# Topology of horocycles on geometrically finite nonpositively curved surfaces

Sergi Burniol Clotet

###### Abstract

We study the closure of horocycles on rank $1$ nonpositively curved surfaces
with finitely generated fundamental group. Each horocycle is closed or dense
on a certain subset of the unit tangent bundle. In fact, we classify each
half-horocycle in terms of the associated geodesic rays. We also determine the
nonwandering set of the horocyclic flow and characterize the surfaces
admitting a minimal set for this flow.

##  1 Introduction

Let $M$ be a complete surface with nonpositive curvature. We say that $M$ is
geometrically finite if the fundamental group is finitely generated. We assume
that $M$ is nonelementary, which means that the fundamental group is not
cyclic, or equivalently, that the surface topologically is not a cylinder. It
is also assumed that the curvature is not identically zero, so $M$ is a rank
$1$ surface. The goal of this article is to study the topological properties
of the horocyclic flow $h_{s}$ on the unit tangent bundle $T^{1}M$. At the
beginning of the next section, we recall the definition of this flow.

In [Ebe79], Eberlein studied the geometry of ends of nonpostively curved
surfaces. When $M$ is geometrically finite there are finitely many ends and
each one is collared, which means that they have a neighborhood homeomorphic
to a cylinder. He suggested two classifications, one metric and the other
dynamical, of the collared ends into four types: simple parabolic, exceptional
parabolic, cylindrical and expanding. Link, Picaud and Peigné proved that the
two classifications are equivalent [LPP06]. In strictly negative curvature,
the simple parabolic end is commonly referred to as a cusp and the expanding
end as a funnel. The cylindrical end has a neighborhood isometric to a flat
cylinder $S^{1}\times\operatorname{\mathbb{R}}_{+}$. The exceptional parabolic
end has a behavior between a cylinder and an exceptional parabolic end.

Let $X$ be the universal cover of $M$ with the lifted metric, and let $\Gamma$
be the group of covering transformations, which is identified to the
fundamental group of $M$. We recall that $X$ can be compactified by adding a
visual boundary $\partial X$ (see Section 2 for the definiton and basic
properties). The limit set $\Lambda$ of $\Gamma$ is the set of accumulation
points in $\partial X$ of an orbit $\Gamma p$, $p\in X$. It is also the
smallest closed $\Gamma$-invariant subset of $\partial X$ [Bal82, Theorem
2.8]. We define the set $\Omega^{+}$ of vectors in $T^{1}M$ which have a lift
in $T^{1}X$ that points in the positive direction to the limit set $\Lambda$.

In [LPP06], it is shown that the horocyclic flow in restriction to
$\Omega^{+}$ is transitive i.e. it has a dense orbit. In fact, they show that
the horocyclic orbit of periodic vectors for the geodesic flow are always
dense in $\Omega^{+}$. Our main result established what is the topology of any
other horocyclic orbit. A geodesic ray $c:\operatorname{\mathbb{R}}_{+}\to M$
is minimizing if the distance between any of its points is realized by the
same geodesic ray $c$. We say that $c$ is eventually minimizing if for some
$T>0$, $c$ is minimizing on $[T,+\infty)$.

######  Theorem 1.1.

Let $M$ be a nonelementary geometrically finite surface with nonpositive
curvature. For $v\in T^{1}M$,

  1. 1.

if the geodesic ray $g_{\operatorname{\mathbb{R}}_{+}}v$ converges to a
parabolic or cylindrical end and is eventually minimizing, then
$h_{\operatorname{\mathbb{R}}}v$ is periodic, hence compact,

  2. 2.

if the geodesic ray $g_{\operatorname{\mathbb{R}}_{+}}v$ converges to an
exceptional parabolic or cylindrical end and is not eventually minimizing,
then
$\overline{h_{\operatorname{\mathbb{R}}}v}=h_{\operatorname{\mathbb{R}}}v\cup\Omega^{+}$,

  3. 3.

if the geodesic ray $g_{\operatorname{\mathbb{R}}_{+}}v$ converges to an
expanding end, then $h_{\operatorname{\mathbb{R}}}v$ is closed and also
converges to that end,

  4. 4.

if the geodesic ray $g_{\operatorname{\mathbb{R}}_{+}}v$ does not converge to
an end, then $\overline{h_{\operatorname{\mathbb{R}}}v}=\Omega^{+}$.

The vectors satisfying any of the conditions of the theorem above also admit a
nice description in terms of their end points at infinity and their relation
with the limit set $\Lambda$ and the group $\Gamma$. The phenomenon presented
by horocyclic orbits of item $2$ does not occur in strictly negative
curvature. We will see that these orbits are nonwandering, but still they
accumulate to the set $\Omega^{+}$.

To prove this classification, it is convenient to understand the behavior of
the positive and the negative half-horocycles
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ and
$h_{\operatorname{\mathbb{R}}_{-}}(v)$, since it is not always the same. We
know the following:

  * •

in item $2$, one half-horocycle accumulates to $\Omega^{+}$ and the other
converges to the corresponding end,

  * •

in item $4$, if the vector $v$ generates a closed geodesic which bounds an
expanding end, or is asymptotic to such a geodesic, then one half-horocycle is
dense in $\Omega^{+}$ and the other converges to the expanding end,

  * •

if $v$ is any other vector in item $4$, each half-horocycle is dense in
$\Omega^{+}$.

In strictly negative curvature, the study of half-horocycles was carried out
by Schapira [Sch11], with the same conclusion for vectors in item $4$. Our
methods are partly based on her work.

A closed $h_{s}$-invariant nonempty subset $\mathcal{M}$ of $T^{1}M$ is called
minimal if there is no proper subset of $\mathcal{M}$ with that property.
Periodic or closed horocycles are trivial minimal sets. In Section 2, we
generalize the notion of horocyclic limit point. One relevant consequence of
our classification is the following.

######  Corollary 1.2.

The horocyclic flow of $M$ has a nontrivial minimal set if and only if all the
points in the limit set $\Lambda$ are horocyclic. If it exists, the nontrivial
minimal set is $\Omega^{+}$.

A vector $v\in T^{1}M$ is nonwandering for the horocyclic flow $h_{s}$ if for
every open subset $U$ of $T^{1}M$ containing $v$ the set

$\\{s\in\operatorname{\mathbb{R}}:h_{s}(U)\cap U\not=\emptyset\\}$

is unbounded. The vectors satisfying the property of item $1$ in Theorem 1.1
are nonwandering, because they are periodic, but they are not in the set
$\Omega^{+}$ in general. We show:

######  Theorem 1.3.

The nonwandering set of the horocyclic flow is the union of $h_{s}$-periodic
vectors and the set $\Omega^{+}$.

##  2 Preliminaries

###  2.1 Boundary at infinity and horocycles

Let $X$ be a simply connected nonpositively curved rank $1$ surface. We say
that two geodesic rays $c_{1},c_{2}:\operatorname{\mathbb{R}}_{+}\to X$ are
asymptotic if there exists a constant $C>0$ such that

$d(c_{1}(t),c_{2}(t))\leq C$

for all $t\geq 0$. The boundary at infinity $\partial X$ is defined as the set
of equivalence classes of asymptotic rays. For a full geodesic
$c:\operatorname{\mathbb{R}}\to X$, let $c(+\infty)$ (resp. $c(-\infty)$)
denote the class of the positive (resp. negative) ray of $c$. For $v\in
T^{1}X$, we denote by $v_{+}$ (resp. $v_{-}$) the asymptotic class of the
positive (resp. negative) geodesic ray generated by $v$. We say that $v_{+}$
is the end point of $v$, and that $v$ points to $v_{+}$. For every $x\in X$
and every $\xi\in\partial X$, there is a unique vector $V(x,\xi)\in
T^{1}_{x}X$ pointing to $\xi$.

