2.1 Organization of the article

In §3, we prove that geodesic flow on the unit tangent bundle of a hyperbolic surface is quasigeodesic. Moreover, Anosov flows on Seifert fibered 3-manifolds are quasigeodesic.

In §4, we describe the construction of generalized Franks–Williams flows and in §5, we describe the Riemannian metric we are going to work with in this article.

Section 6 contains the proof of Theorem 1.2; §6.1 proves the key proposition for the proof and §6.2 completes the proof.

3.1 Geodesic flows on the unit tangent bundle

We note that the flow lines of the geodesic flow on $T\Sigma $ are of the form $(\gamma ,\gamma ')$ . As $\gamma $ is a geodesic on $\Sigma $ , we get ${d}/{dt}\langle \gamma '(t),\gamma '(t)\rangle =0$ , that is, $\|\gamma '(t)\|=$ constant. This property allows us to restrict the flow $\mathfrak {G}_t$ on $T\Sigma $ to the unit tangent bundle $S\Sigma $ , where

$$ \begin{align*}S\Sigma=\{(p,v)|p\in\Sigma,v\in T_p\Sigma, \|v\|=1\}.\end{align*} $$

Similarly, we can restrict the flow $\widetilde {\mathfrak {G}}_t$ on $S\mathbb {H}$ , the unit tangent bundle on $\mathbb {H}$ . It is immediate by Theorem 3.2 that the flow lines of the geodesic flow on $S\mathbb {H}$ are globally length minimizing.

It is clear that $S\mathbb {H}$ is a cover of $S\Sigma $ , though it is not the universal cover. As the flow lines of the geodesic flow on $S\mathbb {H}$ are globally length minimizing and $S\mathbb {H}$ is complete, lifts of the flow lines of the geodesic flow in the universal cover $\widetilde {S\mathbb {H}}=\mathbb {H}\times \mathbb {R}$ are also globally length minimizing, a stronger property than being quasigeodesic.

So far, we have considered the metric $ds^2=({dx^2+dy^2})/{y^2}$ on $\mathbb {H}$ , and the geodesics and geodesic flow on a surface completely depend on the choice of Riemannian metric. However, geodesic flows associated with any two negatively curved metrics on a surface are orbit equivalent [Reference GhysGhy84]. More precisely, there is a homeomorphism between the unit tangent bundles of the respective Riemannian metrics which takes orbits to orbits as described in Definition 2.3. It is easy to check that any homeomorphism between two compact manifolds gives a quasi-isometry when lifted to the universal covers. In particular, as unit tangent bundles of negatively curved closed surfaces are compact, the orbit equivalence maps are quasi-isometries between the universal covers; and quasi-isometries preserve quasigeodesics. This implies geodesic flow with respect to any negatively curved metric on a closed surface is quasigeodesic. We conclude the following theorem.

3.2 Anosov flows in Seifert manifolds

Now, we prove that any Anosov flow on a Seifert fibered 3-manifold is quasigeodesic. The following theorem relates Anosov flows on Seifert fibered 3-manifolds with geodesic flows.

We combine Theorems 3.3 and 3.4 to get the following.

Proof. By Theorem 3.4, $\Phi _t$ is orbit equivalent to a finite lift of the geodesic flow $\mathfrak {G}_t$ on the unit tangent bundle $S\Sigma $ of a hyperbolic surface $\Sigma $ . We denote the finite lift of $S\Sigma $ by $\widehat {S\Sigma }$ and the finite lift of the geodesic flow by $\widehat {\mathfrak {G}}_t$ .

Fix a Riemannian metric g on $\mathcal {N}$ . Let $\widehat {ds}$ be the metric on $\widehat {S\Sigma }$ , $\widehat {ds}$ which is the lift of the metric $ds$ as constructed before using the upper-half-plane $\mathbb {H}$ . This is the metric for which Theorem 3.4 holds. We denote the path metrics induced by the lifts of the metrics $\widetilde {g}$ and $\widetilde {ds}$ on $\widetilde {\mathcal {N}}$ and $\widetilde {S\Sigma }$ respectively by $d_1$ and $d_2$ .

Fix an orbit equivalence $h:\mathcal {N}\rightarrow \widehat {S\Sigma }$ and let $\widetilde {h}:\widetilde {\mathcal {N}}\rightarrow \widetilde {S\Sigma }$ be a lift of h to the universal covers. By the compactness of $\mathcal {N}$ and $\widehat {S\Sigma }$ , we can fix $\eta _1, \eta _2\text> 0$ such that for any $x,y\in \widetilde {\mathcal {N}}$ lying on the same flow line $\gamma $ ,

(3.2) $$ \begin{align} \text{if length}_{\widetilde{g}}(\gamma_{[x,y]})\geq \eta_1\quad\text{then length}_{\widetilde{ds}}(\widetilde{h}(\gamma_{[x,y]}))\geq\eta_2. \end{align} $$

Consider any two points $a_1,a_2\in \widetilde {\mathcal {N}}$ such that they are on the same flow line of $\gamma $ of $\widetilde {\Phi }_t$ . It is an easy exercise to prove the following using equation (3.2) and by Theorem 3.3:

(3.3) $$ \begin{align} \text{length}_{\widetilde{g}}(\gamma_{[a_1,a_2]}) &\leq \frac{\eta_1}{\eta_2} \text{length}_{\widetilde{ds}}(\widetilde{h}_(\gamma_{[\widetilde{h}(a_1),\widetilde{h}(a_2)]})) + \eta_1\nonumber\\ &= \frac{\eta_1}{\eta_2} d_2(\widetilde{h}(a_1),\widetilde{h}(a_2)) + \eta_1 \quad \text{by Theorem }3.3. \end{align} $$

Finally, as $h:\mathcal {N}\rightarrow \widehat {S\Sigma }$ is a homeomorphism between compact manifolds, the lifts to the universal covers induce quasi-isometries between the universal covers. Hence, there exists $\eta _3>1$ and $\eta _4> 0$ such that the map $\widetilde {h}^{-1}:(\widetilde {S\Sigma },d_{\widetilde {ds}})\rightarrow (\widetilde {\mathcal {N}},d_{\widetilde {g}})$ is an $(\eta _3,\eta _4)$ -quasi-isometry.

Applying the quasi-isometry $\widetilde {h}$ on (3.3), we get

$$ \begin{align*}\text{length}_{\widetilde{g}}(\gamma_{[a_1,a_2])})\ \leq \ \frac{\eta_1}{\eta_2}d_2(\widetilde{h}(a_1),\widetilde{h}(a_2)) + \eta_1\ \leq \ \frac{\eta_1}{\eta_2}(\eta_3 d_1(a_1,a_2)+\eta_4) + \eta_1.\end{align*} $$

Finally, let $A_0={\eta _1 \eta _3}/{\eta _2}$ and $A_1=({\eta _1 \eta _4}/{\eta _2}) + \eta _1$ . It follows that every flow line of $\widetilde {\Phi }_t$ is an $(A_0,A_1)$ -quasigeodesic.

4.1 Construction of the Franks–Williams-type hyperbolic plugs

Consider a hyperbolic linear automorphism A on the 2-torus $\mathbb {T}^2$ , which is induced by a linear map ${\widetilde {A}:\mathbb {R}^2\rightarrow \mathbb {R}}$ such that $\widetilde {A}$ has two eigenvalues $\lambda>1$ and $({1}/{\lambda })<1$ . On $\mathbb {T}^2$ , we have a pair of one-dimensional foliations, namely the stable $\mathcal {L}^s$ and unstable $\mathcal {L}^u$ foliations of the hyperbolic map A as described below.

• • Unstable foliation $\mathcal {L}^u: \mathbb {R}^2$ has a foliation $\tilde {\mathcal {L}}^u$ by the slope- $\lambda $ lines and this foliation is $\widetilde {A}$ -invariant. Hence, $\tilde {\mathcal {L}}^u$ on $\mathbb {R}^2$ projects down to a foliation on $\mathbb {T}^2$ and it is the unstable foliation $\mathcal {L}^u$ of A on $\mathbb {T}^2$ .

• • Stable foliation $\mathcal {L}^s$ : similarly, the foliation on $\mathbb {R}^2$ induced by the slope- ${1}/{\lambda }$ lines projects down to the stable foliation $\mathcal {L}^s$ of A on $\mathbb {T}^2$ .

These two foliations are everywhere transversal to each other on $\mathbb {T}^2$ . Hence, they define a two-frame $\{\mathscr {X},\mathscr {Y}\}$ on the tangent bundle $T\mathbb {T}^2$ , where $\mathscr {X}(p)$ is a vector in $T_p\mathbb {T}^2$ tangent to the stable direction and similarly, $\mathscr {Y}(p)$ is a vector tangent to the unstable direction in $T_p\mathbb {T}^2$ . In fact, we can define a new coordinate system $\{x,y\}$ on $\mathbb {R}^2$ .

A new coordinate system $\{x,y\}$ : fix a basis $\{v_{1/\lambda },v_{\lambda }\}$ on $\mathbb {R}^2$ , where the basis vectors are eigenvectors of the two distinct eigenvalues $\lambda $ and $1/\lambda $ . Then, the new coordinate system on $\mathbb {R}^2$ with respect to $\{v_{1/\lambda },v_{\lambda }\}$ is denoted by $\{x,y\}$ . In this coordinate, $\widetilde {A}$ can be written as $\widetilde {A}(x,y)=(({1}/{\lambda }) x,\lambda y)$ . We use this coordinate system extensively in the rest of the article.

The fixed point $(0,0)$ of $\widetilde {A}$ on $\mathbb {R}^2$ projects to a fixed point of A, denoted by $\mathfrak {o}$ , on $\mathbb {T}^2$ . We can change it to a point source or a point sink using the ‘Derived from Anosov(DA)’ bifurcation on a neighborhood of $\mathfrak {o}$ . Here, we give a quick description of the technique, a detailed description can be found in [Reference Katok and HasselblattKH95, §17.2] or in [Reference WilliamsWil70].

