
\section{Quasigeodesic pseudo-Anosov flows}

Pseudo-Anosov flows are a generalization of suspension flows of
pseudo-Anosov surface homeomorphisms. These flows behave much like
Anosov flows, but they have finitely many singular orbits with
a prescribed behavior. In order to define pseudo-Anosov flows, first
we recall singularities of pseudo-Anosov surface homeomorphisms. 

Given $n \ge 2$, the quadratic differential $z^{n-2} dz^2$ on the complex
plane $\complex$ (see \cite{St} for quadratic differentials)
has a horizontal singular foliation $f^u$ with
transverse measure $\mu^u$, and a vertical singular foliation $f^s$
with transverse measure $\mu^s$. These foliations have $n$-pronged
singularities at the origin, and are regular and transverse to each
other at every other point of $\complex$. Given $\lambda > 1$, there is
a map $\psi \from \complex \to \complex$ which takes $f^u$ and $f^s$ to
themselves, preserving the singular leaves, stretching the leaves of
$f^u$ and compressing the leaves of $f^s$ by the factor $\lambda$. Let
$R_\theta$ be the homeomorphism $z \to e^{2 \pi \theta} z$ of
$\complex$. If $0 \le k < n$ the map $R_{k/n} \composed
\psi$ has a unique fixed point at the origin, and this defines the
local model for a {\em pseudohyperbolic fixed point}, with $n$-prongs
and with rotation $k$. Let $d_\Euclidean$ be the singular
Euclidean metric on
$\complex$ associated to the quadratic differential $z^{n-2} dz^2$, given by 
$$d_\Euclidean^2 = \mu_u^2 + \mu_s^2
$$ 
Note that 
$$(R_{k/n} \composed \psi)^* d_\Euclidean^2 = \lambda^{-2} \mu_u^2 +
\lambda^2 \mu_s^2
$$

Now consider the mapping torus $N = \complex \cross \reals / (z,r+1)
\sim (R_{k/n} \composed \psi(z),r)$, with suspension flow $\Psi$
arising from the flow in the $\reals$ direction on $\complex \cross
\reals$. The suspension of the origin defines a periodic orbit
$\gamma$ in $N$, and we say that $(N,\gamma)$ is the local model for a
{\em pseudohyperbolic periodic orbit}, with $n$ prongs and with
rotation $k$. The suspension of the foliations $f^u,f^s$ define
2-dimensional foliations on $N$, singular along $\gamma$, called the
{\em local weak unstable and stable foliations}. Note that there is a
singular Riemannian metric
$ds$ on $\complex \cross \reals$ that is preserved by the gluing
homeomorphism $(z,r+1) \sim (R_{k/n} \composed \psi(z),r)$, given by
the formula
$$ ds^2 = \lambda^{-2t} \mu_u^2 + \lambda^{2t} \mu_s^2 + dt^s
$$ 

The metric $ds$ descends to a metric on $N$ denoted $ds_N$.

Let $\Phi$ be a flow on a closed, oriented 3-manifold $M$. We say
that $\Phi$ is a {\em pseudo-Anosov flow} if the following are
satisfied:

\begin{itemize}

\item For each $x \in M$, the flow line $t \to \Phi(x,t)$ is $C^1$,
and the tangent vector bundle $D_t \Phi$ is $C^0$.

\item There is a finite number of periodic orbits $\{ \gamma_i \}$,
called {\em singular orbits}, such that the flow is smooth off of the
singular orbits.

\item Each singular orbit $\gamma_i$ is locally modelled on a
pseudo-hyperbolic periodic orbit. More precisely, there exist $n,k$
with $n \ge 3$ and $0 \le k < n$, such that if $(N,\gamma)$ is the
local model for an pseudo-hyperbolic periodic orbit with $n$ prongs
and with rotation $k$, then there are neighborhoods $U$ of $\gamma$
in $N$ and $U_i$ of $\gamma_i$ in $M$, and a diffeomorphism $f
\from U \to U_i$, such that $f$ takes orbits of the semiflow $R_{k/n} \composed
\psi \restrictedto U$ to orbits of $\Phi \restrictedto U_i$.

