\newcommand{\flowtime}{\func{\tau}}
\newcommand{\Fr}{\func{Fr}}
\newcommand{\Bc}{\func{Bc}}
\newcommand{\ClFr}{\sigma^f}
\newcommand{\ClBc}{\sigma^b}
\newcommand{\Axis}{\func{Axis}}

\section{Inductive proof of theorem A}
\label{qg}

To set up the induction, apply proposition \ref{hierarchy} to obtain a
hierarchy $B_0\subset\cdots\subset B_N$. We use the following
notation. If $0 \le n \le N$ let $P_n = P(B_{N-n})$, and let $P_{N+1} =
M$, so $P_0 \subset P_1 \subset \cdots \subset P_N \subset P_{N+1}$;
the indexing is reversed to facilitate the induction proof. If $1 \le
n \le N+1$ let $S_n = S(B_{N-n+1}) = B_{N-n+1} \intersect P_n$, so
$S_n$ is properly embedded in $P_n$, and $P_{n-1}$ is obtained from
$P_n$ by removing a regular neighborhood of $S_n$. Since some (and
hence all) compact leaves of $\Fol$ are quasi-fuchsian, the same is
true for components of $B_0$, hence proposition \ref{DiscsAndBalls}
applies.

Given a flow line $x \cdot \reals$ of $\Phi$, define the {\em codepth}
to be the minimal integer $n$ such that $x \cdot \reals \subset P_n$,
and define $\Omega_n$ to be the union of all flow lines of codepth at
most $n$. Note that $\Omega_n$ is the set of all flow lines contained
in $P_n$, and $\Omega_n$ is closed. As special cases, $\Omega_0 =
\emptyset$ since $P_0$ is a product, and $\Omega_{N+1} = M$. 

The program for the proof of theorem A
is to assume that orbits in $\Omega_{n-1}$ 
are uniformly quasigeodesic and then show that orbits in $\Omega_n$
are also uniformly quasigeodesic (with a bigger quasigeodesic
constant). Since $\Omega_0 = \emptyset$ and $\Omega_{N+1} = M$,
induction will show that all orbits are uniformly quasigeodesic.
This will complete the proof of theorem A.

For notational convenience, throughout this section we write $P =
P_n$, $\Omega = \Omega_n$, $S = S_n$, $P' = P_{n-1}$, and $\Omega' =
\Omega_{n-1}$. Let $\pi \from \cover P \to P$ be the universal
covering, and let $\cover \Omega = \pi^\inverse(\Omega)$. 

Fix a connected lift $\cover P' \inject \cover P$, and let $\cover
\Omega' = \pi^\inverse(\Omega') \intersect \cover P'$. The induction
hypothesis says that orbits in $\cover \Omega'$ are uniform
quasigeodesics in $\cover P'$. Since $\cover P' \inject \cover P$ is a
quasi-isometry, then orbits in $\cover \Omega'$ are uniformly
quasigeodesic in $\cover P$; using the action of $\pi_1 P$ by
isometries on $\cover P$, the same is true for orbits in
$\pi^\inverse(\Omega')$.
Recall that the hypothesis in theorem $A$ is
that $\wwp$ has Hausdorff orbit space.
This will be used in verifying conditions (b) and (c) of
theorem \ref{upgrade}.

We will need the following well known simple result \cite{Co}:

\begin{lemma}{}{} Let $W$ be a compact metric space with a nonsingular
semiflow $\varphi$ parameterized by arc length. Let $\Omega$ be the set
of points $x$ for which $\varphi_t(x)$ is defined for all $t \in
\rrrr$. Then given any $\delta > 0$, there is $a > 0$ so that any orbit
$\gamma$ of $\varphi$ is in the $\delta$-neighborhood of $\Omega$
except perhaps for an initial segment of length $< a$ and another
final segment of length $< a$.
\end{lemma}

We will also use the following localization property of quasigeodesics.

\begin{proposition}{}{\cite{Gr,Gh-Ha}} Let $M$ be a compact manifold 
with negatively curved
$\pi_1(M)$. Then for any $K > 0$ there is $L(K)  > 0$ (usually $L(K) >>
K$) satisfying: if $\gamma$ is an embedded curve so that any subarc of
length $\leq L(K)$ is a $K$ quasigeodesic then $\gamma$ is a $2
K$-quasigeodesic.
\label{local}
\end{proposition}



The following essential fact which is a direct
consequence of proposition \ref{DiscsAndBalls} will
be used explicitly or implicitly throughout this section:
if $\gamma$ is a curve contained in $\widetilde P'$, then
$\gamma$ is a quasigeodesic in $\widetilde P'$ if and only if
$\gamma$ is a quasigeodesic in $\widetilde P$ and also
 if and only if
$\gamma$ is a quasigeodesic in $\hhh$.
The quasigeodesic constants may differ.
We caution the reader that some arguments are done in $\hhh$
while others are done in $\widetilde P$. The context makes it clear.