Consider two closed disks $D_1$ and $D_2$ on $\mathbb {T}^2$ centered at $\mathfrak {o}$ such that $\mathfrak {o}\in D_1 \subset \mathring {D}_2$ . On $D_2$ , we consider the local coordinate system $\{x,y\}$ around $\mathfrak {o}$ projected from the coordinates $\{x,y\}$ on $\mathbb {R}^2$ around $(0,0)$ . With respect to those coordinates on $D_2$ , the fixed point $\mathfrak {o}\in \mathbb {T}^2$ is represented by $(0,0)$ . Then we ‘blow-up’ the fixed point $\mathfrak {o}$ using a smooth map $\phi $ as described as follows:

$$ \begin{align*} \begin{split} \phi(x,y)&=(\theta(x,y)x, y)\quad \text{on }D_2\\ \phi &= \textrm{Id}\quad \text{on }\mathbb{T}^2\setminus D_2. \end{split} \end{align*} $$

In the above description, $\theta (x,y):\mathbb {T}^2\to [1,\infty )$ is a smooth map such that, on $\mathbb {T}^2\setminus D_2$ , we have $\theta (x,y)=1$ and near the point $\mathfrak {o}$ , the map $\theta (x,y)$ is large enough to counteract the contraction along the x-lines. Then, $\Phi =A \circ \phi $ is a map with a point source at $\mathfrak {o}$ . Notice that the coordinates $(x,y)$ make sense in a neighborhood of $\mathfrak {o}$ , but clearly one cannot have global coordinates in $\mathbb {T}^2$ . Still the equations above make sense.

Next, consider the suspension manifold

$$ \begin{align*}M=\frac{\mathbb{T}^2\times \mathbb{R}}{(q,t)\sim (\Phi(q), t-1)}\quad\text{for all }q\in\mathbb{T}^2\text{ and }t\in\mathbb{R}.\end{align*} $$

The universal cover of M, denoted by $\widetilde {M}$ , is $\mathbb {R}^2\times \mathbb {R}$ equipped with the coordinate system $\{x,y,t\}$ , where the x-axis and y-axis are as described above and the t-axis is in the vertical direction. The vertical lines induce a natural flow $\psi _t$ on M so that its lift $\widetilde \psi _t$ to the universal cover $\widetilde {M}$ is defined by $\psi _t([q,s])=[q,t+s], q \in \mathbb {R}^2$ . Note that we have a periodic orbit $\mathcal {C}$ of $\psi _t$ homeomorphic to the circle inside M through the fixed point $\mathfrak {o}\in \mathbb {T}^2$ .

To construct a hyperbolic plug, we cut out an open solid torus neighborhood $N(\mathcal {C})$ of the periodic orbit $\mathcal {C}$ , the new manifold $M_1=M\setminus N(\mathcal {C})$ is a manifold with boundary and we denote the boundary by $T_1=\partial M_1$ , the boundary is homeomorphic to a 2-torus. We choose $N(\mathcal {C})$ in such a way that the boundary of $N(\mathcal {C})$ is a smooth torus embedded in M and the flow lines of $\psi _t$ transversally intersect the boundary of $N(\mathcal {C})$ . We will have a further condition on $T_1$ described later. Finally, we can restrict the flow $\psi _t$ on $M_1$ , and the restricted semiflow on $M_1$ will be denoted by $\psi ^1_t$ .

It is clear from the construction that $(M_1,\psi ^1_t)$ is an attracting hyperbolic plug as the flow $\phi _t$ is the suspension flow of a ‘DA’ map with an attractor in the maximal invariant set. To ensure that when another plug is attached to $M_1$ along $\partial M_1$ , the semiflows are matched smoothly along the boundary, we attach a collar neighborhood homeomorphic to ${T_1 \times [0,1]}$ along $\partial M_1=T_1$ such that $\partial M_1$ is glued with $T_1\times \{0\}$ . We call the new manifold $\mathcal {M}_1$ , and the boundary component of $\mathcal {M}_1$ is denoted by $\mathcal {T}_1=\partial \mathcal {M}_1$ . Now propagate $\psi ^1_t$ in $T_1\times [0,1]$ via an isotopy such that the extension of the flow on $T_1\times [0,1]$ is a product flow topologically. We denote the extended flow on $\mathcal {M}_1$ by $\Psi ^1_t$ . In the next proposition, we sum up the description of the above construction.

4.2 Construction of the example manifolds and the flows on them

We consider a finite collection of Franks–Williams-type plugs, say

$$ \begin{align*}\{(\mathcal{M}_1,\Psi^1_t);(\mathcal{M}_2,\Psi^2_t);\cdots;(\mathcal{M}_n,\Psi_t^n)\}\end{align*} $$

along with a diffeomorphism $\Omega $ from the collection of exit boundaries $\mathcal {D}^{\mathrm {out}}=\sqcup _1^n\partial ^{\mathrm {out}}_i$ to the collection of entrance boundaries $\mathcal {D}^{in}=\sqcup _1^n\partial ^{in}_i$ . In this notation, for any plug $(\mathcal {M}_i,\Psi ^i_t)$ , either $\partial ^{\mathrm {out}}_i$ or $\partial ^{in}_i$ is empty, and the other one is non-empty. If $\partial ^{in}_i$ is non-empty, then it may have more than one component, each homeomorphic to a 2-torus, and the weak-stable foliation $\mathcal {L}^{ws}(\Lambda _i)$ of the semiflow $\Psi _t^i$ intersects each component of $\partial ^{in}_i$ in a union of two Reeb annuli. Similarly, if $\partial ^{\mathrm {out}}_i\neq \emptyset $ , each component of $\partial ^{\mathrm {out}}_i$ intersects the weak-unstable foliation $\mathcal {L}^{wu}(\Lambda _i)$ in a one-dimensional foliation with two Reeb annuli.

Using the diffeomorphism $\Omega $ , we can construct the manifold

$$ \begin{align*}\mathcal{N}=\frac{\mathcal{M}_1\sqcup \mathcal{M}_2\sqcup\cdots\sqcup \mathcal{M}_n}{\Omega(q)\sim q}\end{align*} $$

and the semiflows $\{\Psi ^1_t,\Psi ^2_t,\ldots ,\Psi ^n_t\}$ match to produce a flow $\Psi _t$ on $\mathcal {M}$ . If we consider a diffeomorphism $\Omega : \mathcal {D}^{\mathrm {out}}\rightarrow \mathcal {D}^{in}$ that transversally maps the one-foliation on each component of $\partial ^{\mathrm {out}}_i$ to the one-foliation on the respective component of $\mathcal {D}^{in}$ , then by Proposition 4.2, the flow $\Psi _t$ on $\mathcal {N}$ is Anosov.

The original construction by Franks and Williams was to construct a hyperbolic plug $(\mathcal {M}_1,\Psi ^1_t)$ from a DA map and with one exit boundary component, and attach it with $(\mathcal {M}_1,\Psi ^1_{-t})$ (that is, the same manifold equipped with the reversed flow) along the boundaries with a $\pi /2$ -rotation.

5.1 Construction of a Riemannian metric $\mathcal {G}_1$ on $\mathcal {M}_1$

Consider the attracting DA map $\Phi :\mathbb {T}^2\rightarrow \mathbb {T}^2$ and the manifold $M=\mathbb {T}^2\times [0,1]/\sim $ as described in the previous section. The universal cover of M, $\widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ , is equipped with the coordinate system $\{x,y,t\}$ . The lift of the flow $\psi _t$ to the universal cover $\widetilde {\mathcal {M}}=\mathbb {R}^2\times \mathbb {R}$ will be henceforth denoted by $\widehat {\psi }_t$ (we explain the $\widehat { }$ notation later). We can also define a three-frame $\{\mathscr {X},\mathscr {Y},\mathscr {T}\}$ on the tangent bundle $T(\mathbb {R}^2\times \mathbb {R})$ , where the vector fields $\mathscr {X}$ , $\mathscr {Y}$ , and $\mathscr {T}$ are parallel to the x-direction, y-direction, and t-direction, respectively. In addition, we define the vector field $\mathscr {T}$ as $\mathscr {T}=({d}/{dt})\widehat {\psi }_t$ .

We first construct a metric on M and then restrict it to $M_1=M\setminus N(\mathcal {C})$ . Our convention is that $M_1$ is the complement of the interior of $N(\mathcal {C})$ in M, so $M_1$ is compact and with boundary. We will use an intermediate cover of $M_1$ which will be denoted by $\widehat {M}_1$ . Consider the repelling periodic orbit $\mathcal {C}$ of $\psi _t$ in M and $N(\mathcal {C})$ a solid torus neighborhood of $\mathcal {C}$ as described in §3.1. Let $\widehat {N(\mathcal {C})}$ be the collection of lifts of $N(\mathcal {C})$ in $\widetilde {M} = \mathbb {R}^2\times \mathbb {R}$ . Then define $\widehat {M}_1=(\mathbb {R}^2\times \mathbb {R})\setminus \widehat {N(\mathcal {C})}$ , again the convention is that we are removing the interior of the sets. In other words, $\widehat {M}_1$ is the pullback of $M_1$ under the cover $\widetilde M \to M$ . This is an infinite cover of $M_1=M\setminus N(\mathcal {C})$ , but it is not the universal cover of $M_1$ . Later in this article, we do much of the analysis in $M_1$ and $\widehat {M}_1$ instead of $\widetilde {M}$ . For this reason, we will denote the metrics using the hat notation.