\item There exists a path metric $d_M$ on $M$, such that $d_M$ is a
smooth Riemannian metric off of the singular orbits, and for a
neighborhood $U_i$ of a singular orbit $\gamma_i$ as above, the
derivative of the map $f \from (U - \gamma) \to (U_i - \gamma_i)$ has
bounded norm, where the norm is measured using the metrics $ds _N$ on
$U$ and $d_M$ on $U_i$.

\item On $M - \Union \gamma_i$, there is a continuous splitting of the
tangent bundle into three 1-dimensional line bundles $E^u \oplus E^s
\oplus T \Phi$, each invariant under $\Phi$, such that $T \Phi$ is
tangent to flow lines, and for some constants 
$\nu > 1, \theta >  1$ we have

\begin{enumerate}

\item If $v \in E^u$ then $|D\Phi_t(v)| \le 
\theta \nu^t |v|$ for $t<0$
\item If $v \in E^s$ then $|D\Phi_t(v)| \leq \theta \nu^{-t} |v|$ for
$t>0$

\end{enumerate}

{\noindent
where norms of tangent vectors are measured using the metric $d_M$.}

\end{itemize}

With the definition formulated in this manner, the entire theory of
Anosov flows can be mimicked for pseudo-Anosov flows. In particular,
a pseudo-Anosov flow $\Phi$ has a 2-dimensional weak unstable
foliation $\fu$ tangent to $E^u \oplus T \Phi$ away from the singular
orbits, and a 2-dimensional weak stable foliation $\fs$ tangent to
$E^s \oplus T \Phi$. These foliations are singular along the singular
orbits of $\Phi$, and regular everywhere else. In the
neighborhood $U_i$ of an $n$-pronged singular orbit $\gamma_i$, the
images of $\fs$ and $\fu$ in the model manifold $N$ are identical with
the local weak stable and unstable foliations. 

The restriction of
$\fs$ to $M - $ (singular orbits) defines a true foliation; a
complete leaf of this foliation is called a {\em nonsingular leaf} of
$\fs$; an incomplete leaf may be completed by adding a singular orbit
$\gamma$ of $\Phi$, and the result is called a {\em singular leaf} of
$\fs$ abutting $\gamma$. Singular and nonsingular leaves of $\fu$ are
similarly defined. The bare term ``leaf'' means either a nonsingular
or a singular leaf. For some small neighborhood $W$ of any point $x$
lying on a singular orbit $\gamma$, the singular leaves of $\fs$
divide $W$ into $n$ {\em sectors}, each sector parameterized by $D^2
\cross I$, so that the restriction of $\fs$ to the sector agrees with
the foliation by level discs $D^2 \cross t$; and similarly for $\fu$.
All the terms defined here apply as well to the lifted singular
foliations $\fns, \fnu$ in $\mi$.


\begin{proposition}{}{}
If $\Phi$ is a pseudo-Anosov flow in $M^3$ then the
orbit space $\oo$ of $\wwp$ is homeomorphic to $\rrrr^2$.
\end{proposition}

\begin{proof}{}
Let $\fs, \fu$ be the singular stable and unstable foliations of
$\Phi$. A short embedded path $\alpha$ is a {\em quasi-transversal} of
$\fs$ if one of the following happens: either $\alpha$ is transverse
to $\fs$; or there is a singular orbit $\gamma$ and $x \in
\interior(\alpha) \intersect \gamma$ such that the closure of each
half of $\alpha - x$ either is transverse to $\fs$ or lies on a
singular leaf of $\fs$, and the two halves do not both lie in the
closure of any sector of $\fs$ at $x$. Quasitransversals of $\fu$ are
similarly defined. We need the following facts about $\Phi$, $\fs$ and
$\fu$.

\begin{enumerate}

\item Each periodic orbit of $\Phi$ is homotopically nontrivial.

\item A quasitransversal $\rho$ of $\fu$ is not path homotopic into a
leaf of $\fu$; and similary for $\fs$.

\item The inclusion map of every leaf of $\fns$ or $\fnu$ into $\mi$
is proper.

