Proof of Lemma 2.1

We claim that

$$ \begin{align*} \ell:=\inf_{\zeta\in\mathcal{C}} L(\zeta) \geq 2\,\mathrm{inj}(M,g)>0. \end{align*} $$

Indeed, assume by contradiction that $L(\zeta )<2\,\mathrm {inj}(M,g)$ for some $\zeta \in \mathcal {C}$ . This implies that $\zeta $ can be written as

$$ \begin{align*}\zeta(t)=\exp_{\zeta(0)}(V(t))\end{align*} $$

for some smooth function $V:S^1\to T_{\zeta (0)}M$ . We define the smooth homotopy

$$ \begin{align*} \zeta_r:S^1\to M,\ \zeta_r(t):=\exp_{\zeta(0)}(rV(t)),\quad r\in[0,1], \end{align*} $$

which satisfies $\zeta _0\equiv \zeta (0)$ and $\zeta _1=\zeta $ . Since the open subset U is weakly convex, each $\zeta _r$ is contained in U. Therefore, $\zeta $ is contractible in U, which is a contradiction.

We claim that

$$ \begin{align*} \mathcal{W}(\ell,\epsilon)\cap\mathcal{C}\neq\varnothing \quad \text{for all } \epsilon>0. \end{align*} $$

Indeed, assume that $\mathcal {W}(\ell ,\epsilon )\cap \mathcal {C}=\varnothing $ for some $\epsilon>0$ . Consider the constant $\delta>0$ and the continuous function $\tau :\mathrm {Emb}(S^1,M)^{<\ell +\delta }\to (0,\infty )$ provided by property (v). For each $\zeta _0\in \mathcal {C}$ of length $L(\zeta _0)<\ell +\delta $ , property (vi) guarantees that $\zeta _s:=\Psi _s(\zeta _0)\in \mathcal {C}$ for all $s>0$ for which it is well defined. Property (v) then implies that $L(\zeta _{\tau (\zeta )})<\ell -\delta $ , which contradicts the definition of $\ell $ .

We choose an arbitrary $\gamma _n\in \mathcal {W}(\ell ,1/n)$ parameterized with constant speed. Up to extracting a subsequence, we have that $\gamma _n(0)\to x$ and $\dot \gamma _n(0)\to v$ for some $(x,v)\in TM$ with $\|v\|=\ell $ . The curve $\gamma :S^1\to M$ , $\gamma (t):=\exp _x(tv)$ is a closed geodesic of length $\ell $ , and $\gamma _n\to \gamma $ in the $C^1$ topology. Since $\|k_{\gamma _n}-k_\gamma \|_{L^\infty }\leq 1/n$ , actually, $\gamma _n\to \gamma $ in the $C^2$ topology. Finally, since the closed geodesic $\gamma $ is the $C^2$ limit of embedded circles on an orientable surface, $\gamma $ is a simple closed geodesic.

Proof. Let $V\subset U\setminus \pi (K)$ be a path-connected component and consider two arbitrary distinct points $x_1,x_2\in V$ that can be joined by an absolutely continuous curve $\zeta $ contained in V of length strictly less than the injectivity radius $\mathrm {inj}(M,g)$ . Since U is weakly convex, the shortest geodesic segment $\gamma $ joining $x_1$ and $x_2$ is contained in U. We assume, by contradiction, that $\gamma \cap \pi (K)\neq \varnothing $ , and choose a point $v\in K$ such that $x:=\pi (v)\in \gamma $ . Let $B\subset M$ be the Riemannian open ball of radius $\mathrm {inj}(M,g)$ centered at x and let $\eta \subset \pi (K)\cap B$ be the maximal geodesic segment passing through x tangent to v. Notice that $\eta $ separates B and intersects $\gamma $ only in x. In particular, $x_1$ and $x_2$ lie in distinct connected components of $B\setminus \eta $ . This is not possible, since the curve $\zeta $ joining $x_1$ and $x_2$ is contained in $B\setminus \pi (K)\subset B\setminus \eta $ ; indeed, any absolutely continuous curve joining $x_1$ and $x_2$ and not entirely contained in B must have length larger than or equal to $\mathrm {inj}(M,g)$ .

Proof. Let $A\subset M$ be a compact annulus bounded by the waists $\gamma _0,\gamma _1$ , with $L(\gamma _1)\leq L(\gamma _0)$ . Let $W\subset A\setminus \gamma _1$ be a sufficiently small open neighborhood of $\gamma _0$ . If $W\setminus \gamma _0$ contains a simple closed geodesic, such a closed geodesic must be $C^0$ -close to $\gamma _0$ and in, particular, be non-contractible in A, and the proof is complete. Now, assume that $W\setminus \gamma _0$ does not contain any simple closed geodesic. In particular, every absolutely continuous curve $\gamma :S^1\to W$ homotopic to $\gamma _0$ within W and not geometrically equivalent to $\gamma _0$ satisfies the strict inequality $L(\gamma )> L(\gamma _0)$ (see Remark 2.4). Up to shrinking W, there exists $\rho>0$ such that any smooth curve $\gamma :S^1\hookrightarrow \overline W$ homotopic to $\gamma _0$ within $\overline W$ and that intersects $\partial W\setminus \gamma _0$ has length $L(\gamma )\geq L(\gamma _0)+\rho $ . We now detect a simple closed geodesic in $\operatorname {\mathrm {int}}(A)$ by means of a minmax procedure, as follows. Let