If $\widehat {G}_1$ is a Riemannian metric on $\widetilde {M}$ which projects down to M, then the deck transformations on $\widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ have to be isometries with respect to $\widehat {G}_1$ . The deck transformation group on $\widetilde {M}$ is generated by the following diffeomorphisms:

1. (1) $\Gamma : \mathbb {R}^2\times \mathbb {R} \rightarrow \mathbb {R}^2\times \mathbb {R}$ ,

   $$ \begin{align*}\Gamma(x,y,t)=(\widetilde{\Phi}(x,y), t-1),\end{align*} $$
   where $\widetilde {\Phi }(x,y)$ is the lift of the map $\Phi =A\circ \phi $ from $\mathbb {T}^2$ to $\mathbb {R}^2$ so that $\widetilde {\Phi }(0,0) = (0,0)$ ;

2. (2) translations by one unit in two horizontal directions with respect to the Euclidean coordinate system on $\mathbb {R}^2\times \mathbb {R}$ , that is:

   1. (a) $E_1(e_1, e_2,t)=(e_1+1, e_2,t)$ ;

   2. (b) $E_2(e_1, e_2,t)=(e_1, e_2+1,t)$ ,

   where $e_1$ and $e_2$ are given with respect to the Euclidean co-ordinates on $\mathbb {R}^2\times \mathbb {R}$ .

It is enough to construct the metric $\widehat {G}_1$ on $\mathbb {R}^2\times [0,1]$ such that the maps

$$ \begin{align*}\Gamma:\mathbb{R}^2\times\{1\}\rightarrow \mathbb{R}^2\times \{0\}, \ \ \Gamma(x,y,1)=(\widetilde{\Phi}(x,y),0)=(\tilde{A}\circ \tilde{\phi}(x,y),0)\end{align*} $$

and the translations $E_1$ and $E_2$ restricted on $\mathbb {R}^2\times [0,1]$ are isometries. Notice that the first map is between two-dimensional sets, and the other two are between three-dimensional sets.

As before, we consider the coordinate system $\{x,y,t\}$ on $\mathbb {R}^2\times [0,1]$ . The idea to define the metric on $\mathbb {R}^2\times [0,1]$ is as follows. We will pick a suitable metric $g_0$ on the level $t = 0$ and consider a family $h_s$ of maps, smoothly varying with $s\in [0,1]$ , where $h_0=\textrm {Id}$ and $h_1=\widetilde {\Phi }$ . Then we will pull-back the metric $g_0$ from the level $\{t=0\}$ to the level $\{t=s\}$ via the map $h_s$ .

First, we define a family of maps on $\mathbb {R}^2$ . Consider the neighborhood $D_2$ with the local co-ordinate system $\{x,y\}$ as defined in the description of the DA map in the previous section. Let $\widetilde {\theta }$ be the lift to $\mathbb {R}^2$ of $\theta : \mathbb {T}^2 \to \mathbb {R}$ . Now define the family of maps ${\tilde {\eta }_s:\mathbb {R}^2\rightarrow \mathbb {R}^2}$ for $s\in [0,1]$ as follows. First define

$$ \begin{align*}B_s(x,y) \ = \ (\lambda^{-s} x, \lambda^s y).\end{align*} $$

Notice that $B_1 = \widetilde {A}$ . Also define

$$ \begin{align*} \begin{split} \nu_s(x,y) &= \ ((\theta(x,y))^s x, y)\quad \text{on }D_2\\ &= \ (x,y)\quad \text{on }\mathbb{T}^2\setminus D_2. \end{split} \end{align*} $$

Again notice that $\nu _1 = \phi $ . Let $\tilde {\nu }_s: \mathbb {R}^2 \to \mathbb {R}^2$ be the lift of $\nu _s$ so that $\tilde {\nu }_s(0,0) = (0,0)$ . Then define

$$ \begin{align*}\tilde{\eta}_s(x,y) = B_s \circ \tilde{\nu}_s (x,y).\end{align*} $$

Now we are ready to define the family of maps $h_s:\mathbb {R}^2\times \{s\}\rightarrow \mathbb {R}^2\times \{0\}$ :

$$ \begin{align*}h_s(x,y,s)=(\tilde{\eta}_s(x,y),0)\quad\text{for }s\in[0,1].\end{align*} $$

In general, it is not easy to find the exact formula of the lifts $\tilde {\eta }_s$ , except near the point $(0,0)$ , where it is easy to get an explicit formula.

For each s, the map $h_s$ takes the level $\mathbb {R}^2 \times \{ s \}$ to $\mathbb {R}^2 \times \{ 0 \}$ . We start with the metric $g_0^2=dx^2+dy^2$ on the level $\mathbb {R}^2\times \{0\}$ where $dx$ (respectively $dy$ ) measures the length along x-directions (respectively y-directions).

Using the family $h_s, 0 \leq s \leq 1$ , we can pull-back the metric $g_0^2=dx^2+dy^2$ from $\mathbb {R}^2\times \{0\}$ to each level $\mathbb {R}^2 \times \{ s \}$ . Hence, on $\mathbb {R}^2\times \{t\}$ , we define the pull-back metric $g_t=(h_t)^*(g_0)$ , where $t\in [0,1]$ .

With the induced differentiable structure in $\mathbb {R}\times [0,1]$ from inclusion in $\mathbb {R}^2\times \mathbb {R}$ , it is easy to see that the the metrics on $\mathbb {R}\times \{t\}$ vary smoothly with $t\in [0,1]$ because of the smoothness of family $h_s, 0 \leq s \leq 1$ . We now define a metric not only on horizontal vectors, but on all vectors. At the point $q=(q_1,t)\in \mathbb {R}^2\times [0,1]$ , the metric is defined as follows:

(5.4) $$ \begin{align} &\widehat{G}_1(\mathscr{X},\mathscr{Y}) = g_t(\mathscr{X},\mathscr{Y}) \quad\text{on the level }\mathbb{R}^2\times\{t\}\text{ for }t\in[0,1]; \nonumber\\[4pt] & \widehat{G}_{1}(\mathscr{T}, a\mathscr{X}+b\mathscr{Y})= 0\quad\text{for all }a,b\in\mathbb{R};\nonumber\\[4pt] & \widehat{G}_1(\mathscr{T},\mathscr{T})= 1. \end{align} $$

Restrict the metric $G_1$ on $M_1=M\setminus N(\mathcal {C})$ .

Before describing the metric on the whole manifold $\mathcal {N}$ , we prove the following lemma, whose proof is simple but crucial, derived from the above defined Riemannian metric on $(\mathcal {M}_1,\mathcal {G}_1)$ .

Proof. Consider the universal cover $\widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ with the metric $\widehat {G}_1$ and the subset $\widehat {M}_1$ contained in it. It is enough to prove that the flow lines or flow rays in $\widehat {M}_1$ are length minimizing in $\widehat {M}_1$ with respect to $\widehat {G}_1$ . This is because any rectifiable curve $\gamma $ in $\widetilde {M}_1$ joining two points in a flow line or flow ray in $\widetilde {M}_1$ projects to a curve in $\widehat {M}_1$ joining two points in a flow line or flow ray.

What we prove is that the flow lines in $\widetilde {M}$ are globally length minimizing. This implies that that the flow lines or flow rays contained in $\widehat {M}_1$ are length minimizing in $\widehat {M}_1$ . Notice also that we denote the flows in $\widetilde {M}$ or $\widehat {M}_1$ by $\widehat {\psi }_t$ , see previous remark.

We note that the flow lines of the suspension flow $\widehat {\psi }_t$ in $\widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ are the vertical lines $\{*\}\times \mathbb {R}$ in $\mathbb {R}^2\times \mathbb {R}$ . These lines are the integral curves of the vector field ${\mathscr {T}=({d}/{dt})\widehat {\psi }_t}$ in $\mathbb {R}^2\times \mathbb {R}$ . As the vectors in the t-directions are orthogonal to the vectors in the span of $\{\mathscr {X},\mathscr {Y}\}$ (that is, in the horizontal levels $\mathbb {R}^2\times \{t\}$ ), the integral curves of the vector field $\mathscr {T}=({d}/{dt})\widehat {\psi }_t$ are globally length minimizing. More precisely, consider a vertical line $\{a\}\times \mathbb {R}$ and take two points $p_1$ and $p_2$ on it. Suppose $\sigma $ is a curve connecting $p_1$ and $p_2$ , then

$$ \begin{align*} \begin{aligned} \text{length}(\sigma) & =\int_{\text{domain}(\sigma)}\bigg\|\frac{d}{dt}\gamma(t)\bigg\|\,dt \\ & =\int_{\text{domain}(\sigma)}\bigg(\bigg\|\frac{d}{dt}\gamma(t)|_{\mathscr{T}(\gamma(t))}\bigg\|^2+\bigg\|\frac{d}{dt}\gamma(t)|_{\mathscr{X},\mathscr{Y}(\gamma(t))}\bigg\|^2\bigg)^{1/2}dt\\ & \geq \int_{\text{domain}(\sigma)}\bigg\|\frac{d}{dt}\gamma(t)|_{\mathscr{T}(\gamma(t))}\bigg\|\,dt, \end{aligned} \end{align*} $$

where $({d}/{dt})\gamma (t)|_{\mathscr {T}(\gamma (t))}$ and $({d}/{dt})\gamma (t)|_{\mathscr {X},\mathscr {Y}(\gamma (t))}$ denote the components of $\gamma '(t)=({d}/{dt})\gamma (t)$ along the t-direction and in the span of $\{\mathscr {X},\mathscr {Y}\}$ , respectively. Hence, it is clear that the integral curves of the vector field $\mathscr {T}=({d}/{dt})\widehat {\psi }_t$ , that is, the flow lines of $\widehat {\psi }_t$ in $\mathbb {R}^2\times \mathbb {R}$ , are globally length minimizing geodesic in $\widehat {M}=\mathbb {R}^2\times \mathbb {R}$ with respect to the metric $\widehat {G}_1$ .

As the flow lines of $\widehat {\psi }_t$ in $\widetilde {\mathcal {M}}$ are length minimizing geodesic, so is the flow lines or flow rays of the restricted flow $\widehat {\psi }_t|_{\widehat {M}_1}$ on $\widehat {M}_1$ inside $\widetilde {\mathcal {M}}$ .

