\vskip .1in
\section{From continuous extension to quasigeodesic behavior}
\vskip .1in

Here are the basic definitions we need concerning quasi-isometries
and quasigeodesics. A good source for foundational material
on quasi-isometries is \cite{BKS}, especially chapter 10 by
Ghys and De la Harpe on ``Infinite groups as geometric objects (after
Gromov)'' and chapter 11 by Cannon on ``The theory of negatively
curved groups and spaces''. 

\begin{define}{(quasi-isometries and quasigeodesics)}{}
Given metric spaces $(X,d)$ and $(Y,d')$, a map $f \from X \to Y$ is a
quasi-isometric embedding if there is $k \geq 1$ so that for any $z, w \in
X$, we have

$$ \frac{1}{k} \, d'(f(z),f(w)) - 1 \leq d(z,w) \leq k \, d'(f(z),f(w)) +
k.$$

\noindent
Once a metric is fixed we say that $f$ is a $k$-quasi-isometry. 

The spaces $X,Y$ are quasi-isometric if there is a quasi-isometry $f
\from X \to Y$ and a constant $k'$ such that each point of $Y$ is
within distance $k'$ of the image of $f$.

Given a metric space $X$, a quasigeodesic is a map
$f \from \rrrr \to X$ which is a quasi-isometry. If $f$ is a
$k$-quasi-isometry we say that $f$ is a $k$-quasigeodesic.
\end{define}

A {\em flow} on a manifold $X$ is a continuous action of $\reals$ on $X$,
i.e.\ a continuous map $\psi \from X \cross \reals \to X$ written $(x,t) \to
\psi_t(x)$, such that 
$$\psi_{s+t}(x) = \psi_s(\psi_t(x)) \quad\text{for all}\quad x \in X, \,
s,t \in \reals.
$$
A {\em semiflow} on $X$ is defined similarly, except that the domain of
$\psi$ is a closed subset $D \subset X \cross \reals$ such that for each $x
\in X$, $(x\cross \reals) \intersect D = x \cross J$ for some closed,
connected set $J \subset \reals$ containing $0$, and the above equation holds
whenever the two sides are defined.  All flows and semiflows in this article
are generated by nonzero, continuous vector fields, i.e.\ flow lines are
smooth immersions and the tangent vector field is continuous on $X$. When
$\psi$ is understood, we often write $x \cdot t = \psi_t(x)$, and if $J
\subset \reals$ is an interval we write $x \cdot J = \psi_J(x)$. If $y = x
\cdot t$ let $[x,y] = x \cdot [0,t]$ and $(x,y) = [x,y] - \{ x, y \}$; also
let $\tau(x,y) = |t|$.


\begin{define}{(uniformly quasigeodesic flows)}{}
Let $\Phi$ be a flow on a compact manifold $M$. Let $\widetilde \Phi$ be the
lifted flow on the universal cover $\widetilde M$. We say that $\Phi$ is
uniformly quasigeodesic if there exists a constant $k \ge 1$ such that for
each $x\in \widetilde M$, the map $t \to x \cdot t$ is a
$k$-quasigeodesic. 
\end{define}

Since $M$ is compact this property is independent of the parameterization of
$\Phi$. Henceforth flowlines are always parameterized by arc length. Given
this, a flow line in $\widetilde M$ is $k$-quasigeodesic if for any $x,y$ in
the flow line, $\tau(x,y) \le k \, d(x, y) + k$. We sometimes say that flow
lines of $\Phi$ are quasigeodesic.

We first show that there is a weak form of quasigeodesic
behavior which follows from purely topological properties of
the orbit space of $\wwp$.
Given $\epsilon, T > 0$ an $\epsilon, T$ cycle of $\Phi$ is a closed
loop in $M$ obtained from an orbit segment of length $> T$ with endpoints
less than $\epsilon$ apart, closed up by an arc of length $< \epsilon$.
The following lemma does not assume that the manifold is
hyperbolic.
This lemma is not logically necessary for proving the main theorem
and theorems $A, B$;
but it describes additional hypotheses (which are quite common)
under which the proofs of these theorems can be simplified.

\begin{lemma}{(weak quasigeodesics)}{}
Let $Y$ be a closed invariant set of a flow $\Phi$ in a closed manifold $M$.
Let $\widetilde Y$ be the lift of $Y$ to $\mi$. Suppose that 

(a) $\wwp |_{\widetilde Y}$ has Hausdorff orbit space,

(b) There are $\epsilon, T > 0$ so that any
$\epsilon, T$ cycle of $\Phi |_Y$ is not null homotopic in $M$.