(2.2) $$ \begin{align} \ell:=\inf_{\mathcal{F}} \max_{r\in[0,1]} L(\zeta_r), \end{align} $$

where the infimum ranges over the family $\mathcal {F}$ of continuous homotopies of smooth embedded loops $\zeta _r:S^1\hookrightarrow A$ , $r\in [0,1]$ such that $\zeta _0=\gamma _0$ , $\zeta _1=\gamma _1$ , and $\zeta _r\subset \operatorname {\mathrm {int}}(A)$ for all $r\in (0,1)$ . Notice that, for any such homotopy $(\zeta _r)_{r\in [0,1]}$ , there exists a minimal $r_0\in (0,1]$ such that the curve $\zeta _r$ intersects $\partial W\setminus \gamma _0$ ; therefore,

$$ \begin{align*}L(\zeta_{r_0})\geq L(\gamma_0)+\rho.\end{align*} $$

This readily implies that

$$ \begin{align*} \ell \geq L(\gamma_0)+\rho>L(\gamma_0). \end{align*} $$

We claim that $\ell $ is the length of a non-contractible simple closed geodesic in A, which must actually be contained in $\operatorname {\mathrm {int}}(A)$ since $L(\gamma _1)\leq L(\gamma _0)<\ell -\rho $ . Let us assume that this is not the case. In particular, for $\epsilon>0$ small enough, no embedded loop $\gamma \in \mathcal {W}(\ell ,\epsilon )$ is entirely contained in A and non-contractible therein. We apply property (v) of the curve shortening flow, which provides $\delta>0$ and a suitable continuous function $\tau :\mathrm {Emb}(S^1,M)^{<\ell +\delta }\to (0,\infty )$ . Let $(\zeta _r)_{r\in [0,1]}$ be a homotopy in $\mathcal {F}$ that is optimal up to $\delta $ , which means that

$$ \begin{align*}\max_{r\in[0,1]} L(\zeta_r)< \ell+\delta.\end{align*} $$

We push the homotopy by means of the curve shortening flow, defining

$$ \begin{align*} \eta_r:=\Psi_{\tau(\zeta_r)}(\zeta_r),\ r\in[0,1]. \end{align*} $$

Property (v) implies that $\eta _r\in \mathrm {Emb}(S^1,M)^{<\ell -\delta }\cup \mathcal {W}(\ell ,\epsilon )$ for each $r\in [0,1]$ . Since the compact annulus A is bounded by simple closed geodesics, $\operatorname {\mathrm {int}}(A)$ is weakly convex. Property (vi) implies that $\eta _r\subset \operatorname {\mathrm {int}}(A)$ for each $r\in (0,1)$ . Since $\zeta _0=\gamma _0$ and $\zeta _1=\gamma _1$ are simple closed geodesics, we have $\eta _0=\gamma _0$ and $\eta _1=\gamma _1$ . Therefore, the homotopy $(\eta _r)_{r\in [0,1]}$ belongs to $\mathcal {F}$ . Since no curve $\gamma \in \mathcal {W}(\ell ,\epsilon )$ is entirely contained in A and non-contractible therein, we conclude that $\eta _r\in \mathrm {Emb}(S^1,M)^{<\ell -\delta }$ for each $r\in [0,1]$ , which contradicts the definition of the minmax value $\ell $ .

The case of a compact disk $D\subset M$ bounded by a waist $\gamma $ is analogous, except that, in the definition of the minmax (2.2), the infimum ranges over the family of continuous homotopies of smooth embedded loops $\zeta _r:S^1\hookrightarrow D$ , $r\in [0,1]$ , such that $\zeta _0=\gamma $ , $\zeta _r\subset \operatorname {\mathrm {int}}(D)$ for all $r\in (0,1]$ and $L(\zeta _1)<L(\gamma )$ .

Proof of Lemma 2.7

Since $\gamma $ is hyperbolic, the closed orbit $\dot \gamma $ has a Floquet multiplier $\sigma \in (-1,1)$ . We parameterize $\gamma $ with unit speed, and denote by $\ell>0$ its length, so that $\dot \gamma (t)=\dot \gamma (t+\ell )$ . We consider the stable line bundle $E^{s}$ over $\dot \gamma $ , which is given by

$$ \begin{align*} E^{s}(\dot\gamma(t))=\ker(d\phi_\ell(\dot\gamma(t))-\sigma\,\mathrm{id}). \end{align*} $$

The stable bundle is invariant under the linearized geodesic flow: that is,

$$ \begin{align*} d\phi_t(\dot\gamma(0))E^{s}(\dot\gamma(0))=E^{s}(\dot\gamma(t)). \end{align*} $$