We equip $X\cup\partial X$ with the cone topology [EO73, Bal95], which makes
this space compact. The boundary at infinity is homeomorphic to a circle,
which we orient in the counter-clockwise direction. Given two points $\eta$
and $\xi$ in $\partial X$, we denote the set of points on the image of the
curve that goes from $\eta·$ to $\xi$ in the counterclockwise direction by
$[\eta,\xi]$, which is referred to as a closed interval. Similarly, we define
(half-)open intervals.

We recall that the Busemann cocycle $\beta_{\xi}:X\times
X\to\operatorname{\mathbb{R}}$ at $\xi\in\partial X$ is, for $p,q\in X$,

$\beta_{\xi}(q,p)=\lim_{z\to\xi}d(z,q)-d(z,p).$

For fixed $p$, $\beta_{\xi}(\,\cdot\,,p)$ is said to be a Busemann function.
The horocycle with center $\xi\in\partial X$ passing through $p\in X$ is the
set

$H_{\xi}(p)=\\{q\in X,\beta_{\xi}(q,p)=0\\}.$

The closed horoball bounded by $H_{\xi}(p)$ is the set

$B_{\xi}(p)=\\{q\in X,\beta_{\xi}(q,p)\leq 0\\}.$

######  Proposition 2.1.

[HIH77] For every $q\in X$ and $\xi\in\partial X$, the function
$\beta_{\xi}(\cdot,p)$ is convex and of class $C^{2}$. Moreover, the function
$(\xi,q,p)\mapsto\beta_{\xi}(q,p)$ is continuous.

Therefore, horocycles are simple $C^{2}$ curves of $X$. Let $\pi$ be the
canonical projection from $T^{1}X$ to $X$. For $v\in T^{1}X$, the stable
horocycle $H^{s}(v)$ of $v$ is the set of vectors with basepoint on
$H_{v_{+}}(\pi(v))$ normal to this curve and directed towards the horoball
$B_{v_{+}}(\pi(v))$. All the vectors in $H^{s}(v)$ point to $v_{+}$. The
horocyclic flow $h_{s}:T^{1}X\to T^{1}X$ is obtained as the parametrization of
the horocycles on $T^{1}X$ by the arc-length on $X$, where the horocycles are
also oriented in the counter-clockwise direction to match the orientation on
the boundary at infinity (Figure 1). By Proposition 2.1, the resulting flow is
continuous.

Figure 1: Orientation of the boundary at infinity and the horocycles.

The following is known regarding the relation between horocycles and the
boundary at infinity:

######  Proposition 2.2.

[LPP06, Lemme I.7] For $p,q\in X$, we set

$H_{\xi}(\infty)=\\{\eta\in\partial
X,\,t\mapsto\beta_{\xi}(\sigma_{p,\eta}(t),q)\text{ is bounded above}\\}.$

  1. 1.

$H_{\xi}(\infty)$ does not depend on $p$ and $q$.

  2. 2.

$\overline{B_{\xi}(p)}^{\bar{X}}\cap\partial X=H_{\xi}(\infty).$

  3. 3.

$H_{\xi}(\infty)$ is a closed interval $I=[\xi_{1},\xi_{2}]$ in $\partial X$
containing $\xi$.

  4. 4.

If $\xi_{1}\not=\xi_{2}$, the only pair of points in $H_{\xi}(\infty)$ that
may be joined by a geodesic is $\xi_{1},\xi_{2}$. If this two points are
joined by a geodesic, then this geodesic bounds a flat half-plane.

  5. 5.

If $\eta\in\mathring{I}$, then
$\lim_{t\to+\infty}\beta_{\xi}(\sigma_{p,\eta}(t),q)=-\infty$.

  6. 6.

For $p\in X$, the horocycle $H_{\xi}(p)$ accumulates exactly at the points
$\xi_{1}$ and $\xi_{2}$.

###  2.2 Joining points at infinity

We have mentioned that it is always possible to join a point in $X$ to a point
in $X\cup\partial X$ by a unique geodesic. On the one hand, there may be more
than one geodesic joining two different points in the boundary.

######  Theorem 2.3.

(Flat Strip Theorem) If there exist two geodesics $c_{1},c_{2}$ with the same
end points in $\partial X$, then $c_{1}$ and $c_{2}$ bound a flat strip
isometric to $[0,a]\times\operatorname{\mathbb{R}}$, where $a$ is the distance
between $c_{1}$ and $c_{2}$.

On the other hand, it is not always possible to join two distinct points at
infinity by a geodesic. Examples of this are flat half-planes or flat sectors,
or more generally the situation of item 4 in Proposition 2.2. The following
lemma gives a sufficient condition to join points at infinity.

######  Lemma 2.4.

[Bal95, Lemma III.3.1] Let $c$ be a geodesic that does bound a flat strip of
width $R>0$. Then there exist neighborhoods $U$ and $V$ of $c(+\infty)$ and
$c(-\infty)$ in $\partial X$ such that any pair of points in $U\times V$ can
be joined by a geodesic $c^{\prime}$. For any such geodesic $c^{\prime}$,
$d(c^{\prime},c(0))<R$.

Finally, we recall some equivalent conditions to joining points at infinity.

######  Proposition 2.5.

Let $\xi,\eta\in\partial X$ be distinct points at infinity. The following are
equivalent:

  1. 1.

there is a geodesic joining $\xi$ to $\eta$,

  2. 2.

there exist sequences $x_{n}$ ans $y_{n}$ in $X$ converging respectively to
$\xi$ and $\eta$, a point $p\in X$ and a constant $C$ such that the geodesics
$c_{n}$ joining $x_{n}$ to $y_{n}$ satisfy

$d(c_{n},p)\leq C,$

  3. 3.

there exist horocycles $H_{\xi}$ and $H_{\eta}$ centered at $\xi$ and $\eta$
such that $\\#H_{\xi}\cap H_{\eta}\geq 2$,

  4. 4.

if $h_{\xi}$ is a Busemann function centered at $\xi$ and $h_{\eta}$ is a
Busemann function centered at $\eta$, then $h_{\xi}+h_{\eta}$ assumes its
minimum.

Moreover, if there is no geodesic joining $\xi$ to $\eta$,
$H_{\xi}(\infty)\cap H_{\eta}(\infty)\not=\emptyset.$ If there is a geodesic
joining $\xi$ to $\eta$ and $H_{\xi}(\infty)\cap
H_{\eta}(\infty)\not=\emptyset$, then the geodesic bounds a flat half-plane.

###### Proof.

The equivalence between 1 and 2 is standard. The equivalence with 3 and the
final statement are shown in [LPP06, Corollaire I.8]. The equivalence with 4
can be found in the Proof of Proposition 9.35 in [BH99] ∎

###  2.3 Tits metric

It is not essential in this article, but it can be useful to briefly recall
the definition of the Tits distance on the visual boundary $\partial X$. Given
$\xi,\eta\in\partial X$, we first define the angular distance between $\xi$
and $\eta$ as

$\angle(\xi,\eta)=\sup_{p\in X}\angle_{p}(\xi,\eta).$

The Tits distance is defined as the length metric of $(\partial X,\angle)$ and
has values in $[0,+\infty]$.

######  Proposition 2.6.

[BGS85, 4.10] The set $H_{\xi}(\infty)$ is equal to the closed ball of radius
$\pi/2$ and center $\xi$ in the Tits distance.

###  2.4 Isometries

We now study the isometries of $X$. Given an isometry $\gamma$, we consider
the displacement function $d_{\gamma}:X\to\operatorname{\mathbb{R}}$ defined
by $d_{\gamma}(p)=d(p,\gamma p)$. We denote the infimum of this function by
$l_{\gamma}$.