For flow segments disjoint from $S$, the following lemma establishes
the quasigeodesic property directly.

\begin{lemma}{}{}
There is $K > 0$ so that all flow segments, half orbits or full orbits
of $\wwp$ contained in $\cover P'$ are $K$-quasigeodesics of $\cover
P$; translating by the action of $\pi_1(P)$, the same is true in any
lift of $P'$.
\label{uniform}
\end{lemma}

\begin{proof}{}
The full orbits staying in $\cover P'$ are precisely the orbits in
$\cover\Omega'$, which are $k$-quasigeodesic for some uniform $k$. Fix
$\delta_0 > 0$, with $2 k \delta_0 < 1$. Let $L = L(k+1)$ be given by
proposition \ref{local}. Choose $\delta_1 > 0$ so that if $x, y \in
\cover P$ and $d(x,y) < \delta_1$ then $d(x \cdot t, y \cdot t) <
\delta_0$ for any $|t| < L$. Choose $a > 0$ so that any orbit in $P'$
is in the $\delta_1$-neighborhood of $\Omega'$, except perhaps for
initial and final segments of length $< a$.

Let now $x, y \in \cover P'$ with $y = x \cdot t$. Choose a
subsegment $\gamma = [z,w] \subset [x,y]$, with $z = x \cdot t_1$, $w
= y \cdot t_2$, $0 \le t_1 < a$ and $-a < t_2 \le 0$, so
that $\gamma \subset U_{\delta_1} (\cover \Omega')$. Let $\alpha =
[z_0,w_0]$ be a subarc of $\gamma$ with $0 < t_3 = \flowtime(z_0,w_0)
\le L$. By the choice of $\delta_1$ it follows that  there is $z_1 \in
\cover \Omega'$, with $d(z_0 \cdot t, z_1 \cdot t) < \delta_0$, for
any $|t| < L$. Since $z_1 \cdot \reals$ is a $k$-quasigeodesic then
\begin{eqnarray*}
\flowtime(z_0,w_0) = t_3 = \flowtime(z_1,z_1 \cdot t_3) 
\leq k \, d(z_1,z_1 \cdot t_3) + k \\
\leq k d(z_0,w_0) + 2 k \delta_0 +
k < k d(z_0,w_0) + (k + 1).
\end{eqnarray*}
Clearly this also works for any subsegment of $\alpha$, hence $\alpha$
is a $(k+1)$-quasigeodesic. By the previous proposition, one concludes
that $\gamma$ is a $(2k+2)$-quasigeodesic. This implies that 
\begin{eqnarray*}
\flowtime(x,y) & < 2a + \flowtime(z,w) \le (2k+2) d(z,w) + (2k+2+2a) \\
     & \le (2k+2) \bigl( d(x,y) + 2a \bigr) + (2k+2+2a) \\
     & = (2k+2) d(x,y) + \bigl( 2k+2+2a + (2k+2)2a \bigr ) 
\end{eqnarray*}
Hence any piece of orbit of $\wwp$ contained in $\cover P'$ is
a $(4ka + 2k + 6a + 2)$-quasigeodesic of $\widetilde P'$.
Since $\widetilde P'$ is quasi-isometrically embedded in
$\widetilde P'$, there is $K > 0$ so that any
piece of orbit of $\wwp$ contained in $\widetilde P$ is a 
$K$-quasigeodesic 
of $\widetilde P$.
\end{proof}

Now we prepare the ground for applying theorem \ref{upgrade} to
show that full orbits in $\cover\Omega$ are uniformly quasigeodesic. We
must study how orbits in $\cover\Omega - \cover\Omega'$ cross lifts of
$S$ in $\cover P$, so we embark on a study of these lifts.

Let ${\mathcal E}$ be the collection of lifts of $S$ to $\cover P$. Any
element $E \in {\mathcal E}$ is transversely oriented and separates
$\cover P$. The front of $E$ is the component $\Fr(E)$ of $\cover P -
E$ on the positive side of $E$, and the closure of $\Fr(E)$ in
$\widetilde P \cup \partial _{\infty} \widetilde P$,
is denoted $\ClFr(E)$. The back $\Bc(E)$ and its closure
$\ClBc(E)$ are similarly defined. Define a strict partial order on
$\mathcal E$ where $E < E'$ if $\Bc(E) \intersect \Fr(E') = \emptyset$ and
$E \not = E'$, hence in particular $E' \subset \Fr(E)$.