\noindent
Then for any $b > 0$ there is $c_b > 0$ (depending only on $b$)
so that if $x, y$ are in an orbit $\gamma$ of $\wwp$ and
$\tau(x,y) > c$ then $d(x,y) > b$.
\label{weakqg}
\end{lemma}


\begin{proof}{}
Otherwise there is $b > 0$ and $x_i, y_i \in \widetilde Y$,
with $x_i, y_i$ in the same orbit $\gamma_i$ of $\wwp$ and
$\tau(x_i,y_i) > i$ but $d(x_i,y_i) < b$.
Since $M$ is compact, then up to covering translations
and taking subsequences we may assume that $x_i \rightarrow x$
and $y_i \rightarrow y$.
Notice that $x, y \in \widetilde Y$ since $\widetilde Y$ is closed.

By hypothesis (a) it follows that $x$ and $y$ are in the same
orbit of $\wwp$ so $y = x \cdot t$ for some $t \in \rrrr$.
By the local product structure of the flow along compact
orbit segments, there are $t_i \rightarrow t$ so
that $z_i = x_i \cdot t_i \rightarrow y$.
If $i$ is big enough then $\tau(z_i,y_i) > T$
and $d(z_i,y_i) < \epsilon$, thereby producing 
$\epsilon, T$-cycles
of $\wwp$. These project to null homotopic
$\epsilon, T$-cycles of $\Phi$, contradiction.
\end{proof}

\noindent
{\bf Remarks:} (1) Notice in particular that
condition (b) implies that orbits of $\wwp$ are
never periodic and are properly embedded in $\mi$.
When (b) holds we say that $\Phi |_Y$ satisfies the
$\epsilon, T$-cycles condition.

(2) Quasigeodesic behavior is the additional property that
$c_b$ is bounded by an affine function of $b$.
In general conditions (a),(b) are not sufficient to ensure
quasigeodesic behavior. For instance Anosov flows
always satisfy these conditions \cite{Fe1}, but there are
many examples of Anosov flows in hyperbolic manifolds which
are not quasigeodesic \cite{Fe1}.

(3) When $Y=M$ 
conditions (a),(b) together are equivalent to the orbit space $\oo$ of
$\wwp$ being homeomorphic to either the plane $\rrrr^2$
or the sphere ${\bf S}^2$.
(a) implies that $\oo$ is a 2-dimensional manifold,  and it 
is Hausdorff. Since $\oo$ is simply connected and
has no boundary it is either ${\bf S}^2$ or $\rrrr^2$.
This means that the flow $\wwp$ 
is topologically a {\em product} flow in $\mi$.
Lemma \ref{weakqg} means that topological product flows
always satisfy a weak quasigeodesic property.
This is reminiscent of the same situation for codimension
one foliations which was studied in \cite{Fe4}.
Finally we remark that our main interest is in
closed hyperbolic
manifolds, where $\pi_2(M)$ is trivial.
In that case conditions (a),(b) of lemma \ref{weakqg} are
equivalent to $\wwp$ having orbit space homeomorphic to $\rrrr^2$.


\smallskip
For flows in hyperbolic $3$-manifolds, 
we now develop a method to  upgrade
information about asymptotic behavior of flow lines
of $\wwp$ into metric efficiency of flow lines.
This will be the key tool to prove uniform quasigeodesic
behavior for a large class of flows in hyperbolic $3$-manifolds.

\begin{theorem}{}{}
Let $Y$ be a closed invariant set of a non singular flow $\Phi$ 
in $M^3$ closed hyperbolic. Suppose that

%(a) $\wwp |_{\widetilde Y}$ has Hausdorff orbit space.