######  Proposition 2.7.

There are four possibilities:

  1. 1.

$\gamma$ is elliptic if it has a fixed point on $X$.

  2. 2.

$\gamma$ is axial if $d_{\gamma}$ assumes its infimum $l_{\gamma}$ and
$l_{\gamma}>0$; then $\gamma$ leaves invariant the geodesic through $p$ and
$\gamma p$ and acts on it as a translation by $l_{\gamma}$. Such a geodesic is
called an axis. If there are two axis, then they bound a flat strip. The
isometry fixes at least the two end points of their axis.

  3. 3.

$\gamma$ is simple parabolic if $d_{\gamma}$ does not assume its infimum
$l_{\gamma}$ and $l_{\gamma}=0$; then $\gamma$ fixes a single point $\xi$ at
infinity and leaves invariant all the horocycles centered at this point.

  4. 4.

$\gamma$ is exceptional parabolic if $d_{\gamma}$ does not assume its infimum
$l_{\gamma}$ and $l_{\gamma}>0$; then there exists a unique $\xi\in\partial X$
such that $\gamma$ leaves invariant every horocycle centered at $\xi$ and
fixes the nontrivial interval $H_{\xi}(\infty)$ point by point.

If $\gamma$ is a parabolic isometry, the point $\xi$ given by the previous
preposition is called the center of $\gamma$.

###  2.5 Limit set

Let $\Gamma$ be a discrete group of isometries of $X$.

######  Definition 2.8.

The limit set $\Lambda$ of $\Gamma$ is the set of accumulation points in
$\partial X$ of an orbit $\Gamma p$, $p\in X$, i.e.

$\Lambda=\overline{\Gamma p}\cap\partial X.$

We distinguish different types of limit points.

######  Definition 2.9.

Let $\xi\in\Lambda$ be a limit point.

  * •

We say that $\xi$ is radial if there exists a sequence $\gamma_{n}$ in
$\Gamma$ such that, for some $p\in X$ and some geodesic ray $c$ pointing to
$\xi$, $\xi=\lim_{n}\gamma_{n}p$ and $d(\gamma_{n}p,c)$ is bounded for all
$n\in\mathbb{N}$.

  * •

We say that $\xi$ is horocyclic if there exists a sequence $\gamma_{n}$ in
$\Gamma$ such that, for some $p\in X$, $\xi=\lim_{n}\gamma_{n}p$ and
$\lim_{n}\beta_{\xi}(\gamma_{n}p,p)=-\infty$.

  * •

We say that $\xi$ is simple parabolic if it is the fixed point of a simple
parabolic isometry.

  * •

We say that $\xi$ is exceptional parabolic if it is the fixed point of an
exceptional parabolic isometry.

Being a simple parabolic limit point is the same as being the center of a
simple parabolic isometry. For an exceptional parabolic isometry $\gamma$ with
center $\xi$, the associated exceptional parabolic limit points are $\xi_{1}$
and $\xi_{2}$ such that $H_{\xi}(\infty)=[\xi_{1},\xi_{2}]$. We ignore wether
$\xi$ can coincide with one of the points $\xi_{1}$ and $\xi_{2}$.

######  Lemma 2.10.

Two points $\eta$ and $\xi$ in $\partial X$ that are not be joined by a
geodesic bound an interval of $\partial X$ which is disjoint from the limit
set $\Lambda$. In particular, for any $\zeta\in\partial X$, the limit set does
not intersect the interior of $H_{\zeta}(\infty)$.

###### Proof.

Assume that there is no geodesic joining $\eta$ to $\xi$. Let $p$ be any point
in $X$. Necessarily, the non-oriented angle from $p$ between $\eta$ and $\xi$
is stricly less that $\pi$. Let $D$ be the sector of $X$ delimited by the
geodesic $c_{p,\eta}$ and $c_{p,\xi}$ with angle strictly less that $\pi$. It
follows easily by the Gauss-Bonnet formula and Proposition 2.5, that the total
curvature of $D$ is bounded. The intersection of the closure of $D$ in
$X\cup\partial X$ with $\partial X$ is a closed interval of $\partial X$. We
take $I$ as the interior of this interval. Let $q\in X$ be a point where the
curvature is negative and let $\varepsilon>0$. Since the total curvature of
$D$ is bounded, there is a finite number of $\gamma\in\Gamma$ such that
$\gamma B(q,\varepsilon)\subset D$. This implies that no point in $I$ is an
accumulation point of $\Gamma q$.

For the second part, since no two points in the interior of
$H_{\zeta}(\infty)$ can be joined by a geodesic, this set cannot intersect
$\Lambda$. ∎

The lemma implies that if $\xi$ is a limit point fixed by an exceptional
parabolic isometry $\gamma$ and $\zeta$ is the unique point in $\partial X$
such that $\gamma$ fixes every horocycle centered at $\zeta$, then $\xi$ is
one of the extremities of the interval $H_{\zeta}(\infty)$.

An extremity of an interval in $\partial X\setminus\Lambda$ corresponding to a
cylindrical or an expanding end is a radial limit point.

######  Lemma 2.11.

  1. 1.

Every radial point is horocyclic.

  2. 2.

Limit points that are the center of a parabolic isometry are not horocyclic.

  3. 3.

Exceptional parabolic limit points which are not the center of the associated
parabolic isometry are horocyclic but not radial.

###### Proof.

  1. 1.

Let $\gamma_{n}$ be a sequence in $\Gamma$ such that $\lim_{n}\gamma_{n}p=\xi$
and $d(\gamma_{n}p,c)\leq R$, where $c$ is the geodesic ray from $p$ to $\xi$.
Let $c(t_{n})$ be the point in $c$ closest to $\gamma_{n}p$. Since
$\gamma_{n}p$ converges to $\xi$, $t_{n}$ goes to $+\infty$. We have

$\beta_{\xi}(\gamma_{n}p,p)=\beta_{\xi}(\gamma_{n}p,c(t_{n}))+\beta_{\xi}(c(t_{n}),p)\leq
R-t_{n},$

which shows that $\lim_{n}\beta_{\xi}(\gamma_{n}p,p)=-\infty$.

  2. 2.

Let $\xi$ be the center of a parabolic isometry $\gamma_{0}$. By a result of
Eberlein (see Proposition 2.17), there exists a horoball $B$ centered at $\xi$
which satisfies for every $\gamma\in\Gamma$ either $\gamma B=B$ or $\gamma
B\cap B=\emptyset$. This implies that for any point $p$ in the boundary of
$B$, $\beta_{\xi}(\gamma p,p)\geq 0$. So $\xi$ cannot be horocyclic.

  3. 3.

Let $\xi$ be an exceptional parabolic limit point. Necessarily, $\xi$ is the
extremity of an open interval $I$ in $\partial X\setminus\Lambda$ and there
exists an exceptional parabolic isometry $\gamma_{0}$ which fixes every point
in the closure of the interval $I$. The center of $\gamma_{0}$ is different
from $\xi$ by hypothesis. As before, there exist a horoball $B$ centered at
$\xi$ which satisfies for every $\gamma\in\Gamma$ either $\gamma B=B$ or
$\gamma B\cap B=\emptyset$. Let $p$ be any point in the boundary of $B$ an
consider the horoball $B^{\prime}$ centered at $\xi$ whose boundary contains
$p$. The horoball $B^{\prime}$ intersects the horocycle $\partial B$ in a
half-horocycle by Propostion 2.5.