Notice that since $S$ is compact, a bounded subset of $\cover P$
intersects only finitely many $E \in {\mathcal E}$. The following
lemmas strengthen this fact, by showing that sequences in
$\mathcal E$ are limited in how they may accumulate in the ball 
$\widetilde P \cup \partial _{\infty} \widetilde P$;
these lemmas will be useful in analyzing flow lines that cross $S$ many
times.
If $\rho: \pi_1(M) \rightarrow PSL_2({\bf C})$ is the holonomy
representation, then $\rho(\pi_1(P))$ is a Kleinian group
which is convex cocompact.

\begin{lemma}{}{}
There is $J_0 > 0$ such that if $E_1, \ldots, E_{J_0}$ are distinct
elements of $\mathcal E$, then $\bdy_\infinity(E_1) \intersect\cdots
\intersect \bdy_\infinity(E_{J_0}) = \emptyset$. If moreover $E_1 <
\cdots < E_{J_0}$ then $\sigma^b E_1 \intersect \sigma^f E_{J_0} =
\emptyset$.
\label{LimitSetsDisjoint}
\end{lemma}

\begin{proof}{}
First we need the fact that if $H_1, \ldots, H_n$ are geometrically
finite Kleinian groups that generate a discrete group, then $\Lambda
H_1 \intersect \cdots \intersect \Lambda H_n = \Lambda(H_1 \intersect
\cdots \intersect H_n)$. When $n=2$ this is proved in \cite{Su},
%-[p. Susskind,
%Kleinian groups with intersecting limit sets, J. d'Analyze
%Mathematique {\bf 52} (1989), 26--38], 
and the statement for finitely
many subgroups follows by induction.
%: if the fact is true for
%$H_1, \ldots, H_n$, apply Susskind's theorem to the groups $H_1
%\intersect \cdots \intersect H_n$ and $H_{n+1}$ to obtain the
%fact for $H_1, \ldots, H_{n+1}$. 

To prove the first statement, suppose that $\bdy_\infinity(E_1)
\intersect\cdots \intersect \bdy_\infinity(E_J) \ne \emptyset$. Let
$H_i\subset \pi_1 P$ be the stabilizer subgroup of $E_i$, so
$\Lambda(H_i) = \bdy_\infinity(E_i)$, and by the above argument using
Susskind's theorem it follows that $H_1 \intersect \cdots \intersect
H_J \ne \emptyset$. Let $f$ be in the intersection, and let $\Axis_f$
be an axis for $f$ in $\hyperbolic^3$. By conjugation, we may assume
that $\Axis_f$ intersects a fixed fundamental domain $D$ for $\pi_1
M$. 

Since $E$ is quasi-isometrically embedded in $\hhh$, it
is $R$-quasiconvex, that is, for any 
$x, y \in  E \cup \partial _{\infty} E$,
the geodesic in $E$ connecting them is at most $R$ distant
from the hyperbolic geodesic connecting them \cite{Gr,Gh-Ha}.
The $R$ is independent of the lift $E$ of $S$.
Thus each $E_i$ intersects $U_R(D)$, the open $R$-neighborhood
of $D$. This shows that $J \le J_0$ where $J_0$ is an upper bound for
the number of distinct $E \in {\mathcal E}$ intersecting the bounded set
$U_R(D)$.

To prove the second statement, suppose that $\sigma^b E_1 \intersect
\sigma^f E_{J_0} \ne \emptyset$. 
It follows that $\bdy_\infinity E_1
\intersect \bdy_\infinity E_{J_0} \ne \emptyset$; let $\xi$ be a point
in this intersection. 
If for some $1 \leq i \leq n$, $\xi \not \in \bdy_{\infty} E_i$,
let $V$ be a neighborhood of $\xi$ in 
$\widetilde P \cap \partial _{\infty} \widetilde P$ with
$E_i \cap (V \cap \widetilde P) = \emptyset$.
However since $\xi \in \bdy_{\infty} E_0 \cap \bdy_{\infty} E_{J_0}$,
there are $x \in E_0 \cap V$ and $y \in E_{J_0} \cap V$.
Hence $x$ can be connected to $y$ in $V \cap \widetilde P$,
contradicting the fact that $E_i$ separates $E_0$ from $E_{J_0}$.
%Since $E_1$ and $E_{J_0}$ are quasi-isometrically
%embedded we may choose sequences $x_i$ in $E_1$ and $y_i$ in $E_{J_0}$
%such that $x_i \to \xi$, $y_i \to \xi$, and $d(x_i,y_i)$ is bounded.
%Let $\gamma_i \subset \cover P$ be a path from $x_i$ to $y_i$ of
%bounded length. If $1 < i < J_0$ then $E_i$ separates $E_1$ from
%$E_{J_0}$ in $\cover P$, so $\gamma_i$ intersects $E_i$ in some point
%$z_i$. Since $d(x_i,z_i)$ is bounded then $z_i \to \xi$, and hence
%$\xi \in \bdy_\infinity E_i$. 
But then we have proved $\bdy_\infinity
E_1 \intersect \cdots\intersect \bdy_\infinity E_{J_0} \ne \emptyset$,
contradicting the first statement of the lemma.
\end{proof}

The $J_0$ given by the previous lemma is fixed from now on. The next
lemma gives an even stronger accumulation property.