\end{enumerate}

These facts are proved using the theory of essential laminations
\cite{Ga-Oe}; we give the argument for $\fs$. Split $\fs$ along
singular leaves to produce a lamination $\ls$. There is one
complementary component $T_\gamma$ of $\ls$ for each singular orbit
$\gamma$ of $\Phi$, and if $\widehat T_\gamma$ is the metric completion of
$T_\gamma$ then $\widehat T_\gamma$ is homeomorphic to a solid torus with
an $(n,k)$ torus knot removed from the boundary, where $\gamma$ has $n$
prongs and rotation $k$. Since $n \ge 3$ for all $\gamma$ it follows
that $\ls$ is an essential lamination. There is a map $f \from M \to
M$ homotopic to the identity taking $\ls$ onto $\fol^s$,
taking each leaf of $\ls$ not
on the boundary of some $T_\gamma$ homeomorphically onto a
non-singular leaf of $\fs$, and taking each leaf on the boundary of
each $T_\gamma$ onto a union of singular leaves abutting
$\gamma$. Given a leaf $L \subset \bdy T_\gamma$ if gcd$(n,k) \ne
1$ then $L$ is mapped onto a union of two singular leaves and the map
is 1-1 off of $\gamma$; and if gcd$(n,k) = 1$ then $L$ maps onto one
singular leaf and the map is 2-1 off of $\gamma$; in either case,
$f^\inverse(\gamma) \intersect L$ is a core curve of the annulus $L$.

Properties (1--3) follow from results of \cite{Ga-Oe}. Property (1)
follows because each periodic orbit of $\Phi$ is homotopically
nontrivial in a leaf of $\fs$, and because leaves of $\ls$ are
$\pi_1$-injective. Property (2) follows because if $\rho'$ is a path
transverse to $\ls$, and if the closure of each component of $\rho' -
\ls$ is not path homotopic into a leaf of
$\ls$, then $\rho'$ is not path homotopic into a leaf of $\ls$.
Property (3) follows because leaves of $\lns$ and $\lnu$ include
properly in $\mi$.

In order to prove the proposition,
the key facts to prove are that $\oo$ is a two dimensional manifold and
that it is Hausdorff.

Suppose there are $\epsilon, T$-cycles of $\Phi$ for $\epsilon$
arbitrarily small and $T$ arbitrarily big. We can then choose $x_i
\rightarrow x \in \mi$ and $t_i \to +\infinity$ so that $y_i = x_i
\cdot t_i \to x$. Let $\alpha_i = x_i \cdot [0,t_i]$. 
If $x_i, y_i$ are in the same local orbit of $\wwp$ near $x$,
then $\alpha_i$ can be easily completed to a closed orbit,
contradicting  property (1) above. 
%If $y_i, x_i$ are in the same local sheet of $\fns$,
%then they can be connected by an arbitrarily small
%segment in $\ws(x_i)$. But this contradicts
%the fact that flow lines of $\wwp$  
%form a product foliation
%in $\ws(x_i)$. 
If the endpoints of $\alpha_i$ are not in the same local
orbit and $\epsilon$ is sufficiently small, 
then the endpoints of $\alpha_i$ may be joined by a path $\beta$
which is a quasi-transversal to either $\fns$ or $\fnu$; but
$\beta$ is path homotopic to $\alpha_i$, contradicting property (2)
above.


%First consider the case that $\pi(x)$ is not  in a singular
%orbit of $\Phi$. Close up $\alpha_i$ with a local transversal to
%$\fns$, to produce $\beta_i$. After splitting $\fns$ to $\lns$ the
%resulting loop can easily be perturbed to produce a  closed
%transversal to $\lns$ (we may need two $\gamma_i$ in order to insure
%local orientations near $x$ match). This produces a null homotopic
%transversal to $\ls$, contradiction.

%If on the other hand $\pi(x)$ is on a singular
%orbit of $\Phi$ , let $\alpha$ be the lift of
%this orbit to $\mi$ with $x \in \alpha$.
%The splitting operation breaks $\ws(\alpha)$ into finitely
%many components, each of which 
%separates $\mi$ (Gabai-Oertel). It follows that $x_i$ and $y_i$ have
%all to be in the same sector of $\mi$ defined by $\ws(\alpha)$,
%where a
%sector is the closure of a component of $\mi - \ws(\alpha)$.
%Therefore if $x_i$ and $y_i$ are near enough
%each other we can still close the arc with a
%transversal to $\fns$ and apply the arguments above.