If we had $E^{s}(\dot \gamma (t))=\ker (d\pi (\dot \gamma (t)))$ for some $t\in \mathbb {R}/\ell \mathbb {Z}$ , the endpoints of the geodesic segment $\gamma |_{[t,t+\ell ]}$ would be conjugate. Therefore, since $\gamma $ has no conjugate points, we infer that

$$ \begin{align*} E^{s}(\dot\gamma(t))\cap\ker(d\pi(\dot\gamma(t)))=\{0\}\quad\text{for all } t\in\mathbb{R}/\ell\mathbb{Z}. \end{align*} $$

This, together with the fact that

$$ \begin{align*} T_{\dot\gamma(t)}W^{s}(\dot\gamma)=\mathrm{span}\{\dot\gamma(t)\}\oplus E^{s}(\dot\gamma(t))\quad\text{for all } t\in\mathbb{R}/\ell\mathbb{Z}, \end{align*} $$

readily implies that the local stable manifold of $\dot \gamma $ is a graph over the base manifold M. Namely, for any sufficiently small neighborhood $V\subset SM$ of the closed orbit $\dot \gamma $ , if $W\subset V\cap W^{s}(\dot \gamma )$ is the path-connected component containing $\dot \gamma $ , then the base projection $\pi |_W:W\to M$ is a diffeomorphism onto a neighborhood of $\gamma $ . For each $z\in W^{s}(\dot \gamma )\setminus \dot \gamma $ , for all $t>0$ large enough, we have $\phi _t(z)\in W\setminus \dot \gamma $ , and therefore $\pi \circ \phi _t(z)\in \pi (W)\setminus \gamma $ .

Proof. We denote by $\Upsilon $ the union of the Birkhoff annuli of the closed geodesics in $\mathcal {G}$ , that is,

$$ \begin{align*} \Upsilon := \bigcup_{\gamma\in\mathcal{G}} ( A(\dot\gamma)\cup A(-\dot\gamma) ). \end{align*} $$

Since $\mathcal {K}\hspace{-1pt}=\hspace{-1pt}\varnothing $ , we have empty trapped sets $\mathrm {trap}_\pm (SU)\hspace{-1pt}=\hspace{-1pt}\varnothing $ . Therefore, for each $z\hspace{-1pt}\in\hspace{-1pt} SM\setminus \Upsilon $ , there exists $\ell _z>1$ such that the orbit segment $\phi _{[1,\ell _z-1]}(z)$ intersects $\operatorname {\mathrm {int}}(\Upsilon )$ transversely. The transversality guarantees that there exists an open neighborhood $N_z\subset SM$ of z such that, for each $z'\in N_z$ , the orbit segment $\phi _{[0,\ell _z]}(z')$ intersects $\operatorname {\mathrm {int}}(\Upsilon )$ . Now consider the constant $\ell>0$ and the open neighborhood N of R given by property (iii) above. Since $SM$ is compact, there exist finitely many points $z_1,\ldots ,z_n\in SM$ such that $N\cup N_{z_1}\cup \cdots \cup N_{z_n}=SM$ . For $\ell ':=\max \{\ell ,\ell _{z_1},\ldots ,\ell _{z_n}\}$ , we conclude that, for each $z\in SM$ , the orbit segment $\phi _{[0,\ell ']}(z)$ intersects $\Upsilon $ . Namely, every geodesic segment of length $\ell '$ intersects some geodesic in $\mathcal {G}$ .

Proof. We first show that, for each $\gamma \in \mathcal {K}$ and for each connected component $W\subseteq W^{u}(\dot \gamma )\setminus \dot \gamma $ , there exists a heteroclinic

(4.1) $$ \begin{align} W\cap W^{s}(K)\neq\varnothing. \end{align} $$

We prove this by contradiction, assuming that $W\cap W^{s}(K)=\varnothing $ .

The path-connected component W is an immersed cylinder in $SM$ with one end equal to $\dot \gamma $ . Let $S\subset W$ be an embedded essential circle that is $C^1$ -close to $\dot \gamma $ and transverse to the geodesic vector field, so that its base projection $\pi (S)$ does not intersect $\gamma $ (Lemma 2.7) nor any other closed geodesic in the collection $\mathcal {G}$ . We consider the union of the Birkhoff annuli

$$ \begin{align*} \Upsilon':=\bigcup_{\zeta\in\mathcal{G}\setminus\{\gamma\}} ( A(\dot\zeta)\cup A(-\dot\zeta) ) \end{align*} $$

and an open neighborhood $V\subset SM$ of $\Upsilon '$ such that $V\cap \phi _{(-\infty ,0]}(S)=\varnothing $ . We apply Fried surgery, as explained in §4.1, to resolve the self-intersection points of $\Upsilon '$ with its interior (if there are any) and produce surfaces of section $\Sigma '\subset V$ such that

(4.2) $$ \begin{align} \phi_{(-1,1)}(z)\cap\Sigma'\neq\varnothing\quad \text{for all } z\in\Upsilon'. \end{align} $$

We set

$$ \begin{align*}\Upsilon=\Upsilon'\cup A(\dot\gamma)\cup A(-\dot\gamma),\quad \Sigma=\Sigma'\cup A(\dot\gamma)\cup A(-\dot\gamma).\end{align*} $$