%\break

\begin{lemma}{}{}
Given an infinite sequence $E_1, E_2, \ldots \in {\mathcal E}$ such that
$E_i\ne E_j$ if $i \ne j$, suppose there exists $E_0 \in {\mathcal E}$ such
that $E_0 < E_i$ for all $i \ge 1$. Then there is a subsequence
$E_{i_n}$ such that $\ClFr(E_{i_n})$ converges to a single point $\xi
\in\bdy_\infinity \cover P$, in the Hausdorff topology on closed
subsets of $\widetilde P \cup \partial _{\infty} \widetilde P$.
\label{BoundedBelow}
\end{lemma}

\begin{proof}{} Choose covering translations $f_i \in \pi_1 P$ such
that $f_i(E_0) = E_i$. Then $f_i \ne f_j$ when $i \ne j$, so by the
convergence group property for Kleinian groups applied to
$\rho(\pi_1(M))$ (see pg. 22 of Maskit's book
\cite{Ma}), we may pass to a subsequence so that there is a pair of
points $\xi_-, \xi_+ \in \bdy_\infinity P$ called a {\em source} and
{\em sink} for the sequence of functions $f_i$, meaning that $f_i
\restrictedto \bigl( \widetilde P \union \bdy_\infinity \widetilde P
\bigr) - \xi_-$ converges uniformly on compact sets to the constant
map with value $\xi_+$, and $f_i^\inverse \restrictedto \bigl(
\widetilde P \union \bdy_\infinity \widetilde P
\bigr) - \xi_+$ converges similarly to $\xi_-$. Since
$f_i^\inverse(\ClBc(E_i)) = \ClBc(E_0)$ for all $i$, it follows
that $f_i^\inverse(\ClBc(E_0)) \subset \ClBc(E_0)$, and hence $\xi_-
\in \ClBc(E_0)$. 

If $\xi_- \in \sigma^b E_0 - \bdy_\infinity E_0$ then we are done,
because then $\xi_- \notin \ClFr(E_0)$, so the sequence of maps $f_i
\restrictedto \ClFr(E_0)$ converges uniformly to the constant map with
value $\xi_+$, and hence the images $\ClFr(E_i)$ of these maps converge
in the Hausdorff topology to $\xi_+$.

Suppose on the other hand that $\xi_- \in \bdy_\infinity E_0$. By the
previous lemma, for all but finitely many $E_i$ we have $\xi_- \notin
\bdy_\infinity E_i$; in particular this is true for some $i = i_0$.
We also have $\xi_- \notin \ClFr(E_{i_0})$, because $\xi_- \in
\sigma^b(E_0)$. Note that $\xi_-, \xi_+$ is a source, sink pair for the
sequence $f_i \composed f_{i_0}^\inverse$, so the maps $f_i \composed
f_{i_0}^\inverse \restrictedto \ClFr(E_{i_0})$ converge uniformly to
the constant map with value $\xi_+$, and hence their images
$\ClFr(E_i)$ converge in the Hausdorff topology to $\xi_+$.
\end{proof}

Now we show that all flow lines in $\cover \Omega$ extend continuously
to $\bdy_\infinity \widetilde P$, verifying condition (a) of theorem
\ref{upgrade}.

\begin{proposition}{(extension of orbits)}{}
If $x \in \cover \Omega$, then the following limits exist:
$$\lim_{t \rightarrow +\infty} x \cdot t = \eta_+(x) \in
\bdy_\infinity \cover P \quad {\text and } \quad
\lim_{t \rightarrow -\infty} x \cdot t = \eta_-(x) \in
\bdy_\infinity \cover P
$$
\label{ext}
\end{proposition}

\begin{proof}{} We only consider forward limits. There are two cases:

\smallskip
\noindent
{\underline {Case 1}} --- $x \cdot \Rplus$  eventually stops
intersecting $\pi^{-1}(S)$ (this includes $x \in \cover \Omega'$).

Let $y$ be the last intersection of $x \cdot \RPlus$ with
$\pi^{-1}(S)$ if any exists, otherwise let $y = x$. Then $\pi(y) \cdot
\RPlus$ is contained in $P'$ hence $y \cdot \RPlus$ is a
$K$-quasigeodesic in $\cover P$ by lemma \ref{uniform}. Therefore it
has a unique limit point.