%(b) There are $\epsilon, T > 0$ so that any $\epsilon, T$
%cycle of $\Phi |_Y$ is not null homotopic in $M$.

(a) half orbits converge: for any $x \in \widetilde Y$
each of the two rays of $x \cdot \rrrr$ accumulate in a single 
point of $\si$, that is,
the following limits exist:

$$\lim_{t \rightarrow +\infty} x \cdot t = \eta_+(x) \in \si
\ \ {\rm and} \ \ 
\lim_{t \rightarrow -\infty} x \cdot t = \eta_-(x) \in \si.$$

(b) For each $x \in \widetilde Y$, $\eta_+(x) \not = \eta_-(x)$.


(c) The maps $\eta_+, \eta_-: \widetilde Y \rightarrow \si$
are continuous.

\noindent
Then the orbits  in $\wwp |_{\widetilde Y}$ are uniformly
quasigeodesics: there is $k > 0$ so that for any 
orbit $\gamma$ of $\wwp$ in $\widetilde Y$ and for
any $x, y \in \gamma$, 
$\tau(x,y) < k d(x,y) + k$.
\label{upgrade}
\end{theorem}

Notice that conditions (a), (b) and (c) are necessary to
get quasigeodesic behavior. Conditions (a) and (b) follow
directly from the fact that single flow lines are quasigeodesics
\cite{Th1,Gr,Gh-Ha}. Condition (c) is not true for an 
arbitrary collection of quasigeodesics, but holds for 
uniformly quasigeodesic closed invariant sets of flows.
Since we will use this last fact in the proof of theorem $A$, we
provide a proof in section \ref{qg}.

\begin{proof}{}
Since $\eta_+(x) \not = \eta_-(x)$ for  any $x \in \widetilde Y$,
let $g_x$ be the unique geodesic in $\hhh$ with endpoints
$\eta_+(x), \eta_-(x)$  and
let $\bdy g_x \subset \si$ be the ideal points of $g_x$.
Given $a > 0$ let $U_a(g_x) \subset \hhh$ be the neighborhood
of radius $a$ around $g_x$.
We first show:

\centerline{$(*)$ there exists $a > 0$ so that for any $x \in
\widetilde Y$ we have $x \cdot \rrrr \subset U_a(g_x)$.}

\noindent
Otherwise, let $x_i \in \widetilde Y$ with $d(x_i, g_{x_i}) \to +\infinity$.
Up to covering translations and taking a subsequence assume that $x_i \to x$.
Hence $x \in \widetilde Y$ and by (b), (c) we have 
$$\lim \eta_+(x_i) = \eta_+(x) \ne \eta_-(x) = \lim \eta_-(x_i)
$$ 
But $d(x,g_{x_i}) \to +\infinity$,
so up to taking a subsequence we may assume
that there exists $p
\in S^2_\infinity$ such that $\lim (g_{x_i} \cup \partial g_{x_i})  = p$ 
in the Hausdorff topology
on closed subsets of $\hhh \union S^2_\infinity \homeo B^3$. 
Therefore $p = \lim \bdy g_{x_i} = \lim \{ \eta_+(x_i), \eta_-(x_i) \}$,
contradicting that $\lim \eta_+(x_i) \ne \lim \eta_-(x_i)$.

\smallskip
We now assume that $\wwp | _{\widetilde Y}$ is not uniformly 
quasigeodesic and derive a contradiction.
There are two steps in the argument:

\begin{enumerate} 

\item[] Step(1) Using $(*)$ we show that there are  $x_i, y_i$ in the
same orbit of $\widetilde Y$ with  $d(x_i,y_i)$ bounded but
$\tau(x_i,y_i) \rightarrow +\infty$. 