For each $z\in S$ , we have $\phi _{(-\infty ,0]}(z)\subset SU$ , whereas $\phi _{(-\infty ,\infty )}(z)$ is not entirely contained in $SU$ and thus intersects $\Upsilon $ . This, together with (4.2), implies that there exists a minimal $t_z>0$ such that the orbit $t\mapsto \phi _{t}(z)$ intersects $\Sigma $ transversely for $t=t_z$ . This transversality, together with the compactness of the circle S, implies that the function $z\mapsto t_z$ is smooth on S, and we obtain an embedded circle

$$ \begin{align*} S_0:=\{\phi_{t_z}(z)\ |\ z\in S\}\subset W\cap\Sigma. \end{align*} $$

We consider the first return time

$$ \begin{align*} \tau:\operatorname{\mathrm{int}}(\Sigma)\to(0,+\infty], \quad \tau(z):=\inf\{ t>0\ |\ \phi_{t}(z)\in\Sigma \}, \end{align*} $$

which is smooth on the open subset $\Sigma _0:=\tau ^{-1}(0,\infty )$ . The first return map

$$ \begin{align*} \psi:\Sigma_0\to\operatorname{\mathrm{int}}(\Sigma),\quad\psi(z)=\phi_{\tau(z)}(z) \end{align*} $$

is a diffeomorphism onto its image that preserves the area form $d\unicode{x3bb} |_{\operatorname {\mathrm {int}}(\Sigma )}$ , where $\unicode{x3bb} $ is the Liouville contact form of $SM$ .

Since $S_0\cap W^{s}(K)=\varnothing $ , we have $S_0\cap \mathrm {trap}_+(SU)=\varnothing $ , and therefore $\psi ^n(S_0)\subset \Sigma _0$ for all $n\geq 0$ . We obtained an infinite sequence $S_n:=\psi ^n(S_0)$ of pairwise disjoint embedded circles in the interior of the surface of section $\Sigma $ . Since the unstable manifold $W^{u}(\dot \gamma )$ is an immersed surface tangent to the geodesic vector field, the 2-form $d\unicode{x3bb} |_W$ vanishes identically. If $S_n$ bounds a disk $B_n\subset \operatorname {\mathrm {int}}(\Sigma )$ , then we denote by $A_n\subset W$ the compact annulus with boundary $\partial A_n=\dot \gamma \cup S_n$ , and Stokes’ theorem implies that

$$ \begin{align*} \mathrm{area}(B_n,d\unicode{x3bb})=\int_{B_n} d\unicode{x3bb}=\int_{A_n\cup B_n} d\unicode{x3bb} = \int_{\dot\gamma} \unicode{x3bb} = L(\gamma), \end{align*} $$

where $L(\gamma )>0$ is the length of the simple closed geodesic $\gamma $ . In particular, if $S_{n_1},S_{n_2}$ bound disks $B_{n_1},B_{n_2}\subset \operatorname {\mathrm {int}}(\Sigma )$ for some distinct $n_1,n_2\geq 0$ , we have $B_{n_1}\cap B_{n_2}=\varnothing $ . This, together with the finiteness of the area

$$ \begin{align*} \mathrm{area}(\operatorname{\mathrm{int}}(\Sigma),d\unicode{x3bb}) = \int_{\Sigma} d\unicode{x3bb} = \int_{\partial\Sigma}\unicode{x3bb}, \end{align*} $$

readily implies that there exist at most finitely many $n\geq 0$ such that $S_n$ is contractible in $\operatorname {\mathrm {int}}(\Sigma )$ . Therefore, since the embedded circles $S_n$ are pairwise disjoint, there exist distinct $n_1,n_2\geq 0$ and an embedded compact annulus $A\subset \operatorname {\mathrm {int}}(\Sigma )$ with boundary $\partial A=S_{n_1}\cup S_{n_2}$ . Let $A'\subset W$ be the embedded compact annulus with boundary $\partial A'=S_{n_1}\cup S_{n_2}$ , so that the union $A\cup A'$ is a piecewise smooth embedded torus in $SM$ . By Stokes theorem,

$$ \begin{align*} \int_A d\unicode{x3bb} = \int_{A\cup A'} d\unicode{x3bb} = 0, \end{align*} $$

which contradicts the fact that $d\unicode{x3bb} |_{A}$ is an area form. This proves the existence of heteroclinics (4.1).

Our transversality assumption on the heteroclinics, together with the shadowing lemma from hyperbolic dynamics [Reference Fisher and HasselblattFH19, Theorem 5.3.3], implies that, for all $\gamma _1,\gamma _2,\gamma _3\in \mathcal {K}$ and for each path-connected component $W\subset W^{u}(\dot \gamma _1)\setminus \dot \gamma _1$ , if there are heteroclinics $W\cap W^{s}(\dot \gamma _2)\neq \varnothing $ and $W^{u}(\dot \gamma _2)\cap W^{s}(\dot \gamma _3)\neq \varnothing $ , then there are also heteroclinics ${W\cap W^{s}(\dot \gamma _3)\neq \varnothing }$ .