Up to taking the inverse we know that $\gamma_{0}^{n}p$,
$n\in\operatorname{\mathbb{N}}$, stays in the horoball $B^{\prime}$ and
$\lim_{n}\gamma_{0}^{n}p=\xi$. We have

$\beta_{\xi}(\gamma_{0}^{n}p,p)=n\beta_{\xi}(\gamma_{0}p,p)$

because $\gamma_{0}$ fixes $\xi$. Since $\gamma_{0}$ does not fix the
horocycles centered at $\xi$, we must have $\beta_{\xi}(\gamma_{0}p,p)<0$.
This implies that $\lim_{n}\beta_{\xi}(\gamma_{0}^{n}p,p)=-\infty$, showing
that $\xi$ is horocyclic.

Let us explain why $\xi$ is not radial. The orbit $\Gamma p$ is contained in
the union of the boundary of $B$ with its complement. Let $c$ be the geodesic
ray starting at $p$ pointing to $\xi$. This ray is included in the horoball
$B$. We will show that the distance between $c(t)$ and $\partial B$ goes to
infinity, when $t\to+\infty$ in Lemma 2.12 below. The same will be true for
any geodesic ray asymptotic to $c$. So there is no sequence $\gamma_{n}p$
converging to $\xi$ which stays within a bounded distance from a ray pointing
to $\xi$.

∎

######  Lemma 2.12.

Let $\gamma$ be an exceptional parabolic isometry with center $\xi$. Let
$H_{\xi}(\infty)=[\xi_{1},\xi_{2}]$. Let $c$ be a geodesic ray pointing to
$\xi_{1}$ and let $H$ be the horocycle centered at $\xi$ and passing through
$c(0)$. Then

$\lim_{t\to+\infty}d(c(t),H)=+\infty.$

###### Proof.

We observe that $c$ stays inside the horoball bounded by $H$ and does not
intersect $H$ for positive time. Otherwise, $c$ and $H$ would coincide and
they would bound a flat-half plane.

Consider the convex function $f$ defined by $f(t):=\beta_{\xi}(c(t),c(0))$. We
observe that

$d(c(t),H)=|f(t)|$

It satisfies $f(0)=0$ and $f^{\prime}(0)<0$. By the observation made above,
$f$ does not vanish again for $t>0$. By the convexity it is non-increasing.
Let $a=\lim_{t\to+\infty}f(t)\in[-\infty,0)$. We show that $a=-\infty$ to
finish the proof. Assume that $a$ is finite and we reason by contradiction. We
consider the horocycle $H^{\prime}$ centered at $\xi$ such that
$\beta_{\xi}(H^{\prime},H)=a$. Then $d(c(t),H^{\prime})=f(t)-a\to 0$ when
$t\to+\infty$. In the quotient $M$, $H^{\prime}$ projects to a periodic
horocycle $\overline{H^{\prime}}$ and $c$ projects to a geodesic ray that
acumulates to $\overline{H^{\prime}}$. By the continuity of the geodesic flow,
$\overline{H^{\prime}}$ is also a geodesic. But this can only happen if
$\overline{H^{\prime}}$ bounds a flat cylinder. This contradicts the fact that
$\overline{H^{\prime}}$ bounds an exceptional parabolic end.

∎

###  2.6 Ends of nonpositively curved surfaces

######  Definition 2.13.

An end of a surface $M$ is a map $\varepsilon$ which, to each compact subset
$K$ of $M$, associates a connected component of $M\setminus K$ with the
property that, for any two compact subsets $K_{1}\subset K_{2}$,
$\varepsilon(K_{2})\subset\varepsilon(K_{1})$. A subset $U\subset M$ is a
neighborhood of $\varepsilon$ if, for some compact subset $K$, $U$ contains
$\varepsilon(K)$. A sequence of points or curves $(c_{n})_{n}$ converges to
the end $\varepsilon$ if, for any neighborhood $U$ of $\varepsilon$, $c_{n}$
is contained in $U$ for $n$ large enough.

We notice that we can associate an end $\varepsilon$ of $M$ to each divergent
geodesic ray $c:\operatorname{\mathbb{R}}_{+}\to M$ by choosing as
$\varepsilon(K)$ the connected component of $M\setminus K$ where
$c([T,\infty))$ is contained for $T$ large enough. We will also say that $c$
converges to the end $\varepsilon$.

The simplest topology that an end can have is that of a cylinder.

######  Definition 2.14.

An end $\varepsilon$ is collared if it has a neighborhood homeomorphic to a
cylinder $S^{1}\times(0,+\infty)$.

Eberlein described the metric of collared ends.

######  Theorem 2.15.

Let $\varepsilon$ be a collared end of $M$. Then there exist an open
neighborhood $U$ of $\varepsilon$ with local $C^{1}$ coordinates
$(r,\theta)\in(0,+\infty)\times S^{1}$ such that the metric in $U$ has the
form

$dg^{2}=dr^{2}+G(r,\theta)^{2}d\theta^{2},$ (1)

where $G:(0,+\infty)\times S^{1}\to(0,+\infty)$ is a continuous function with
$G(\cdot,\theta)$ twice differentiable and convex, for each $\theta\in S^{1}$.

Moreover, if there exists a sequence of continuous picewise smooth curves
$(c_{n})_{n}$ converging to $\varepsilon$ such that $(c_{n})_{n}$ belong to
the same nontrivial free homotopy class, then $\varepsilon$ is collared.

In a parametrization of a collared end as above, the $r$-curves are minimizing
geodesic rays. The curvature $K$ at the point with coordinates $(r,\theta)$ is
equal to

$K(r,\theta)=-G(r,\theta)^{-1}\frac{\partial^{2}G}{\partial r^{2}}(r,\theta)$

When the function $G$ does not depend on the coordinate $\theta$, we obtain a
surface of revolution. In order to distinguish different behaviors of the
ends, we consider the function $L:(0,+\infty)\to(0,+\infty)$ giving the length
of the corresponding $\theta$-curve, that is,

$L(r)=\int_{S^{1}}G(r,\theta)\,\mathrm{d}\theta,$

For an end $\varepsilon$, we denote the set of vectors in $T^{1}M$ whose
positive geodesic ray converges to $\varepsilon$ by $V(\varepsilon)$.

The following result derives from the equivalence of the two classifications
of collared ends proposed by Eberlein, and it was proved by Link, Peigné and
Picaud.

######  Theorem 2.16.

[LPP06] Let $\varepsilon$ be a collared end of a nonelementary nonpositively
curved surface $M$. Then $\varepsilon$ satisfies one of the following:

  1. 1.

(simple parabolic) The set $V(\varepsilon)$ has empty interior and there
exists a sequence of smooth curves $(c_{n})_{n}$ converging to $\varepsilon$
with uniformly bounded length. For any parametrization of a neighborhood of
$\varepsilon$ of the form (1), the associated function $L$ decreases strictly
to $0$.

  2. 2.

(exceptional parabolic) The set $V(\varepsilon)$ has nonempty interior but it
is not open and there exists a sequence of smooth curves $(c_{n})_{n}$
converging to $\varepsilon$ with uniformly bounded length. For any
parametrization of a neighborhood of $\varepsilon$ of the form (1), the
associated function $L$ decreases strictly to some $l>0$.

  3. 3.

(cylindrical) The set $V(\varepsilon)$ is open and there exists a sequence of
smooth curves $(c_{n})_{n}$ converging to $\varepsilon$ with uniformly bounded
length. For any parametrization of a neighborhood of $\varepsilon$ of the form
(1), the associated function $L$ is constant for $r\geq r_{0}$.

  4. 4.

(expanding) Any sequence of smooth curves $(c_{n})_{n}$ converging to
$\varepsilon$ has unbounded length. The set $V(\varepsilon)$ is open. For any
parametrization of a neighborhood of $\varepsilon$ of the form (1), the
associated function $L$ tends to $+\infty$.

Moreover, when $\varepsilon$ is parabolic or cylindrical, the $\theta$-curves
of the parametrization of a neighborhood of $\varepsilon$ can be choosen to be
closed horocycles. If the end $\varepsilon$ is expanding, $U$ can be chosen so
that its boundary $\partial U$ is a closed geodesic.