Note that $\widehat {M}_1$ is not the universal cover of $M_1$ , but an infinite subcover of $M_1$ . As the flow rays are globally length minimizing geodesic in a subcover, the same has to be true in the universal cover $\widetilde {M}_1$ . This completes the proof.

We end this subsection with a crucial remark which will be used later.

Remark 5.6. The construction of our metric $G_1$ is motivated by the Solv metric ${dS^2=\lambda ^{-2s}\,dx^2+\lambda ^{2s}\,dy^2+ds^2}$ on $\mathbb {R}^2\times \mathbb {R}$ . We observe that there is another way to see the metric $\widehat {G}_1$ on $M_1$ , we can find a map $\mathcal {H}$ such that the family of map $h_s$ can be written as $h_s=\overline {B}_s\circ \mathcal {H}$ , where $\overline {B}_s(x,y,s)=(\lambda ^{-s}x,\lambda ^s y,0)$ . To see that such an $\mathcal {H}$ exists, it is enough to determine the map near the t-axis and it is easy to check that the map $\mathcal {H}(x,y,s)=((\tilde {\theta }(x,y))^s x,y,s)$ near $\{(0,0,t)|t\in \mathbb {R}\}$ serves the purpose. We can extend it to all of $\widetilde {M}$ using the definition of $\theta $ in $\mathbb {T}^2$ . Hence, the pull-back metric on the level $\mathbb {R}^2\times \{s\}$ is the same with $\mathcal {H}^*( \overline {B}_s^*(g_0))$ .

If we consider the family of pull-back metrics $\overline {B}_s^*(g_0)$ where $g_0^2=dx^2+dy^2$ on the level $\mathbb {R}^2\times \{0\}$ , it is easy to check that we get the Solv metric $dS$ on $\widetilde {\mathcal {M}}=\mathbb {R}^2\times \mathbb {R}$ ,

$$ \begin{align*}dS^2=\lambda^{-2s}\,dx^2+\lambda^{2s}\,dy^2+dt^2\quad\text{for }s\in[0,1].\end{align*} $$

Hence, the metric we defined, $\widehat {G}_1=\mathcal {H}^*(dS)$ , is a bounded perturbation of the Solv metric on each level $\mathbb {R}^2\times \{s\}$ .

5.2 Defining the metric on the whole manifold

As before, we consider a collection of attracting and repelling Franks–Williams-type hyperbolic plugs

$$ \begin{align*}\{(\mathcal{M}_1,\Psi_t^1);(\mathcal{M}_2,\Psi_t^2);(\mathcal{M}_3,\Psi_t^3);\cdots;(\mathcal{M}_n,\Psi_t^n\}.\end{align*} $$

By our construction, we have a metric $G_i$ on $M_i\subset \mathcal {M}_i$ for each i. The (possibly disconnected) surface $\partial M_1$ is smooth in the metric in M and hence inherits a Riemannian metric. Now, we smoothly extend the metrics $G_i$ on the union collar neighborhoods $\bigcup _i \partial M_i\times [0,1]$ to get a Riemannian metric $\widetilde {\mathcal {G}}$ on the whole manifold $\widetilde {\mathcal {N}}$ , in particular, $\widetilde {\mathcal {G}}|_{M_i}=G_i$ for all i. Note that there are only finitely many collar neighborhoods in $\bigcup _i \partial M_i\times [0,1]$ . Any flowline which enters in a collar neighborhood exits it after a finite time. Hence, the length of any flow segment contained in a collar neighborhood is bounded above by a global upper bound. For this reason, the choices of metrics on the collar neighborhoods do not affect the quasigeodesic behavior.

The restriction of $\widetilde {\mathcal {G}}$ to a lift $\widetilde {\mathcal {M}}_i$ of a single hyperbolic plug is denoted by $\widetilde {\mathcal {G}}_i$ , and the path metric induced by $\widetilde {\mathcal {G}}_i$ on $\widetilde {\mathcal {M}}_i$ is denoted by $d_{\widetilde {\mathcal {G}}_i}$ .

Notation:

• • for any $\mathcal {M}_i$ , $\mathcal {G}|_{\mathcal {M}_i} =\mathcal {G}_i$ ;

• • for any two points $p_1$ and $p_2$ in $\widetilde {\mathcal {M}}_i$ ,

  $$ \begin{align*}d_{\widetilde{\mathcal{G}}_i}(p,q)=\text{minimum}\{\text{length}_{\widetilde{\mathcal{G}}_i}(\sigma)|\sigma\text{ is a curve contained in }\widetilde{\mathcal{M}}_i\text{ connecting }p_1,p_2\}.\end{align*} $$

We finish this section showing that the flow lines or flow rays contained in the lift $\widetilde {\mathcal {M}}_i$ of a single hyperbolic plug in the universal cover $\widetilde {\mathcal {N}}$ is a quasigeodesic with respect to the restricted metric $d_{\widetilde {\mathcal {G}}_i}$ .

Lemma 5.8. There is an $\epsilon \geq 1$ and $\epsilon '\geq 0$ such that if $\gamma $ is a flow ray or flow line fully contained in some $\widetilde {\mathcal {M}}_i$ , and $p_1$ , $p_2$ are two points on $\gamma \subset \widetilde {\mathcal {M}}_i$ , then

$$ \begin{align*}\text{length}_{\widetilde{\mathcal{G}}_i}(\gamma_{[p_1,p_2]})\leq \epsilon d_{\widetilde{\mathcal{G}}_i}(p_1,p_2)+\epsilon'.\end{align*} $$

Proof. Without loss of generality, we prove the lemma in $\widetilde {\mathcal {M}}_1$ . As both $M_1$ and $\mathcal {M}_1$ are compact and the metric $G_1$ is the restriction of the metric $\mathcal {G}_1$ on $M_1$ , it is easy to check that $(\widetilde {M_1},d_{\widetilde {G}_1})$ is quasi-isometric to $(\widetilde {\mathcal {M}}_1, d_{\widetilde {\mathcal {G}}_1})$ . Hence, there exists $\epsilon _1\geq 1$ and $\epsilon _2 \geq 0$ such that for any $p_1,\ p_2\in \widetilde {M}_1$ ,

$$ \begin{align*} \begin{split} \text{length}_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_1,p_2]})&=\text{length}_{\widetilde{G}_1}(\gamma_{[p_1,p_2]}) \quad\text{as }\widetilde{G}_1=\widetilde{\mathcal{G}}_1|_{\widetilde{M_1}}\\ &=d_{\widetilde{G}_1}(\gamma_{[p_1,p_2]}) \quad\text{by Lemma 5.5}\\ &\leq \epsilon_1 d_{\widetilde{\mathcal{G}}_1}(p_1,p_2)+\epsilon_2 \quad\text{as }(\widetilde{M}_1,d_{\widetilde{G}_1})\text{ is quasi-isometric to}(\widetilde{\mathcal{M}}_1,d_{\widetilde{\mathcal{G}}_1}).\\ \end{split} \end{align*} $$

Now, we assume that $p_2\in \widetilde {\mathcal {M}}_1\setminus \widetilde {M}_1$ and $p_1 \in \widetilde {M}_1$ .

Here, $\widetilde {\mathcal {M}}_1$ can be written as $\widetilde {M_1}\cup (\partial \widetilde {M}_1\times [0,1])$ . By the compactness of $\partial M_1\times [0,1]$ , we can consider $\epsilon _3>0$ such that for any flow ray $\gamma $ ,

$$ \begin{align*}\text{length}_{\widetilde{\mathcal{G}}_1}(\gamma\cap (\partial \widetilde{M}_1\times [0,1]))\leq \epsilon_3.\end{align*} $$

Now consider the flow segment $\gamma _{[p_1,p_2]}$ , as $p_2\in \widetilde {\mathcal {M}}_1\setminus \widetilde {M}_1$ , the flow segment must intersect $\partial \widetilde {M}_1$ at a single point, say $p_3$ . Then,

$$ \begin{align*} \begin{split} \text{length}_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_1,p_2]})&=\text{length}_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_1,p_3]})+\text{length}_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_3,p_2]})\\ &\leq \epsilon_1 d_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_1,p_3]})+\epsilon_2+\text{length}_{\widetilde{\mathcal{G}}_1}(\gamma_{[p_3,p_2]})\quad\text{as }p_1,p_3\in \widetilde{M}_1\\ & \leq \epsilon_1 d_{\widetilde{\mathcal{G}}_1}(p_1,p_3)+\epsilon_2+\epsilon_3\\ &\leq \epsilon_1 (d_{\widetilde{\mathcal{G}}_1}(p_1,p_2)+d_{\widetilde{\mathcal{G}}_1}(p_2,p_3))+\epsilon_2+\epsilon_3\\ &\leq \epsilon_1 (d_{\widetilde{\mathcal{G}}_1}(p_1,p_2)+\epsilon_3)+\epsilon_2+\epsilon_3 \quad\text{as }p_2,p_3\in\partial\widetilde{M}_1\times [0,1]\\ & \leq \epsilon_1 d_{\widetilde{\mathcal{G}}_1}(p_1,p_2)+\epsilon_1\epsilon_3+\epsilon_2+\epsilon_3. \end{split} \end{align*} $$

We redefine $\epsilon _1$ and $\epsilon _1\epsilon _3+\epsilon _2+\epsilon _3$ as $\epsilon $ and $\epsilon '$ .

As there are only finitely many plugs, we can choose $\epsilon $ and $\epsilon '$ big enough such that it works for all $\partial \widetilde {M}_i\times [0,1]$ and the result follows.

6.1 Flowlines in the plug $(\mathcal {M}_1,\mathcal {G}_1,\Psi ^{1}_t)$

Recall that $\mathcal {M}_1$ is made from $M=({\mathbb {T}^2\times \mathbb {R}})/{(q,t)\sim (\Phi (q),t-1)}$ as in §4.1, where $\Phi =A \circ \phi $ is a repelling DA map. The universal cover $\widetilde {M}$ can be considered as $\mathbb {R}^2\times \mathbb {R}$ equipped with the coordinate system $\{x,y,t\}$ , where x-directions and y-directions are parallel to the strong stable and the strong unstable directions of the hyperbolic map $\widetilde {A}$ on $\mathbb {R}^2$ and t-directions are along the flow lines of the suspension flow $\widehat {\psi }^1_t$ .