For each $\gamma \in \mathcal {K}$ , we fix arbitrary path-connected components $W_{\pm \dot \gamma }\subset W^{u}(\pm \dot \gamma )\setminus \pm \dot \gamma $ . We have already proved that every such path-connected component must contain a heteroclinic to K. Therefore, there exists a sequence of oriented waists $\gamma _i\in \mathcal {K}$ with heteroclinics $W_{\dot \gamma _i}\cap W^{s}(\dot \gamma _{i+1})\neq \varnothing $ . The same waist may appear at different times with opposite orientations in the sequence. Nevertheless, since the collection $\mathcal {G}$ is finite, there exists $n\leq 2\,\#\mathcal {G}+1$ such that $\gamma _1=\gamma _n$ as oriented waists. This implies that $W_{\dot \gamma _1}\cap W^{s}(\dot \gamma _1)\neq \varnothing $ .

Proof. By Lemma 4.3, there exists $\gamma \in \mathcal {K}$ whose associated periodic orbit $\dot \gamma $ has homoclinics in all path-connected components of $W^{u}(\dot \gamma )\setminus \dot \gamma $ . Since $\gamma $ is a non-degenerate waist, Lemma 2.7 implies that there exists a tubular neighborhood $A\subset M$ of $\gamma $ and an open neighborhood $V\subset SM$ of $\dot \gamma $ such that, if we denote by $W\subset V\cap W^{u}(\dot \gamma )$ the path-connected component containing $\dot \gamma $ , the restriction of the base projection $\pi |_W:W\to ~A$ is a diffeomorphism. An analogous statement holds for the stable manifold $W^{s}(\dot \gamma )$ . Therefore, for any homoclinic point $z\in W^{u}(\dot \gamma )\cap W^{s}(\dot \gamma )\setminus \dot \gamma $ , the corresponding geodesic $\zeta (t):=\pi \circ \phi _t(z)$ is contained in $A\setminus \gamma $ provided $|t|$ is large enough.

The complement $A\setminus \gamma $ is the disjoint union of two open annuli $A_1$ and $A_2$ , and therefore $W\setminus \dot \gamma $ is the disjoint union of the open annuli $W_1:=\pi |_W^{-1}(A_1)$ and $W_2:=\pi |_W^{-1}(A_2)$ . Our assumption on $\gamma $ implies that both $W_1$ and $W_2$ intersect the stable manifold $W^{s}(\dot \gamma )$ , and we fix homoclinic points $z_i\in W_i\cap W^{s}(\dot \gamma )$ such that, if we denote by $\zeta _i(t):=\pi \circ \phi _t(z)$ the associated geodesics, we have $\zeta _i(t)\in A_i$ for all $t\leq 0$ . We have two possible cases.

Case 1: There exists $i\in \{1,2\}$ such that $\zeta _i(t)\in A_{3-i}$ for all $t>0$ large enough. For each $\delta>0$ , there exist arbitrarily large $a,b>0$ such that the points $\phi _{-a}(z_i)$ and $\phi _{b}(z_i)$ are $\delta $ -close (where the distance on $SM$ is the one induced by the Riemannian metric g). We consider the orbit segment

$$ \begin{align*} \Gamma_\delta:[-a,b]\to SM,\quad \Gamma_\delta(t)=\phi_{t}(z_i), \end{align*} $$

and we extend it as a discontinuous periodic curve of period $\tau :=a+b$ .

Case 2: For all $i\in \{1,2\}$ , we have $\zeta _i(t)\in A_{i}$ for all $t>0$ large enough. For each $\delta>0$ , there exist arbitrarily large $a_1,b_1,a_2,b_2>0$ such that the points $\phi _{-a_i}(z_i)$ and $\phi _{b_i}(z_i)$ are $\delta $ -close. We consider the orbit segments

$$ \begin{align*}\Gamma_{\delta,i}:[-a_i,b_i]\to SM,\quad \Gamma_{\delta,i}(t)=\phi_{t}(z_i),\end{align*} $$

and we define $\Gamma _\delta $ to be the discontinuous periodic curve of period $\tau :=a_1+b_1+a_2+b_2$ obtained by extending periodically the concatenation of $\Gamma _{\delta ,1}$ and $\Gamma _{\delta ,2}$ .

In both cases, for each $\delta>0$ , we obtained a periodic $\delta $ -pseudo orbit $\Gamma _\delta $ of the geodesic flow with arbitrarily large minimal period. Moreover, the projection $\pi \circ \Gamma _\delta $ makes a $\delta $ -jump from $A_1$ to $A_2$ . The shadowing lemma from hyperbolic dynamics [Reference Fisher and HasselblattFH19, Theorem 5.3.3] implies that, for each $\epsilon>0$ , there exist $\delta>0$ and $z\in SM$ such that the orbit $t\mapsto \phi _t(z)$ is periodic and pointwise $\epsilon $ -close to $\Gamma _\delta $ up to a time reparameterization with Lipschitz constant $\epsilon $ . Up to choosing $\epsilon>0$ small enough, the corresponding closed geodesic $\zeta (t):=\pi \circ \phi _t(z)$ must intersect $\gamma $ transversely.