\smallskip
\noindent
{\underline {Case 2}} --- 
$x \cdot \RPlus$ keeps intersecting $\pi^{-1}(S)$.

 Let $E_0 < E_1 < \cdots \in {\mathcal E}$ be the elements that $x
\cdot \RPlus$ intersects. Thus, all accumulation points of $x
\cdot \RPlus$ are contained in
$\ClFr(E_1) \intersect \ClFr(E_2) \intersect \cdots $. Since this is
a nested intersection, it is the same as the Hausdorff limit of the
sets $\sigma^f(E_1), \sigma^f(E_2), \cdots$, which by
lemma \ref{BoundedBelow} is a single point.
\end{proof}


Next we verify condition (b) of theorem \ref{upgrade}.
We use the following facts: if $\alpha$ and $\beta$ are
two quasigeodesic rays in $\hhh$ with the same ideal point,
then there is $R > 0$ (depending on $\alpha$ and $\beta$)
so that $\alpha$ is in the $R$ neighborhood of $\beta$ and
vice versa. For a pair of bi-infinite quasigeodesics with 
the same ideal points there is also a bound, which
depends only on the quasigeodesic constant.
We also need the following very useful result:



\begin{lemma}{}{}
Suppose that $\gamma_i = x_i \cdot [0,t_i)$ and $\gamma = x \cdot
\Rplus$ are $k_1$-quasigeodesic flow segments or rays, where $x_i \to
x$ in $\cover P$ and $t_i \to +\infinity$ in $[0,+\infinity]$; we allow
$t_i = +\infinity$. Then $x_i \cdot t_i \to\eta_+(x)$ in 
$\widetilde P \cup \partial _{\infty} \widetilde P$,
where $x_i \cdot t_i$ means $\eta_+(x_i)$ if $t_i = +\infinity$.
\label{conv}
\end{lemma}

\begin{proof}{}
Otherwise, passing to a subsequence we have $x_i \cdot t_i \to w \in
\widetilde P \cup \partial _{\infty} \widetilde P$
with $w \ne \eta_+(x)$. Choose $R > 0$ so that any
$k_1$-quasigeodesic is at most $R$ distant from a corresponding minimal
geodesic. 

Choose disjoint small neighborhoods $W_0$ of $\eta_+(x)$ in 
$\compact P$,
$W_1$ of $w$ in $\compact P$ and $W_2$ of $x$ in $\cover P$ so that
for any minimal geodesic segment or ray $\beta$ starting in $W_2$ and
with endpoint in $W_1$ then

$$U_R(\beta) \cap (W_0 \cap \cover P) = \emptyset.$$ 

\noindent
For $i$ big enough $x_i \in W_2$ and $x_i \cdot t_i \in W_1$, so
since $\gamma_i$ is at most $R$ distant from the corresponding minimal
geodesic, then $\gamma_i \cap (W_0 \cap \cover P) = \emptyset$.
But since $x_i \rightarrow x$ then by continuity of the flow 
$\gamma_i \cap (W_0 \cap \cover P) \ne \emptyset$ for $i$ big
enough, contradiction.
\end{proof}




\begin{proposition}{(distinct limit points)}{} For any $x \in 
\cover \Omega$, $\eta_+(x) \not = \eta_-(x)$.
\label{distinct}
\end{proposition}

\begin{proof}{}
There are two cases:

\smallskip
\noindent
{\underline {Case 1}} --- $x \cdot \reals$ intersects $\pi^{-1}(S)$
only finitely many times. 

Then there are $t_0 < t_1 \in \rrrr$, with
$\gamma_0 = x \cdot (-\infty,t_0]$ and $\gamma_1 = x \cdot
[t_1,+\infty)$, so that $\gamma_0 \cap \pi^{-1}(S) = \gamma_1 \cap
\pi^{-1}(S) = \emptyset$. By lemma \ref{local}, $\gamma_0$ and
$\gamma_1$ are $K$-quasigeodesics in $\hhh$. Assume they have the same
ideal point $p$. Then there are sequences $v_i \in \gamma_0$, $u_i \in
\gamma_1$, such that $u_i \to p$, $v_i \to p$, and $d(u_i,v_i)$ is
bounded. But $\tau(u_i,v_i) \to \infinity$.



Let $y = x \cdot t_0$ and $z = x \cdot t_1$.
Up to taking a subsequence assume that $\pi(u_i)$ 
converges in $M$. Then there are covering translations $h_i$
so that $h_i(u_i) \rightarrow u$. Since $u_i \in \widetilde P'$,
then $\pi(u) \in P'$.
We can also assume that $h_i(v_i) \rightarrow v$ and hence
$\pi(v) \in P'$.