This implies that $\oo$ is locally $2$-dimensional. We next prove that
it is Hausdorff. The proof given for Anosov flows in \cite{Fe1} goes
through almost verbatim, with slight changes to take pseudo-Anosov
behavior into account, or we may proceed as follows.

Let $x_i \rightarrow x$, $y_i = x_i \cdot t_i \rightarrow y$ in
$\mi$.  We first want to show that $t_i$ is bounded. Assume then
up to taking a subsequence that $t_i \rightarrow +\infty$. If $x$ lies
on a singular orbit $\gamma$ of $\Phi$, pass to a subsequence so that
$x_i$ lies in a single sector of $\fns$ near $x$. Now vary $x_i$
to nearby $z_i$, so that $z_i \rightarrow x$, $z_i \in \ws(x_i) \cap
\wu(x)$. Since flow lines in the stable foliation converge together
exponentially it follows that there are $s_i \rightarrow +\infty$ with
$w_i = z_i \cdot s_i \rightarrow y$. The sequence $z_i$ forms a bounded
subset of a leaf $L$ of $\wu(x)$, and hence the sequence $z_i \cdot
s_i$ leaves every compact subset of $L$,
because flow lines are properly embedded in
the leaf $L$ --- the flow induces a product
structure  in $L$. But by (3) above the inclusion
map $L \inject \mi$ is proper, contradicting that $z_i \cdot s_i$
converges in $\mi$. This contradiction shows that the $t_i$ are
bounded.

As the $t_i$ are bounded, we may assume up to subsequence
that $t_i \rightarrow t_0$. Then $y_i \rightarrow x \cdot t_0 = y$.
This shows that $\oo$ is Hausdorff and it follows that
it is homeomorphic to $\rrrr^2$.
\end{proof}

Given any closed, irreducible,  orientable $3$-manifold $M$
with  $H_2(M) \not = 0$, Gabai \cite{Ga1} constructed many
finite depth foliations in $M$: for any $\zeta \not = 0$ in
$H_2(M)$ there is a taut, finite depth foliation $\fol$
in $M$ so that the compact leaves of $\fol$ represent $\zeta$.

If in addition $M$ is atoroidal, then Mosher \cite{Mo4} constructed
pseudo-Anosov flows in $M$ which are {\em almost tranverse} to $\fol$.
These flows become transverse to $\fol$ after blow up of finitely
many singular orbits $\gamma_i, 1 \leq i \leq i_0$ of $\Phi$.
The blown up flow is denoted by $\Phi^b$. The blow up transforms the
orbit $\gamma_i$ into  $Z_i = (T_i \times [0,1])/f$, where $T_i$ is a
finite simplicial tree, and $f$ is a homeomorphism of $T_i \times \{ 1
\}$ to $T_i \times \{ 0 \}$ which sends edges to edges.
Each edge $E$ of $T_i$ eventually returns to itself, producing an
annulus $A$ in $Z_i$. The flow in $A$ is as follows: the boundary
consists of two orbits coherently oriented, and the interior orbits
spiral from one boundary circle to the other without forming a two
dimensional Reeb component. Clearly there is a global section to
$\Phi^b$ restricted to $Z_i$. The flow $\Phi^b$ is transverse to
$\fol$.

In addition $\Phi^b$ is semiconjugate to $\Phi$: there is a continuous
map $q \from M \to M$, so that 

\begin{itemize}
\item $q$ is homotopic to the identity,
\item 
$q$ takes orbits of $\Phi^b$ to orbits of $\Phi$ preserving
orientation,
\item
$q$ is $C^1$ along orbits of $\Phi^b$,
%continuous non
%zero derivative,
%locally one to one along all orbits,
%and so that 
\item
$q$ is one to one
except in $\cup_{1 \leq i \leq i_0} Z_i$,
\item $q(Z_i) = \gamma_i$.
\end{itemize}


\begin{proposition}{}{}
The orbit space $\oo^b$ of $\wwp^b$ is homeomorphic to $\rrrr^2$.
\label{orbit}
\end{proposition}

\begin{proof}{}
Since $\Phi^b$ is transverse to $\fol$, then if $\epsilon>0$ is very
small and $T>0$ very large, it follows that an $\epsilon, T$ cycle
can be perturbed to be transverse to $\fol$, hence it is not null
homotopic by Novikov's theorem \cite{No}. 
This shows that $\oo^b$ is locally two
dimensional. We now show that $\oo^b$ is Hausdorff.

Lift $q$ to a map $\widetilde q \from \mi \rightarrow \mi$,
which is boundedly homotopic to the identity map. Let $\tau(x,y)$ be
the difference in flow parameter between $x,y$ on the same flow line
of $\wwp$, and let $\tau^b(x,y)$ be similarly defined for $\wwp^b$.

\smallskip
\noindent
{\underline {Claim}} ---
There are $m_0, m_1 > 0$ so that if $x, y$ are in the same flow line
of $\wwp^b$, then  $m_0 \tau^b(x,y) < \tau(\tilde q(x), \tilde q(y)) <
m_1 \tau^b(x,y)$.

Given $z \in M$, let $g(z)$ be the derivative at $z$ of $q$ restricted
to the flow line through $z$. Then since $g(z)$ is continuous and
positive and $M$ compact, there are $m_0$ and $m_1 > 0$ which are the
maxima and minima of $g$ in $M$. The claim follows.

%Else there are $x_i \in \mi, y_i = \wwp^b_{t_i}(x_i)$,
%with $t_i > i$ and
%$d_{\wwp}(\tilde q(x_i), \tilde q(y_i))
%\rightarrow 0$. 
%Up to covering translations and taking a subsequence 
%we may assume that both $\tilde q(x_i),
%\tilde q(y_i) \rightarrow z$.
%
%If $\pi(z)$ is not in  a singular orbit of $\Phi$,
%then there is a neighborhood $V_0$
%of $z$ so that $\tilde q$ is one to one
%in $\tilde q ^{-1}(V_0)$. Then the above is impossible.
%
%Suppose that $\pi(z) \in \gamma_1$.
%Let $V$ be a tubular neighborhood of $\gamma_1$ in $M$
%and let $W = \tilde q ^{-1}(V)$.
%Let $\tilde V$  be the lift of $V$ to $\mi$ containing
%$z$ and let $\widetilde W = 
%\tilde q ^{-1}(\widetilde V)$.
%
%Let $A$ be a meridian disk in $V$ which is transverse to $\Phi$  and
%let $B= q ^{-1}(A)$, transverse to $\Phi^b$.
%Notice that $B$ provides a topological cross section for 
%the restriction of $\Phi^b$ to $T_1 \times {\bf S}^1$.
%Hence if $\pi(x_i)$ is sufficiently near $T_i \times {\bf S}^1$,
%then there is a bound on how long it takes for $\Phi^b_{\rrrr}(\pi(x))$
%to get to $B$.
%Hence $\Phi^b{[0,t_i]}(\pi(x_i))$ will intersect $B$ at least
%twice. This implies that  the flow segment of $\Phi$ from
%$\pi(\tilde q(x_i))$ to $\pi(\tilde q(y_i))$
%will go around $\gamma_1$ at least once.
%This implies that the lengths of these segments 
%are not going to zero, contradiction.
%This finishes the proof of the claim.
%
\smallskip

Let now $x_i \in \mi, y_i = \wwp^b_{t_i}(x_i)$ so that $y_i
\rightarrow y$ and $x_i \rightarrow x$. Then $\tilde q(x_i) \rightarrow
\tilde q(x)$ and $\tilde q(y_i) \rightarrow \tilde q(y)$.  Since $\Phi$
is pseudo-Anosov, the previous proposition shows that 
the orbit space $\oo$ of $\wwp$ is
Hausdorff, hence $\tilde q(y) = \wwp_s(\tilde q(x))$. Since $\oo$ is
homeomorphic to $\rrrr^2$, it follows that $\tilde q(y_i) = 
\wwp_{s_i}(\tilde q(x_i))$ where $s_i \rightarrow s$.