\end{enumerate}

\noindent
If in addition one knows that $\Phi |_Y$ satisfies the
$\epsilon,T$-cycles condition and
$\wwp | _{\widetilde Y}$ has 
Hausdorff orbit space, then the conclusion of step
(1) is disallowed by lemma \ref{weakqg}, finishing the proof.
However, these additional assumptions
are not necessary because:

\begin{enumerate}
\item[] Step (2) The conclusion of step (1) is disallowed by conditions
(a), (b) and (c).
\end{enumerate}

We need the following definitions: given $w \in \hhh$ and an oriented
geodesic $g \subset \hhh$, let $P(w,g)$ be the hyperplane of $\hhh$ 
containing $w$ and perpendicular to $g$. Let $F(w,g)$ be the component of
$\hhh - P(w,x)$ containing the positive endpoint of $g$ in its closure and
let $B(w,x)$ be the other component. For $w \in g$ and $b > 0$ let $w+b$
be the point of $g$ with $d(w, w+b) = b$ and $w+b$ in the ray from $w$ to
the positive endpoint of $g$.

For any $x \in \widetilde Y$ we define $\rho_x \from \hhh \to g_x$ to be the
closest point projection onto $g_x$. Then for any $x, y$ in the same orbit of
$\wwp$ in $\widetilde Y$,
$$d(\rho_x (x), \rho_x (y)) \le d(x,y) \le 
d(\rho _x (x), \rho _x (y)) + 2a \eqno(1)
$$ 
because $x,y \in U_a(g_x)$.

To prove step (1), because we have assumed that orbits are not
uniformly quasigeodesic, for any
$C > 0$ there is an orbit segment $\gamma = [x,y]$ in $\widetilde Y$
which satisfies
$\Length(\gamma) / d(x,y) > 2C$ and $\Length(\gamma) > C$. 
Assume first that $d(x,y) \ge 1 + 2a$.
Hence $d(\rho _x (x), \rho _x (y)) \ge 1$ by (1).
Also $d(x,y) \ge
d(\rho _x (x), \rho _x (y))$, so

\begin{eqnarray*}
{\frac{\Length(\gamma)}{d(\rho _x (x), \rho _x (y))}} \ 
	\geq \ \frac{\Length(\gamma)}{d(x,y)} \  > \ 2C \
\ge \ C + \frac{C}{d(\rho _x (x), \rho _x (y))} \\
\end{eqnarray*}

\noindent
and therefore

\begin{eqnarray*}
\frac{\Length(\gamma)}{C} \ >  \ d(\rho _x (x),\rho _x (y)) + 1 \
  > \ \leftceiling d(\rho _x (x),\rho _x (y)) \rightceiling \\
\end{eqnarray*}

\noindent
where $\leftceiling x \rightceiling$ denotes the ``ceiling'' function, 
that is, the
least integer $\ge x$. Set $n_0 = \leftceiling d(\rho _x (x),\rho _x (y))
\rightceiling$, so
$\Length(\gamma) > n_0C$. Also, 
$(n_0-1) < d(\rho _x (x),\rho _x (y)) \le n_0$,
so we can find consecutive points 
$\rho _x (x) = z_0, z_1, \ldots, z_{n_0} = \rho _x (y)$
on $g_x$ such that $d(z_{n-1},z_n) = 1$ if $1 \le n < n_0$ and
$0 < d(z_{n_0-1},z_{n_0}) \le 1$. 
Let $x=x_0$ and for $1 \le n \le n_0$ let $x_n$
be the last point on $\gamma = [x,y]$ such that $\rho _x (x_n) = z_n$, so
the points $x_0,x_1,\ldots,x_{n_0}$ are consecutive on $\gamma$ and they
partition $\gamma$ into subsegments $\gamma = \gamma_1 * \cdots * \gamma_{n_0}$
with $\gamma_n = [x_{n-1},x_n]$. Since 
$$\Sum_{n=1}^{n_0} \Length(\gamma_n) =
\Length(\gamma) > n_0C
$$ 
then for some $n$ we have $\Length(\gamma_n) >
C$.



This shows that regardless of whether $d(x,y) \geq 1 + 2a$ or not,
we produce a pair
$x', y'$ so that
$d(x',y') < 1 + 2a$ but $\tau(x',y') > C$.
This finishes the proof of step 1.

We now prove step 2.
Let then $x_i, y_i$ with $d(x_i,y_i) < 1 + 2a$ but
$\tau(x_i,y_i) \rightarrow +\infty$.
Without loss of generality assume that $y_i = x_i \cdot t_i$
with $t_i > 0$.