Consider the repelling periodic orbit $\mathcal {C}$ of $\psi _t$ in M and $N(\mathcal {C})$ is an open solid torus neighborhood of $\mathcal {C}$ as described in §3.1. Let $\widehat {N(\mathcal {C})}$ be the collection of lifts of $N(\mathcal {C})$ in $\mathbb {R}^2\times \mathbb {R}$ , then $\widehat {M}_1=(\mathbb {R}^2\times \mathbb {R})\setminus \widehat {N(\mathcal {C})}$ is an infinite cover of $M_1=M\setminus N(\mathcal {C})$ , but not the universal cover.

Here, $M_1$ is equipped with the Riemannian metric $G_1$ as constructed in the previous section, similarly, we denote the lifted metric on $\widehat {M}_1$ by $\widehat {G}_1$ .

Additionally, $\widehat {M}_1$ is a manifold with boundary, where $\partial \widehat {M}_1$ is the lift of $\partial {M_1}=T_1$ . Furthermore, $\partial \widehat {M}_1$ is a collection of disjoint infinite cylinders in $\mathbb {R}^2\times \mathbb {R}$ which are transverse to the flow lines. Suppose $\partial \widehat {M}_1=\bigcup _{i\in \mathbb {N}}\widehat {T}_i$ , where $\widehat {T}_i$ are the infinite cylindrical boundary components of $\partial \widehat {M}_1$ .

There are exactly two types of flow lines of the lifted semiflow $\widehat {\psi }^1_t|_{\widehat {M}_1}$ in $\widehat {M}_1$ ; if $\gamma $ is a flow line in $\widehat {M}_1$ , then:

1. (1) either $\gamma $ is contained in $\widehat {\mathcal {A}}$ , the lift of the attractor $\mathcal {A}\subset M_1$ ;

2. (2) or $\gamma $ intersects a lift of $\partial M_1=T_1$ . Moreover, $\gamma $ intersects exactly one such lift of $T_1$ .

In this subsection, we show that almost all flow rays which intersect the boundary $\partial \widehat {M}_1$ ‘go away’ from the boundary component it intersects in an efficient manner as time t goes to positive infinity. In the next subsection, we extend the result in the universal cover $\widetilde {\mathcal {M}}_1$ . To state the precise statement, we first need to fix some notation.

Consider the repelling fixed point $\mathfrak {o}$ of the DA map $\Phi =A \circ \phi $ on $\mathbb {T}^2$ . There are also two hyperbolic fixed points, denoted by $p_1$ and $p_2$ , on the attractor in $\mathbb {T}^2$ , as shown in Figure 1(a). In the suspension manifold $M=\mathbb {T}^2\times [0,1]/\sim $ , there are two periodic orbits, $C_{p_1}$ and $C_{p_2}$ , coming from $p_1$ and $p_2$ , and these two orbits are contained in the attractor $\mathcal {A}$ of $\psi ^1_t$ . Now we consider the repelling orbit $\mathcal {C}$ and the open solid torus neighborhood $N(\mathcal {C})$ around it, as described in §3.1. The weak stable leaves of $C_{p_1}$ and $C_{p_2}$ intersect the boundary torus $T_1=\partial N(\mathcal {C})=\partial M_1$ in two circles, denoted by $\mathcal {C}_1$ and $\mathcal {C}_2$ , as shown in Figure 1(b).

Definition 6.2. Take a point $q\in \widehat {M}_1$ such that the flow line $\gamma _q$ through q intersects $\partial \widehat {M}_1$ at exactly one point. The boundary component is denoted by $\widehat {T}_q$ .

Figure 1 (a) Two-dimensional image of the blow up of a hyperbolic point, here ‘O’ denotes the origin. (b) Three-dimensional image of the blow-up of a hyperbolic orbit.

Then, let:

• • $\mathcal {D}_{\widehat {G}_1}(q,\widehat {T}_q)$ (respectively $\mathcal {D}_{S}(q,\widehat {T}_q)$ ) denote the distance between q and $\widehat {T}_q$ with respect to the path metric induced by $\widehat {G}_1$ (respectively the Solv metric $dS$ ) on $\widehat {M}_1$ ;

• • $\ell _{\widehat {G}_1}(q)$ (respectively $\ell _{S}(q)$ ) is the length of the flow line segment connecting q and $\widehat {T}_q$ with respect to $\widehat {G}_1$ (respectively $dS$ ).

Next, we fix a small number $\delta>0$ and consider the open $\delta $ -neighborhood $N_{\delta }(\mathcal {C}_1\cup \mathcal {C}_2)$ on $T_1$ and let $N_{\delta }(\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2)$ denote the lift of $N_{\delta }(\mathcal {C}_1\cup \mathcal {C}_2)$ in $\widehat {M}_1$ . Notice that $\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2$ is a countable, infinite collection of properly embedded lines in $\widehat {M}_1$ . The following is a key technical result used in this article.

Proposition 6.3. There exists $C> 1$ and $c> 0$ satisfying the following. Let $q\in \widehat {M}_1$ such that the flow ray through q intersects $\partial \widehat {M}_1$ , say at the boundary component $\widehat {T}_q$ . If $\gamma _q$ intersects $\widehat {T}_q$ on the region $\widehat {T}_q\setminus N_{\delta }(\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2)$ , then

$$ \begin{align*}\ell_{\widehat{G}_1}(q)\leq C\mathcal{D}_{\widehat{G}_1}(q, \widehat{T}_q) +c.\end{align*} $$

Proof. We will prove the lemma in the manifold $\widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ using the coordinate system $\{x,y,t\}$ on it. This is enough, because we have $\widehat {M}_1\subset \widetilde {M}=\mathbb {R}^2\times \mathbb {R}$ and the distance between two points in $\widehat {M}_1$ with respect to the metric $d_{\widehat {G}_1}$ is bigger than the distance in $\mathbb {R}^2\times \mathbb {R}$ with respect to $d_{\widehat {G}}$ . In addition, the length of a flow segment is the same in both $\widehat {M}_1$ and $\widetilde {M} = \mathbb {R}^2\times \mathbb {R}$ , as the Riemannian metric in $\widehat {M}_1$ is the one induced from the inclusion $\widehat {M}_1 \subset \widetilde {M} = \mathbb {R}^2\times \mathbb {R}$ .

It is enough to prove the result for one component of the boundary $\partial \widehat {M}_1$ as we can permute the boundary components using the translation isometries. In addition, up to changing the coordinates $(x,y)$ , we can assume the orbit passes through $(0,0,0)$ . In other words, we fix the lift of the repelling orbit $\mathcal {C}$ passing through $(0,0,0)\in \mathbb {R}^2\times \mathbb {R}$ , that is, the line $\{(0,0,t)|t\in \mathbb {R}\}$ and the lift of the $\partial M_1=T_1$ around this line, which we denote by  $\widehat {T}_0$ .

We will first prove the result for the points q on the $yt$ -plane $\{x=0\}$ . The reason behind it is that the Riemannian metric on the $yt$ -plane is almost $\lambda ^{2t}\,dy^2+dt^2$ . We make this more precise in the following remark.

Remark 6.4.

1. (1) Note that the map $\tilde {\phi }:\mathbb {R}^2\rightarrow \mathbb {R}^2$ perturbs the y-directions only near the lifts in $\mathbb {R}^2$ of the repelling fixed point $\frak {o}$ , and outside of those neighborhoods, the map $\widetilde {\phi }$ is the identity map. Hence, it is also true that that map $\mathcal {H}$ distorts the $yt$ -plane boundedly in $\mathbb {R}^2\times \mathbb {R}$ . By Remark 5.6, we know that $\widehat {G}_1=\mathcal {H}^{*}(dS)$ , where $dS$ is the Solv metric. In particular, the Solv metric restricted on the $yt$ -plane is $\lambda ^{2t}\,dy^2+dt^2$ , and the x-directions are everywhere perpendicular to the $yt$ -plane. Hence, we can assume that the Riemannian metric on the $yt$ -plane induced by $\widehat {G}_1$ is boundedly distorted from and very close to the Solv metric. In other words, we can find two constants $a_0>1$ and $a_1>0$ such that if $\sigma $ is a curve on the $yt$ -plane, then

   $$ \begin{align*}\frac{1}{a_0}\text{length}_{\widehat{G}_1}(\sigma)-a_1\leq\text{length}_{S}(\sigma)\leq a_0\text{ length}_{\widehat{G}_1}(\sigma)+a_1,\end{align*} $$
   where $\text {length}_S(\sigma )$ and $\text {length}_{\widehat {G}_1}(\sigma )$ mean the length of $\sigma $ with respect to the Solv metric $dS$ and $\widehat {G}_1$ , respectively.

2. (2) Note that the $yt$ -plane is the unstable leaf through $(0,0,0)$ for the suspension of the linear Anosov map $\tilde {A}$ . Notice that the $yt$ -plane is not invariant under the map $\widetilde {A}\circ \widetilde {\phi }$ .

Now we can state the precise result we want to prove on the $yt$ -plane.

Lemma 6.5. There exists $K>1$ and $k>0$ such that for any q in the intersection of the $yt$ -plane with $\widehat {M}_1$ for which the flow line $\gamma _q$ intersects $\widehat {T}_0$ , we have

$$ \begin{align*}\ell_{\widehat{G}_1}(q)\leq K\mathcal{D}_{\widehat{G}_1}(q, \widehat{T}_0)+k.\end{align*} $$

Proof. Consider a point q on the $yt$ -plane such that the flow ray $\gamma _q$ passing through q intersects $\widehat {T}_0$ .

As the manifolds are complete, every distance can be realized by a curve. Suppose $\sigma :[0,1]\rightarrow \widetilde {M}$ denotes a shortest path connecting q and $\widehat {T}_0$ in $\widehat {M}_1$ .

Claim 6.6. Any shortest path $\sigma $ between q and $\widehat {T}_0$ lies on the $yt$ -plane.

Proof. Suppose $\sigma $ is parameterized as $\sigma (t)=(\sigma _1(t),\sigma _2(t),\sigma _3(t))$ in $\mathbb {R}^2\times \mathbb {R}$ . Consider the projection of $\sigma $ on the $yt$ -plane, $\Pi (\sigma (t))=(0,\sigma _2(t),\sigma _3(t))$ . Suppose $\Pi (\sigma (a))$ is the first intersection point of $\Pi (\sigma )$ and $\widehat {T}_0$ . We show that

$$ \begin{align*}\text{length}(\Pi(\sigma))|_{[0,a]}\leq \text{length}(\sigma)\end{align*} $$

and the claim follows from that.

In Remark 5.6, we have seen that the metric $\widehat {G}_1$ can be expressed as the pull-back of the Solv metric $dS$ by the map $\mathcal {H}$ , that is, $\widehat {G}_1=\mathcal {H}^*(dS)$ , where ${dS=\lambda ^{-2s}\,dx^2+\lambda ^{2s}\,dy^2+dt^2}$ .

Consider the map $\mathcal {H}:\mathbb {R}^2\times \mathbb {R}\rightarrow \mathbb {R}^2\times \mathbb {R}$ as in Remark 5.6. As $\widehat {G}_1=\mathcal {H}^*(dS)$ , for any curve $\zeta $ ,

$$ \begin{align*}\text{length}(\zeta)\text{ with respect to the metric }\widehat{G}_1=\text{length}(\mathcal{H}(\zeta))\text{ with respect to the metric }dS.\end{align*} $$

As the directions x, y, and t are orthogonal to each other with respect to the Solv metric $dS$ , it is easy to check that for any curve $\zeta $ in $\mathbb {R}^2\times \mathbb {R}$ and its projection $\Pi (\zeta )$ on $yt$ -plane, we have

$$ \begin{align*}\text{length}(\Pi(\delta))\leq \text{length}(\sigma)\text{ with respect to the metric }dS.\end{align*} $$

Moreover, the projection map $\Pi $ commutes with $\mathcal {H}$ because $\mathcal {H}$ does not change the y-coordinate of a point and $\Pi $ is the projection on the $yt$ -plane. This can also be easily verified by the formulas.

Hence, we conclude

$$ \begin{align*} & \text{length}(\Pi(\mathcal{H}(\sigma)))\leq \text{length}(\mathcal{H}(\sigma))\text{ with respect to the metric }dS \\ \implies &\text{length}(\mathcal{H}(\Pi(\sigma)))\leq \text{length}(\mathcal{H}(\sigma))\text{ with respect to the metric }dS \\ \implies &\text{length}(\Pi(\sigma)))\leq \text{length}(\sigma)\text{ with respect to the metric }\widehat{G}_1. \end{align*} $$

This completes the claim.

In the rest of the proof of Lemma 6.5, we will use the Solv metric on the $yt$ -plane, that is, the metric $\lambda ^{2t}\,dy^2+dt^2$ for simplicity of calculations. In Remark 5.2, we have seen that the Solv metric on the $yt$ -plane is quasi-isometric to the metric induced by $\widehat {G}_1$ .

Consider the point $q=(0,c,0)$ with $c>0$ on the $\widehat {T}_0\cap yt$ -plane and the forward flow ray $\gamma _{q}=\widetilde {\psi }^1_{[0,\infty )}(p)$ , as shown in Figure 2, $\gamma _q=\{(0,c,t)|t\in [0,\infty )\}$ .

Figure 2 (a) A minimal path in the $yt$ -plane from $q'$ to the lift of the boundary torus and (b) several geometric quantities that are used in the analysis. In particular, $R_q(\overline {t}') \lambda ^{\overline {t}'}$ is the length in the solvable metric of the horizontal segment depicted at height $\overline {t}'$ .

We prove the lemma for the flow ray $\gamma _q$ first. Later, we explain how to derive the result for all other flow lines in the $yt$ plane intersecting $\widehat {T}_0$ from this particular result on $\gamma _q$ .

Proof for the flow line $\boldsymbol {\gamma _q}$ : suppose $q'=(0,c,t')$ is a point on $\gamma _q$ and let $\sigma _{q'}$ be a length minimizing curve on the $yt$ -plane joining $q'$ and $\widehat {T}_0$ . We define three functions as follows.

1. (1) Let $\Pi _t(\sigma _{q'})$ denote the projection of $\sigma _{q'}$ on the vertical line $\gamma _q=\{(0,c,t)|t\in \mathbb {R}\}$ along y-directions—same as horizontal directions (notice that $\sigma _{q'}$ is contained in the $yt$ plane).

2. (2) Suppose $\bar {t'}$ is the lowest t-value attained by the curve $\sigma _{q'}$ as shown in Figure 2(a). Then we denote the projection of $\sigma _{q'}$ on the line $t=\bar {t'}$ along t-directions—same as vertical directions—by $\Pi _y(\sigma _{q'})$ .

3. (3) For a point $(0,c,t")$ on $\gamma _{q'}$ , suppose the line $t=t"$ intersects $\widehat {T}_0$ in the positive side of y-direction at $(0,c',t")$ as shown in Figure 2(b). Then we define $R_{q}(t")=|c-c'|$ . Note that, with respect to the Solv metric $dS$ , the length of the segment on the line $t=t"$ connecting $(0,c,t")$ and $(0,c',t")$ is $\lambda ^{t"}|c-c'|$ or $R_{q}(t")\lambda ^{t"}$ . Clearly, $R_{q}$ is an increasing function of t on $[0,\infty )$ .

Now we are ready to state two lower estimates of $\mathcal {D}_{S}(q',\widehat {T}_0)$ .

Claim 6.7. Suppose $\bar {t'}$ is the lowest t-value in the projection $\Pi _t(\sigma _{q'})$ as shown in Figure 2(a). Then:

1. (1) $t'-\bar {t'}\leq \mathcal {D}_{S}(q',\widehat {T}_0)$ ;

2. (2) in addition, $R_{q}(\bar {t'})\lambda ^{\bar {t'}}\leq \mathcal {D}_{S}(q',\widehat {T}_0)$ .

Proof. As the t-directions and y-directions are everywhere orthogonal with respect to the metric $\lambda ^{2t}\,dy^2+dt^2$ , it is easy to check that $\text {length}_{S}(\Pi _t(\sigma _{q'}))\leq \text {length}_{S}(\sigma _{q'})$ and, in addition, we have $\text {length}_S(\Pi _y(\sigma _{q'}))\leq \text {length}_S(\sigma _{q'})=\mathcal {D}_S(q',\widehat {T}_0)$ .

1. (1) As the curve $\Pi _t(\sigma _{q'})$ connects the points $q'=(0,c,t')$ and $(0,c,\bar {t'})$ , it is clear that $|t'-\bar {t'}|\leq \text {length}_S(\Pi _t(\sigma _{q'}))\leq \mathcal {D}_{S}(q',\widehat {T}_0)$ .

   Note that $\bar {t'}$ cannot be a negative number, because if $\bar {t'}<0$ , then

   $$ \begin{align*}t'-\bar{t'}>t'=\ell_S(q')\geq \mathcal{D}_{S}(q',\widehat{T}_0),\end{align*} $$
   which cannot be true as we have just proved $t'-\bar {t'}\leq \mathcal {D}_{S}(q',\widehat {T}_0)$ .

2. (2) As $\bar {t'}\geq 0$ , we observe that the line segment $\{(0,y,\bar {t'})|y\in [c-R_{q}(\bar {t'}),c]\}$ is contained in the curve $\Pi _y(\sigma _{q'})$ as shown in Figure 2(b). Hence,

   $$ \begin{align*}R_{q}(\bar{t'})\lambda^{\bar{t'}}\leq\text{length}_S(\Pi_t(\sigma_{q'}))\leq\text{length}_S(\sigma_{q'}) =\mathcal{D}_{S}(q',\widehat{T}_0).\end{align*} $$

This proves the claim.

Consider the function $P_{q}(t)={R_{q}({t}/{2})\lambda ^{{t}/{2}}}/{t}$ on $t\in (0,\infty )$ .

As the function $R_{q}(t/2)$ is increasing and $\lambda ^{t/2}/t$ is strictly increasing for large t values, we can fix a value $k'>0$ such that $P_{q}(t)>1$ when $t>k'$ .

For the point $q'=(0,c,t')$ , if $t'>k'$ , then $R_q(t'/2)\lambda ^{t'/2}>t'$ . However, $t'=\ell _{S}(q')\geq \mathcal {D}_{S}(q',\widehat {T}_0)$ . Combining these two inequalities, we get

(6.5) $$ \begin{align} \mathcal{D}_{S}(q',\widehat{T}_0)\leq \ell_{S}(q')=t' < R_{q}(t'/2)\lambda^{t'/2}. \end{align} $$

Hence, by Claim 6.7(2), $t'/2$ cannot be smaller than the lowest t-value attained by the curve $\Pi _t(\sigma _{q'})$ , that is, $t'/2>\bar {t'}$ . Otherwise, we get

$$ \begin{align*}R_{q}(t'/2)\lambda^{t'/2}\leq R_{q}(\bar{t'})\lambda^{\bar{t'}}\leq \mathcal{D}_{S}(q',\widehat{T}_0),\end{align*} $$

which contradicts equation (6.5).

Finally, as $t'/2>\bar {t'}$ , applying Claim 6.7(1), we deduce

(6.6) $$ \begin{align} &t'-t'/2\leq t'-\bar{t'}\leq \mathcal{D}_{S}(q',\widehat{T}_0)\nonumber\\ &\quad\!\implies\! t'/2\leq \mathcal{D}_{S}(q',\widehat{T}_0) \nonumber\\ &\quad\!\implies\! t'\leq 2\mathcal{D}_{S}(q',\widehat{T}_0)\nonumber\\ &\quad\!\implies\! {\ell_S(q')\leq 2\mathcal{D}_{S}(q',\widehat{T}_0)\quad\text{when }q'=(0,c,t')\text{ and }t'>k'.} \end{align} $$

If $t'\leq k'$ , then for a point $q'=(0,c,t')$ , we get

(6.7) $$ \begin{align} \ell_{S}(q')\leq k'\leq k'+2\mathcal{D}_{S}(q',\widehat{T}_0). \end{align} $$

Combining equations (6.6) and (6.7), we conclude

(6.8) $$ \begin{align} \ell_{S}(q')\leq2\mathcal{D}_{S}(q',\widehat{T}_0)+k'\quad\text{for all } q'\in\gamma_q=\{(0,c,t)|t\in[0,\infty)\}. \end{align} $$

Finally, by using Remark 6.4, we can deduce $\mathcal {D}_{S}(q',\widehat {T}_0)\leq a_0\mathcal {D}_{\widehat {G}}(q',\widehat {T}_0)+a_1$ and replacing it in the previous equation,

$$ \begin{align*}\ell_{S}(q')&\leq2\mathcal{D}_{S}(q',\widehat{T}_0)+k'\\&\leq 2a_0\mathcal{D}_{\widehat{G}_1}(q',\widehat{T}_0)+2a_1+k'\quad\text{for all } q'\in\gamma_q=\{(0,c,t)|t\in[0,\infty)\}.\end{align*} $$

By renaming $2a_0=K$ and $2a_1+k'=k$ , and replacing $\ell _{S}(q')=\ell _{\widehat {G}_1}(q')$ ,

$$ \begin{align*}{\ell_{\widehat{G}_1}(q')\leq K\mathcal{D}_{\widehat{G}_1}(q',\widehat{T}_0)+k\quad\text{for all } q'\in\gamma_q=\{(0,c,t)|t\in[0,\infty)\}.}\end{align*} $$

This completes the proof of Lemma 6.5 only on the flow ray $\gamma _q$ , where $q=(0,c,0)$ . To deal with the other flow rays in the $yt$ -plane, we do the following: consider a family of maps

$$ \begin{align*}\mu_a: \widetilde{M} \to \widetilde{M},\quad \mu_a(x,y,t) = (\lambda^a x, \lambda^{-a} y, t + a),\end{align*} $$

where a is an arbitrary real number. It is easy to see that any $\mu _a$ is an isometry of the Solv metric $dS$ . In addition, we choose the original torus $T_0$ transverse to the flow so that $\mu _a$ leaves invariant the fixed lift $\widehat {T}_0$ of $T_0$ to $\widetilde {M}$ for any $a \in \mathbb {R}$ . Notice that $\mu _a$ also fixes the $yt$ plane. In the $yt$ plane, $\mu _a$ sends flow lines to flow lines. If $p = \mu _a(q)$ , then p is also in $\widehat {T}_0$ . Now for any $p'$ in the forward flow line of p, one obtains equation (6.8) for $p'$ as well, since $\mu _a$ is an isometry of the Solv metric. This obtains all flow lines in the $yt$ -plane, except for the flow line through $(0,0,0)$ , but this one does not intersect $\widehat {T}_0$ . Now use Remark 6.4 to finish the proof of Lemma 6.5.

Now we are ready to prove Proposition 6.3.

Proof of Proposition 6.3

The maximal invariant set of the semiflow $\psi ^1_t$ is a hyperbolic set $\mathcal {A}$ in $M_1$ , which is a two-dimensional attractor. Consider the weak-stable foliation $\mathcal {L}^{ws}$ of $\mathcal {A}$ associated with the semiflow $\psi ^1_t$ in $M_1$ . Every point in $M_1$ is in $\mathcal {L}^{ws}$ , as every point in $M_1$ is attracted to $\mathcal {A}$ . The foliation $\mathcal {L}^{ws}$ intersects the boundary $\partial M_1$ in a one-dimensional foliation which has two Reeb components [Reference Franks and WilliamsFW80]; we denote this foliation on $\partial M_1$ by $\mathcal {F}$ . As described before, the circular leaves of $\mathcal {F}$ are denoted by $\mathcal {C}_1$ and $\mathcal {C}_2$ . They are the common boundary circles of the Reeb annuli of $\mathcal {F}$ .

Suppose $\widehat {\mathcal {F}}$ is the lift of $\mathcal {F}$ on $\widehat {T}_0$ . The intersection of the $yt$ -plane and $\widehat {T}_0$ has two components, say $D_1$ and $D_2$ , as shown in Figure 3. Here, $\widehat {\mathcal {C}}_1$ and $\widehat {\mathcal {C}}_2$ are on two different sides of the $yt$ -plane. As any leaf L of $\widehat {\mathcal {F}}$ (except $\widehat {\mathcal {C}}_1$ and $\widehat {\mathcal {C}}_2$ ) is asymptotic to both $\widehat {\mathcal {C}}_1$ and $\widehat {\mathcal {C}}_2$ , it must intersect the $yt$ -plane in a single point either on $D_1$ or on $D_2$ , as shown in Figure 3.

Figure 3 (a) Foliation $\widehat {\mathcal {F}}$ in a lift of a torus; (b) leaf of $\widehat {\mathcal {F}}$ through q intersects $D_1$ at $\bar {q}$ . The shaded region represents the ‘bad region’.

Consider $q\in \widehat {T}_0\setminus \{\widehat {\mathcal {C}}_1, \widehat {\mathcal {C}}_2\}$ and the leaf $L_q$ of $\widehat {\mathcal {F}}$ passing through q. As described above, $L_q$ intersects either $D_1$ or $D_2$ at a single point, without loss of generality we assume that $L_q$ intersects $D_1$ and let $\bar {q}=D_1\cap L_q$ , as shown in Figure 3. Note that q and $\bar {q}$ lie on the same leaf of the weak stable foliation of $\widehat {\mathcal {A}}$ , and hence there exists $s\in \mathbb {R}$ such that $\widehat {\psi }^1_s(q)$ and $\bar {q}$ lie on the same strong stable leaf of $\hat {\psi }^1_t$ . It follows that

$$ \begin{align*}d_{\widehat{G}_1}(\widehat{\psi}^1_{t+s}(q),\widehat{\psi}^1_t(\bar{q}))\to 0\quad\text{as }t\to \infty.\end{align*} $$

Let $s_1 = d_{\widehat {G}_1}(q,\bar {q})$ . Fix $\delta _1>0$ , $\delta _1 << 1, \delta _1 < s_1$ . By the above limit, we can find ${s_2>0}$ such that $d_{\widehat {G}_1}(\widehat {\psi }^1_{t+s}(q),\widehat {\psi }^1_t(\bar {q}))\leq \delta _1$ whenever $t>s_2$ .

For any $t'>s_2$ , consider the four points $q,q_{t'}=\widehat {\psi }^1_{t'+s}(q),\bar {q}$ and $\bar {q}_{t'}=\widehat {\psi }^1_{t'}(\bar {q})$ . By the triangle inequality,

(6.9) $$ \begin{align} d_{\widehat{G}_1}(q,q_{t'})\leq d_{\widehat{G}_1}(q,\bar{q})+d_{\widehat{G}_1}(\bar{q},\bar{q}_{t'})+d_{\widehat{G}_1}(\bar{q}_{t'},q_{t'}). \end{align} $$

Note that $d_{\widehat {G}_1}(q,q_{t'})=\ell _{\widehat {G}_1}(q_{t'})$ as q and $q_{t'}$ lie on the same flow ray and by Lemma 5.5, flow segments are length minimizing in $\widehat {M}$ with respect to the path metric of $\widehat {G}_1$ , similarly $d(\bar {q},\bar {q}_{t'})=\ell _{\widehat {G}_1}(q_{t'})$ . Moreover, by our assumption, $d_{\widehat {G}_1}(q,\bar {q}) = s_1$ , $d_{\widehat {G}_1}(q_{t'},\bar {q}_{t'})<\delta _1$ . Replacing in the above inequality (6.9), we get

(6.10) $$ \begin{align} \ell_{\widehat{G}_1}(q_{t'}) & \leq d_{\widehat{G}_1}(q,\bar{q})+d_{\widehat{G}_1}(\bar{q},\bar{q}_{t'})+d_{\widehat{G}_1}(\bar{q}_{t'},q_{t'}) \nonumber\\ & \leq s_1+\ell_{\widehat{G}_1}(\bar{q}_{t'})+\delta_1\nonumber\\ & = \ell_{\widehat{G}_1}(\bar{q}_{t'})+2s_1\quad\text{for any }t'>s_2. \end{align} $$

As $\bar {q}\in \widehat {T}_0\cap \{yt\}-$ plane, by Lemma 6.5, we know there are global $K, k> 0$ , so that $\ell _{\widehat {G}_1}(\bar {q}_{t'})\leq K\mathcal {D}_{\widehat {G}_1}(\bar {q}_{t'},\widehat {T}_0)+k$ . Applying it in equation (6.10), we have

(6.11) $$ \begin{align} \ell_{\widehat{G}_1}(q_{t'})\leq \ell_{\widehat{G}_1}(\bar{q}_{t'})+2s_1 \leq K\mathcal{D}_{\widehat{G}_1}(\bar{q}_{t'},\widehat{T}_0)+k+2s_1 \quad \text{when }t'>s_2. \end{align} $$

Finally, suppose $a\in \widehat {T}_0$ is a point in $\widehat {T}_0$ that is closest to $q_{t'}$ , that is, $d_{\widehat {G}_1}(q_{t'},a)=\mathcal {D}_{\widehat {G}_1}(q_{t'},\widehat {T}_0)$ . By the triangle inequality, we get $d_{\widehat {G}_1}(\bar {q}_{t'},a)\leq d_{\widehat {G}_1}(q_{t},a)+d_{\widehat {G}_1}(q_{t'},\bar {q}_{t'})$ . Moreover, $\mathcal {D}_{\widehat {G}_1}(\bar {q}_{t'},\widehat {T}_0)\leq d_{\widehat {G}_1}(\bar {q}_{t'},a)$ as $a\in \widehat {T}_0$ . Combining all these facts, we conclude that when $t'>s_2$ ,

(6.12) $$ \begin{align} \mathcal{D}_{\widehat{G}_1}(\bar{q}_{t'},\widehat{T}_0) & \leq d_{\widehat{G}_1}(\bar{q}_{t'},a) \leq d_{\widehat{G}_1}(q_{t'},a)+d_{\widehat{G}_1}(\bar{q}_{t'},q_{t'}) \nonumber\\ &= \mathcal{D}_{\widehat{G}_1}(q_{t'},\widehat{T}_0)+d(\bar{q}_{t'},q_{t'})\quad\text{as }d_{\widehat{G}_1}(q_{t'},a)=\mathcal{D}_{\widehat{G}_1}(q_{t'},\widehat{T}_0)\nonumber\\ &= \mathcal{D}_{\widehat{G}_1}(q_{t'},\widehat{T}_0)+s_1\quad\text{as }d_{\widehat{G}_1}(\bar{q}_{t'},q_{t'}) = s_1 \text{ by assumption}. \end{align} $$

Combining equations (6.11) and (6.12), we get

(6.13) $$ \begin{align} \ell_{\widehat{G}_1}(q_{t'})\leq K\mathcal{D}_{\widehat{G}_1}(\bar{q}_{t'},\widehat{T}_0)+k+2s_1\leq K\mathcal{D}_{\widehat{G}_1}(q_{t'})+Ks_1+k+2s_1\quad\text{when }t'>s_2. \end{align} $$

The above inequality proves Proposition 6.3 for $t'>s_2$ . If $t'\leq s_2$ , then $\ell (q_{t'})=t'<s_2$ . Adding the case when $t'\leq s_2$ in inequality (6.13), we conclude

(6.14) $$ \begin{align} \ell_{\widehat{G}_1}(q_{t'})\leq K\mathcal{D}_{\widehat{G}_1}(q_{t'})+Ks_1+k+2s_1+s_2\quad\text{for all }t'\in[0,\infty). \end{align} $$

By renaming $C=K$ and $c=Ks_1+k+2S_1+s_2$ , we rewrite the above inequality (6.14) as

$$ \begin{align*}{\ell_{\widehat{G}_1}(q_{t'})\leq C\mathcal{D}_{\widehat{G}_1}(q_{t'})+c\quad\text{for all }t'\in[0,\infty),}\end{align*} $$

which completes the proof of Proposition 6.3 for the point q.

We still need to argue why we can find constants $C>1$ and $c>0$ which work for all $q\in \widehat {T}_0\setminus N_{\delta }(\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2)$ . We need to find $s_1$ and $s_2$ big enough, such that inequality (6.14) holds for all flow rays intersecting $\widehat {T}_0\setminus N_{\delta }(\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2)$ .

Consider a fundamental domain in $\widehat {T}_0$ which quotient downs on the torus in $M_1$ . Without loss of generality, we assume that the fundamental domain is bounded by the planes $t=0$ and $t=1$ , and we call it $\overline {T}_{0,1}$ . Next, we consider the compact set $\mathcal {S}$ which is the closure of the set $\widehat {T}_{0,1}\setminus N_{\delta }(\widehat {\mathcal {C}}_1\cup \widehat {\mathcal {C}}_2)$ .

As $\mathcal {S}$ is compact, it has finite radius with respect to the metric $d_{\widehat {G}_1}$ . Recall the definition of $s_1$ . It is $s_1 = d_{\widehat {G}_1}(q, \bar {q})$ . Here, $\bar {q} = (D_1 \cup D_2) \cap L_q$ , where $L_q$ is a leaf of $\widehat {\mathcal {F}}$ and $D_1 \cup D_2$ are the intersections of the $yt$ -plane with $\widehat {T}_0$ . Since q is in a compact set $\mathcal {S}$ , it follows that $\bar {q}$ is also in a compact set. It follows that $s_1$ is globally bounded.

Now we consider $s_2$ . Given q, the value s was defined so that $\widehat {\psi }^1_s(q)$ and $\bar {q}$ lie in the same strong stable leaf of $\widehat {\psi }^1_t$ . Again, since $\mathcal {S}$ is compact and $\bar {q}$ is in a compact set, it follows that the values of s as a function of q are also globally bounded in $\mathcal {S}$ . Then, there is a global $s_2> 0$ so that $d_{\widehat {G}_1}(\widehat {\psi }^1_{t+s}(q), \widehat {\psi }^1_t(\bar {q})) < \delta _1$ for all $t> s_2$ .

This shows that $s_1, s_2$ can be chosen globally bounded for q in the fundamental domain $\mathcal {S}$ . Since it is a fundamental domain, this shows that $s_1, s_2$ can be chosen globally bounded.

This finishes the proof of Proposition 6.3.

Now we extend Proposition 6.3 in the universal cover of $\widetilde {M}_1$ with respect to the lifted Riemannian metric $\widetilde {G}_1$ . Note that $\partial \widetilde {M}_1$ is the lift of the torus $\partial M_1$ , and it is a collection of infinitely many planes homeomorphic to $\mathbb {R}^2$ . We first re-define the notation as follows.

For a point $q\in \widetilde {M}_1$ , suppose the flow line through q intersects a component of $\partial \widetilde {M}$ , say $\widetilde {T}_q$ , then:

• • $\mathcal {D}_{\widetilde {G}_1}(q,\widetilde {T}_q)$ denotes the distance between q and $\widetilde {T}_q$ with respect to the path metric induced by $\widetilde {G}_1$ on $\widetilde {M}_1$ ;

• • $\ell _{\widetilde {G}_1}(q)$ is the length of the flow line segment connecting q and $\widetilde {T}_q$ with respect to $\widetilde {G}_1$ .

Consider the lift of the neighborhood $N_{\delta }(\mathcal {C}_1\cup \mathcal {C}_2)$ in $\widetilde {M}_1$ , we denote it as $N_{\delta }(\widetilde {\mathcal {C}}_1\cup \widetilde {\mathcal {C}}_2)$ . We can restate Lemma 6.3 in $\widetilde {M}_1$ as follows.

Lemma 6.9. There exists $C> 1$ and $c> 0$ satisfying the following. Let $q\in \widetilde {M}_1$ such that the flow ray through q intersects $\partial \widetilde {M}_1$ , say at the boundary component $\widetilde {T}_q$ . If $\gamma _q$ intersects $\widetilde {T}_q$ on the region $\widetilde {T}_q\setminus N_{\delta }(\widetilde {\mathcal {C}}_1\cup \widetilde {\mathcal {C}}_2)$ , then

$$ \begin{align*}\ell_{\widetilde{G}_1}(q)\leq C\mathcal{D}_{\widetilde{G}_1}(q, \widetilde{T}_q) +c.\end{align*} $$

Proof. Note that $\widehat {M}_1$ is an intermediate cover of $M_1$ , and hence $\widetilde {M}_1$ is the universal cover of $\widehat {M}_1$ . For any two points $b_1$ and $b_2$ in $\widetilde {M}_1$ , if $\bar {b}_1,\bar {b}_2 $ denotes the projection of $b_1, b_2$ in $\widehat {M}_1$ , then

$$ \begin{align*}d_{\widehat{G}_1}(\bar{b}_1,\bar{b}_2)\leq d_{\widetilde{G}_1}(b_1,b_2).\end{align*} $$

As $\widetilde {M}_1$ is the universal cover of a compact manifold, for any point $q\in \widetilde {M}_1$ , there exists a point $q^*\in \widetilde {T}_q$ such that $\mathcal {D}_{\widetilde {G}_1}(q,\widetilde {T}_0)=d_{\widetilde {G}_1}(q,q^*)$ . In addition, there is a path in $\widetilde {M}_1$ from q to $q^*$ which realizes this distance. This implies that if $\bar {q}\in \widehat {M}_1$ is the projection of the point $q\in \widetilde {M}_1$ , then $\mathcal {D}_{\widehat {G}_1}(\bar {q},\widehat {T}_q)\leq \mathcal {D}_{\widetilde {G}_1}(q,\widetilde {T}_q)$ . Moreover, the t-directions are unchanged in $\widehat {M}_1$ and $\widetilde {M}_1$ , and hence $\ell _{\widehat {G}_1}(\bar {q})=\ell _{\widetilde {G}_1}(q)$ . Combining all the information, we get

$$ \begin{align*}\ell_{\widetilde{G}_1}(q)=\ell_{\widehat{G}_1}(\bar{q})\leq C\mathcal{D}_{\widehat{G}_1}(\bar{q},\widehat{T}_{\bar{q}})+c\leq C\mathcal{D}_{\widetilde{G}_1}(q,\widetilde{T}_q)+c.\\[-37pt]\end{align*} $$

Now we extend Lemma 6.9 to $\mathcal {M}_1= M_1\cup (\partial {M_1}\times [0,1])$ . The manifold $\mathcal {N}$ is the union of the collection of $\{(\mathcal {M}_1, \Psi _t^1, \mathcal {G}_1);(\mathcal {M}_2, \Psi _t^2, \mathcal {G}_2);(\mathcal {M}_3, \Psi _t^3, \mathcal {G}_3);\cdots ;(\mathcal {M}_n, \Psi _t^n, \mathcal {G}_n)\}$ , where the plugs intersect each other along their boundary components. As described in §4, we extend Riemannian metrics $G_i$ from $M_i$ to $\mathcal {M}_i$ and the extended metric is $\mathcal {G}_i$ . The induced Riemannian metric in $\mathcal {N}$ is $\mathcal {G}$ . In particular,