We consider a neighborhood $U$ of the end $\varepsilon$ given by the theorem,
which can be thought as the quotient of the universal cover $\tilde{U}$ of $U$
by the group generated by an isometry $\gamma$ of infinite order. If
$\varepsilon$ is simple (resp. exceptional) parabolic, then $\tilde{U}$ is a
horoball and $\gamma$ is simple (resp. exceptional) parabolic. Conversely, a
parabolic isometry produces an end of the same type. In fact:

######  Proposition 2.17.

[Ebe79, Proposition 3.6] Let $\gamma$ be a parabolic isometry and let $\xi$ be
the point in $\partial X$ such that $\gamma$ fixes the horocycles centered at
$\xi$. Then, there exists a horoball $B$ centered at $x$ such tha t, for all
$g\in\Gamma$, $gB\cap B=\emptyset$ if $gx\not=x$ and $gB=B$ if $gx=x$.

######  Remark 2.18.

If $\varepsilon$ is the parabolic end associated to the point $\xi$, then
$U=B/\operatorname{Stab}_{\Gamma}(x)$ is a neighborhood of $\varepsilon$. A
neighborhood of $\varepsilon$ obtained in this way will be called a _standard
neighborhood_ of $\varepsilon$.

By Proposition 2.17, there is a correspondence between the conjugacy classes
of parabolic isometries in $\Gamma$ and the parabolic ends of the surface $M$,
preserving the fact of being simple or exceptional.

For a cylindrical end $\varepsilon$ with neighborhood $U$ given by the
theorem, the universal cover $\tilde{U}$ is a flat half-plane and $\gamma$ is
an axial isometry, hence, a translation in the flat-half plan. If
$\varepsilon$ is expanding, then $\tilde{U}$ is a nonflat half-plane and
$\gamma$ is an axial isometry. The set $\tilde{U}$ can be thought as follows:

######  Proposition 2.19.

Let $\varepsilon$ be a cylindrical or expanding end of $M$. Then there exists
an axial isometry $\gamma$ which leaves invariant an interval $I$ of $\partial
X\setminus\Lambda$ associated to the end $\varepsilon$. If a geodesic $c$ is
an axis of $\gamma$, then $c(-\infty)$ and $c(+\infty)$ are the extremities of
$I$ and the half-plane $\tilde{U}$ delimited by $c$ and $I$ satisfies, for all
$g\in\Gamma$, $g\tilde{U}\cap\tilde{U}=\emptyset$ if $g$ does not fix $c$ and
$g\tilde{U}=\tilde{U}$ otherwise.

######  Remark 2.20.

The end $\varepsilon$ has a neighborhood of the form
$U=\tilde{U}/\operatorname{Stab}_{\Gamma}(c)$. This neighborhood is bounded by
a closed geodesic unique up to homotopy. A neighborhood of $\varepsilon$
obtained in this way will be called a _standard neighborhood_ of
$\varepsilon$.

###  2.7 Description of horocycles

From now on, we consider a nonelementary nonpositively curved surface $M$ and
we denote its universal cover by $X$. We equip $X$ with the lifted metric and
we let $\Gamma$ denote the covering group of $M$ acting on $X$ by isometries.
The goal of this section is to describe the horocycles on $X$. We begin with a
direct consequence of Lemma 2.10.

######  Proposition 2.21.

Let $\eta\in\Lambda$ a limit point which does not bound a connected component
of $\partial X\setminus\Lambda$. Then any horocycle centered at $\eta$ only
accumulates to infinity at $\eta$.

To understand the horocycles centered at points which bound a connected
component of $\partial X\setminus\Lambda$ or which are not in the limit set,
we will use the classification of ends.

######  Proposition 2.22.

Let $I=(\eta_{1},\eta_{2})$ be a connected component of $\partial
X\setminus\Lambda$ associated to a collared end $\varepsilon$ of $M$. Then,

  1. 1.

if $\varepsilon$ is expanding, $Td(\eta_{1},\eta_{2})=+\infty$. For $i=1,2$,
the horocycle centered at $\eta_{i}$ satisfies
$H_{\eta_{i}}(\infty)=\\{\eta_{i}\\}$. Moreover, if $\xi\in I$, then
$H_{\xi}(\infty)\subset I$,

  2. 2.

if $\varepsilon$ is cylindric, $Td(\eta_{1},\eta_{2})=\pi$ and there exists
$\eta\in I$ such that $H_{\eta}(\infty)=[\eta_{1},\eta_{2}]$,

  3. 3.

if $\varepsilon$ is parabolic exceptional, $Td(\eta_{1},\eta_{2})\leq\pi$ and
there exists $\eta\in I$ such that $H_{\eta}(\infty)=[\eta_{1},\eta_{2}]$.

###### Proof.

First we observe that the closed interval $\bar{I}=[\eta_{1},\eta_{2}]$ is the
set at infinity of the lift $\tilde{U}$ of a neighborhood of $\varepsilon$
obtained from Theorem 2.16. If the end $\varepsilon$ is cylindrical,
$\tilde{U}$ is a flat half-plane, so the Tits distance between $\eta_{1}$ and
$\eta_{2}$ is exactly $\pi$ and $\eta$ is the direction perpendicular to the
geodesic bounding the flat half-plane. If the end $\varepsilon$ is exceptional
parabolic, the there is no geodesic joining $\eta_{1}$ to $\eta_{2}$, so the
Tits distance between them is less or equal than $\pi$ by [BH99, Proposition
9.21]. The point $\eta$ is the center of the horocycle that bounds the
neighborhood $\tilde{U}$.

Now assume that $\varepsilon$ is expanding. Let $c$ be the geodesic that
bounds $\tilde{U}$, which joins $\eta_{1}$ to $\eta_{2}$. We consider a
geodesic ray $h$ starting at a point on $c$, perpendicular to it and pointing
to the interior of $\tilde{U}$. Let $\gamma$ be the translation isometry of
$c$. The curves $c$, $h$ and $\gamma h$ bound a fundamental domain of the
expanding end. The isometry $\gamma$ does not fix the point at infinity of
$h$, which we denote by $\xi$. Hence, the Tits distance between $\xi$ and
$\gamma\xi$ is positive. The translates of $\xi$ by $\gamma$ form a sequence
in $I$ of points which are at a positive constant Tits distance, so the
distance between the two end points of $I$ is infinite. In fact, the same
argument shows that the Tits distance between any point in $I$ and one of the
extremities is infinite. As a consequence, for any $\eta\in I$, the set
$H_{\eta}(\infty)={\bar{B}_{Td}}(\xi,\pi/2)$ is contained in $I$. Also, it
implies that the intervals $H_{\eta_{i}}(\infty)$ cannot intersect $I$. Since
$\eta_{i}$ is no isolated in $\Lambda$, we have in fact
$H_{\eta_{i}}(\infty)=\\{\eta_{i}\\}$.

∎

######  Remark 2.23.

Link, Peigné and Picaud observed that for exceptional parabolic ends obtained
as surfaces of revolution, Clairaut’s relation implies that
$Td(\eta_{1},\eta_{2})=\pi$. It is not known if this property holds in
general.

We can now describe precisely the accumulations points of horocycles centered
at a cylindrical or an exceptional parabolic end.