Let $\mathcal {K}'$ be the subcollection of those simple closed geodesics $\eta \in \mathcal {K}$ that intersect $\zeta $ , which is non-empty since it contains $\gamma $ . The collection $\mathcal {G}\cup \{\zeta \}$ is a complete system of closed geodesics with limit subcollection $\mathcal {K}\setminus \mathcal {K}'$ .

Proof. We first consider non-contractible embedded loops $\zeta _1,\ldots ,\zeta _{2G}\subset M$ such that $\zeta _i\cap \zeta _{i+1}$ is a singleton for all $i\in \{1,\ldots ,2G-1\}$ , $\zeta _i\cap \zeta _j=\varnothing $ if $|i-j|\geq 2$ , and $M\setminus (\zeta _1\cup \cdots \cup \zeta _{2G})$ is simply connected (Figure 9). We denote by $\mathcal {C}_i\subset \mathrm {Emb}(S^1,M)$ the connected component containing $\zeta _i$ . Notice that the $\mathcal {C}_i$ are pairwise distinct and the embedded loops $\zeta _1,\ldots ,\zeta _{2G}$ are in minimal position: that is,

$$ \begin{align*} \#(\eta_i\cap\eta_j) \geq \#(\zeta_i\cap\zeta_j)\quad\text{for all } i<j,\ \eta_i\in\mathcal{C}_i,\ \eta_j\in\mathcal{C}_j. \end{align*} $$

By Lemma 2.1, for each $i=1,\ldots ,2G$ , there exists a waist $\gamma _i\in \mathcal {C}_i$ such that

(4.3) $$ \begin{align} L(\gamma_i)=\min_{\gamma\in\mathcal{C}_i} L(\gamma). \end{align} $$

Figure 9 The embedded loops $\zeta _1,\ldots ,\zeta _{2G}$ .

We recall that a geodesic bigon is a compact disk $D\subset M$ whose boundary is the union of two geodesic arcs. For each $i<j$ , there is no geodesic bigon bounded by an arc in $\alpha _i\subset \gamma _i$ and an arc $\alpha _j\subset \gamma _j$ . Indeed, if there were such a geodesic bigon with $L(\alpha _i)\leq L(\alpha _j)$ , we could further shrink the simple closed geodesics $\gamma _j$ and obtain a shorter embedded loop in the same homotopy class $\mathcal {C}_j$ , which would contradict (4.3). By the bigon criterium [Reference Farb and MargalitFM12, Proposition 1.7], the waists $\gamma _1,\ldots ,\gamma _{2G}$ are in minimal position. Therefore,

$$ \begin{align*} \#(\gamma_i\cap\gamma_j) = \#(\zeta_i\cap\zeta_j)\quad\text{for all } i<j. \end{align*} $$

To prove that $B:=M\setminus (\gamma _1\cup \cdots \cup \gamma _{2G})$ is simply connected, we cut open the surface M at the simple closed geodesics $\gamma _2,\gamma _4,\ldots ,\gamma _{2G}$ , and we thus obtain a compact surface $M'$ of genus zero with $2G$ boundary components. Each waist $\gamma _{2i}$ , with $i=1,\ldots ,G$ , corresponds to two boundary components $\gamma _{2i}^+,\gamma _{2i}^-\subset \partial M'$ . Each waist $\gamma _{2i+1}$ , with $i=1,\ldots ,G-1$ , splits into two embedded arcs $\gamma _{2i+1}^+,\gamma _{2i+1}^-$ ; the arc $\gamma _{2i+1}^\pm $ joins the boundary components $\gamma _{2i}^\pm $ and $\gamma _{2i+2}^\pm $ . Finally, the waist $\gamma _1$ corresponds to an embedded arc $\gamma _1^+$ joining the boundary components $\gamma _2^+$ and $\gamma _2^-$ . We now cut open $M'$ along the embedded arcs $\gamma _{2i+1}^\pm $ and obtain a compact disk $M"$ . The interior $\operatorname {\mathrm {int}}(M")$ is diffeomorphic to B.

Proof. We see $\Upsilon $ as a non-injectively immersed surface in $SM$ with boundary and interior

$$ \begin{align*}\partial\Upsilon=\bigcup_{i=1}^{2G} (\dot\gamma_i\cup-\dot\gamma_i), \quad \operatorname{\mathrm{int}}(\Upsilon)=\bigcup_{i=1}^{2G}(\operatorname{\mathrm{int}}(A_i^+)\cup\operatorname{\mathrm{int}}(A_i^-)), \end{align*} $$

where $A_i^\pm :=A(\pm \dot \gamma _i)$ . We set $\ell _0:=\max \{L(\gamma _1),\ldots ,L(\gamma _{2G})\}$ . Since $\gamma _i$ intersects $\gamma _{i-1}$ and $\gamma _{i+1}$ for all $2\leq i\leq 2G-1$ , we have $\phi _{[2,\ell _0+2]}(z)\cap \operatorname {\mathrm {int}}(\Upsilon )\neq \varnothing $ for all $z\in \partial \Upsilon $ . Therefore, there exists an open neighborhood $N\subset SM$ of $\partial \Upsilon $ such that