%Since $[z,u_i]$ and $[v_i,y]$ are 
%is contained in $\widetilde P'$,
%they are all $K$-quasigeodesics by lemma \ref{uniform}.
%Since $\tau(z,u_i), \tau(v_i,y) \rightarrow +\infty$,
%it then follows that $d(z,u_i), d(v_i,y) \rightarrow +\infty$,
%so $d(h_i(z), h_i(u_i)) \rightarrow +\infty$.
Since $[z,u_i]$ is contained in $\widetilde P'$,
it is a $K$-quasigeodesic by lemma \ref{uniform}.
Since $\tau(z,u_i) \rightarrow +\infty$,
it then follows that $d(z,u_i) \rightarrow +\infty$,
so $d(h_i(z), h_i(u_i)) \rightarrow +\infty$.
Hence up to taking a further subsequence we may assume
that $h_i(z)$ converges to some point in $\si$. Since
$\tau(h_i(z),h_i(u_i))
\rightarrow +\infty$ and $h_i(u_i) \rightarrow u$, lemma \ref{conv}
implies that $h_i(z) \rightarrow \eta_-(u)$.
In addition since $h_i(z) \cdot [0,+\infty)$
contains $h_i(u_i) \rightarrow u \in \mi$ 
and $h_i(u_i) = h_i(z) \cdot r_i$ with $r_i \rightarrow +\infty$,
it follows that 
every point in $u \cdot \rrrr$ is obtained as a limit
of points $q_i \in h_i(z) \cdot [0,+\infty)$.
Since $\pi^{-1}(P')$ is a closed set in $\mi$, 
this shows that $\pi(u \cdot \rrrr)$ 
is contained in $P'$ and therefore $u \cdot \rrrr$ is a 
$K$-quasigeodesic.
In the same way $h_i(y) \rightarrow \eta_+(v)$ and 
$v \cdot \rrrr$ is obtained as a limit of $h_i(y) \cdot (-\infty,0]$,
hence $v \cdot \rrrr$ is also a $K$-quasigeodesic,
see fig. \ref{dist}.

\begin{figure}
\centeredepsfbox{qg2.eps}
\caption{Identification of ideal points in an orbit produces two
quasigeodesic orbits with same pair of ideal points.}
\label{dist}
\end{figure}

As $d(h_i(z), h_i(y))$ is a constant, it follows that
$\eta_+(v) = \eta_-(u)$.
Since both $v \cdot \rrrr$ and $u \cdot \rrrr$ are $K$-quasigeodesics,
this equality implies that $u$ and $v$ are not in the
same flow line of $\wwp$. 
But for each $i$, $h_i(v_i)$ and $h_i(u_i)$ are in the same
orbit of $\wwp$, $h_i(v_i) \rightarrow v$, and $h_i(u_i) \rightarrow
u$. This contradicts the Hausdorff orbit space condition.
We conclude that $\gamma_0$ and $\gamma_1$ do not have
the same ideal point.


%\break


\medskip
\noindent
{\underline {Case 2}} --- 
$x \cdot \reals$ intersects $\pi^{-1}(S)$ infinitely many times. 

Then there is a sequence $E_1 < \cdots < E_{J_0} \in {\mathcal E}$
such that $x \cdot \reals$ intersects each, and by
lemma \ref{LimitSetsDisjoint} 
we have $\ClBc(E_1) \intersect \ClFr(E_{J_0})
= \emptyset$. But $\eta_-(x) \in \ClBc(E_1)$ and $\eta_+(x) \in
\ClFr(E_{J_0})$ so $\eta_-(x) \ne \eta_+(x)$.
\end{proof}


Finally we prove property (c) of \ref{upgrade}.

\begin{proposition}{(continuity of extension)}{}
The map $\eta_+: \cover \Omega \rightarrow \si$
is continuous and similarly for $\eta_-$.
\label{cont}
\end{proposition}

\begin{proof}{}
We only consider $\eta_+$.


\smallskip
\noindent
{\underline {Case 1}} --- $x \cdot \RPlus$  intersects
$\pi^{-1}(S)$ infinitely often.

Let $E_1 < E_2 < \cdots$ be the elements of $\mathcal E$ that $x \cdot
\RPlus$ intersects. From lemma \ref{BoundedBelow},  $\Intersection_{i
\in {\bf N}} \ClFr(E_i)$ is a single point $p$ in $\bdy_\infinity
\cover P$. Since $\ClFr(E_i) \cap \bdy_\infinity\cover P$ is a
decreasing sequence of compact sets in $\bdy_\infinity\cover P$, then
for each $\delta > 0$, there is $i_0$ so that $\forall i > i_0$, the
diameter of $\ClFr(E_i) \cap \bdy_\infinity\cover P < \delta$. If $z$
is sufficiently near $x$ then $z \cdot \RPlus$ will intersect
$E_{i_0}$ therefore $\eta_+(z) \in \ClFr(E_i) \cap \bdy_\infinity\cover
P$. Hence the distance from $\eta_+(z)$ to $\eta_+(x)$ $< \delta$.
This implies that $\eta_+$ is continuous at $x$.