\smallskip
\noindent
{\underline {Case 1}} --- 
The sequence of intervals 
$[x_i,y_i]$ has a subsequence with bounded diameter, which
we may assume is the original sequence.

Let $v_i$ be the middle point in $[x_i,y_i]$ (with respect to
the parametrization) and up
to subsequence and covering translations assume that $v_i
\rightarrow v \in \widetilde Y$.
For any $t \in \rrrr$, $v_i \cdot t \rightarrow v \cdot t$
and for $i$ big enough $v_i \cdot t \in [x_i,y_i]$.
Since $d(v_i \cdot t_i, v_i)$ is bounded, this shows that
$d(v \cdot t,v)$ is also bounded.
It follows that $v \cdot \rrrr$ is contained in a bounded set
in $\hhh$, hence it accumulates in $\hhh$ contradicting
condition (a) of the theorem.

\smallskip
\noindent
{\underline {Case 2}} --- 
There exist $x_i,y_i \in \widetilde Y$ such that $d(x_i,y_i) < 1 + 2a$
and diameter$([x_i,y_i]) \rightarrow +\infty$. 

Let $g_i = g_{x_i}$ and let
$\rho_i \from \hhh \to g_i$ be the closest point projection. Choose $w_i \in
[x_i,y_i]$ so that


$$d(\rho_i (x_i), \rho_i (w_i))
= \max _{w \in (x_i, y_i)} d(\rho_i(x_i), \rho_i(w)).$$ 

\noindent
Since diam$[x_i,y_i] \rightarrow +\infty$ and $d(x,y) < 1 + 2a$, then
$d(w_i,x_i) >> d(x_i,y_i)$. Assume first that $w_i \in F(x_i, g_i)$, hence
$w_i \in F(y_i, g_i)$. If $x_i \in B(y_i,g_i)$, then as
$P(y_i,g_i)$ separates $x_i$ from $w_i$, we can find $z_i,v_i
\in [x_i,y_i]$ so that:
$$
w_i \in (z_i,v_i), \ \
\rho_i(z_i) = \rho_i(v_i) = \rho_i(y_i)
\ \ {\rm and } \ \ \rho_i(w) \not = \rho_i(y_i), \ 
\forall w \in (z_i,v_i).
$$
\noindent
Similarly if $x_i \in F(y_i,g_i) \cup P(y_i,g_i)$, we can find $z_i, v_i$
satisfying the conditions above except that $\rho_i(z_i) = \rho_i(v_i) =
\rho_i(x_i)$. We obtain a similar statement if $w_i \in B(x_i,g_i)$.





%If $x_i \in B(\rho_{x_i}(y_i), x_i) \cup
%P(\rho_{x_i}(y_i), x_i)$ then since $\eta_+(x)$ is in the closure of
%the set $F(\rho_ {x_i} (y_i) + 1, x_i)$,
%there is
%$x'_i = x_i \cdot t'_i$ with $0 < t'_i \leq t_i$ and
%either $\rho_{x_i}(x'_i) = \rho_{x_i} (x_i)$ (*) or
%$\rho_{x_i}(x'_i) = \rho_{x_i} (y_i)$ (**).
%In addition  we can choose $x'_i$ so that 
%in situation (*), $\tau(x'_i,x_i) \rightarrow +\infty$ and
%in (**), $\tau(x'_i,y_i) \rightarrow +\infty$,
%otherwise we would be able to reduce the analysis to case 1.
%


These arguments show that there are $x_i, y_i$ so that
$\rho_i(x_i) =
\rho_i(y_i)$, 
$\rho_i(w) \not =
\rho_i(x_i)$ for any $w \in (x_i, y_i)$ and 
diam$[x_i,y_i] \rightarrow +\infty$.
%Hence either $(x_i,y_i) \subset B(\rho_{x_i}(x_i), x_i)$
%or $(x_i,y_i) \subset F(\rho_{x_i}(x_i), x_i)$.
If $(x_i,y_i) \subset F(x_i, g_i)$, 
let $w_i$ be defined as above.
Since $\eta_+(x_i)$ is in the closure of $F(\rho_i(x_i) + 1, g_i)$
in $\hhh \cup \si$ (where $\rho_i(x_i) + 1$ is computed in $g_i$)
and $y_i = w_i \cdot r'_i$ with $r'_i > 0$,
there must be a point  $u_i = w_i \cdot s_i$, with
$s_i > 0$ and $\rho_i(u_i) = \rho_i(w_i)$.
In addition we may assume that $(w_i,u_i) \subset B(w_i,g_i)$
and diam$[w_i,u_i] \rightarrow +\infty$.
%which then reduces the analysis to the situation
%$(x_i,y_i) \subset B(\rho_{x_i}(x_i),x_i)$.