Partitioning $s_i$ into a uniformly bounded number of subsegments of
length $\leq 1$, and using the claim above, it follows that $y_i =
\wwp^b_{r_i}(x_i)$ where $r_i$ is uniformly bounded. 
Up to taking a 
subsequence, assume that $r_i$ converges to some $r$, and hence
$y_i\rightarrow\wwp^b_r(x)$. Consequently
$y = \wwp^b_r(x)$. This shows that
$\oo^b$ is Hausdorff. It follows that $\oo^b$ is homeomorphic to
$\rrrr^2$.
\end{proof}

Now we prove the main theorem:

\medskip
\noindent{\bf Main theorem} \ 
{\em Let $M^3$ be closed, orientable, irreducible with non zero
second betti number. Let $\zeta \not = 0 \in H_2(M)$ and 
$\fol$ be a taut finite depth foliation with compact
leaves representing $\zeta$. Let $\Phi$ be a pseudo-Anosov flow
which is almost transverse to $\fol$. Then $\Phi$ is quasigeodesic.}
\smallskip

\begin{proof}{} 
By the previous proposition the flow $\widetilde\Phi^b$ on $\widetilde M$
has Hausdorff orbit space. 

Suppose first that $\fol$ has a compact leaf which is not a fiber.
Then $\Phi^b$ is quasigeodesic by theorem A. The semiconjugacy $\tilde
q$ from $\widetilde \Phi^b$ to $\widetilde \Phi$ moves points a
uniformly bounded distance in $\mi$. Since there is also a bound on
how much the map $\tilde q$ expands the lengths of flow lines (given
by the claim in the previous proposition), it follows that flow lines
of $\wwp$ are uniform quasigeodesics.

Now suppose that each compact leaf of $\fol$ is a fiber. If $\fol$ is
a fibration over the circle, then $\Phi^b$ is quasigeodesic by
\cite{Ze}, and the arguments of the previous paragraph show that
$\Phi$ is quasigeodesic. 

Suppose $\fol$ is not a fibration over the circle. We claim that
$\fol$ can be replaced by another finite depth foliation of smaller
depth that is still transverse to $\Phi^b$; continuing by induction,
eventually $\Phi^b$ is transverse to a fibration over the circle, and
we are done.

We sketch a proof of the claim. Let $\fol$ have depth $n$. Note that
each component of $\fol - \fol^0$ is homeomorphic to the product of a
depth 0 leaf crossed with an interval; similarly each component of
$\fol - \fol^k$ is homeomorphic to the product of a depth $k$ leaf
crossed with an interval. Note also that the restriction of $\fol$ to a
component of $\fol - \fol^{n-1}$ is a fibration over the circle with
fiber a depth $n$ leaf $L$. It follows that if $f
\from L \to L$ is the first return map of $\Phi^b$, then there is a
{\em translation map} $g \from L \to L$, i.e.\ a map which generates a
properly discontinuous, free action of
$\integers$, such that $f$ is isotopic to $g$ by a compactly supported
isotopy. Any compact subset of $L$
invariant under $f$ is contained in the support of the isotopy from
$f$ to $g$, and hence there is a maximal compact invariant set $C$ of
$f$. Using the fact that $\Phi^b$ is a blown up pseudo-Anosov flow,
the set $C$ is nonempty if and only if $f$ has periodic points.
Moreover, any periodic points are non-removable in the proper
isotopy class of $f$; the proof given for pseudo-Anosov surface
homeomorphisms in \cite{Bi-Ki} works just as well in the present
context. Since $g$ has no periodic points, it follows that $C =
\emptyset$, and therefore $f$ is itself a translation map. The
component of $\fol - \fol^{n-1}$ containing $L$ may therefore be
refoliated by leaves of depth $n-1$, staying transverse to $\Phi^b$.
Doing this for each component of $\fol - \fol^{n-1}$ proves the claim.
\end{proof}