######  Proposition 2.24.

Let $I=(\eta_{1},\eta_{2})$ be a connected component of $\partial
X\setminus\Lambda$ associated to a cylindrical or an exceptional parabolic
end, and let $\eta\in[\eta_{1},\eta_{2}]$ be such that
$H_{\eta}(\infty)=[\eta_{1},\eta_{2}]$. We parametrize $[\eta_{1},\eta_{2}]$
by a geodesic $c:[a,b]\to[\eta_{1},\eta_{2}],\,-\pi/2\leq a\leq 0\leq
b\leq\pi/2$, in the Tits distance such that $c(a)=\eta_{1}$, $c(0)=\eta$,
$c(b)=\eta_{2}$. Then, for every $t\in[a,b]$,

$H_{c(t)}(\infty)=[c(\max(a,t-\pi/2)),c(\min(b,t+\pi/2))]$

###### Proof.

We recall that two points in the boundary $\partial X$ within finite Tits
distance can be joined by a geodesic of $\partial X$. The proposition follows
then from the fact that intervals of horocycles at infinity are balls in the
Tits distance of radius $\pi/2$. ∎

In general, there are some restrictions for two horocycles at infinity to
intersect.

######  Proposition 2.25.

Let $\eta,\xi\in\partial X$ and denote $H_{\eta}(\infty)=[\eta_{1},\eta_{2}]$
and $H_{\xi}(\infty)=[\xi_{1},\xi_{2}]$. If $\xi_{2}\in[\eta_{1},\eta_{2})$,
then $\xi\in[\xi_{1},\eta)$.

###### Proof.

The distance between $\xi_{1}$ and $\eta_{2}$ is less or equal than $2\pi$. So
there exists a geodesic in the Tits distance joining $\xi_{1}$ to $\eta_{2}$.
The distance between $\eta_{2}$ and $\xi_{2}$ must be strictly positive and
the point $\xi_{2}$ is a distance exactly $\pi/2$ from $\xi$. This shows that
the Tits distance between $\eta_{2}$ and $\xi$ is strictly bigger than
$\pi/2$, while $Td(\eta_{2},\eta)\leq\pi/2$. So $\xi$ cannot be in the
interval $[\eta,\xi_{2}]$. ∎

Two horocycles with distinct centers at infinity can accumulate at the same
point at infinity. We next show that, even when this happens, the horocycles
diverge.

######  Proposition 2.26.

Let $v,w\in T^{1}X$ two vectors whose geodesics are not asymptotic. Then the
horocycles of $v$ and $w$ diverge, that is, if we consider the
parametrizations $\alpha,\beta:\operatorname{\mathbb{R}}\to X$ given by
$\alpha(s):=\pi(h_{s}(v))$ and $\beta(s):=\pi(h_{s}(w))$, then

$d(\alpha(s),\beta(\operatorname{\mathbb{R}}_{+}))\xrightarrow[s\to+\infty]{}+\infty,$
$d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(s))\xrightarrow[s\to+\infty]{}+\infty.$

###### Proof.

First we show that both formulas imply each other. Assume that

$\lim_{s\to+\infty}d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(s))=+\infty.$
(2)

For every $s\in\operatorname{\mathbb{R}}_{+}$, let
$t_{s}\in\operatorname{\mathbb{R}}_{+}$ such that
$d(\alpha(s),\beta(\operatorname{\mathbb{R}}_{+}))=d(\alpha(s),\beta(t_{s}))$.
We have

$d(\alpha(s),\beta(\operatorname{\mathbb{R}}_{+}))\geq
d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(t_{s})).$ (3)

Let $s_{k}$ be a sequence going to $+\infty$ such that

$\liminf_{s\to+\infty}d(\alpha(s),\beta(\operatorname{\mathbb{R}}_{+}))=\lim_{k}d(\alpha(s_{k}),\beta(\operatorname{\mathbb{R}}_{+})).$

Up to taking a further subsequence, we can assume that $t_{s_{k}}$ converges
to a point $t$ in $\operatorname{\mathbb{R}}_{+}\cup\\{+\infty\\}$. If
$t=+\infty$, then $d(\alpha(s),\beta(\operatorname{\mathbb{R}}_{+}))$ tends to
infinity by (2) and (3). If $t\in\operatorname{\mathbb{R}}_{+}$, the sequence
$\beta(t_{s_{k}})$ is bounded, so

$\lim_{k}d(\alpha(s_{k}),\beta(\operatorname{\mathbb{R}}_{+}))=\lim_{k}d(\alpha(s_{k}),\beta(t_{s_{k}}))=+\infty.$

We can also reduce the statement to the case that $\pi(v)=\pi(w)$. In the
general case we can replace $w$ by
$w^{\prime}:=g_{-\beta_{w_{+}}(\pi(v),\pi(w))}w$, so that $v$ is based one the
horocycle of $w^{\prime}$. But the horocycle
$\beta(\operatorname{\mathbb{R}})$ of $w$ and that of $w^{\prime}$ are
equidistant, so $\alpha(s)$ diverges from one half-horocycle if and only if it
diverges from the other.

Let $\theta$ be the signed angle between $v$ and $w$. Without loss of
generality, we assume that $\theta>0$. The function defined by
$f(t):=\beta_{w_{+}}(c_{v}(t),\pi(w))$ is convex. Moreover, $f(0)=0$ and
$f^{\prime}(0)=-\cos\theta$.

We distinguish two cases. Assume first that $f$ vanishes again for $t_{0}>0$,
which means that the geodesic of $v$ escapes the horoball bounded by $\beta$.
Therefore, there exists $s_{0}$ such that for $s\geq s_{0}$, $\beta(s)$ stays
in the negative half-plane $X_{v}^{-}$. But the half-horocycle
$\alpha(\operatorname{\mathbb{R}}_{+})$ lies in the half-plane $X_{w}^{+}$ and
the distance from $\alpha(t)$ to $c_{v}$ goes to infinity. This shows that
$d(\alpha(t),\beta([s_{0},+\infty)))$ goes to infinity when $t\to+\infty$,
which implies the other limit by the equivalence made at the beginning.

The second case is when $f$ does not vanish for positive time, which implies
that $f$ is strictly negative. For $t>0$, the distance between $c_{v}(t)$ and
$\beta(\operatorname{\mathbb{R}})$ satisfies

$d(c_{v}(t),\beta(\operatorname{\mathbb{R}}))=|f(t)|\leq\cos\theta\cdot t.$

For $s>0$, we consider the geodesic ray from $\beta(s)$ pointing to the end
point of $w$, which intersects the geodesic ray of $v$ at a point
$c_{v}(t_{s})$. We have $d(c_{v}(t_{s}),\beta(s))\leq\cos\theta\cdot t_{s}$.
In particular, $t_{s}$ must be unbounded. Now,

$t_{s}\leq d(\alpha(\operatorname{\mathbb{R}}_{+}),c_{v}(t_{s}))\leq
d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(s))+d(\beta(s),c_{v}(t_{s}))\leq
d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(s))+\cos\theta\cdot t_{s}.$

Passing to the limit, we see
$\lim_{s\to+\infty}d(\alpha(\operatorname{\mathbb{R}}_{+}),\beta(s))=+\infty$.
We apply again the equivalence to obtain the other limit.

∎

###  2.8 Geometrically finite surfaces

We are concerned with geometrically finite surfaces $M$. Recall that this
means that the fundamental group of $M$ is finitely generated. The topology
and the geometry of the ends of such surface is easier to understand than for
arbitrary surfaces.

######  Theorem 2.27.

[Ker23] Let $M$ be a (topological, connected, complete) geometrically finite
surface. Then $M$ has a finite number of ends and each of them is collared.

In fact $M$ can be obtained by removing a finite number of points of a compact
surface. We will now describe a way of decomposing our nonpositively curved
surface $M$ into a compact part and some neighborhoods of the finitely many
ends. We start by choosing suitable neighborhoods of the ends.

Propositions 2.17 and 2.19 provide an almost canonical way of choosing
pairwise disjoint neighborhoods of the collared ends of $M$. When $M$ is
geometrically finite, the number of ends is finite and each one is collared.
We choose standard neighborhoods $E_{1},\dots,E_{r}$, $r\geq 1$, of the ends
of $M$. The remaining part of $M$,

$K:=M\setminus\bigcup_{i=1}^{r}E_{i},$

will be called the _compact part_ of $M$.