(4.5) $$ \begin{align} \phi_{[1,\ell_0+3]}(z)\cap\operatorname{\mathrm{int}}(\Upsilon)\neq\varnothing\quad\text{for all } z\in N. \end{align} $$

Let $V\subset SM$ be an open neighborhood of $\Upsilon $ . We apply Fried surgeries in an arbitrarily small neighborhood of the self-intersection points of $\Upsilon $ , as explained in §4.1, and obtain a surface of section $\iota :\Sigma \looparrowright V$ with the same boundary $\partial \Sigma =\partial \Upsilon $ and satisfying point (iv) in the statement of the lemma. This, together with (4.5), implies point (iii) for $\ell :=\ell _0+4$ .

We claim that the interior $\operatorname {\mathrm {int}}(\Sigma )$ is path connected. Indeed, near the point in $\operatorname {\mathrm {int}}(A_1^+)\cap \partial A_2^+\cap \partial A_2^-$ , Fried surgery glues together the intersecting annuli $A_1^+$ , $A_2^-$ and $A_2^+$ in the same connected component of $\operatorname {\mathrm {int}}(\Sigma )$ , as is clear from Figures 6 and 8; analogously, near the point in $\operatorname {\mathrm {int}}(A_1^-)\cap \partial A_2^+\cap \partial A_2^-$ , Fried surgery glues together the intersecting annuli $A_1^-$ , $A_2^-$ and $A_2^+$ . Therefore, the surgery sends the four annuli $A_1^+$ , $A_1^-$ , $A_2^+$ and $A_2^-$ to the same path-connected component of $\operatorname {\mathrm {int}}(\Sigma )$ . Assume, by induction, that Fried surgery sends the annuli $A_1^+,A_1^-,\ldots ,A_{i}^+,A_{i}^-$ to the same path-connected component $W\subset \operatorname {\mathrm {int}}(\Sigma )$ . Near the point in $A_i^+\cap \partial A_{i+1}^+\cap \partial A_{i+1}^-$ , Fried surgery glues together the intersecting annuli $A_{i}^+$ , $A_{i+1}^-$ and $A_{i+1}^+$ . Therefore, the surgery sends $A_{i+1}^-$ and $A_{i+1}^+$ to the connected component W as well, and we conclude that $W=\operatorname {\mathrm {int}}(\Sigma )$ .

We now determine the number of connected components of the boundary $\partial \Sigma $ and how they cover the closed orbits $\pm \dot \gamma _i$ . The simple closed geodesic $\gamma _1$ intersects $\gamma _2$ in one point, but does not intersect any of the other simple closed geodesics $\gamma _3,\ldots ,\gamma _{2G}$ ; therefore, $\dot \gamma _1$ intersects the interior of the Birkhoff annulus $\operatorname {\mathrm {int}}(A_2^+)$ in one point, but does not intersect any of the other Birkhoff annuli $A_i^\pm $ . Analogously, $-\dot \gamma _1$ intersects $\operatorname {\mathrm {int}}(A_2^-)$ in one point but none of the other $A_i^\pm $ . Therefore, there is a unique connected component of $\partial \Sigma $ that covers $\dot \gamma _1$ , and such a connected component winds around $\dot \gamma _1$ twice; analogously, there is a unique connected component of $\partial \Sigma $ that covers $-\dot \gamma _1$ , and such a connected component winds around $-\dot \gamma _1$ twice (Figure 10). The same conclusions hold for $\gamma _{2G}$ . For each $i\in \{2,\ldots ,2G-1\}$ , $\gamma _i$ intersects $\gamma _{i-1}$ and $\gamma _{i+1}$ in one point each, but does not intersect any other $\gamma _j$ ; therefore, $\dot \gamma _i$ intersects the interiors of the Birkhoff annuli $\operatorname {\mathrm {int}}(A_{i-1}^-)$ and $\operatorname {\mathrm {int}}(A_{i+1}^+)$ in one point each, but does not intersect any of the other Birkhoff annuli $A_j^\pm $ ; analogously, $-\dot \gamma _i$ intersects $A_{i-1}^+$ and $A_{i+1}^-$ in one point each, but does not intersect any other $A_j^\pm $ . Therefore, there are two connected components of $\partial \Sigma $ that are mapped diffeomorphically to $\dot \gamma _i$ and two other connected components of $\partial \Sigma $ that are mapped diffeomorphically to $-\dot \gamma _i$ (Figure 11). All together, $\partial \Sigma $ has $8G-4$ boundary components.

Figure 10 The surface $\Sigma $ near the boundary component $-\dot \gamma _1$ .

Figure 11 The surface $\Sigma $ near the boundary component $-\dot \gamma _i$ , for $2\leq i\leq 2G-1$ . Notice that there are two open annuli in $\operatorname {\mathrm {int}}(\Sigma )$ having boundary on $-\dot \gamma _i$ .