\begin{figure}
\centeredepsfbox{qg1.eps}
\caption{Too much backtracking eventually traps orbits in
the negative direction.}
\label{backtrack}
\end{figure}






Since $d(w_i,x_i) \rightarrow +\infty$ these arguments show that in any case
there are $x_i, y_i \in \widetilde Y$, with $\rho_i(x_i) =
\rho_i(y_i)$, $(x_i,y_i) \subset B(x_i,g_i)$ and
diam$[x_i,y_i] \rightarrow +\infty$, see
figure \ref{backtrack}. Up to covering translations and
taking a subsequence we may assume that $x_i \rightarrow x \in \widetilde Y$.
Since $g_i = g_{x_i} \rightarrow g_x$ and $\rho_i(x_i) = \rho_{x_i}(x_i)
\rightarrow \rho_x(x)$, then
$${\overline {P(\rho_{x_i}(x_i) + 2,g_i)}} \ \rightarrow \
{\overline {P(\rho_{x}(x) + 2,g_x)}}$$
\noindent
in the topology of closed sets of $\hhh \cup \si$.
Then
$$\eta_+(x) \in {\overline{F(\rho_{x_i}(x_i) + 1,g_i)}}$$
\noindent 
for $i$ big enough.
Hence there is $t_0 > 0$ so that for $i$ big enough
$x \cdot [t_0,+\infty) \subset F(\rho_{x_i}(x_i),g_i)$. It therefore follows
that for $i$ big enough there is $r_i$ with $r_i < t_0 + 1$ and $x_i \cdot
r_i \in F(\rho_{x_i}(x_i),g_i)$. This contradicts the facts $(x_i,y_i)
\subset B(\rho_{x_i}(x_i), g_i)$ with $y_i = x_i \cdot t_i$ and $t_i
\rightarrow +\infty$. This finally finishes the proof of theorem
\ref{upgrade}.
%But $d(x_{n-1},x_n) \le d(z_{n-1},z_n) + 2a \le
%1+2a$, contradicting lemma \ref{weakqg}.
%
%We conclude that for any  two points $x, y$ in a flow line of
%$\wwp$ in $\widetilde Y$,
%$\tau(x,y) \leq 2C d(x,y) + C$, hence they
%are all $2C$ quasigeodesics.
\end{proof}

\noindent
{\bf Remarks:} 

(1) The above proof works verbatim
for $M^n$ closed hyperbolic.

%(2) Without much more work, one can also prove this result when
%$M^3$ is irreducible and
%$\pi_1(M)$ is word hyperbolic in the sense of Gromov \cite{Gr,Gh-Ha}.

%ALTERNATIVE TO (2):

(2) Without much more work, one can also prove this result 
under more general assumptions: 

%\begin{itemize}

%\item[(i)] 
(i) $M$ is any manifold and $\pi_1(M)$ is word
hyperbolic in the sense of Gromov \cite{Gr}. In that case instead of
$\si$ one uses the appropriate boundary at infinity $\partial
_{\infty} \mi$ --- notice that if $M$ is an oriented, irreducible
3-manifold then $\partial_{\infty} \mi$ is still a $2$-dimensional
sphere \cite{Be-Me}.

%\item[(ii)] 
(ii) $Y$ is a compact space on which a flow $\Phi$ is defined,
and instead of $Y$ being a subset of $M$ we have instead a continuous
function $f \from Y \to M$. By the pullback construction we have $f
\from \widetilde Y \to \widetilde M$ and a flow $\widetilde \Phi$ on
$\widetilde Y$, and theorem \ref{upgrade} applies.

%\end{itemize}