######  Proposition 2.28.

Let $M$ be a nonpositively curved surface with finitely generated fundamental
group. Then the limit set $\Lambda$ only contains radial limit points and
finitely many $\Gamma$-orbits of parabolic limit points.

###### Proof.

We consider the maps which associates a parabolic limit point to the
corresponding end the manifold $M$. Each fiber of this map consists of one or
two $\Gamma$-orbits (one if the end is simple parabolic and two if the end is
exceptional parabolic). Since the number of ends is finite, the parabolic
limit points are in a finite union of $\Gamma$-orbits.

Let $\xi\in\Lambda$ and let $c$ be a geodesic ray on $X$ such that
$c(+\infty)=\xi$. We consider the lift to $X$ of the standard neighborhoods of
$M$,

$Y:=\Pi^{-1}(E_{1}\cup\dots\cup E_{r}).$

There are two possibilities. If $c$ leaves every connected component of the
set $Y$, then there exists a sequence $t_{n}\to\infty$ such that
$c(t_{n})\in\tilde{K}:=\Pi^{-1}(K)$. Let $o\in\tilde{K}$. There exists
isometries $\gamma_{n}$ in $\Gamma$ such that
$d(c(t_{n}),\gamma_{n}o)\leq\operatorname{diam}(K)<+\infty$. This means that
$\xi$ is radial by definition.

The other possibility is that $c$, after some time, stays in a connected
component of $Y$, say $Y_{0}$. This component $Y_{0}$ is a lift of a standard
neighborhood $E_{i}$ of and end of $M$. If the end is parabolic, then $Y_{0}$
is a horoball and there is a parabolic isometry $\gamma$ associated to the end
which fixes $Y_{0}$ and such that $E_{i}=Y_{0}/\langle\gamma\rangle$. But
$\xi\in\Lambda\cap\overline{Y_{0}}$, so it can only be a simple parabolic or
an exceptional parabolic point. If the end is cylindrical or expanding, then
$Y_{0}$ is a half-plane and there exists an axial isometry $\gamma$ leaving
invariant $Y_{0}$. Again, $\xi\in\Lambda\cap\overline{Y_{0}}$, so it is the
end point of an axis of $\gamma$, which implies that it is radial.

∎

The complement of the limit set $\Lambda$ is formed by a countable union of
intervals. Each of these intervals corresponds to an end of the surface, which
can be either expanding, cylindrical or exceptional parabolic.

##  3 Proofs of the results

Recall that we chose an orientation of $X$ so that the induced orientation on
$\partial X$ is counter-clockwise. This gives the horocycles of $X$ a natural
orientation which allowed us to define the horocyclic flow $h_{s}$ on
$T^{1}X$. We assume that $M$ is orientable, so every element of $\Gamma$ is
orientation-preserving and the orientation of $X$ descends to an orientation
of $M$. Hence, the horocyclic flow $h_{s}$ is defined on $T^{1}M$
unambigously. A positive half-horocycle on $T^{1}M$ is a positive orbit of the
horocyclic flow $h_{\operatorname{\mathbb{R}}_{+}}(v)$.

###  3.1 Horocyclic limit points and dense orbits

We will first establish a relation between vectors whose full horocylic orbit
is dense on a certain set and horocyclic limit points. The result we prove
will also follow from the study of half-horocycles later, but here we present
a direct proof following [Dal00].

We fix a reference point $o\in X$. A horocycle $H$ is uniquely determined by
its point at infinity $\xi\in\partial X$ and the Busemann cocycle
$t=\beta_{\xi}(H,o)\in\operatorname{\mathbb{R}}$. We use the data
$(\xi,t)\in\partial X\times\operatorname{\mathbb{R}}$ as coordinates on the
set of horocycles of $X$. The natural action of $\Gamma$ on the set of
horocycles is given in these coordinates by

$\gamma(\xi,t)=(\gamma\xi,t+\beta_{\xi}(o,\gamma^{-1}o)),\quad\forall\gamma\in\Gamma,\,(\xi,t)\in\partial
X\times\operatorname{\mathbb{R}}.$

We denote by $\Omega^{+}$ the set of vectors $v$ in $T^{1}M$ such that for any
lift $\tilde{v}$ to $T^{1}X$, the positive end point $v_{+}$ is in the limit
set $\Lambda$. The set $\Omega^{+}$ is invariant by the horocyclic flow. The
set of horocycles contained in $\Omega^{+}$ is identified with the set
$\Lambda\times\operatorname{\mathbb{R}}$ in the coordinates defined above.

The following result completely characterizes dense horocycles in
$\Omega^{+}$.

######  Proposition 3.1.

Let $M$ be a geometrically finite nonpositively curved surface. Let
$v\in\Omega^{+}$. The stable horocycle of $v$ is dense in $\Omega^{+}$ if and
only if for any lift $\tilde{v}$ of $v$ the end point $\tilde{v}_{+}$ is
horocyclic.

In the proof of this proposition, we will use the following standard fact.

######  Lemma 3.2.

Let $v\in T^{1}M$. The stable horocycle of $v$ is dense in $\Omega^{+}$ if and
only if for any lift $\tilde{v}$ of $v$ the $\Gamma$-orbit of horocycle
determined by $\tilde{v}$ is dense in
$\Lambda\times\operatorname{\mathbb{R}}$.

_Proof of Proposition 3.1._ We begin by proving that $\tilde{v}_{+}$ is
horocyclic, provided that the stable horocycle of $v$ is dense in
$\Omega^{+}$. In view of Lemma 3.2, we know that the $\Gamma$-orbit of the
horocycle $h_{\operatorname{\mathbb{R}}}\tilde{v}$ is dense in the set of
horocycles with coordinates in $\Lambda\times\operatorname{\mathbb{R}}$. For
$n\geq 1$, the horocycle $h_{\operatorname{\mathbb{R}}}g_{n}\tilde{v}$ has
coordinates $(\tilde{v}_{+},\beta_{\tilde{v}_{+}}(\pi(\tilde{v}),o)+n)$.

By the density of $\Gamma h_{\operatorname{\mathbb{R}}}\tilde{v}$, we can find
$\gamma_{n}\in\Gamma$ such that $\gamma_{n}\tilde{v}_{+}\to\tilde{v}_{+}$ and

$|\beta_{\tilde{v}_{+}}(\pi(\tilde{v}),o)+n-\beta_{\gamma_{n}\tilde{v}_{+}}(\gamma_{n}\pi(\tilde{v}),o)|=|n-\beta_{\tilde{v}_{+}}(o,\gamma_{n}^{-1}o)|\leq
1.$

This shows that $\lim_{n}\beta_{\tilde{v}_{+}}(\gamma_{n}^{-1}o,o)=-\infty$.

Now we extract a subsequence $\gamma^{\prime}_{n}$ of $\gamma_{n}^{-1}$ such
that $\gamma^{\prime}_{n}o$ converges to a point $\xi\in\Lambda$. If we can
show that $\xi=\tilde{v}_{+}$, we obtain that $\tilde{v}_{+}$ is horocyclic as
we wanted. The point $\xi$ must be in $H_{\tilde{v}_{+}}(\infty)$. Since
$\tilde{v}_{+}\in H_{\tilde{v}_{+}}(\infty)$ lies in $\Lambda$, the set
$H_{\tilde{v}_{+}}(\infty)$ must be of the form $\\{\tilde{v}_{+}\\}$,
$[\tilde{v}_{+},\eta]$ or $[\eta,\tilde{v}_{+}]$ for some $\eta\in\partial
X\setminus\\{\tilde{v}^{+}\\}$. In the first case, we have $\xi=\tilde{v}_{+}$
as desired. In the second case, the interval $(\tilde{v}_{+},\eta)$ is in a
connected component of $\partial X\setminus\Lambda$. This component is
associated to an end of the manifold.