\smallskip
\noindent
{\underline {Case 2}} --- $x \cdot \RPlus$ has finite intersection
with $\pi^{-1}(S)$.

There is $s_1 \geq 0$ with $x \cdot [s_1,+\infty) \cap
\pi^\inverse(S) =
\emptyset$. Let $x' = x \cdot s_1$. Since 

$$x_i \rightarrow x  \ \ \Longleftrightarrow  \ \ 
x_i \cdot s_1 \rightarrow x',$$  

\noindent we may assume that $x \cdot \Rplus \cap \pi^{-1}(S)  =
\emptyset$. Consider a sequence $x_i \to x$, and let
$$ m_i = \bigl| x_i \cdot \Rplus  \intersect \pi^\inverse(S) \bigr|
$$
where $\bigl| V \bigr|$ denotes cardinality of $V$. Passing to a
subsequence, either $m_i$ takes a constant value $\le J_0$ for all
$i$, or $m_i > J_0$ for all $i$.

\smallskip
\noindent
{\underline {Case 2.1}} --- $m_i = 0$ for all $i$.

We may assume all $\gamma_i = x_i \cdot \RPlus$ and $\gamma_0 =
x \cdot \RPlus$ are in the $\delta_1$-neighborhood of $\Omega$. By
lemma \ref{uniform}, $\gamma_0$ and each $\gamma_i$ are
$K$-quasigeodesic. The proof in this case is finished by lemma
\ref{conv}.


Before addressing the case $m_i > 0$ we  need the following lemma.


\begin{lemma}{}{}
Suppose $m_i \geq j-1$ for all $i$. Let $v_i$ be the $j$-th
intersection of $x_i~\cdot \RPlus$ with $\pi^{-1}(S)$, if it exists,
otherwise let $v_i = \eta_+(x_i)$. Then $v_i$ converges to 
$\eta_+(x)$ in $\widetilde P \cup \partial _{\infty} \widetilde P$.
\label{inter}
\end{lemma}

\begin{proof}{} First suppose $j=1$, and let $v_i = x_i \cdot t_i$
with $t_i \in [0,+\infinity]$. By lemma \ref{uniform} the segments
or rays $[x_i,v_i)$ are all $K$-quasigeodesics. In addition
$t_i\rightarrow +\infty$, since $x \cdot \RPlus \cap \pi^{-1}(S) =
\emptyset$ and $x_i \rightarrow x$. By lemma \ref{conv} we conclude
that $z_i \rightarrow \eta_+(x)$.


\begin{figure}
\centeredepsfbox{qg3.eps}
\caption{Successive intersections with ${\mathcal E}$.}
\label{induction}
\end{figure}


Suppose now that $m_i \ge j-1$ for all $i$, and let $w_i$ be the
$(j-1)$-th intersection. We assume by induction that $w_i \to
\eta_+(x)$. Let $\zeta_i = [w_i,v_i]$, so each $\zeta_i$ is a
$K$-quasigeodesic, see fig. \ref{induction}.
If $\zeta_i$ is not escaping compact sets in $\cover
P$, then passing to a subsequence $\zeta_i$ accumulates on some $u \in
\cover P$, leading to a contradiction as follows: if $u\in
x\cdot\rrrr$, then since each $\zeta_i - \{ w_i \}$ is separated
from $x$ by at least one  element of ${\mathcal E}$, this would
imply that $x \cdot \rrrr_+$ has to intersect ${\mathcal E}$ contradiction.
%we obtain $\epsilon, T$-cycles for $\epsilon$
%arbitrarily small and $T$ arbitrarily big, a contradiction; otherwise
In the other case,
$x$ and $u$ are not on the same flow line, contradicting the
Hausdorff orbit space condition.

The paths $\zeta_i$ therefore escape to infinity in $\cover P$. These
paths are $K$-quasigeodesics,
so their diameters are shrinking to zero
in a topological metric on 
$\widetilde P \cup \partial _{\infty} \widetilde P$.
Since $w_i \rightarrow
\eta_+(x)$ it then follows that $v_i \rightarrow \eta_+(x)$.
Induction now completes the proof of the lemma.
\end{proof}


\smallskip
\noindent
{\underline {Case 2.2}} --- $m_i = m$ is a constant with $0 < m \le
J_0$.