Since $\operatorname {\mathrm {int}}(\Sigma )$ is transverse to the geodesic vector field X, $\Sigma $ is orientable. Indeed, the Liouville contact form $\unicode{x3bb} $ of $SM$ induces an area form $d\unicode{x3bb} |_{\operatorname {\mathrm {int}}(\Sigma )}$ . To compute the genus of $\Sigma $ , we first triangulate every Birkhoff annulus $A_i^\pm $ with 6 vertices, 14 edges and 8 faces. We choose such triangulations so that, along the intersections $A_i^+\cap A_i^-$ , $A_i^\pm \cap A_{i+1}^\pm $ and $A_i^\pm \cap A_{i+1}^\mp $ , the vertices and edges of $A_{i}^\pm $ and $A_{i+1}^\pm $ match as in Figure 12. All together, $\Upsilon $ is triangulated with $24\,G$ vertices, $56\,G$ edges and $32\,G$ faces (once again, we stress that $\Upsilon $ must be seen as an immersed compact surface with boundary, and therefore distinct Birkhoff annuli do not share vertices, edges or faces). The same number of edges and faces can be used to triangulate $\Sigma $ as well; however, the triangulation of $\Upsilon $ provides one extra vertex for each of the points $x_i,y_i,w_i,z_i$ depicted in Figure 12, where $i=1,\ldots ,2G-1$ , and therefore we need to throw away $8\,G-4$ vertices. All together, $\Sigma $ is triangulated with $16\,G+4$ vertices, $56\,G$ edges and $32\,G$ faces, and therefore has Euler characteristic

$$ \begin{align*} \chi(\Sigma)=16\,G+4 - 56\,G + 32\,G = -8\,G + 4. \end{align*} $$

Since

$$ \begin{align*} -8\,G+4=\chi(\Sigma)=2-2\,\mathrm{genus}(\Sigma) - \#\pi_0(\partial\Sigma)=2-2\,\mathrm{genus}(\Sigma) - 8\,G + 4, \end{align*} $$

we conclude that $\Sigma $ has genus one.

Proof. With the terminology of Example 3.2, the open disk $B:=M\setminus (\gamma _1\cup \cdots \cup \gamma _{2G})$ is a convex geodesic polygon and, in particular, satisfies Assumption 3.1. Since B does not contain any simple closed geodesic without conjugate points, Theorem 3.4 implies that we have empty trapped sets $\mathrm {trap}_+(SB)=\mathrm {trap}_-(SB)=\varnothing $ . Therefore, the collection of waists $\{\gamma _1,\ldots ,\gamma _{2G}\}$ is a complete system of closed geodesics with empty limit subcollection. Let $\Upsilon $ be the union of the Birkhoff annuli of the waists $\gamma _1,\ldots ,\gamma _{2G}$ , as in (4.4). By Lemma 4.2, there exists $\ell>0$ such that, for each $z\in SM$ , the orbit segment $\phi _{[0,\ell ]}(z)$ intersects $\Upsilon $ . This, together with Lemma 4.6(iv), implies that, for each $z\in SM$ , the orbit segment $\phi _{(0,\ell +2)}(z)$ intersects $\Sigma $ . Therefore, $\Sigma $ is a Birkhoff section.

Proof. Both connected components $B_1$ and $B_2$ of the complement $S^2\setminus \gamma $ satisfy the convexity Assumption 3.1. Since $B_1$ and $B_2$ do not contain any simple closed geodesic without conjugate points, Theorem 3.4 implies that we have empty trapped sets $\mathrm {trap}_\pm (S(B_1\cup B_2))=\varnothing $ . Therefore, $\{\gamma \}$ is a complete system of closed geodesics with empty limit subcollection. By Lemma 4.2, there exists $\ell>0$ such that any geodesic segment of length $\ell $ intersects $\gamma $ . This implies that both Birkhoff annuli $A(\dot \gamma )$ and $A(-\dot \gamma )$ are Birkhoff sections.

Proof. First, assume that M has genus $G\geq 1$ , so that $\mathcal {G}'=\{\gamma _1,\ldots ,\gamma _{2G}\}$ . The complement $B\setminus (\gamma _1\cup \cdots \cup \gamma _{2G})$ satisfies the convexity Assumption 3.1. Since every closed geodesic $\gamma \in \mathcal {G}'$ intersects some other closed geodesic in $\mathcal {G}'$ , if we fix $\ell> L(\gamma )$ , for every $z\in SM$ sufficiently close to $\dot \gamma \cup -\dot \gamma $ , then the orbit segment $\phi _{(0,\ell ]}(z)$ is not contained in $SB$ . If B does not contain simple closed geodesics, then Theorem 3.4 implies that $\mathrm {trap}_\pm (SB)=\varnothing $ , and therefore $\mathcal {G}'$ is a complete system of closed geodesics with empty limit subcollection $\mathcal {K}=\varnothing $ . Now assume that B contains at least a closed geodesic. By Theorem 3.5, B contains a finite collection of pairwise disjoint simple closed geodesics $\mathcal {G}"$ such that: