# New Classes of Quasigeodesic Anosov Flows in $3$-Manifolds

Anindya Chanda and SÉRGIO R. FENLEY  Department of Mathematics
Florida State University
1017 Academic Way
Tallahassee, FL, 32304, United States<EMAIL_ADDRESS>Department of
Mathematics
Florida State University
1017 Academic Way
Tallahassee, FL, 32304, USA and Institute for Advanced Study, Princeton, NJ
08540, USA<EMAIL_ADDRESS>
###### Abstract.

Quasigeodesic behavior of flow lines is a very useful property in the study of
Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In
fact up to orbit equivalence, the only previously known examples of
quasigeodesic Anosov flows were suspension flows. In this article, we prove
that a new class of examples are quasigeodesic. These are the first examples
of quasigeodesic Anosov flows on three manifolds that are neither Seifert, nor
solvable, nor hyperbolic. In general, it is very hard to show that a given
flow in quasigeodesic, and in this article we provide a new method to prove
that an Anosov flow is quasigeodesic.

###### Key words and phrases:

Anosov flows, Quasigeodesics, Geometric properties of flow lines. Large scale
properties of flows

###### 2020 Mathematics Subject Classification:

Primary: 57R30, 37E10, 37D20, 37C85. Secondary: 53C12, 37C27, 37D05, 37C86.

S. Fenley’s research partially supported by Simons foundation 637554, by
National Science Foundation grant DMS-2054909, and by the Institute for
Advanced Study.

## Introduction

A flow on a manifold is called quasigeodesic if its orbits are uniformly
efficient up to a bounded multiplicative and additive error in measuring
distances when lifted to the universal cover. Quasigeodesics are extremely
important for example in hyperbolic manifolds. This is because the Morse Lemma
says that on a hyperbolic space, any quasigeodesic is within a bounded
distance from a length minimizing geodesic (when lifted to the universal
cover) [Gro87, Thu82]. The distance depends on the quasigeodesic constants.

From the dynamical systems point of view, there are several important reasons
to study hyperbolic flows, and in three manifolds these are Anosov and pseudo-
Anosov flows [Ano69, KH95, Mos92b]. The question as to whether an Anosov or a
pseudo-Anosov flow in a closed hyperbolic three manifold is quasigeodesic has
been studied a lot [Fen94, Fen95, Fen16, Fen22, FM01, Mos92a]. In addition any
suspension flow is a quasigeodesic flow, in any manifold [Zeg93].

The question of quasigeodesic behavior for Anosov flows on non-hyperbolic
three manifolds, which are not orbit equivalent to suspensions, has not been
studied at all. The goal of this article is to start the study of the
quasigeodesic property for Anosov flows in more general three manifolds.

First we analyze Anosov flows in Seifert fibered three manifolds and prove:

######  Theorem 0.1.

Let $\Phi$ be an Anosov flow in a closed, Seifert fibered $3$-manifold. Then
$\Phi$ is a quasigeodesic flow.

To prove this theorem we first show that, under an appropriate natural metric,
flow lines of the geodesic flow on the unit tangent bundle of a hyperbolic
surface are globally length minimizing (in the universal cover). An Anosov
flow on a Seifert fibered manifold is orbitally equivalent to a finite lift of
the geodesic flow on the unit tangent bundle of a hyperbolic surface. By using
the orbit equivalence we show that the flow lines of the Anosov flow on a
Seifert fibered manifold are length minimizing up to finite bounds when lifted
to the universal cover.

We remark that not every Anosov flow is quasigeodesic: in [Fen94], the second
author proved that there exist infinitely many examples of Anosov flows on
three dimensional hyperbolic manifolds which are not quasigeodesic. In
addition for a flow on a general manifold $\mathcal{M}$, there might exist a
Riemannian metric such that all the flow lines are geodesic (a differential
geometric condition), but that does not guarantee that the flow lines are
quasigeodesic.

The main result of this article proves the quasigeodesic property for Anosov
flows in new classes of three manifolds. These manifolds have non trivial JSJ
decomposition [Hem76]. First recall that the DA operation [Wil70] transforms a
hyperbolic periodic orbit into either an attracting or repelling periodic
orbit. Franks and Williams [FW80] used this operation to produce the first
examples of non transitive Anosov flows in dimension three as follows: they
did a DA operation on a closed orbit of a suspension, producing a repelling
orbit and an attractor. They removed a solid torus neighborhood of the
periodic orbit to produce a flow in a manifold with boundary so that the flow
is incoming along the boundary. They carefully glued this with a copy with a
time reversed flow. Under certain homotopy types of gluings, the resulting
flow is Anosov. These examples revolutionized the study of Anosov flows in
dimension three.

In modern terminology, the manifold with boundary, equipped with the incoming
flow, is called a hyperbolic plug [BBY17]. In their article Beguin, Bonatti
and Yu prove that under very general conditions, gluing hyperbolic plugs
produces Anosov flows. In this article we consider Anosov flows which we call
generalized Franks-Williams flows. They are obtained as gluings of hyperbolic
plugs as follows: start with a suspension Anosov flow and do a DA operation on
finitely many periodic orbits. The operations are done so that either they all
produce attracting orbits, or they all produce repelling orbits. Then remove a
solid torus neighborhood of each DA orbit. Each plug contains either an
attractor or a repeller. Glue finitely many of these plugs using the
techniques of [BBY17]. The resulting flow is an Anosov flow [BBY17].

The main result of this article is the following:

######  Theorem 0.2.

Let $\Phi$ be a generalized Franks-Williams Anosov flow in a closed
$3$-manifold $M$. Then $\Phi$ is a uniform quasigeodesic flow.

######  Remark 0.3.

This result is new in the sense that the manifolds in question are neither
Seifert, nor hyperbolic, nor solvable. The JSJ decomposition is not trivial.
It is easy to prove that the supporting manifolds of the hyperbolic plugs in
question are atoroidal. In addition, by a result of Leeb [Lee95], a Haken
manifold with at least one atoroidal piece in the JSJ decomposition can be
given a Rimennnian metric with non-positive sectional curvature, and hence the
universal cover is $CAT(0)$ w.r.t. the induced path metric. Therefore the
manifolds considered in Theorem 0.2 admit $CAT(0)$ metrics.

###  0.1. Some ideas on the proof of Theorem 0.2.

First we mention a big difference from the case that the manifold is
hyperbolic. As we remarked previously, a lot of study has been done on the
quasigeodesic property for Anosov flows in hyperbolic $3$-manifolds. In these
manifolds a quasigeodesic satisfies that in the universal cover it is at a
bounded Hausdorff distance from a geodesic. On hyperbolic manifolds geodesics
are globally length minimizing curves in the universal cover. The bounded
distance property is strongly connected with the quasigeodesic property, and
in certain situations it is an intermediate step in proving quasigeodesic
behavior.

The manifolds in this article are not hyperbolic. In particular a
quasigeodesic in the universal cover may not be at a bounded Hausdorff
distance from a minimal geodesic. This happens for example in Euclidean space.
In the examples we study, there might exist quasi-flats in the universal cover
$-$ possible examples are lifts of the gluing tori, and they behave like the
Euclidean plane.

In this article, we analyze the flow in each individual block, and then
analyze how the blocks are assembled together. In each block the flow is
obtained from a blow up of a suspension Anosov flow, hence it satisfies the
quasigeodesic property when restricted to the block. The much more complicated
property is to analyze orbits that cross the tori gluing different blocks.
This turns out to be very intricate. We produced our flows so that we glue a
collection of repellers to a collection of attractors. It follows that an
orbit not contained in a plug intersects one (and only one) of the gluing
tori, and goes from being near a repeller to being near an attractor. One
potential problem is the following: it could happen that the segments of an
orbit on either side of the gluing torus may track the torus for a long time
and go in opposite directions. Lifting to the universal cover one produces a
big length along an orbit, but the distance between two points on the orbit
may not be large enough compared to the length of orbit segment connecting
them. In fact if one considers arbitrary gluings on the gluing tori (and not
just the ones generating Anosov flows as in [BBY17]), then this problem can
occur and the flow is not quasigeodesic.

We do a very careful analysis to show that when the resulting flow is Anosov,
then the flow lines are quasigeodesics. One crucial step is related to the
potential problem above: we show that if a flow line intersect certain regions
of a gluing tori, then the forward half orbit ‘moves away efficiently’ from
the lift of the torus when lifted to the universal cover. In other words, in
the manifold, the forward orbit cannot keep tracking close to the torus for a
long time. This result is Proposition 5.2 and it is the key component of the
main result. Of course this good behavior is not true for all orbits
intersecting the torus and there is a bad region as well. In order to prove
the quasigeodesic behavior we have to consider both forward and backwards half
orbits from points in the gluing tori, and how they are pierced together.

##  1\. Preliminaries

A map
$f\mathrel{\mathop{\ordinarycolon}}(X_{1},d_{1})\rightarrow(X_{2},d_{2})$
between two metric spaces $(X_{1},d_{1})$ and $(X_{2},d_{2})$ is a quasi-
isometric embedding if there exist two constants $C>1$ and $c>0$ such that,
for any points $p_{1},p_{2}\in X_{1}$,

$\frac{1}{C}d_{1}(p_{1},p_{2})-c\leq d_{2}(f(p_{1}),f(p_{2}))\leq
Cd_{1}(p_{1},p_{2})+c$

A quasigeodesic in $(X_{1},d_{1})$ is a quasi-isometric embedding of an
interval in $\mathbb{R}$ (with the standard metric) in $(X_{1},d_{1})$; the
interval can be any of the forms $[a,b],[a,b),(a,b]$ or
$(a,b)\subset\mathbb{R}$ where $a,b\in\mathbb{R}\cup\\{+\infty,-\infty\\}$. If
$a$ or $b$ is contained in the interval, then we assume it is not either of
$+\infty,-\infty$.

Suppose $\mathcal{N}$ is a closed Riemannian manifold with a Riemannian metric
$g$ and let the path metric induced by $g$ be denoted by $d_{g}(\ ,\ )$. A
flow $\Phi_{t}$ on $\mathcal{N}$ with $C^{1}$-orbits is called quasigeodesic,
if each flow line $\gamma$ of the lifted flow $\widetilde{\Phi}_{t}$ in the
universal cover $\widetilde{\mathcal{N}}$ is a quasi-isometric embedding of
$\mathbb{R}$ for some constants $C>1$ and $c>0$. The metric in $\mathbb{R}$ is
the path distance along the flow line. It is immediate that the quasigeodesic
property of a flow line $\gamma$ in $\widetilde{\mathcal{N}}$ is equivalent to
the following: there exists $C>1$ and $c>0$ such that for any two points
$a,b\in\gamma$,

$\text{length}_{\widetilde{g}}(\gamma_{[a,b]})\leq Cd_{\widetilde{g}}(a,b)+c$

where $\gamma_{[a,b]}$ is the flow segment connecting $a,b\in\gamma$,
$\widetilde{g}$ is the lift of the Riemannian metric $g$ on
$\widetilde{\mathcal{N}}$ and $d_{\widetilde{g}}$ is the path metric on
$\widetilde{\mathcal{N}}$ induced from $\widetilde{g}$.

The definition is independent of the metric as the quasigeodesic property is
preserved under quasi-isometric embeddings and as our underlying manifold is
compact, any two metrics in $\widetilde{\mathcal{N}}$ which are lifts from
metrics in $\mathcal{N}$ are quasi-isometric to each other. However the quasi-
isometry constants may change.

If the same quasi-isometry constants $C>1$ and $c>0$ work for all the flow
lines then we say that flow is uniformly quasigeodesic. It is not true that
every quasigeodesic flow is uniform. Notice however that in closed hyperbolic
manifolds, Calegari proved in [Cal06] that every quasigeodesic flow is
uniformly quasigeodesic.

The focus of this article is to study the quasigeodesic behavior of Anosov
flows.

######  Definition 1.

A $C^{1}$-flow
$\Phi_{t}\mathrel{\mathop{\ordinarycolon}}\mathcal{M}\rightarrow\mathcal{M}$
on a Riemannian manifold $\mathcal{M}$ is $Anosov$ if the tangent bundle
$T\mathcal{M}$ splits into three $D\Phi_{t}$-invariant sub-bundles
$T\mathcal{M}=E^{s}\oplus E^{0}\oplus E^{u}$ and there exists two constants
$B,b>0$ such that

  * •

$E^{0}$ is generated by the non-zero vector field defined by the flow
$\Phi_{t}$;

  * •

For any $v\in E^{s}$ and $t>0$,

$||D\Phi_{t}(v)||\leq Be^{-bt}||v||$

  * •

For any $w\in E^{u}$ and $t>0$,

$||D\Phi_{t}(w)||\geq Be^{bt}||w||$

The definition is independent of the choice of the Riemannian metric $||.||$
as the underlying manifold $\mathcal{M}$ is compact.

For a point $p\in\mathcal{M}$, we will denote the flow line through $p$ by
$\gamma_{p}$, i,e $\gamma_{p}=\\{\Phi_{t}(p)|t\in\mathbb{R}\\}$. The
collection of all flow lines of a flow defines an one-dimensional foliation on
$\mathcal{M}$. For an Anosov flow there are several flow invariant foliations
associated to the flow and these foliations play a key role in the study of
Anosov flows.

######  Property 1.1 ([Ano69]).

For an Anosov flow $\Phi_{t}$ on $\mathcal{M}$, the distributions $E^{u}$,
$E^{s}$, $E^{0}\oplus E^{u}$ and $E^{0}\oplus E^{s}$ are uniquely integrable.
The associated foliations are denoted by $\mathcal{F}^{u}$, $\mathcal{F}^{s}$,
$\mathcal{F}^{wu}$ and $\mathcal{F}^{ws}$ respectively and they are called the
strong unstable, strong stable, weak unstable and weak stable foliation on
$\mathcal{M}$.

We conclude this section by introducing the notion of orbit equivalence
between two flows $\Phi_{t}^{1}$ and $\Phi_{t}^{2}$.

######  Definition 2.

Two flows
$\Phi_{t}^{1}\mathrel{\mathop{\ordinarycolon}}\mathcal{M}\rightarrow\mathcal{M}$
and
$\Phi_{t}^{2}\mathrel{\mathop{\ordinarycolon}}\mathcal{N}\rightarrow\mathcal{N}$
are said to be orbit equivalent if there exists a homeomorphism
$h\mathrel{\mathop{\ordinarycolon}}\mathcal{M}\rightarrow\mathcal{N}$ such
that there exists a continuous map
$\tau\mathrel{\mathop{\ordinarycolon}}\mathcal{N}\times\mathbb{R}\rightarrow\mathbb{R}$
such that $h\circ\Phi^{1}_{t}\circ h^{-1}(x)=\Phi^{2}_{\tau(x,t)}(x)$ for all
$x\in\mathcal{N}$.

An orbit equivalence maps orbits to orbits with a possible time change.

### Organization of the article:

In section 2, we prove that geodesic flow on the unit tangent bundle of a
hyperbolic surface is quasigeodesic. Moreover, Anosov flows on Seifert Fibered
three manifolds are quasigeodesic.

In Section 3, we describe the construction of generalized Franks-Williams
flows and in section 4, we describe the Riemannian metric we are going to work
with in this article.

Section 5 contains the proof of Theorem 0.2; Subsection 5.1 proves the key
proposition for the proof and Subsection 5.2 completes the proof.

##  2\. Anosov flows in Seifert manifolds and quasigeodesic behavior

Suppose $\Sigma$ is a hyperbolic surface and $T\Sigma$ denotes its tangent
bundle, i,e $T\Sigma=\\{(p,v)|p\in\Sigma,v\in T_{p}\Sigma\\}$. The universal
cover of $\Sigma$ is the hyperbolic plane, we will consider the  Póincare
upper-half plane model $\mathbb{H}$ in this article, i,e.

$\mathbb{H}=\\{(x,y)\in\mathbb{R}^{2}|y>0\\}\text{ with the Riemannian metric
}ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}$

On the tangent bundle $T\Sigma$ we can define the geodesic vector field w.r.t.
the metric $ds$; this is a classical construction, in this article we follow
the notations and the detailed description as in [doC88, Chapter 3].

######  Definition 3.

The geodesic field is defined to be the unique vector field $G$ on $T\Sigma$
whose trajectories are of the form $(\gamma(t),\gamma^{\prime}(t))$ where
$\gamma$ is a geodesic on $\Sigma$ w.r.t. $ds$.

The flow $\mathfrak{G}_{t}$ of the geodesic field is called the geodesic flow
on $T\Sigma$.

Suppose $\widetilde{\mathfrak{G}}_{t}$ is the lift of the geodesic flow on the
universal cover
$\widetilde{T\Sigma}=T\mathbb{H}=\mathbb{H}\times\mathbb{R}^{2}$. We show that
if $(\gamma(t),\gamma^{\prime}(t))$ is a flow line of
$\widetilde{\mathfrak{G}}_{t}$ on $T\mathbb{H}=\mathbb{H}\times\mathbb{R}^{2}$
then it is a quasigeodesic in $T\mathbb{H}$.

First we choose an appropriate metric on $\widetilde{T\Sigma}$ which projects
down to $T\Sigma$. Consider the projection map
$\pi\mathrel{\mathop{\ordinarycolon}}T\mathbb{H}\rightarrow\mathbb{H}$. We can
define a metric on $T\mathbb{H}$ using the projection $\pi$ and the metric
$ds$ on $\mathbb{H}$ as described in [doC88, Chapter 3, Exercise 2], here is
the detailed description:

Suppose $(p,v)\in T\mathbb{H}$, consider
$\alpha_{1}\mathrel{\mathop{\ordinarycolon}}t\rightarrow(p_{1}(t),v_{1}(t))$
and
$\alpha_{2}\mathrel{\mathop{\ordinarycolon}}t\rightarrow(p_{2}(t),v_{2}(t))$
where $p_{1}(0)=p_{2}(0)=p\in\mathbb{H}$ and $v_{1}(0)=v_{2}(0)=v\in
T_{p}(\mathbb{H})$. Let $V_{1}=\alpha_{1}^{\prime}(0)$ and
$V_{2}=\alpha_{2}^{\prime}(0)$. Then define the inner product as

$\langle V_{1},V_{2}\rangle_{(p,v)}=\langle
d\pi(V_{1}),d\pi(V_{2})\rangle_{p}+\langle\frac{Dv_{1}}{dt}(0),\frac{Dv_{2}}{dt}(0)\rangle_{p}$

where $\langle,\rangle_{p}$ is given by the metric $ds$ on $\mathbb{H}$ and
$\frac{D}{dt}$ denotes the covariant derivative as defined in [doC88,
Proposition 2.2].

Clearly the metric on $T\mathbb{H}$ as defined above projects down to
$T\Sigma$ as the metric $ds$ on $\mathbb{H}$ projects down to $\Sigma$.
Abusing the notation, we denote metric on $T\Sigma$ by $ds$.

Next we prove that $\mathfrak{G}_{t}$ on $T\Sigma$ is a quasigeodesic flow.
Suppose $(\gamma(t),\gamma^{\prime}(t))$ is a flow line of
$\widetilde{\mathfrak{G}}_{t}$ on $T\mathbb{H}$, let $(p,v)$ and $(q,w)$ be
two points on $(\gamma,\gamma^{\prime})$. Consider a curve
$\zeta\mathrel{\mathop{\ordinarycolon}}t\rightarrow(\zeta_{1}(t),\zeta_{2}(t))\in
T\mathbb{H}$ on $t\in[0,1]$ such that $\zeta(0)=(p,v)$ and $\zeta(1)=(q,w)$
and $\zeta_{2}(t)\in T_{\zeta_{1}(t)}\mathbb{H}$ for all $t\in[0,1]$. Then

(2.1)
$\begin{split}\text{length}(\zeta)&=\int_{0}^{1}||\zeta^{\prime}(t)||dt\\\
&=\int^{1}_{0}\sqrt{||\zeta_{1}^{\prime}(t)||^{2}+||\frac{D\zeta_{2}(t)}{dt}||^{2}}dt\\\
&\geq\int^{1}_{0}\sqrt{||\zeta_{1}^{\prime}(t)||^{2}}dt=\text{length}(\zeta_{1})\end{split}$

Note that $\zeta_{1}$ is a curve on $\mathbb{H}$ connection
$p,q\in\mathbb{H}$. But $\gamma$ is a geodesic on $\mathbb{H}$ and geodesics
on $\mathbb{H}$ are globally length minimizing on $\mathbb{H}$, which means

$\text{length}(\zeta_{1})\geq\text{length}(\gamma)\text{ between
}p,q\in\mathbb{H}$

As $\gamma$ is a geodesic, the covariant derivative of $\gamma$ vanishes by
definition, i,e $\frac{D\gamma^{\prime}}{dt}=0$. Using the fact that
$\frac{D\gamma^{\prime}}{dt}=0$ and the Riemannian metric on $T\mathbb{H}$, it
is easy to verify that,

$\text{length}(\gamma,\gamma^{\prime})=\text{length}(\gamma)\text{ between
}p,q\in\mathbb{H}$

Replacing in 2.1 we conclude that between $(p,v),(q,w)\in T\mathbb{H}$,

$\text{length}(\zeta)\geq\text{length}(\zeta_{1})\geq\text{length}(\gamma)=\text{length}(\gamma,\gamma^{\prime})$

As the choice of $(\gamma,\gamma^{\prime})$ and $\zeta$ were arbitrary, the
above inequality implies that the flow lines of $\widetilde{\mathfrak{G}}_{t}$
are globally length minimizing in $T\mathbb{H}$, a stronger property than
being a quasigeodesic. In other words, we proved the following:

######  Theorem 2.1.

The flow lines of the geodesic flow $\widetilde{\mathfrak{G}}_{t}$ on
$T\mathbb{H}$ are globally length minimizing.

If $\Sigma$ is a hyperbolic surface then the geodesic flow on $T\Sigma$ is a
quasigeodesic flow.

###  2.1. Geodesic flows on unit tangent bundle

We note that the flow lines of the geodesic flow on $T\Sigma$ are of the form
$(\gamma,\gamma^{\prime})$. As $\gamma$ is a geodesic on $\Sigma$ we get
$\frac{d}{dt}\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle=0$ i,e
$||\gamma^{\prime}(t)||$=constant. This property allows us to restrict the
flow $\mathfrak{G}_{t}$ on $T\Sigma$ to the unit tangent bundle $S\Sigma$
where

$S\Sigma=\\{(p,v)|p\in\Sigma,v\in T_{p}\Sigma,||v||=1\\}$

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 2.1 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
flowlinws 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 $ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}$ on
$\mathbb{H}$, and the geodesics and geodesic flow on a surface completely
depend on the choice of Riemannian metric. But geodesic flows associated with
any two negatively curved metric on a surface are orbit equivalent [Ghy84].
More precisely, there is a homeomorphism between the unit tangent bundles of
the respective Riemannian metrices which takes orbits to orbit as described in
Definition 2. 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 w.r.t. any negatively curved metric on a closed surface is quasigeodesic.
We conclude the following theorem,

######  Theorem 2.2.

If $\Sigma$ is a hyperbolic surface, the geodesic flows in the unit tangent
bundle $S\Sigma$ is quasigeodesic. More precisely, the flow lines in the
universal cover are globally length minimizing w.r.t. the metric $ds$.

###  2.2. Anosov flows in Seifert manifolds

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

######  Theorem 2.3.

[Bar96] Any Anosov flow on a closed Seifert fibered space is orbit equivalent
to a finite lift of a geodesic flow on a hyperbolic surface.

We combine Theorem 2.2 and Theorem 2.3 to get the following:

######  Theorem 2.4.

If $\Phi_{t}$ is an Anosov flow on a Seifert fibered three manifold
$\mathcal{N}$, then $\Phi_{t}$ is quasigeodesic.

###### Proof.

By Theorem 2.3, $\Phi_{t}$ 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 2.3 holds. We denote the path metrics induced by the lifts
of the metrices, $\widetilde{g}$ and $\widetilde{ds}$ on
$\widetilde{\mathcal{N}}$ and $\widetilde{S\Sigma}$ resp., by $d_{1}$ and
$d_{2}$.

Fix an orbit equivalence
$h\mathrel{\mathop{\ordinarycolon}}\mathcal{N}\rightarrow\widehat{S\Sigma}$,
let
$\widetilde{h}\mathrel{\mathop{\ordinarycolon}}\widetilde{\mathcal{N}}\rightarrow\widetilde{S\Sigma}$
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 $\gamma$,

(2.2) $\text{if length}_{\widetilde{g}}(\gamma_{[x,y]})\geq\eta_{1},\ \text{
then length}_{\widetilde{ds}}(\widetilde{h}(\gamma_{[x,y]}))\geq\eta_{2}$

Consider any two points $a_{1},a_{2}\in\widetilde{\mathcal{N}}$ such that they
are on the same flow line of $\gamma$ of $\widetilde{\Phi}_{t}$. Let $n\ =\
[\text{length}_{\widetilde{g}}(\gamma_{[a_{1},a_{2}]})]$.

Let $b_{0}=a_{1},b_{2},...,b_{n+1}=a_{2}$ be the points in $\gamma$ such that
$\text{length}_{\widetilde{g}}(\gamma_{[b_{i},b_{i+1}]})=\eta_{1}$ for all
$0\leq i\leq n-1$ and
$\text{length}_{\widetilde{g}}(\gamma_{[b_{n},b_{n+1}]})<\eta_{1}$. Here
$b_{n}=b_{n+1}$ if $\text{length}_{\widetilde{g}}(\gamma_{[a_{1},a_{2}]})$ is
an integer and $b_{n}\not=b_{n+1}$ otherwise. Then we get,

(2.3)
$\begin{split}\text{length}_{\widetilde{g}}(\gamma_{[a_{1},a_{2}]})&=\sum_{0\leq
i\leq n}\text{length}_{\widetilde{g}}(\gamma_{[b_{i},b_{i+1}]})\\\
&=\left(\sum_{0\leq i\leq
n-1}\eta_{1}\right)+\text{length}_{\widetilde{g}}(\gamma_{[b_{n},b_{n+1}]})\\\
&\leq\left(\sum_{0\leq i\leq n-1}\eta_{1}\right)+\eta_{1}\ \
=\left(\frac{\eta_{1}}{\eta_{2}}\sum_{0\leq i\leq
n-1}\eta_{2}\right)+\eta_{1}\\\
&\leq\left(\frac{\eta_{1}}{\eta_{2}}\sum_{0\leq i\leq
n-1}\text{length}_{\widetilde{ds}}(\widetilde{h}(\gamma_{[b_{i},b_{i+1}]})\right)+\eta_{1}\\\
&\leq\frac{\eta_{1}}{\eta_{2}}\text{length}_{\widetilde{ds}}(\widetilde{h}_{(}\gamma_{[\widetilde{h}(a_{1}),\widetilde{h}(a_{2})]}))+\eta_{1}\\\
&=\frac{\eta_{1}}{\eta_{2}}d_{2}(\widetilde{h}(a_{1}),\widetilde{h}(a_{2}))+\eta_{1},\
\ \text{ by Theorem }\ref{UT}\end{split}$

Finally, as
$h\mathrel{\mathop{\ordinarycolon}}\mathcal{N}\rightarrow\widehat{S\Sigma}$ 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
$\widetilde{h}^{-1}\mathrel{\mathop{\ordinarycolon}}(\widetilde{S\Sigma},d_{\widetilde{ds}})\rightarrow(\widetilde{\mathcal{N}},d_{\widetilde{g}})$
is an $(\eta_{3},\eta_{4})$-quasi-isometry.

Applying the quasi-isometry $\widetilde{h}$ on 2.3, we get

$\text{length}_{\widetilde{g}}(\gamma_{[a_{1},a_{2}])})\ \leq\
\frac{\eta_{1}}{\eta_{2}}d_{2}(\widetilde{h}(a_{1}),\widetilde{h}(a_{2}))+\eta_{1}\
\leq\ \frac{\eta_{1}}{\eta_{2}}(\eta_{3}d_{1}(a_{1},a_{2})+\eta_{4})+\eta_{1}$

Finally, let $A_{0}=\frac{\eta_{1}\eta_{3}}{\eta_{2}}$ and
$A_{1}=\frac{\eta_{1}\eta_{4}}{\eta_{2}}+\eta_{1}$. It follows that every flow
line of $\widetilde{\Phi}_{t}$ is a $(A_{0},A_{1})$-quasigeodesic. ∎

##  3\. Construction of Generalized Franks-Williams Flows

A common way to construct Anosov flows is to assemble building blocks. In
general a building block is a compact three manifold with boundary equipped
with a non-singular vector field transverse to the boundary. In their article
[BBY17], the authors have combined many known ‘assembling building blocks’
techniques under a broad general criteria. The building blocks of these type
of examples are called hyperbolic plugs (defined below). The first example of
a non-transitive Anosov flow, i,e the Franks-Williams flow [FW80] is a
classical example of this type of construction.

A $plug$ is a pair $(\mathcal{M},V)$ where $\mathcal{M}$ is a compact three
manifold with boundary and $V$ is a non-singular $C^{1}$-vector field on
$\mathcal{M}$ transverse to the boundary of $\mathcal{M}$. The vector field
induces a flow, denoted by $\mathcal{V}_{t}$, on $\mathcal{M}$. If
$\mathcal{M}$ has non-empty boundary, the flow is not complete, i,e, every
orbit is defined on a closed time interval of $\mathbb{R}$, but not every flow
line is defined on the whole $\mathbb{R}$. We consider the maximal invariant
set $\Lambda$ of $\mathcal{V}_{t}$, defined as
$\Lambda\mathrel{\mathop{\ordinarycolon}}=\bigcap_{t\in\mathbb{R}}\mathcal{V}_{t}(\mathcal{M})$.
In other words, $\Lambda$ is the collection of all orbits which are defined
for the whole $\mathbb{R}$, equivalently these orbits do not intersect
$\partial\mathcal{M}$. If $\Lambda$ is a hyperbolic set, we say
$(\mathcal{M},V)$ is a $hyperbolic$ plug. Here is the precise definition:

######  Definition 4.

A hyperbolic plug $(\mathcal{M},V)$ is a plug whose maximal invariant set
$\Lambda$ is hyperbolic, which means, for every $x\in\Lambda$,
$T_{x}\mathcal{M}$ splits into three one-dimensional sub-bundles

$T_{x}\mathcal{M}=E^{s}(x)\oplus\mathbb{R}V(x)\oplus E^{u}(x)$

The bundle $E^{s}(x)$ (resp. $E^{u}(x)$) is called the stable (resp. unstable)
bundle and there exists a Riemannian metric such that the differential of the
time-one map of the flow uniformly contracts (resp. uniformly expands) the
vectors of the stable bundle (resp. unstable bundle). The splitting varies
continuously on $x\in\Lambda$ and is invariant under the derivative of the
flow $\mathcal{V}_{t}$.

The study of hyperbolic dynamics is an enormous area of study, here we recall
few of the preliminary properties (as in [BBY17]) required for this article,
we refer to [KH95] for details:

  * •

for every $x\in\mathcal{M}$, the strong stable manifold $W^{ss}(x)$ is defined
as follows :

$W^{ss}(x)=\\{y\in\mathcal{M}|\
d(\mathcal{V}_{t}(x),\mathcal{V}_{t}(y))\rightarrow 0\text{ as
}t\rightarrow+\infty)\\}$

The strong unstable manifold is defined as the strong stable of the reversed
flow $-\mathcal{V}_{t}$.

  * •

The weak stable manifold $W^{s}$( resp. weak unstable manifold $W^{u}$) of a
point $x\in\mathcal{M}$ is defined as the union of the strong stable manifolds
(resp. strong unstable manifolds) of all points on the orbit of $x$.

  * •

There exists two 2-laminations, denoted by $W^{s}(\Lambda)$ and
$W^{u}(\Lambda)$, whose leaves are the weak stable and weak unstable
manifolds, respectively, of the points of $\Lambda$. The leaves of the
laminations are $C^{1}$-immersed manifolds tangent to continuous plane fields
$E^{s}\oplus\mathbb{R}V(x)$ and $E^{u}\oplus\mathbb{R}V(x)$.

The boundary of $\mathcal{M}$ is partitioned into two disjoint subsets, namely
the exit boundary $\partial^{out}$ and the entrance boundary $\partial^{in}$,
where $\mathcal{V}_{t}$ points outwards on $\partial^{out}$ and inwards on
$\partial^{in}$. If $\partial^{out}=\emptyset$ then $(\mathcal{M},V)$ is an
$attracting$ plug and similarly $\partial^{in}=\emptyset$ implies a
$repelling$ plug.

  * •

If both $\partial^{in}\neq\emptyset$ and $\partial^{out}\neq\emptyset$ then
$\Lambda$ is a ‘saddle’. In that case, the weak stable lamination
$W^{s}(\Lambda)$ intersects $\partial^{in}$ transversally and forms a one
dimensional lamination $\mathcal{L}^{s}_{V}=\partial^{in}\cap W^{s}$ on
$\partial^{in}$. Similarly, the weak unstable lamination $W^{wu}(\Lambda)$
intersects $\partial^{out}$ in a one dimensional lamination
$\mathcal{L}^{u}_{V}=\partial^{out}\cap W^{u}$.

  * •

For an attracting plug, $\Lambda$ is an $attractor$. In this case the weak-
stable lamination $W^{s}(\Lambda)$ intersects $\partial^{in}$ in an one
dimensional lamination $\mathcal{L}^{s}_{V}=\partial^{in}\cap W^{s}$.

  * •

For a repelling plug, $\Lambda$ is a $repeller$. In this case the weak-
unstable lamination $W^{u}(\Lambda)$ intersects $\partial^{out}$ in an one
dimensional lamination $\mathcal{L}^{u}_{V}=\partial^{out}\cap W^{u}$.

######  Proposition 3.1 ([BBY17]).

Consider a finite collection of hyperbolic plugs, denoted by
$(\mathcal{M}_{1},V_{1}),(\mathcal{M}_{2},V_{2}),...,(\mathcal{M}_{n},V_{n})$.
Assume that each of these plugs is either an attracting or a repelling plug.
Let $\mathcal{D}^{out}=\sqcup_{1}^{n}\partial^{out}(\mathcal{M}_{i})$ and
$\mathcal{D}^{in}=\sqcup_{1}^{n}\partial^{in}(\mathcal{M}_{i})$. Suppose that
the laminations
$\mathcal{L}^{s}_{\mathcal{M}_{i}},\mathcal{L}^{u}_{\mathcal{M}_{j}}$ (if they
are non empty) are filling laminations in the respective boundary components.
Suppose there exists a diffeomorphism
$\Omega\mathrel{\mathop{\ordinarycolon}}\mathcal{D}^{out}\rightarrow\mathcal{D}^{in}$
such that $\Phi_{*}(\mathcal{L}^{s}_{\mathcal{M}_{i}})$ is transversal to
$\mathcal{L}^{u}_{\mathcal{M}_{j}}$ on each appropriate component. Then the
quotient vector field $\frac{V_{1}\sqcup V_{2}\sqcup...\sqcup V_{n}}{\Omega}$
is Anosov on the quotient manifold
$\frac{\mathcal{M}_{1}\sqcup\mathcal{M}_{2}\sqcup...\sqcup\mathcal{M}_{n}}{\Omega}$.

Since the plugs are either attractors or repellers, the laminations in the
boundary are actually foliations. The result above is then Proposition 1.1 of
[BBY17].

In this article we consider a special type of $attracting$ and $repelling$
plugs, which we call Fanks-Williams type hyperbolic plug. This type of of
construction was first introduced by Franks and Williams in [FW80]. We
construct our plugs using DA bifurcations of hyperbolic automorphisms on two-
torus $\mathbb{T}^{2}$. Details of the construction are described below:

###  3.1. Construction of the Franks-Williams Type Hyperbolic Plugs:

Consider a hyperbolic linear automorphism $A$ on the two-torus
$\mathbb{T}^{2}$, which is induced by a linear map
$\widetilde{A}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\rightarrow\mathbb{R}$
such that $\widetilde{A}$ has two eigenvalues $\lambda>1$ and
$\frac{1}{\lambda}<1$. 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 $\mathcal{L}^{u}\mathrel{\mathop{\ordinarycolon}}$
$\mathbb{R}^{2}$ has a foliation $\tilde{\mathcal{L}}^{u}$ by the lines
parallel to the eigenvalue direction $\lambda$ 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 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 lines on the $\frac{1}{\lambda}$-direction
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 $\mathbb{R}^{2}$ w.r.t.
$\\{v_{1/\lambda},v_{\lambda}\\}$ is denoted by $\\{x,y\\}$. In this
coordinate, $\widetilde{A}$ can be written as
$\widetilde{A}(x,y)=(\frac{1}{\lambda}x,\lambda y)$. 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 neighbourhood of $\mathfrak{o}$. Here we give a
quick description of the technique, a detailed description can be found in
[KH95, section 17.2] or in [Wil70].

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 $(0,0)$. W.r.t that 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 below:

$\begin{split}\phi(x,y)&=(\theta(x,y)x,y)\text{ on }D_{2}\\\ \phi&=Id\text{ on
}\mathbb{T}^{2}\setminus D_{2}\end{split}$

In the above description,
$\theta(x,y)\mathrel{\mathop{\ordinarycolon}}\mathbb{T}^{2}\to[1,\infty)$ 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.

######  Property 3.2 ([Wil70, Sma67]).

The new map satisfies the following properties:

  1. (1)

$\Phi=A\circ\phi$ is homotopic to $A$.

  2. (2)

The maximal invariant set of $\Phi$ consists of a point source and a one-
dimensional hyperbolic attractor, denoted by $\Lambda$.

  3. (3)

$A\circ\phi$ preserves the stable foliation $\mathcal{L}^{s}$ of $A$. More
precisely, the attractor of $\Phi$ on $\mathbb{T}^{2}$ is an attracting
hyperbolic set, denoted by $\Lambda$. This induces a stable foliation in
$\mathbb{T}^{2}-\mathfrak{o}$, denoted by $\mathcal{L}^{s}(\Lambda)$. The
construction is done so that the leaves of $\mathcal{L}^{s}(\Lambda)$ are
contained in leaves of $\mathcal{L}^{s}$. Only the stable leaf of
$\mathfrak{o}$ is split into two stable leaves of $\mathcal{L}^{s}(\Lambda)$.
All the other leaves are the same.

######  Remark 3.3.

The usual form of blow up is to first apply the hyperbolic map $A$ and then
the blow up $\phi$. It is equivalent to what we do here: the inverse
$\Phi^{-1}=\phi^{-1}\circ A^{-1}$ and the contraction under $\phi^{-1}$ in a
neighborhood of $\mathfrak{o}$ is stronger than the expansion of $A^{-1}$.
Hence $\mathfrak{o}$ is an attractor for $\Phi^{-1}$ and there is a one
dimensional repeller $\Lambda$ for $\Phi^{-1}$. $\Lambda$ is the attractor for
$\Phi$. We do it in this form, since it is easier to prove later on some
invariance properties of a metric we will be interested in.

######  Remark 3.4.

We have described above the DA bifurcation to get a point source. Similarly,
we can change the fixed point $\mathfrak{o}$ to a sink. In that case the
maximal invariant set will consist of a point sink and an one-dimensional
repeller and the map $A\circ\phi$ would preserve the foliation
$\mathcal{L}^{u}$.

Next consider the suspension manifold

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

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
$t$-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 neighbourhood
$N(\mathcal{C})$ 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
2-torus. We choose $N(\mathcal{C})$ in such a way that boundary of
$N(\mathcal{C})$ is a smooth torus embeded 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 a 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 neighbourhood 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:

######  Proposition 3.5.

The pair $(\mathcal{M}_{1},\Psi^{1}_{t})$ as constructed above is a hyperbolic
plug where $\partial\mathcal{M}_{1}=\partial^{in}\mathcal{M}_{1}$ and its
maximal invariant set in $\mathcal{M}_{1}$ is a hyperbolic attractor. In
fact,we can make the blow up operation on finitely many periodic orbits to
obtain an attracting hyperbolic plug with a finite number of boundary
components. The stable foliation of the hyperbolic attractor inside
$\mathcal{M}_{1}$ intersects each component of $\partial\mathcal{M}_{1}$ in an
one-dimensional foliation with two Reeb annuli.

If we consider a repelling ‘DA’ map in the previous construction, we would get
a repelling hyperbolic plug where the unstable foliation of the repelling set
inside $\mathcal{M}_{1}$ intersects each component of
$\partial\mathcal{M}_{1}$ in an one foliation with Reeb components.

######  Definition 5.

Any attracting or repelling plug with finite number of boundary components as
constructed in proposition 5.1 is called a Franks-Williams type hyperbolic
plug in this article.

######  Remark 3.6.

The explicit description of the blow up is done for the orbit which is the
suspension of the point $(0,0)$. Later computations and results will be done
relative to this orbit. Given a different periodic orbit, one can chose a
different coordinate system so that this different orbit is the one through
the suspension of $(0,0)$. Hence the arguments in this article work for any
collection of blow ups as described above.

###  3.2. Construction of the example manifolds and the flows on them

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

$\\{(\mathcal{M}_{1},\Psi^{1}_{t});(\mathcal{M}_{2},\Psi^{2}_{t});...;(\mathcal{M}_{n},\Psi_{t}^{n})\\}$

along with a diffeomorphism $\Omega$ from the collection of exit boundaries
$\mathcal{D}^{out}=\sqcup_{1}^{n}\partial^{out}_{i}$ 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
$\partial^{out}_{i}$ or $\partial^{in}_{i}$ is empty, and the other one is
non-empty. If $\partial^{in}_{i}$ is nonempty, then it may have more than one
component, each homeomorphic to a two-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
$\partial^{out}_{i}\neq\emptyset$, each component of $\partial^{out}_{i}$
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

$\mathcal{N}=\frac{\mathcal{M}_{1}\sqcup\mathcal{M}_{2}\sqcup...\sqcup\mathcal{M}_{n}}{\Omega(q)\sim
q}$

and the semiflows $\\{\Psi^{1}_{t},\Psi^{2}_{t},...,\Psi^{n}_{t}\\}$ match to
produce a flow $\Psi_{t}$ on $\mathcal{M}$. If we consider a diffemorphism
$\Omega\mathrel{\mathop{\ordinarycolon}}\mathcal{D}^{out}\rightarrow\mathcal{D}^{in}$
that transversally maps the one foliations on each component of
$\partial^{out}_{i}$ to the one foliations on the respective components of
$\mathcal{D}^{in}$, then by the proposition 3.1, the flow $\Psi_{t}$ on
$\mathcal{N}$ is Anosov.

######  Definition 6.

An Anosov flow constructed in the way described above from Franks-Williams
type plugs will be called Generalized Franks-Williams (GFW) flow in this
article.

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

##  4\. Riemannian Metric on the Plugs and the whole manifold

To analyze quasigeodesic behaviour of the flow lines, we will first define a
suitable Riemannian metric on our manifold. As $\mathcal{N}$ is a compact
manifold, for any two Riemannian metrics $\tilde{g}_{1}$ and $\tilde{g}_{2}$
on the universal cover $\widetilde{\mathcal{N}}$ (which project down on
$\mathcal{N}$), the identity map on $\widetilde{\mathcal{N}}$ is a quasi-
isometry w.r.t. the induced path metrics. As quasigeodesic behavior is a
property that is invariant under quasi-isometries, it is enough to work with
one fixed metric.

We construct a Riemannian metric $\mathcal{G}_{i}$ on the hyperbolic plugs
$\mathcal{M}_{i}$ for each $i$, and then attach them along the boundary
components of the $\mathcal{M}_{i}$’s using the map $\Omega$ to construct the
metric $\mathcal{G}$ on the whole manifold $\mathcal{M}$. We describe the
details of the construction of the Riemannian metric $\mathcal{G}_{1}$ on
$\mathcal{M}_{1}$ and the same process works for all other plugs
$\mathcal{M}_{i}$’s.

###  4.1. Construction of a Riemannian metric $\mathcal{G}_{1}$ on
$\mathcal{M}_{1}$

Consider the attracting DA map
$\Phi\mathrel{\mathop{\ordinarycolon}}\mathbb{T}^{2}\rightarrow\mathbb{T}^{2}$
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
$\mathscr{T}=\frac{d}{dt}\widehat{\psi}_{t}$.

######  Remark 4.1.

The $y$-direction on $\mathbb{R}^{2}$ were parallel to the strong unstable
foliation of $\widetilde{A}$ on $\mathbb{R}^{2}$, but the blow-up does not
preserve the unstable direction. Hence the $y$-direction does not represent
the unstable leaves of the attractor of the $DA$-map
$\widetilde{A}\circ\tilde{\phi}$, though the $x$-direction is parallel to the
stable leaves of the attractor.

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 neighbourhood of
$\mathcal{C}$ as described in subsection 2.1. 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 a lot 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 w.r.t
$\widehat{G}_{1}$. The deck transformation group on $\widetilde{M}$ is
generated by the following diffeomorphisms:

  1. (1)

$\Gamma\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\times\mathbb{R}\rightarrow\mathbb{R}^{2}\times\mathbb{R}$,

$\Gamma(x,y,t)=(\widetilde{\Phi}(x,y),t-1)$

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 w.r.t the Euclidean
coordinate system on $\mathbb{R}^{2}\times\mathbb{R}$, i,e

    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 w.r.t 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

$\Gamma\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\times\\{1\\}\rightarrow\mathbb{R}^{2}\times\\{0\\},\
\ \Gamma(x,y,1)=(\widetilde{\Phi}(x,y),0)=(\tilde{A}\circ\tilde{\phi}(x,y),0)$

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 $h_{0}=Id$ 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
neighbourhood $D_{2}$ 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
$\theta\mathrel{\mathop{\ordinarycolon}}\mathbb{T}^{2}\to\mathbb{R}$. Now
define the family of maps
$\tilde{\eta}_{s}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
for $s\in[0,1]$ as follows: First define

$B_{s}(x,y)\ =\ (\lambda^{-s}x,\lambda^{s}y)$

Notice that $B_{1}=\widetilde{A}$. Also define

$\begin{split}\nu_{s}(x,y)&=\ ((\theta(x,y))^{s}x,y)\text{ on }D_{2}\\\ &=\
(x,y)\text{ on }\mathbb{T}^{2}\setminus D_{2}.\end{split}$

Again notice that $\nu_{1}=\phi$. Let
$\tilde{\nu}_{s}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\to\mathbb{R}^{2}$
be the lift of $\nu_{s}$ so that $\tilde{\nu}_{s}(0,0)=(0,0)$. Then define

$\tilde{\eta}_{s}(x,y)\ =\ B_{s}\circ\tilde{\nu}_{s}(x,y)$

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

$h_{s}(x,y,s)=(\tilde{\eta}_{s}(x,y),0)\text{ for }s\in[0,1]$

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
$dx$( resp. $dy$) measures the length along $x$-directions (resp.
$y$-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:

(4.1)
$\begin{split}&\widehat{G}_{1}(\mathscr{X},\mathscr{Y})=g_{t}(\mathscr{X},\mathscr{Y})\text{
on the level }\mathbb{R}^{2}\times\\{t\\}\text{ for }t\in[0,1];\\\
&\widehat{G}_{1}(\mathscr{T},a\mathscr{X}+b\mathscr{Y})=0\text{ for all
}a,b\in\mathbb{R};\\\ &\widehat{G}_{1}(\mathscr{T},\mathscr{T})=1;\end{split}$

######  Observation 4.2.

Then metric $\widehat{G}_{1}$ on $\mathbb{R}^{2}\times[0,1]$ as defined above
in 4.1 is invariant under the the maps $\Gamma$, $E_{1}$ and $E_{2}$ as
follows:

  1. (1)

Clearly the map
$\Gamma\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\times\\{1\\}\rightarrow\mathbb{R}^{2}\times\\{0\\}$
defined by $\Gamma(x,y,1)=(\tilde{A}\circ\tilde{\phi}(x,y),0)$ is an isometry
as

$\widehat{G}_{1}|_{\mathbb{R}^{2}\times\\{1\\}}=(\tilde{A}\circ\tilde{\phi})^{*}(\widehat{G}_{1}|_{\mathbb{R}^{2}\times\\{0\\}})$

  2. (2)

On the level $\mathbb{R}^{2}\times\\{0\\}$ the metric
$g_{0}^{2}=dx^{2}+dy^{2}$ is invariant under the translations. On each level
$\mathbb{R}^{2}\times\\{t\\}$, the metric $(h_{t})^{*}(g_{0})$ can be written
as

$(B_{t}\circ\tilde{\nu_{t}})^{*}(g_{0})\ =\
(\tilde{\nu}_{t})^{*}((B_{t})^{*}(g_{0}))$

Notice first that $B_{t}$ does not leave invariant the integer lattice.
However $(B_{t})^{*}$ leaves invariant the metric $-$ it is the solv metric in
this setting. It is clear that $\tilde{\nu_{t}}$ leaves invariant the metric
under integer translations, since it came from a map in $\mathbb{T}^{2}$. The
$B^{*}_{t}$ invariance of $g_{0}$ was the reason for choosing
$\Phi=A\circ\phi$, rather than $\phi\circ A$.

It follows that the translations $E_{1}$ and $E_{2}$ are isometries w.r.t. the
pull-back metrices on each level $\mathbb{R}^{2}\times\\{t\\}$.

######  Definition 7.

The description of the metric $\widehat{G}_{1}$ on $\mathbb{R}^{2}\times[0,1]$
as in 4.1 together with Observation 4.2 defines a metric on $M$. We denote
this metric on $M$ by $G_{1}$.

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 result derived from the
above defined Riemannian metric on $(\mathcal{M}_{1},\mathcal{G}_{1})$

######  Remark 4.3.

We start with the flow $\psi_{1}$ in $M_{1}$ or $M$. In the following proof
and in other situations in this article we analyze flow lines in the lift
$\widehat{M}_{1}$. This is contained in the universal cover $\widetilde{M}$.
Some arguments are done in $\widetilde{M}$. For simplicity of notation we
denote the lift of $\psi_{t}$ to $\widehat{M}_{1}$ and the lift of $\psi_{t}$
to $\widetilde{M}$ by the same notation $\widehat{\psi}_{t}$.

######  Lemma 4.4.

If $\gamma$ is a flow line or flow ray of $\psi_{t}$ contained $M_{1}$ then
the lift of $\gamma$ in the universal cover $\widetilde{M}_{1}$ is a globally
length minimizing geodesic.

###### 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 $\widehat{M}_{1}$ w.r.t.
$\widehat{G}_{1}$. 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
$\mathscr{T}=\frac{d}{dt}\widehat{\psi}_{t}$ in
$\mathbb{R}^{2}\times\mathbb{R}$. As the vectors in the $t$-directions are
orthogonal to the vectors in the span of $\\{\mathscr{X},\mathscr{Y}\\}$ (i,e.
in the horizontal levels $\mathbb{R}^{2}\times\\{t\\}$), the integral curves
of the vector field $\mathscr{T}=\frac{d}{dt}\widehat{\psi}_{t}$ 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

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

Where $\frac{d}{dt}\gamma(t)|_{\mathscr{T}(\gamma(t))}$ and
$\frac{d}{dt}\gamma(t)|_{\mathscr{X},\mathscr{Y}(\gamma(t))}$ denote the
components of $\gamma^{\prime}(t)=\frac{d}{dt}\gamma(t)$ along $t$-direction
and in the span of $\\{\mathscr{X},\mathscr{Y}\\}$ respectively. Hence it is
clear that the integral curves of the vector field
$\mathscr{T}=\frac{d}{dt}\widehat{\psi}_{t}$, i,e the flow lines of
$\widehat{\psi}_{t}$ in $\mathbb{R}^{2}\times\mathbb{R}$, are globally length
minimizing geodesic in $\widehat{M}=\mathbb{R}^{2}\times\mathbb{R}$ w.r.t. 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, 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 4.5.

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 an 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
$\mathcal{H}$ exits 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 same with
$\mathcal{H}^{*}(\overline{B}_{s}^{*}(g_{0}))$.

If we consider the family of pull-back metrices $\overline{B}_{s}^{*}(g_{0})$
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
$\widetilde{\mathcal{M}}=\mathbb{R}^{2}\times\mathbb{R}$,

$dS^{2}=\lambda^{-2s}dx^{2}+\lambda^{2s}dy^{2}+dt^{2}\text{ for }s\in[0,1]$

Hence the metric we defined, $\widehat{G}_{1}=\mathcal{H}^{*}(dS)$, is a
bounded perturbation of the Solv metric on each level
$\mathbb{R}^{2}\times\\{s\\}$.

###  4.2. Defining the metric on the whole manifold:

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

$\\{(\mathcal{M}_{1},\Psi_{t}^{1});(\mathcal{M}_{2},\Psi_{t}^{2});(\mathcal{M}_{3},\Psi_{t}^{3});...;(\mathcal{M}_{n},\Psi_{t}^{n}\\}$

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. With a little bit
of care we can assume without loss of generality that all boundary components
of all $\partial M_{i}$ are pairwise isometric. Now we smoothly extend the
metrics $G_{i}$ on the union collar neighbourhoods $\cup_{i}\partial
M_{i}\times[0,1]$ 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 $i$. This metric in $\partial
M_{i}\times[0,1]$ is a product metric, so that projection onto the first
factor is a length decreasing map.

######  Remark 4.6.

The manifolds $\mathcal{M}_{i}$ have boundary which is $\pi_{1}$-injective in
$\mathcal{N}$ for any $i$, because each boundary component of each
$\mathcal{M}_{i}$ is $\pi_{1}$-injective in $\mathcal{M}_{i}$. It follows that
any lift of $\mathcal{M}_{i}$ to the universal cover $\widetilde{\mathcal{N}}$
is itself a universal cover of $\mathcal{M}_{i}$. So we can think of these
lifts as copies $\widetilde{\mathcal{M}}_{i}$ of the universal cover of
$\mathcal{M}_{i}$ which are contained in $\widetilde{\mathcal{N}}$.

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 point $p_{1}$ and $p_{2}$ in $\widetilde{\mathcal{M}}_{i}$,

$d_{\widetilde{\mathcal{G}}_{i}}(p,q)=\text{minimum}\\{\text{length}_{\widetilde{\mathcal{G}}_{i}}(\sigma)|\sigma\text{
is a curve contained in }\widetilde{\mathcal{M}}_{i}\text{ connecting
}p_{1},p_{2}\\}$

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 w.r.t. the
restricted metric $d_{\widetilde{\mathcal{G}}_{i}}$.

######  Lemma 4.7.

There is an $\epsilon>0$ 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

$\text{length}_{\widetilde{\mathcal{G}}_{i}}(\gamma_{[p_{1},p_{2}]})\leq
d_{\widetilde{\mathcal{G}}_{i}}(p_{1},p_{2})+\epsilon$

###### Proof.

W.l.o.g. we prove the lemma in $\widetilde{\mathcal{M}}_{1}$. Note that
$\widetilde{M}_{1}\subset\widetilde{\mathcal{M}}_{1}$ and if
$p_{1},p_{2}\in\widetilde{M}_{1}$, then the result is true because inside
$\widetilde{M}_{1}$, flow segments are length minimizing in
$\widetilde{M_{1}}$ by Lemma 4.4. Furthermore by construction of the metric in
$\partial M_{1}\times[0,1]$, there is a length decreasing retraction
$\mathcal{M}_{1}$ to $M_{1}$, so any minimal path in $\mathcal{M}_{1}$ between
points in $M_{1}$ is actually contained in $M_{1}$. It follows that the result
works if $p_{1},p_{2}\in\widetilde{M}_{1}$.

Hence we assume that
$p_{2}\in\widetilde{\mathcal{M}}_{1}\setminus\widetilde{M}_{1}$, and
$p_{1}\in\widetilde{M}_{1}$.

$\widetilde{\mathcal{M}}_{1}$ 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 $\epsilon>0$ such
that for any flow ray $\gamma$,

$\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma\cap(\partial\widetilde{M}_{1}\times[0,1]))\leq\epsilon/2$

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

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

$\begin{split}\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{1},p_{2}]})&=\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{1},p_{3}]})+\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{3},p_{2}]})\\\
&=d_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{1},p_{3}]})+\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{3},p_{2}]})\\\
&\leq d_{\widetilde{\mathcal{G}}_{i}}(p_{1},p_{3})+\epsilon/2\\\ &\leq
d_{\widetilde{\mathcal{G}}_{i}}(p_{1},p_{2})+d(p_{2},p_{3})+\epsilon/2\\\
&\leq
d_{\widetilde{\mathcal{G}}_{i}}(p_{1},p_{2})+\epsilon/2+\epsilon/2\end{split}$

∎

##  5\. Analysis of the Flowlines

In this section we show that every flow line of the flow
$\widetilde{\Psi}_{t}$ on $\widetilde{\mathcal{N}}$ is quasigeodesic with
respect to the path metric induced by Riemannian metric
$\widetilde{\mathcal{G}}$ as constructed in the previous section. As before,
$\mathcal{N}$ is made of the collection of hyperbolic plugs

$\\{(\mathcal{M}_{1},\Psi_{t}^{1},\mathcal{G}_{1});(\mathcal{M}_{2},\Psi_{t}^{2},\mathcal{G}_{2});(\mathcal{M}_{3},\Psi_{t}^{3},\mathcal{G}_{3});...;(\mathcal{M}_{n},\Psi_{t}^{n},\mathcal{G}_{n})\\}$

First we consider a single hyperbolic plug, say
$(\mathcal{M}_{1},\Psi^{1}_{t},\mathcal{G}_{1})$, and analyze the properties
of the flow lines of the semi-flow $\Psi^{1}_{t}$.

###  5.1. Flowlines in the plug
$(\mathcal{M}_{1},\mathcal{G}_{1},\Psi^{1}_{t})$

Recall that $\mathcal{M}_{1}$ is made from
$M=\frac{\mathbb{T}^{2}\times\mathbb{R}}{(q,t)\sim(\Phi(q),t-1)}$ as in
Subsection 3.1, 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 neighbourhood of $\mathcal{C}$ as
described in subsection 2.1. 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.

$M_{1}$ 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 $\widehat{G}_{1}$.

$\widehat{M}_{1}$ is a manifold with boundary, where $\partial\widehat{M}_{1}$
is the lift of $\partial{M_{1}}=T_{1}$. $\partial\widehat{M}_{1}$ is a
collection of disjoint infinite cylinders in $\mathbb{R}^{2}\times\mathbb{R}$
which are transverse to the flow lines. Suppose
$\partial\widehat{M}_{1}=\cup_{i\in\mathbb{N}}\widehat{T}_{i}$, where
$\widehat{T}_{i}$’s 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 $\mathcal{A}\subset M_{1}$, or

  2. (2)

$\gamma$ intersects a lift of $\partial M_{1}=T_{1}$. Moreover $\gamma$
intersects exactly one such lift of $T_{1}$.

######  Remark 5.1.

When we are dealing only with $M_{1},\mathcal{M}_{1}$ we will simplify
notation and denote $\psi^{1}_{t}$ and $\Psi^{1}_{t}$ by $\psi_{t}$ and
$\Psi_{t}$.

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 notations.

(a)

(b)

Figure 1. (A) The two dimensional image of the blow up of a hyperbolic point,
(B) The three dimensional image of the blow up of a hyperbolic orbit.

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 the attractor $\mathcal{A}$ of $\psi^{1}_{t}$. Now we consider the
repelling orbit $\mathcal{C}$ and the open solid torus neighbourhood
$N(\mathcal{C})$ around it. 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 8.

Suppose $q\in\widehat{M}_{1}$ is such that the flow line $\gamma_{q}$ through
$q$ intersects $\partial\widehat{M}_{1}$ at the boundary component denoted by
$\widehat{T}_{q}$. We know that the flow ray $\gamma_{q}$ can intersect
exactly one such component of $\partial\widehat{M}_{1}$. Then let

  * •

$\mathcal{D}_{\widehat{G}_{1}}(q,\widehat{T}_{q})$ ( resp.
$\mathcal{D}_{S}(q,\widehat{T}_{q})$) denote the distance between $q$ and
$\widehat{T}_{q}$ w.r.t. the path metric induced by $\widehat{G}_{1}$ ( resp.
the Solv metric $dS$ ) on $\widehat{M}_{1}$.

  * •

$\ell_{\widehat{G}_{1}}(q)$( resp. $\ell_{S}(q)$) is the length of the flow
line segment connecting $q$ and $\widehat{T}_{q}$ w.r.t. $\widehat{G}_{1}$(
resp. $dS$).

Next we fix a small number $\delta>0$ and consider the open
$\delta$-neighbourhood $N_{\delta}(\mathcal{C}_{1}\cup\mathcal{C}_{2})$ 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 propertly embedded lines in $\widehat{M}_{1}$. The
following is a key technical result used in this article.

######  Proposition 5.2.

There exists $C>1$ and $c>0$ satisfying the following: Let
$q\in\widehat{M}_{1}$ such that the flow ray through $q$ which 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

$\ell_{\widehat{G}_{1}}(q)\leq
C\mathcal{D}_{\widehat{G}_{1}}(q,\widehat{T}_{q})+c$

###### 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
distance between two points in $\widehat{M}_{1}$ w.r.t. the metric
$d_{\widehat{G}_{1}}$ is bigger than distance in
$\mathbb{R}^{2}\times\mathbb{R}$ w.r.t. $d_{\widehat{G}}$. In addition 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
$(0,0,0)\in\mathbb{R}^{2}\times\mathbb{R}$, i,e 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 on the
$yt$-plane is almost $\lambda^{2t}dy^{2}+dt^{2}$. We make this more precise in
the following remark:

######  Remark 5.3.

  1. (1)

Note that the map
$\tilde{\phi}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
perturbs the $y$-directions only near the lifts in $\mathbb{R}^{2}$ of the
repelling fixed point $\mathfrak{o}$, and outside those neighbourhoods 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 4.5, 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

$\frac{1}{a_{0}}\text{length}_{\widehat{G}_{1}}(\sigma)-a_{1}\leq\text{length}_{S}(\sigma)\leq
a_{0}\text{ length}_{\widehat{G}_{1}}(\sigma)+a_{1}$

where $\text{length}_{S}(\sigma)$ and
$\text{length}_{\widehat{G}_{1}}(\sigma)$ means length of $\sigma$ w.r.t. 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 5.4.

There exists $K>1$ and $k>0$ such that for any $q$ on the intersection of the
$yt$-plane with $\widehat{M}_{1}$ for which the flow line $\gamma_{q}$
intersects $\widehat{T}_{0}$, we have

$\ell_{\widehat{G}_{1}}(q)\leq
K\mathcal{D}_{\widehat{G}_{1}}(q,\widehat{T}_{0})+k$

###### 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 $\sigma\mathrel{\mathop{\ordinarycolon}}[0,1]\rightarrow\widetilde{M}$
denotes a shortest path connecting $q$ and $\widehat{T}_{0}$ in
$\widehat{M}_{1}$.

######  Claim 5.5.

Any shortest path $\sigma$ between $q$ and $\widehat{T}_{0}$ lies on
$yt$-plane.

###### Proof.

Suppose $\sigma$ is parametrized 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

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

and the claim follows from that.

In Remark 4.5 we have seen that the metric $\widehat{G}_{1}$ can be expressed
as the pull-back of the $Solv$ metric $dS$ by the map $\mathcal{H}$, i,e
$\widehat{G}_{1}=\mathcal{H}^{*}(dS)$, where
$dS=\lambda^{-2s}dx^{2}+\lambda^{2s}dy^{2}+dt^{2}$

Consider the map
$\mathcal{H}\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{2}\times\mathbb{R}\rightarrow\mathbb{R}^{2}\times\mathbb{R}$
as in Remark 4.5 . As $\widehat{G}_{1}=\mathcal{H}^{*}(dS)$, for any curve
$\zeta$,

$\text{length}(\zeta)\text{ w.r.t. the metric
}\widehat{G}_{1}=\text{length}(\mathcal{H}(\zeta))\text{ w.r.t. the metric
}dS$

As the directions $x$, $y$ and $t$ are orthogonal to each other w.r.t. 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

$\text{length}(\Pi(\delta))\leq\text{length}(\sigma)\text{ w.r.t. the metric
}dS$

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

Hence we conclude

$\displaystyle\text{length}(\Pi(\mathcal{H}(\sigma)))\leq\text{length}(\mathcal{H}(\sigma))\text{
w.r.t. the metric }dS$ $\displaystyle\implies$
$\displaystyle\text{length}(\mathcal{H}(\Pi(\sigma)))\leq\text{length}(\mathcal{H}(\sigma))\text{
w.r.t. the metric }dS$ $\displaystyle\implies$
$\displaystyle\text{length}(\Pi(\sigma)))\leq\text{length}(\sigma)\text{
w.r.t. the metric }\widehat{G}_{1}$

This completes the claim. ∎

In the rest of the proof of Lemma 5.4 we will use the $Solv$ metric on
$yt$-plane, i,e the metric $\lambda^{2t}dy^{2}+dt^{2}$ for simplicity of
calculations. In Remark 4.2 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 $\widehat{T}_{0}\cap yt$-plane
and the forward flow ray $\gamma_{q}=\widetilde{\psi}^{1}_{[0,\infty)}(p)$ as
shown in Figure 2, $\gamma_{q}=\\{(0,c,t)|t\in[0,\infty)\\}$.

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 $\gamma_{q}$: Suppose $q^{\prime}=(0,c,t^{\prime})$ is
a point on $\gamma_{q}$ and let $\sigma_{q^{\prime}}$ is a length minimizing
curve on the $yt$-plane joining $q^{\prime}$ and $\widehat{T}_{0}$. We define
three functions as follows:

Figure 2. (A) This shows a minimal path in the $yt$-plane from $q^{\prime}$ to
the lift of the boundary torus, (B) This figure shows several geometric
quantities that are used in the analysis. In particular
$R_{q}(\overline{t}^{\prime})\lambda^{\overline{t}^{\prime}}$ is the length in
the solvable metric of the horizontal segment depicted at height
$\overline{t}^{\prime}$.

  1. (1)

Let $\Pi_{t}(\sigma_{q^{\prime}})$ denotes the projection of
$\sigma_{q^{\prime}}$ on the vertical line
$\gamma_{q}=\\{(0,c,t)|t\in\mathbb{R}\\}$ along $y$-directions $-$ same as
horizontal directions (notice that $\sigma_{q^{\prime}}$ is contained in the
$yt$ plane).

  2. (2)

Suppose $\bar{t^{\prime}}$ is the lowest $t$-value attained by the curve
$\sigma_{q^{\prime}}$ as shown in the Figure 2(A). Then we denote the
projection of $\sigma_{q^{\prime}}$ on the line $t=\bar{t^{\prime}}$ along
$t$-directions $-$ same as vertical directions $-$ by
$\Pi_{y}(\sigma_{q^{\prime}})$.

  3. (3)

For a point $(0,c,t^{\prime\prime})$ on $\gamma_{q^{\prime}}$, suppose the
line $t=t^{\prime\prime}$ intersects $\widehat{T}_{0}$ in the positive side of
$y$-direction at $(0,c^{\prime},t^{\prime\prime})$ as shown in the Figure
2(B). Then we define $R_{q}(t^{\prime\prime})=|c-c^{\prime}|$. Note that,
w.r.t. the $Solv$ metric $dS$ the length of the segment on the line
$t=t^{\prime\prime}$ connecting $(0,c,t^{\prime\prime})$ and
$(0,c^{\prime},t^{\prime\prime})$ is
$\lambda^{t^{\prime\prime}}|c-c^{\prime}|$ or
$R_{q}(t^{\prime\prime})\lambda^{t^{\prime\prime}}$. Clearly $R_{q}$ is an
increasing function of $t$ on $[0,\infty)$.

Now we are ready to state two lower estimates of
$\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$.

######  Claim 5.6.

Suppose $\bar{t^{\prime}}$ is the lowest $t$-value in the projection
$\Pi_{t}(\sigma_{q^{\prime}})$ as shown in Figure 2(a). Then

  1. (1)

$t^{\prime}-\bar{t^{\prime}}\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

  2. (2)

In addition,
$R_{q}(\bar{t^{\prime}})\lambda^{\bar{t^{\prime}}}\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

###### Proof.

As the $t$-directions and $y$-directions are everywhere orthogonal w.r.t the
metric $\lambda^{2t}dy^{2}+dt^{2}$, it is easy to check that
$\text{length}_{S}(\Pi_{t}(\sigma_{q^{\prime}}))\leq\text{length}_{S}(\sigma_{q^{\prime}})$
and in addition we have
$\text{length}_{S}(\Pi_{y}(\sigma_{q^{\prime}}))\leq\text{length}_{S}(\sigma_{q^{\prime}})=\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$.

  1. (1)

As the curve $\Pi_{t}(\sigma_{q^{\prime}})$ connects the points
$q^{\prime}=(0,c,t^{\prime})$ and $(0,c,\bar{t^{\prime}})$ it is clear that
$|t^{\prime}-\bar{t^{\prime}}|\leq\text{length}_{S}(\Pi_{t}(\sigma_{q^{\prime}}))\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$.

Note that $\bar{t^{\prime}}$ can not be a negative number, because if
$\bar{t^{\prime}}<0$, then

$t^{\prime}-\bar{t^{\prime}}>t^{\prime}=\ell_{S}(q^{\prime})\geq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

which can not be true as we have just proved
$t^{\prime}-\bar{t^{\prime}}\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$.

  2. (2)

As $\bar{t^{\prime}}\geq 0$, we observe that the line segment
$\\{(0,y,\bar{t^{\prime}})|y\in[c-R_{q}(\bar{t^{\prime}}),c]\\}$ is contained
in the curve $\Pi_{y}(\sigma_{q^{\prime}})$ as shown in Figure 2(b). Hence

$R_{q}(\bar{t^{\prime}})\lambda^{\bar{t^{\prime}}}\leq\text{length}_{S}(\Pi_{t}(\sigma_{q^{\prime}}))\leq\text{length}_{S}(\sigma_{q^{\prime}})=\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

This proves the claim. ∎

Consider the function
$P_{q}(t)=\frac{R_{q}(\frac{t}{2})\lambda^{\frac{t}{2}}}{t}$ 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 $k^{\prime}>0$ such that
$P_{q}(t)>1$ when $t>k^{\prime}$.

For the point $q^{\prime}=(0,c,t^{\prime})$, if $t^{\prime}>k^{\prime}$ then
$R_{q}(t^{\prime}/2)\lambda^{t^{\prime}/2}>t^{\prime}$. On the other hand
$t^{\prime}=\ell_{S}(q^{\prime})\geq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$.
Combining these two inequalities we get,

(5.1)
$\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\leq\ell_{S}(q^{\prime})=t^{\prime}<R_{q}(t^{\prime}/2)\lambda^{t^{\prime}/2}$

Hence by the Claim item 5.6(2), $t^{\prime}/2$ can not be smaller than the
lowest $t$-value attained by the curve $\Pi_{t}(\sigma_{q^{\prime}})$, i,e
$t^{\prime}/2>\bar{t^{\prime}}$. Otherwise we get

$R_{q}(t^{\prime}/2)\lambda^{t^{\prime}/2}\leq
R_{q}(\bar{t^{\prime}})\lambda^{\bar{t^{\prime}}}\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

which contradicts Equation 5.1.

Finally, as $t^{\prime}/2>\bar{t^{\prime}}$, applying Claim 5.6(1) we deduce

(5.2) $\begin{split}&t^{\prime}-t^{\prime}/2\leq
t^{\prime}-\bar{t^{\prime}}\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\\\
\implies&t^{\prime}/2\leq\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\\\
\implies&t^{\prime}\leq 2\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\\\
\implies&\boxed{\ell_{S}(q^{\prime})\leq
2\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\text{ when
}q^{\prime}=(0,c,t^{\prime})\text{ and }t^{\prime}>k^{\prime}}\end{split}$

If $t^{\prime}\leq k^{\prime}$, then for a point $q^{\prime}=(0,c,t^{\prime})$
we get

(5.3) $\ell_{S}(q^{\prime})\leq k^{\prime}\leq
k^{\prime}+2\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})$

Combining 5.2 and 5.3, we conclude

(5.4) $\ell_{S}(q^{\prime})\leq
2\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})+k^{\prime}\text{ for all
}q^{\prime}\in\gamma_{q}=\\{(0,c,t)|t\in[0,\infty)\\}$

Finally, by using Remark 5.3, we can deduce
$\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})\leq
a_{0}\mathcal{D}_{\widehat{G}}(q^{\prime},\widehat{T}_{0})+a_{1}$ and
replacing it in the previous equation,

$\ell_{S}(q^{\prime})\leq
2\mathcal{D}_{S}(q^{\prime},\widehat{T}_{0})+k^{\prime}\leq
2a_{0}\mathcal{D}_{\widehat{G}_{1}}(q^{\prime},\widehat{T}_{0})+2a_{1}+k^{\prime}\text{
for all }q^{\prime}\in\gamma_{q}=\\{(0,c,t)|t\in[0,\infty)\\}$

By renaming, $2a_{0}=K$ and $2a_{1}+k^{\prime}=k$, and replacing
$\ell_{S}(q^{\prime})=\ell_{\widehat{G}_{1}}(q^{\prime})$,

$\boxed{\ell_{\widehat{G}_{1}}(q^{\prime})\leq
K\mathcal{D}_{\widehat{G}_{1}}(q^{\prime},\widehat{T}_{0})+k\text{ for all
}q^{\prime}\in\gamma_{q}=\\{(0,c,t)|t\in[0,\infty)\\}}$

This completes the proof of Lemma 5.4 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

$\mu_{a}\mathrel{\mathop{\ordinarycolon}}\widetilde{M}\to\widetilde{M},\ \
\mu_{a}(x,y,t)\ =\ (\lambda^{a}x,\lambda^{-a}y,t+a)$

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 $p^{\prime}$ in the forward flow line of $p$,
one obtains Equation 5.4 for $p^{\prime}$ 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 5.3 to finish the proof of Lemma 5.4. ∎

(a)

(b)

Figure 3. (A) The foliation $\widetilde{\mathcal{F}}$ in a lift of a torus;
(B) Leaf of $\widetilde{\mathcal{F}}$ through $q$ intersect $D_{1}$ at
$\bar{q}$. The shaded region represents the ‘bad region.

Now we are ready to prove Proposition 5.2.

Proof of Proposition 5.2: The semiflow $\psi^{1}_{t}$ has a hyperbolic set
$\mathcal{A}$ in $M_{1}$, it 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 [FW80], 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.
$\widehat{\mathcal{C}}_{1}$ 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 $\widehat{\mathcal{C}}_{2}$ 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.

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, w.l.o.g 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 $\widehat{\mathcal{A}}$,
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

$d_{\widehat{G}_{1}}(\widehat{\psi}^{1}_{t+s}(q),\widehat{\psi}^{1}_{t}(\bar{q}))\to
0\text{ as }t\to\infty$

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 $t^{\prime}>s_{2}$, consider the four points
$q,q_{t^{\prime}}=\widehat{\psi}^{1}_{t^{\prime}+s}(q),\bar{q}$ and
$\bar{q}_{t^{\prime}}=\widehat{\psi}^{1}_{t^{\prime}}(\bar{q})$. By the
triangle inequality,

(5.5) $d_{\widehat{G}_{1}}(q,q_{t^{\prime}})\leq
d_{\widehat{G}_{1}}(q,\bar{q})+d_{\widehat{G}_{1}}(\bar{q},\bar{q}_{t^{\prime}})+d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},q_{t^{\prime}})$

Note that
$d_{\widehat{G}_{1}}(q,q_{t^{\prime}})=\ell_{\widehat{G}_{1}}(q_{t^{\prime}})$
as $q$ and $q_{t^{\prime}}$ lie on the same flow ray and by Lemma 4.4, flow
segments are length minimizing in $\widehat{M}$ w.r.t. the path metric of
$\widehat{G}_{1}$, similarly
$d(\bar{q},\bar{q}_{t^{\prime}})=\ell_{\widehat{G}_{1}}(q_{t^{\prime}})$.
Moreover by our assumption $d_{\widehat{G}_{1}}(q,\bar{q})=s_{1}$,
$d_{\widehat{G}_{1}}(q_{t^{\prime}},\bar{q}_{t^{\prime}})<\delta_{1}$.
Replacing in the above inequality 5.5, we get

(5.6) $\begin{split}\ell_{\widehat{G}_{1}}(q_{t^{\prime}})&\leq
d_{\widehat{G}_{1}}(q,\bar{q})+d_{\widehat{G}_{1}}(\bar{q},\bar{q}_{t^{\prime}})+d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},q_{t^{\prime}})\\\
&\leq s_{1}+\ell_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}})+\delta_{1}\\\
&=\ell_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}})+2s_{1}\text{ for any
}t^{\prime}>s_{2}\end{split}$

As $\bar{q}\in\widehat{T}_{0}\cap\\{yt\\}-$plane, by Lemma 5.4 we know there
are global $K,k>0$, so that $\ell_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}})\leq
K\mathcal{D}_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},\widehat{T}_{0})+k$.
Applying it in 5.6,

(5.7)
$\ell_{\widehat{G}_{1}}(q_{t^{\prime}})\leq\ell_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}})+2s_{1}\leq
K\mathcal{D}_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},\widehat{T}_{0})+k+2s_{1}\
\ \text{ when }t^{\prime}>s_{2}$

Finally, suppose $a\in\widehat{T}_{0}$ is a point in $\widehat{T}_{0}$ that is
closest to $q_{t^{\prime}}$, i,e
$d_{\widehat{G}_{1}}(q_{t^{\prime}},a)=\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}},\widehat{T}_{0})$.
By the triangle inequality, we get,
$d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},a)\leq
d_{\widehat{G}_{1}}(q_{t},a)+d_{\widehat{G}_{1}}(q_{t^{\prime}},\bar{q}_{t^{\prime}})$.
Moreover
$\mathcal{D}_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},\widehat{T}_{0})\leq
d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},a)$ as $a\in\widehat{T}_{0}$.
Combining all these facts we conclude when $t^{\prime}>s_{2}$,

(5.8)
$\begin{split}\mathcal{D}_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},\widehat{T}_{0})&\leq
d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},a)\leq
d_{\widehat{G}_{1}}(q_{t^{\prime}},a)+d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},q_{t^{\prime}})\\\
&=\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}},\widehat{T}_{0})+d(\bar{q}_{t^{\prime}},q_{t^{\prime}})\text{
as
}d_{\widehat{G}_{1}}(q_{t^{\prime}},a)=\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}},\widehat{T}_{0})\\\
&=\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}},\widehat{T}_{0})+s_{1}\text{ as
}d_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},q_{t^{\prime}})=s_{1}\text{ by
assumption}\end{split}$

Combining 5.7 and 5.8 we get,

(5.9) $\ell_{\widehat{G}_{1}}(q_{t^{\prime}})\leq
K\mathcal{D}_{\widehat{G}_{1}}(\bar{q}_{t^{\prime}},\widehat{T}_{0})+k+2s_{1}\leq
K\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}})+Ks_{1}+k+2s_{1}\text{ when
}t^{\prime}>s_{2}$

The above inequality proves Proposition 5.2 for $t^{\prime}>s_{2}$. If
$t^{\prime}\leq s_{2}$, then $\ell(q_{t^{\prime}})=t^{\prime}<s_{2}$. Adding
the case when $t^{\prime}\leq s_{2}$ in Inequality 5.9 we conclude,

(5.10) $\ell_{\widehat{G}_{1}}(q_{t^{\prime}})\leq
K\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}})+Ks_{1}+k+2s_{1}+s_{2}\text{ for
all }t^{\prime}\in[0,\infty)$

By renaming $C=K$ and $c=Ks_{1}+k+2S_{1}+s_{2}$, we rewrite the above
Inequality 5.10 as

$\boxed{\ell_{\widehat{G}_{1}}(q_{t^{\prime}})\leq
C\mathcal{D}_{\widehat{G}_{1}}(q_{t^{\prime}})+c\text{ for all
}t^{\prime}\in[0,\infty)}$

It completes the proof of Proposition 5.2 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 5.10 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 $M_{1}$. w.l.o.g. 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 w.r.t 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 5.2. ∎

######  Remark 5.7.

The reason behind considering the $\delta$-neighbourhood of
$\mathcal{C}_{1}\cup\mathcal{C}_{2}$ is to use the compactness of the set
$\mathcal{S}$, the compactness is used to determine the universality of the
constants $s_{1}$ and $s_{2}$.
$\widehat{T}_{0,1}\setminus{\widehat{\mathcal{C}}_{1}\cup\widehat{\mathcal{C}}_{2}}$
is not compact.

Now we extend Proposition 5.2 in the universal cover of $\widetilde{M}_{1}$
w.r.t. 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 notations as follows:

For a point $q\in\widetilde{M}_{1}$, suppose the flow line through $q$
intersect 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 $\widetilde{T}_{q}$ w.r.t. 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 $\widetilde{T}_{q}$ w.r.t. $\widetilde{G}_{1}$.

Consider the lift of the neighbourhood
$N_{\delta}(\mathcal{C}_{1}\cup\mathcal{C}_{2})$ in $\widetilde{M}_{1}$, we
denote it as
$N_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$. We
can restate the Lemma 5.2 in $\widetilde{M}_{1}$ as follows:

######  Lemma 5.8.

There exists $C>1$ and $c>0$ satisfying the following: Let
$q\in\widetilde{M}_{1}$ such that the flow ray through $q$ which 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

$\ell_{\widetilde{G}_{1}}(q)\leq
C\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{T}_{q})+d$

###### Proof.

Note that $\widehat{M}_{1}$ is an intermediate cover of $M_{1}$, 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

$d_{\widehat{G}_{1}}(\bar{b}_{1},\bar{b}_{2})\leq
d_{\widetilde{G}_{1}}(b_{1},b_{2})$

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
$\widetilde{M}_{1}$, hence
$\ell_{\widehat{G}_{1}}(\bar{q})=\ell_{\widetilde{G}_{1}}(q)$. Combining all
the information we get,

$\ell_{\widetilde{G}_{1}}(q)=\ell_{\widehat{G}_{1}}(\bar{q})\leq
C\mathcal{D}_{\widehat{G}_{1}}(\bar{q},\widehat{T}_{\bar{q}})+c\leq
C\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{T}_{q})+c$

∎

Now we extend Lemma 5.8 to
$\mathcal{M}_{1}=M_{1}\cup(\partial{M_{1}}\times[0,1])$. The manifold
$\mathcal{N}$ is the union of the collection of
$\\{(\mathcal{M}_{1},\Psi_{t}^{1},\mathcal{G}_{1});(\mathcal{M}_{2},\Psi_{t}^{2},\mathcal{G}_{2});(\mathcal{M}_{3},\Psi_{t}^{3},\mathcal{G}_{3});...;(\mathcal{M}_{n},\Psi_{t}^{n},\mathcal{G}_{n})\\}$
where the plugs intersect each other along their boundary components. As
described in Section $3$, we extend the 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

$\mathcal{G}|_{M_{i}}=G_{i}|_{M_{i}}$

In the rest of the article we consider the path metric
$d_{\widetilde{\mathcal{G}}}$ induced from the Riemannian metric
$\widetilde{\mathcal{G}}$ on the whole manifold $\widetilde{\mathcal{N}}$. For
two points $p_{1},p_{2}\in\widetilde{\mathcal{N}}$,

$d_{\widetilde{\mathcal{G}}}(p_{1},p_{2})=\text{minimum}\\{\text{ length}\
\sigma|\ \sigma\text{ is a path
connecting}p_{1},p_{2}\in\widetilde{\mathcal{N}}\\}$

We extend the flow $\psi^{i}_{t}$ from $M_{i}$ to
$\mathcal{M}_{i}=M_{i}\cup(M_{i}\times[0,1])$ as a product flow
(topologically) on $\partial M_{1}\times[0,1]$, and the extended flow is
denoted by $\Psi^{i}_{t}$. It is clear that the foliation
$\mathcal{L}^{ws}(\mathcal{A}_{i})\cap\partial M_{i}$ on $\partial M_{i}$ also
extends to $\partial M_{1}\times[0,1]$, and hence to
$\partial\mathcal{M}_{i}$. The neighbourhood
$N_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$ is
also carried by the extended flow on the new boundary
$\partial\widetilde{\mathcal{M}}_{1}$. In the universal cover
$\widetilde{\mathcal{N}}$, we denote this new set by
$N^{\prime}_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$.

As before we define,

######  Definition 9.

Suppose $q\in{\widetilde{\mathcal{M}}}_{1}$ such that the flow line
$\gamma_{q}$ through $q$ intersects $\partial\widetilde{\mathcal{M}}_{1}$ at
the boundary component $\widetilde{\mathcal{T}}_{q}$. Then let

  * •

$\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{q})$ denote
the distance between $q$ and $\widetilde{\mathcal{T}}_{q}$ w.r.t. the path
metric induced by $\widetilde{\mathcal{G}}_{1}$.

  * •

$L_{\widetilde{\mathcal{G}}}(q)=L_{\widetilde{\mathcal{G}}_{1}}(q)$ denote the
length of the flow line segment connecting $q$ and
$\widetilde{\mathcal{T}}_{q}$ w.r.t. $\widetilde{\mathcal{G}}$.

######  Lemma 5.9.

Let $q\in\widetilde{\mathcal{M}}_{1}$ such that the flow line through $q$
intersects $\partial\widetilde{\mathcal{M}}_{1}$ at the boundary component
$\widetilde{\mathcal{T}}_{q}$. If $\gamma_{q}$ intersects
$\widetilde{\mathcal{T}}_{q}$ on the region
$\widetilde{\mathcal{T}}_{q}\setminus
N^{\prime}_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$,
then there exists $C_{1}>1$ and $c_{1}$ such that,

$L_{\widetilde{\mathcal{G}}}(q)\leq
C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{q})+c_{1}$

Moreover the constants $C_{1}$ and $c_{1}$ do not depends on $q$ or
$\widetilde{\mathcal{T}}_{q}$.

###### Proof.

Again we will prove the lemma only for a component of
$\partial\widetilde{\mathcal{M}}_{1}$, which we will denote by
$\widetilde{\mathcal{T}}_{0}$. The same result holds for all other components
of $\partial\widetilde{\mathcal{M}}_{i}$.

Since the flow is a product in $\partial M_{1}\times[0,1]$ then up to changing
$c_{1}$ to a bigger constant, we can assume that $q\in\widetilde{M}_{1}$.

If $\gamma$ intersects $\widetilde{\mathcal{T}}_{0}$ then it also intersects a
component $\widetilde{T}_{0}$ of $\partial\widetilde{M}_{1}$, because the flow
$\Psi^{1}_{t}$ is a product in $\partial M_{1}\times[0,1]$. As the distances
are measured as minimum lengths of paths for both $\widetilde{G}_{1}$ and
$\widetilde{\mathcal{G}}$ and
$\widetilde{\mathcal{G}}|_{\widetilde{M}_{1}}=\widetilde{G}_{1}$, we conclude

$\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{T}_{0})<\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})$

Notice we are assuming that $q$ is in $\widetilde{M}_{1}$. By the compactness
of $\partial M_{1}\times[0,1]$, we can find $\epsilon>0$ (same as in Lemma
4.7) such that for all $\gamma$ which intersects
$\widetilde{\mathcal{T}}_{0}$,
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma\cap(\widetilde{\widehat{T}_{0}}\times[0,1]))\leq\epsilon$.

Fix a flow ray $\gamma$, suppose $\gamma$ intersects $\widetilde{T}_{0}$ at
$q_{1}$ and $\widetilde{\mathcal{T}}_{0}$ at $q_{2}$. Then for any
$p\in\gamma$, as
$\widetilde{G}_{1}|_{\widetilde{\mathcal{M}}_{1}}=\widetilde{\mathcal{G}}|_{\widetilde{\mathcal{M}}_{1}}$,

$L_{\widetilde{\mathcal{G}}}(q)=\ell_{\widetilde{G}_{1}}(q)+\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q_{2}]})$

By Lemma 5.8, we know $\ell_{\widetilde{G}_{1}}(q)\leq
K\mathcal{D}_{\widetilde{\mathcal{G}}_{1}}(q,\widetilde{T}_{0})+k$ and we have
$\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{\mathcal{T}}_{0})<\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{T}_{0})$.
By the definition of $\epsilon$,
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q_{2}]})\leq\epsilon$.
Hence we conclude

(5.11)
$\begin{split}L_{\widetilde{\mathcal{G}}}(q)&=\ell_{\widetilde{G}_{1}}(q)+\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q_{2}]})\\\
&\leq\ell_{\widetilde{G}_{1}}(q)+\epsilon\text{ as
length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q_{2}]})\leq\epsilon\\\ &\leq
C\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{T}_{0})+c+\epsilon\text{ by
Lemma \ref{key1} }\\\ &\leq
C\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+c+\epsilon\text{
as
}\mathcal{D}_{\widetilde{G}_{1}}(q,\widetilde{T}_{0})\leq\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0}))\end{split}$

As all the constants $C,c$ and $\epsilon$ are independent of the flow line, we
conclude that, for all $q$ in $\gamma$ such that $\gamma$ intersects
$\widetilde{\mathcal{T}}_{q}\setminus
N^{\prime}_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$,

$L_{\widetilde{\mathcal{G}}}(q)\leq
C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{q})+c_{1}$

where $C_{1}=C$ and $c_{1}=d+\epsilon$ as defined above. ∎

######  Remark 5.10.

Lemma 5.8 says that every flow line which intersects the boundary components
of $\widetilde{M}_{1}$ outside the ‘bad region’ goes away from the boundary
component at a uniformly efficient rate as $t\to\infty$.

All other hyperbolic plugs, irrespective of ‘attracting’ or ‘repelling’, have
the same type of property that if a flow ray intersects a boundary component
outside the ’bad region’, it goes ’away’ at a uniformly efficient rate from
the boundary component when $t\to\infty$ (in case of attracting plugs) or
$t\to-\infty$ (in case of repelling plugs). Moreover by taking the maximum
over all the constants, we can fix global additive and multiplicative
constants which work for all of the hyperbolic plugs.

We conclude this section with two remarks on the separating tori at the
boundaries of the hyperbolic plugs, these tori play an important role in this
article.

######  Remark 5.11.

  1. (1)

As remarked before, the boundary tori are incompressible, i,e they are two-
sided and injectively included in the fundamental group
$\pi_{1}(\mathcal{N})$.

  2. (2)

If $\widetilde{\mathcal{T}}$ is a component of the lift of some
$\partial\widetilde{\mathcal{M}}_{i}$, then by Theorem 1.1 of [KL98], see also
Section 3.1 of [Ngu19], the following happens: $\widetilde{\mathcal{T}}$ is
quasi-isometrically embeded in the universal cover $\widetilde{\mathcal{N}}$.
We make it more precise as follows: Consider the lift of a separating torus,
say $\widetilde{T}$. By restricting the Riemannian metric on
$\widetilde{\mathcal{T}}$, we can consider the path metric on
$\widetilde{\mathcal{T}}$ induced by the restriction, we call it
$d_{\widetilde{\mathcal{T}}}$. Then there exists $k_{0},k_{1}$ such that the
inclusion map
$i\mathrel{\mathop{\ordinarycolon}}(\widetilde{\mathcal{T}},d_{\widetilde{\mathcal{T}}})\rightarrow(\widetilde{\mathcal{N}},d_{\widetilde{\mathcal{G}}})$
is a $(k_{0},k_{1})$-quasi-isometric embedding. We can choose $k_{0},k_{1}$ so
that it works for any such $\widetilde{\mathcal{T}}$.

###  5.2. Quasigeodesic behavior in the whole manifold:

Now we are ready to prove the main theorem, that is, the flow lines of
$\widetilde{\Psi}_{t}$ are uniform quasigeodesics in $\widetilde{\mathcal{N}}$
w.r.t. $d_{\widetilde{\mathcal{G}}}$, the path metric induced by
$\widetilde{\mathcal{G}}$. As before, we assume that $\mathcal{N}$ is made of
the collection of hyperbolic plugs
$\\{\mathcal{M}_{1},\mathcal{M}_{2},...,\mathcal{M}_{n}\\}$. Each
$\widetilde{\mathcal{M}}_{i}$ is a manifold with boundary such that
$\partial\widetilde{\mathcal{M}}_{i}$ is a collection of separating planes
homeomorphic to $\mathbb{R}^{2}$ and properly embeded into
$\widetilde{\mathcal{N}}$.

We first prove that if a flow line or flow ray is fully contained in the
universal lift of a single plug $\mathcal{M}_{i}$ then it is a quasigeodesic.

######  Lemma 5.12.

There exits $a_{3}>1$ and $a_{4}>0$ such that if $\gamma$ is a flow ray or
flow line fully contained in a single hyperbolic plug
$\widetilde{\mathcal{M}}_{i}$ then it is a $(a_{3},a_{4})$-quasigeodesic
w.r.t. the metric $d_{\widetilde{\mathcal{G}}}$.

###### Proof.

As before, w.l.o.g. we prove the result only on $\widetilde{\mathcal{M}}_{1}$.

In Lemma 4.7, we have proved that the flow lines or flow rays are
quasigeodesic w.r.t. $d_{\widetilde{\mathcal{G}}_{1}}$, the path metric of the
restriction of $\widetilde{\mathcal{G}}$ on $\widetilde{\mathcal{M}}_{1}$. In
this lemma we need to extend the result on the whole manifold
$\widetilde{\mathcal{N}}$ w.r.t. $d_{\widetilde{\mathcal{G}}}$.

Consider two points $p_{1}$ and $p_{2}$ on a flow ray or flow line
$\gamma\subset\widetilde{\mathcal{M}}_{1}$. Suppose
$\sigma\mathrel{\mathop{\ordinarycolon}}[0,1]\rightarrow\widetilde{\mathcal{N}}$
is a minimal path w.r.t. $\widetilde{\mathcal{G}}$ in
$\widetilde{\mathcal{N}}$ connecting $p_{1}$ and $p_{2}$. In other words
$length(\sigma)=d_{\widetilde{\mathcal{G}}}(p_{1},p_{2})$. We argue that the
length of $\sigma$ can be approximated by a curve fully contained in
$\widetilde{\mathcal{M}}_{1}$, without too much increase in length. Suppose
$\sigma$ exits $\widetilde{\mathcal{M}}_{1}$ through a boundary component
$\widetilde{\mathcal{T}}$ at $0<v_{1}<1$, i,e
$\sigma(v_{1})\in\widetilde{\mathcal{T}}$. As $\widetilde{\mathcal{T}}$ is a
separating plane in $\widetilde{\mathcal{N}}$, $\sigma$ must re-enter
$\widetilde{\mathcal{M}}_{1}$ at some first $v_{2}$ with $0<v_{1}<v_{2}<1$.

By Remark 5.11, $\widetilde{\mathcal{T}}$ is $(k_{0},k_{1})$-quasi-
isometrically embeded in $\widetilde{\mathcal{N}}$, hence we can find a path
$\bar{\sigma}\mathrel{\mathop{\ordinarycolon}}[v_{1},v_{2}]\rightarrow\widetilde{\mathcal{T}}$
such that

$\frac{1}{k_{0}}\text{length}_{\widetilde{\mathcal{G}}}(\bar{\sigma}_{[v_{1},v_{2}]})-k_{1}\leq\text{length}_{\widetilde{\mathcal{G}}}(\sigma|_{[v_{1},v_{2}]})\leq
k_{0}\
\text{length}_{\widetilde{\mathcal{G}}}(\bar{\sigma}_{[v_{1},v_{2}]})+k_{1}$

In the definition of $\sigma$, we can replace $\sigma|_{[v_{1},v_{2}]}$ with
$\bar{\sigma}|_{[v_{1},v_{2}]}$. Now fix $a_{6}>0,a_{6}<<1$ so that any
segment in the image of $\sigma$ with endpoints in
$\partial\widetilde{\mathcal{M}}_{1}$ and interior outside
$\widetilde{\mathcal{M}}_{1}$, and length $<a_{6}$ can be pushed into
$\partial\widetilde{\mathcal{M}}_{1}$ to a segment of length at most $2a_{6}$.
On $[0,1]$, there can exist only finitely many closed intervals on which
$\sigma$ goes out of $\widetilde{\mathcal{M}}_{1}$ and with length $>a_{6}$.
Also replace all of these intervals with minimal curve on respective boundary
components as described before, using the quasi-isometry constants
$k_{0},k_{1}$. Choosing $k_{0}>2$, then we get a curve
$\sigma^{\prime}\mathrel{\mathop{\ordinarycolon}}[0,1]\rightarrow\widetilde{\mathcal{M}}_{1}\subset\widetilde{\mathcal{N}}$
with same endpoints as $\sigma$, and such that

$\frac{1}{k_{0}}\text{length}_{\widetilde{\mathcal{G}}}(\sigma^{\prime})-k_{1}\leq\text{length}_{\widetilde{\mathcal{G}}}(\sigma)\leq
k_{0}\ \text{length}_{\widetilde{\mathcal{G}}}(\sigma^{\prime})+k_{1}$

Note that
$\widetilde{\mathcal{G}}_{1}=\widetilde{\mathcal{G}}|_{\widetilde{\mathcal{M}}_{1}}$.
In particular, as $\sigma^{\prime}$ is a path contained in
$\widetilde{\mathcal{M}}_{1}$ and connecting $p_{1}$ and $p_{2}$, hence

$d_{\widetilde{\mathcal{G}}_{1}}(p_{1},p_{2})\leq\text{length}_{\widetilde{\mathcal{G}}_{1}}(\sigma^{\prime})=\text{length}_{\widetilde{\mathcal{G}}}(\sigma^{\prime})$

Hence combining Lemma 4.7 and the above inequality,

$\begin{split}\text{length}_{\widetilde{\mathcal{G}}_{1}}(\gamma_{[p_{1},p_{2}]})&\leq
d_{\widetilde{\mathcal{G}}_{1}}(p_{1},p_{2})+\epsilon\\\
&\leq\text{length}_{\widetilde{\mathcal{G}}_{1}}(\sigma^{\prime})+\epsilon\\\
&=\text{length}_{\widetilde{\mathcal{G}}}(\sigma^{\prime})+\epsilon\\\ &\leq
k_{0}\text{length}_{\widetilde{\mathcal{G}}}(\sigma)+k_{0}k_{1}+\epsilon\\\
&=k_{0}d_{\widetilde{\mathcal{G}}}(p_{1},p_{2})+k_{0}k_{1}+\epsilon\end{split}$

As
$\widetilde{\mathcal{G}}_{1}=\widetilde{\mathcal{G}}|_{\widetilde{\mathcal{M}}_{1}}$,
$\text{length}_{\widetilde{\mathcal{G}}_{1}}\gamma_{[p_{1},p_{2}]}=\text{length}_{\widetilde{\mathcal{G}}_{1}}\gamma_{[p_{1},p_{2}]}$;
by renaming $a_{3}=k_{0}$ and $a_{4}=k_{0}k_{1}+\epsilon$, we conclude the
proof of the lemma. ∎

The following is the main result of this article:

######  Theorem 5.13.

There exists $C_{0}>1$ and $c_{0}>0$ such that each flow line of
$\widetilde{\Psi}_{t}$ in $\widetilde{\mathcal{N}}$ is a
$(C_{0},c_{0})$-quasigeodesic w.r.t. the metric $\widetilde{\mathcal{G}}$.

###### Proof.

There exists exactly two types of flow lines $\gamma$ of
$\widetilde{\Phi}_{t}$ in $\widetilde{\mathcal{N}}$:

  1. (1)

either $\gamma$ is contained in one of $\widetilde{\mathcal{M}}_{i}$s. In this
case $\gamma$ is contained in the attractor (or repeller) inside
$\widetilde{\mathcal{M}}_{i}$.

  2. (2)

or $\gamma$ intersects the boundary of one of $\widetilde{\mathcal{M}}_{i}$s.
As every boundary component is shared by exactly two plugs, $\gamma$
intersects two adjacent plugs, say $\widetilde{\mathcal{M}}_{i}$ and
$\widetilde{\mathcal{M}}_{j}$. In this case $\gamma$ is subdivided in two rays
and each of $\widetilde{\mathcal{M}}_{i}$ and $\widetilde{\mathcal{M}}_{j}$
contains exactly one subray, say $\gamma^{+}$ and $\gamma^{-}$.

By Lemma 5.12 we know that each flow lines which are entirely contained in one
of $\widetilde{\mathcal{M}}_{i}$s are uniform quasigeodesics, i,e all the
first type of flow lines in the above list are $(a_{3},a_{4})$-quasigeodesics.

If $\gamma$ intersects a common boundary component of
$\widetilde{\mathcal{M}}_{i}$ and $\widetilde{\mathcal{M}}_{j}$, Lemma 5.12
says that both of the forward subray $\gamma^{+}$ and the backward subray
$\gamma^{-}$ are quasigeodesics. But concatenation of two quasigeodesic is not
necessarily a geodesic and that is the main obstacle in this proof. Next we
show that, in our case concatenation of two quasigeodesic flow ray is a
quasigeodesic and the key ingredient of the proof is Lemma 5.9.

######  Lemma 5.14.

Every flow line which intersects a boundary component of a hyperbolic plug is
a quasigeodesic.

###### Proof.

Up to reindexing the $\widetilde{\mathcal{M}}_{i}$, we can assume that there
are lifts $\widetilde{\mathcal{M}}_{1},\widetilde{\mathcal{M}}_{2}$ of
$\mathcal{M}_{1},\mathcal{M}_{2}$ respectively, so that $\gamma$ intersects
$\partial\widetilde{\mathcal{M}}_{1}$ at the boundary component
$\widetilde{\mathcal{T}}_{0}$ and suppose $\widetilde{\mathcal{T}}_{0}$ is a
common boundary component of $\widetilde{\mathcal{M}}_{1}$ and
$\widetilde{\mathcal{M}}_{2}$. We can assume that
$\widetilde{\mathcal{M}}_{1}$ is an attracting plug, so it contains the
forward ray $\gamma^{+}$ and $\widetilde{\mathcal{M}}_{2}$ must be a repelling
plug and contains the backward flow ray $\gamma^{-}$.

######  Claim 5.15.

At least one of $\gamma^{+}$ or $\gamma^{-}$ intersects
$\widetilde{\mathcal{T}}_{0}$ outside the ‘bad region’. More precisely, at
least one of the following is true:

  * •

For all $q\in\gamma^{+}\subset\widetilde{\mathcal{M}}_{1}$

$L_{\widetilde{\mathcal{G}}}(q)\leq
C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+c_{1}$

  * •

Or, $\gamma^{-}$ satisfies the same property, i,e. for all
$q\in\gamma^{-}\subset\widetilde{\mathcal{M}}_{2}$,

$L_{\widetilde{\mathcal{G}}}(q)\leq
C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+c_{1}$

It is possible that both of the flow rays $\gamma^{+}$ and $\gamma^{-}$
satisfy the property.

###### Proof.

We can view $\widetilde{\mathcal{T}}_{0}$ as a boundary component of
$\widetilde{\mathcal{M}}_{1}$, and we denote it by
$\mathcal{T}_{\widetilde{\mathcal{M}}_{1}}$ to emphasize this. In the same way
we can view $\widetilde{\mathcal{T}}_{0}$ as a boundary component of
$\widetilde{\mathcal{M}}_{2}$, which we denote by
$\mathcal{T}_{\widetilde{\mathcal{M}}_{2}}$. These boundary components are
attached to each other by a map
$\widetilde{\Omega}\mathrel{\mathop{\ordinarycolon}}\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}\rightarrow\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{2}}$.

Consider the neighbourhood
$N^{\prime}_{\delta}(\widetilde{\mathcal{C}}_{1}\cup\widetilde{\mathcal{C}}_{2})$
on $\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}$ as described in
Lemma 5.9, lets rename it $\mathfrak{N}_{\delta}$, this is the ‘bad region’ on
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}$. Similarly there is
another bad region, say $\mathfrak{N}_{\delta^{\prime}}$ on
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{2}}$.

Note that we can choose $\delta$ (and $\delta^{\prime}$) in the Lemma 5.2
small enough, so that
$\widetilde{\Omega}(\mathfrak{N}_{\delta})\cap\mathfrak{N}_{\delta^{\prime}}=\emptyset$.
In other words, every flow line $\gamma$ which intersects
$\widetilde{\mathcal{T}}_{0}=\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}\sqcup\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{2}}/\sim$
intersects at least one of the regions
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}\setminus\mathfrak{N}_{\delta}$
or
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{2}}\setminus\mathfrak{N}_{\delta^{\prime}}$.
If $\gamma$ intersects
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{1}}\setminus\mathfrak{N}_{\delta}$
then the subray $\gamma^{+}$ satisfies the claim by Lemma 5.9, and similarly
if $\gamma$ intersects
$\widetilde{\mathcal{T}}_{\widetilde{\mathcal{M}}_{2}}\setminus\mathfrak{N}_{\delta^{\prime}}$
the subray $\gamma^{-}$ satisfies a similar property. ∎

Continuation of the proof of Lemma 5.14.

As before, let $\gamma^{+}\subset\widetilde{\mathcal{M}}_{1}$ and
$\gamma^{-}\subset\widetilde{\mathcal{M}}_{2}$. We assume w.l.o.g, that
$\gamma^{+}\subset\widetilde{\mathcal{M}}_{1}$ satisfies Claim 5.15.

Take two points $q,q^{\prime}\in\gamma$. As each of the subrays of
$\gamma^{+}$ and $\gamma^{-}$ are uniform quasigeodesics by Lemma 5.12, we can
conclude that the flow segment joining $q$ and $q^{\prime}$ is a quasigeodesic
if either $q,q^{\prime}\in\gamma^{+}$ or $\bar{q},q^{\prime}\in\gamma^{-}$.
Hence we assume $q\in\gamma^{+}$ and $q^{\prime}\in\gamma^{-}$.

As $\widetilde{\mathcal{T}}_{0}$ is a separating plane on
$\widetilde{\mathcal{N}}$, $\gamma$ must intersect
$\widetilde{\mathcal{T}}_{0}$. We can conclude that that for fixed
$q^{\prime}\in\gamma^{-}$:

$\mathcal{D}_{\widehat{\mathcal{G}}}(q,\widehat{\mathcal{T}}_{0})<d_{\widehat{\mathcal{G}}}(q,q^{\prime})\text{
for all }q\in\gamma^{+}$

Let $\gamma$ intersect $\widetilde{\mathcal{T}}_{0}$ at $q_{1}$. Now fix
$q\in\gamma^{+}$. We break the flow segment $\gamma_{[q,q^{\prime}]}$ as
$\gamma_{[q,q^{\prime}]}=\gamma_{[q,q_{1}]}*\gamma_{[q_{1},q^{\prime}]}$.
There are two possible cases:

Case I: Suppose
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})\geq\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q^{\prime}]})$.

Note that
$L_{\widetilde{\mathcal{G}}}(q)=\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})$.
By our assumption

$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q^{\prime}]})\leq
2\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})=2L_{\widetilde{\mathcal{G}}}(q).$

As we have assumed that $\gamma^{+}$ satisfies Claim 5.15,

(5.12) $\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q^{\prime}]})\leq
2L_{\widetilde{\mathcal{G}}}(q)\leq
2C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2c_{1}$

We have observed that
$d_{\widetilde{\mathcal{G}}}(q,q^{\prime})>\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})$
as any curve joining $q$ and $q^{\prime}$ also intersects
$\widetilde{\mathcal{T}}_{0}$. Hence replacing in the previous Inequality
5.12, we conclude if $q,q^{\prime}\in\gamma$ and
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})\geq\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q^{\prime}]})$,
then

(5.13)
$\boxed{\text{length}\gamma_{[q,q^{\prime}]}<2L_{\widetilde{\mathcal{G}}}(q)\leq
2C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2c_{1}\leq
2C_{1}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2c_{1}}$

This finishes the proof of uniform quasigeodesic behavior in this case.

Case II: Now we assume
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})<\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q^{\prime}]})$.

Figure 4. This depicts the situation in the universal cover. The curve with
arrows in it is the flow line segment from $q$ to $q^{\prime}$. The other two
curves from $q$ to $q^{\prime}$ and from $q$ to $q^{\prime}$ are curves
realizing the distance between these pairs of points.

We first break $\gamma_{[q,q^{\prime}]}$ as
$\gamma_{[q,q_{1}]}*\gamma_{[q_{1},q^{\prime\prime}]}*\gamma_{[q^{\prime\prime},q^{\prime}]}$
such that
$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q_{1}]})=\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q_{1},q"]})$
as in Figure 4.

Then by 5.13 in case I, we conclude

(5.14)
$d_{\widetilde{\mathcal{G}}}(q,q^{\prime\prime})\leq\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q^{\prime\prime}]})\leq
2C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2c_{1}$

Consider the points $q,q^{\prime}$ and $q^{\prime\prime}$. The flow segment
$\gamma_{[q^{\prime\prime},q^{\prime}]}$ is an $(a_{3},a_{4})$ quasigeodesic
segment by Lemma 5.12 as it is contained in a single plug
$\widetilde{\mathcal{M}}_{2}$. Hence

$\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q^{\prime\prime},q^{\prime}]})\leq
a_{3}d_{\widetilde{\mathcal{G}}}(q^{\prime},q^{\prime\prime})+a_{4}$

By the triangle inequality, we get
$d_{\widetilde{\mathcal{G}}}(q^{\prime},q^{\prime\prime})\leq
d_{\widetilde{\mathcal{G}}}(q^{\prime},q)+d_{\widetilde{\mathcal{G}}}(q,q^{\prime\prime})$.
For the flow segment $\gamma_{[q^{\prime\prime},q^{\prime}]}$ we conclude

(5.15)
$\begin{split}\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q^{\prime\prime},q^{\prime}]})&\leq
a_{3}d_{\widetilde{\mathcal{G}}}(q^{\prime},q^{\prime\prime})+a_{4}\\\ &\leq
a_{3}[d_{\widetilde{\mathcal{G}}}(q^{\prime},q)+d_{\widetilde{\mathcal{G}}}(q,q^{\prime\prime})]+a_{4}\text{
\ \ by the triangle inequality}\\\ &\leq
a_{3}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+a_{3}[2C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2c_{1}]+a_{4}\text{
\ \ by \ref{inq9}}\\\
&=a_{3}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2a_{3}C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\mathcal{T})+2a_{3}c_{1}+a_{4}\end{split}$

Finally adding 5.14 and 5.15 we conclude,

(5.16)
$\text{length}(\gamma_{[q,q^{\prime\prime}]})+\text{length}(\gamma_{[q^{\prime\prime},q^{\prime}]})\leq
2C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2c_{1}+\\\
a_{3}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2a_{3}C_{1}\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})+2a_{3}c_{1}+a_{4}$

As every path connecting $q$ and $q^{\prime}$ intersects
$\widetilde{\mathcal{T}}_{0}$, we get
$\mathcal{D}_{\widetilde{\mathcal{G}}}(q,\widetilde{\mathcal{T}}_{0})\leq
d_{\widetilde{\mathcal{G}}}(q,q^{\prime})$. Replacing in the previous equation
we conclude

(5.17)
$\boxed{\begin{split}\text{length}_{\widetilde{\mathcal{G}}}(\gamma_{[q,q^{\prime}]})&\leq
2C_{1}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2a_{3}C_{1}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+a_{3}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2c_{1}+2a_{3}c_{1}+a_{4}\\\
&=(2C_{1}+2a_{3}C_{1}+a_{3})d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+2c_{1}+2a_{3}c_{1}+a_{4}\end{split}}$

We rename $C_{0}=2C_{1}+2a_{3}C_{1}+a_{3}$ and
$c_{0}=2c_{1}+2a_{3}c_{1}+a_{4}$ and Equations 5.13 and 5.17 together imply
that for any two points $q$ and $q^{\prime}$ on $\gamma$,

$\text{length}_{\widetilde{\mathcal{G}}}{\gamma_{[q,q^{\prime}]}}\leq
C_{0}d_{\widetilde{\mathcal{G}}}(q,q^{\prime})+c_{0}$

As $\gamma$ was chosen arbitrarily, this completes the proof that any flow
line which intersects the common boundary component
$\widetilde{\mathcal{T}}_{0}$ between $\widetilde{\mathcal{M}}_{1}$ and
$\widetilde{\mathcal{M}}_{2}$ is a quasigeodesic. Moreover the same
multiplicative and additive constants work for all flow line intersecting
$\widetilde{\mathcal{T}}_{0}$.

In the beginning of the proof we fixed $\widetilde{\mathcal{M}}_{1}$ and
$\widetilde{\mathcal{M}}_{2}$ and their common boundary component
$\widetilde{\mathcal{T}}_{0}$, but the same method works for all other plugs
which intersects along boundaries. The quasigeodesic constants differ for
different choices of boundary components. Finally, as there are only finitely
many plugs in $\mathcal{N}$, we can take maximums over all possibilities of
boundary components in $\widetilde{\mathcal{N}}$ and we can choose global
quasigeodesic constants for the flow lines which intersect any of the boundary
components.

This ends the proof of Lemma 5.14. ∎

Lemma 5.12 and Lemma 5.14 together imply that any flow line of
$\widetilde{\Psi}_{t}$ in $\widetilde{\mathcal{N}}$ are uniformly
quasigeodesic.

This completes the proof of Theorem 5.13. ∎

######  Remark 5.16.

In the proof of the final theorem in Section $5.2$, we used two key ideas;

  1. (1)

every flow ray of flow line contained in a single plug is quasigeodesic;

  2. (2)

every flow ray which intersects a boundary component outside a narrow region
eventually goes away uniformly efficiently from the boundary component it
intersects.

This suggests that the same general techniques used in this proof can be
applied to study quasigeodesic behavior of flows in different contexts where
these two properties hold.

##  6\. Comments on the Franks-Williams Manifolds $\mathcal{N}$ and a question

As before we consider the decomposition of $\mathcal{N}$ as the union

$\mathcal{N}=\mathcal{M}_{1}\cup\mathcal{M}_{2}\cup...\cup\mathcal{M}_{n}$

where the components $\mathcal{M}_{i}$s intersects each other along their
boundaries and the boundary components of $\mathcal{M}_{i}$ are homeomorphic
to tori. We denote the collection of boundary components as
$\\{\mathcal{T}_{j}|j\in J\\}$ where each $\mathcal{T}_{j}$ is a common
boundary of two plugs from the collection
$\mathcal{M}_{1},\mathcal{M}_{2},...,\mathcal{M}_{n}$. Now we again emphasize
some properties of the manifold $\mathcal{N}$ and the collection of torus
$\\{\mathcal{T}_{j}|j\in J\\}$:

  1. (1)

$\mathcal{N}$ is irreducible: It is easy to obtain from the construction that
the pieces $\mathcal{M}_{i}$ are irreducible, i,e every $2$-sphere bounds a
$2$-ball in $\mathcal{M}_{i}$, and each boundary component is incompressible.
We are attaching the pieces along boundaries homeomorphic to $2$-torus to get
$\mathcal{N}$. Each such torus is incompressible in the particular piece
containing it. Hence the manifold $\mathcal{N}$ is irreducible.

  2. (2)

JSJ-decomposition of $\mathcal{N}$: Every irreducible orientable three-
manifold supports a JSJ-decomposition, that is, a collection of tori
separating the manifold into atoroidal and/or Seifert fibered pieces. In our
case we can show that the collection $\\{\mathcal{T}_{j}|j\in J\\}$ is a
minimal JSJ-decomposition on $\mathcal{N}$. The orientability condition can be
satisfied by moving to a finite cover of $\mathcal{N}$ (if required). Note
from the construction of $\mathcal{N}$ that each of the tori $\mathcal{T}_{j}$
is separating in $\mathcal{N}$. As explained in the previous item, each torus
$\mathcal{T}_{j}$ is incompressible, i,e closed, 2-sided and canonically
injects in the fundamental group of $\mathcal{N}$. To see that the collection
$\\{\mathcal{T}_{j}|j\in J\\}$ is minimal, we claim that each piece in the
decomposition w.r.t. $\\{\mathcal{T}_{j}|j\in J\\}$ is atoroidal, see detail
in the item $3$ below.

  3. (3)

$\mathcal{N}$ is non hyperbolic and non Seifert fibered: As there exits
incompressible tori $\mathcal{T}_{j}$ in $\mathcal{N}$, if follows that
$\mathcal{N}$ cannot be a hyperbolic manifold. To see that $\mathcal{M}$ is
also non Seifert fibered we consider the plugs
$\mathcal{M}_{1},\mathcal{M}_{2},...,\mathcal{M}_{n}$ which are also the
components of the torus decomposition, we note that each of the plugs are
constructed by removing solid torus neighbourhoods from mapping tori of
hyperbolic maps on $\mathbb{T}^{2}$, i,e
$\mathbb{T}^{2}\times[0,1]/((x,1)\sim(\Phi(x),0))$ for some
$\Phi\mathrel{\mathop{\ordinarycolon}}\mathbb{T}^{2}\to\mathbb{T}^{2}$
homotopic to a hyperbolic map on $\mathbb{T}^{2}$. They are mapping tori of
pseudo-Anosov homeomorphims of a torus minus finitely many disks. And it is
well known that such mapping tori are atoroidal, and hence non Seifert
fibered. This now implies that these are the pieces of the JSJ decomposition