as follows. Let $r_{t}: B \to \mathbb R$ be the smooth one-parameter family of smooth functions $C^{k}$ close to $1$ satisfying $(g_{t})_{*}\mu = r_{t}\mu $ . Then, for $s \in [0,1]$ , we solve $(g_{t,s})_{*} \mu = r_{t}^{s} \mu $ , namely div $Z_{t,s} = r^{s}_{t}\log r_{t}^{s}$ , where $Z_{t,s} = \partial _{s} g_{t,s}$ . It is an easy consequence of the proof of the Poincaré lemma that the family $Z_{t,s}$ may be taken to be smooth in t with small t derivatives, because $t \mapsto r_{t}$ as a one-parameter family has the same properties. Moreover, we can take supp $Z_{t,s} \subseteq \text {supp} \, (r_{t}-1)$ . The $C^{k}$ norm of the $Z_{t,s}$ is a continuous function of the $C^{k}$ norm of $r^{s}_{t}\log r_{t}^{s}$ , so that taking $Y_{t} = \partial _{t} g_{t,1}$ finishes the proof.