Applying lemma \ref{inter} with $j=m-1$ it follows that $\eta_+(x_i)
\to \eta_+(x)$.


\vskip .15in
\noindent
{\underline {Case 2.3}} --- $m_i > J_0$ for every $i$.

Let $y_i$ be the corresponding $(J_0 + 1)$-th intersection. Then $y_i
\rightarrow \eta_+(x)$. Let $C_i \in {\mathcal E}$ be the lift of $S$
containing $y_i$.

If there is an infinite subsequence $i_l$ with $C_{i_l} = C$ for all
$l$, then because $z_{i_l} \rightarrow \eta_+(x)$, it follows that
$\eta_+(x) \in \bdy_{\infty} C$. But for any $i$, the flow line through
$x_i$ intersects a sequence $E_1 < \cdots < E_{J_0} \in \mathcal E$ before
intersecting $C_i$. Since $x \cdot \Rplus \subset \sigma^b E_1$ then
$\eta_+(x) \in \sigma^b E_1$, but also $\eta_+(x) \in \bdy_\infinity
C \subset \sigma^f E_{J_0}$, and therefore $\sigma^b E_1
\intersect \sigma^f E_{J_0} \ne \emptyset$ contradicting lemma
\ref{LimitSetsDisjoint}. We may therefore assume up to taking a
further subsequence that all $C_i$ are distinct from each other.

Note that there is $E_0 \in \mathcal E$ such that $E_0 < C_i$ for all
$i$; take any $E_0$ so that $x \in \Fr(E_0)$. Applying lemma
\ref{BoundedBelow}, there is a subsequence $i_l$ so that 
that $\sigma^f C_{i_l}$ approaches a single point of
$\bdy_\infinity P$ in the Hausdorff topology. This point must be
$\eta_+(x)$, since $z_{i_l} \in \sigma^f(C_{i_l})$ and $z_{i_l} \to \eta_+(x)$.
Therefore the sequence $\eta_+(x_{i_l}) \in \sigma^f(C_{i_l})$ approaches
$\eta_+(x)$.


\smallskip
The  arguments of cases 2.1, 2.2 and 2.3,
show that given any sequence $y_i \rightarrow x$,
there is always a  subsequence $y_{i_l}$ for which $\eta_+(y_{i_l})
\rightarrow \eta_+(x)$. This implies that the original sequence
$\eta_+(y_i) \rightarrow \eta_+(x)$. This finally finishes the proof of
proposition \ref{cont}.
\end{proof}

Propositions \ref{ext}, \ref{distinct} and \ref{cont}
show that orbits in $\Omega_{n+1}$
satisfy conditions (a,b,c) of theorem \ref{upgrade}
as orbits in $\cover P \cup \partial _{\infty} \widetilde P$.
Since $\cover P \cup \partial _{\infty} \widetilde P$ embeds
continuously in $\hhh \cup \si$, it follows that they also satisfy
these conditions as seen in $\hhh \cup \si$.
Theorem \ref{upgrade} then implies that orbits of $\wwp$ in $\widetilde P$
are uniformly quasigeodesic in $\hhh$ (hence also in $\widetilde P$).
Induction now finishes the proof
of theorem $A$.


\medskip
\noindent
{\bf Remark:} \ 
In this article we use the inductive step of this section to
study
flows transverse to Reebless finite depth foliations.
By Novikov's theorem \cite{No}, $\Phi$ satisfies the 
$\epsilon, T$-cycles condition,
hence lemma \ref{weakqg}  can be applied, 
simplifying the proof of case 1 in proposition \ref{distinct}.
However, the inductive 
proof we give in this section has more general hypothesis, namely:
(1) $\wwp$ has Hausdorff orbit space, (2) $\Phi$ is well adapted to
a partial sutured manifold hierarchy, that is,
the smallest sutured manifold in the hierarchy, call it $P$ 
may not be a product sutured manifold; and (3) orbits entirely
contained in $P$ are uniform quasigeodesics.
There may be flows satisfying these properties,
which are not transverse to finite depth foliations.


%One possible way to produce new quasigeodesic flows in this manner
%is to start with a quasigeodesic flow adapted to a
%(partial) sutured manifold and do surgery
%along an embedded annulus transverse to the flow, so that
%the final flow is still adapted to the (partial) sutured manifold
%hierarchy and the orbits in the smallest sutured manifold
%$P$ are still quasigeodesic. 
%This surgery, unlike Dehn surgery, does not change
%the homeomorphism type of the manifold, but changes the dynamics
%of the flow - in particular it creates a new closed orbit.
%If one is careful
%it is quite possible the new flow still has Hausdorff orbit
%space in the universal cover and therefore it is quasigeodesic.