We can exclude expanding ends because the Tits distance between the extremity
$\tilde{v}_{+}$ of $H_{\tilde{v}_{+}}(\infty)$ and a point in the interior of
$H_{\tilde{v}_{+}}(\infty)$ is infinity, but here
$Td(\tilde{v}_{+},\eta)\leq\pi/2$. So it must be cylindrical or exceptional
parabolic. In the first case,
$\xi\in\Lambda\cap[\eta,\tilde{v}_{+}]=\\{\tilde{v}_{+}\\}$, so we are done.
Let us deal with the case that the end is exceptional parabolic. Recall that
$\Lambda$ does not intersect the the interior of $H_{\tilde{v}_{+}}(\infty)$,
so the only way for $\xi$ not being equal to $\tilde{v}_{+}$ is that
$\xi=\eta\in\Lambda$.

We assume that the latter happens and we show that $\tilde{v}_{+}$ is
horocyclic anyway (there is a sequence different from $\gamma_{n}^{\prime}$
satisfying the definition). Let $\gamma$ be the isometry associated to the
exceptional parabolic end and let $\bar{\xi}$ be the point at infinity whose
horocycles are left invariant by $\gamma$. Necessarily $\bar{\xi}$ is
different from $\tilde{v}_{+}$ because vectors in $T^{1}M$ whose lift points
to $\bar{\xi}$ have periodic horocycles and here the horocycle of $v$ is dense
by hypothesis. Moreover $H_{\bar{\xi}}(\infty)=[\tilde{v}_{+},\eta]$. Up to
taking the inverses, $\gamma^{n}\pi(\tilde{v})$ belongs to
$H_{\bar{\xi}}(\pi(\tilde{v}))$ and converges to $\tilde{v}_{+}$. Let
$\tilde{w}$ be the vector with base point $\pi(\tilde{v})$ that points to
$\bar{\xi}$. The half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}\tilde{w}$
stays in the horoball bounded by $h_{\operatorname{\mathbb{R}}}\tilde{v}$ and,
by Proposition 2.26, they diverge one from the other. This implies that
$\beta{\tilde{v}_{+}}(\gamma^{n}o,o)$ tends to infinity, showing that
$\tilde{v}_{+}$ is horocyclic.

For $H_{\tilde{v}_{+}}(\infty)=[\eta,\tilde{v}_{+}]$ the proof is analogous.

We now assume that $\tilde{v}_{+}$ is horocyclic and show that the horocycle
$h_{\operatorname{\mathbb{R}}}v$ is dense in $\Omega^{+}$, or equivalently,
that the $\Gamma$-orbit of the horocycle determined by $\tilde{v}$ is dense in
the space $\Lambda\times\operatorname{\mathbb{R}}$. This is already known when
$v$ is periodic for the geodesic flow (see the proof of Theorem A $i)$
$\Rightarrow$ $ii)$ in [LPP06]).

For the general case, it is enough to show that if $\tilde{v}_{+}$ is
horocyclic the closure of the $\Gamma$-orbit of
$h_{\operatorname{\mathbb{R}}}\tilde{v}$ contains a horocycle with a periodic
vector. Assume that $\tilde{v}_{+}$ is horocyclic: there exists a sequence
$\gamma_{n}$ such that $\gamma_{n}o$ tends to $\tilde{v}_{+}$ and
$\beta_{\tilde{v}_{+}}(\gamma_{n}o,o)$ tends to $-\infty$. Up to passing to a
subsequence we can assume that $\gamma_{n}^{-1}\tilde{v}_{+}$ converges to
$\eta\in\partial X$. We choose an axial isometry $\gamma$ whose axis does not
bound a flat half-plane such that $\eta\not\in\\{\gamma^{\pm}\\}$. This is
possible because the pairs of end points of isometries not bounding a flat
half-plane are dense in $\Lambda\times\Lambda$. Up to taking another
subsequence of $\gamma_{n}$, we can also choose a sequence $r_{n}$ of integers
going to $+\infty$ such that

$\beta_{\tilde{v}_{+}}(\gamma_{n}o,o)+r_{n}l(\gamma)$

converges to some $\lambda\in\operatorname{\mathbb{R}}$.

We claim that the images under $\gamma^{r_{n}}\gamma_{n}^{-1}$ of the
horocycle $h_{\operatorname{\mathbb{R}}}\tilde{v}$ tend to a horocycle with
coordinates $(\gamma^{+},\alpha)$, $\alpha\in\operatorname{\mathbb{R}}$. We
first observe that, since $\gamma$ is axial and its axis does not bound a flat
half-plane, compact subsets of $\partial X\setminus\\{\gamma^{-}\\}$ converge
uniformly to $\gamma^{+}$ [Bal95, Lemma III.3.3]. So
$\gamma^{r_{n}}\gamma_{n}^{-1}\tilde{v}_{+}$ tends to $\gamma^{+}$. The
isometries $\gamma^{r_{n}}\gamma_{n}^{-1}$ act on the second coordinate of
$\partial X\times\operatorname{\mathbb{R}}$ by adding the quantity
$\beta_{\tilde{v}_{+}}(o,(\gamma^{r_{n}}\gamma_{n}^{-1})^{-1}o)$. We write

$\beta_{\tilde{v}_{+}}(o,(\gamma^{r_{n}}\gamma_{n}^{-1})^{-1}o)=\beta_{\tilde{v}_{+}}(o,\gamma_{n}o)+\beta_{\tilde{v}_{+}}(\gamma_{n}o,\gamma_{n}\gamma^{-r_{n}}o),$

where the second term is equal to

$\beta_{\tilde{v}_{+}}(\gamma_{n}o,\gamma_{n}\gamma^{-r_{n}}o)=\beta_{\gamma_{n}^{-1}\tilde{v}_{+}}(o,\gamma^{-r_{n}}o)=2(\gamma_{n}^{-1}\tilde{v}_{+}|\gamma^{-})_{o}-r_{n}l(\gamma)-2(\gamma^{r_{n}}\gamma_{n}^{-1}\tilde{v}_{+}|\gamma^{-})_{o}.$

The Gromov product taking values in
$\operatorname{\mathbb{R}}_{+}\cup\\{\infty\\}$ is continuous on
$\Lambda\times\Lambda$. Taking the limit in the previous equations, we obtain
that $\beta_{\tilde{v}_{+}}(o,(\gamma^{r_{n}}\gamma_{n}^{-1})^{-1}o)$
converges to

$-\lambda+2(\eta|\gamma^{-})_{o}-2(\gamma^{+}|\gamma^{-})_{o}.$

Since $\gamma^{-}$ is the enpoint of an axis not bounding a flat half-plane,
there is a geodesic joining it to any other point at infinity, which implies
that the Gromov products above are finite. $\square$

###  3.2 Half-horocycles

This section is devoted to understanding when the half-horocycles are dense in
$\Omega_{+}$. It is divided in two parts: half-horocycles of vectors in
$\Omega^{+}$ and half-horocycles of vectors outside of $\Omega^{+}$. The first
part is inspired by the article [Sch11], which does the same for hyperbolic
surfaces. The pinched negative curvature case can be treated in the same way,
but nonpositively curved surfaces are much more subtle.

####  3.2.1 Dense half-horocycles in $\Omega^{+}$

Given a vector $\tilde{w}\in T^{1}X$, the geodesic $c_{\tilde{w}}$ generated
by $\tilde{w}$ divides $X$ into two half-planes. One of the half-planes is
oriented such that the orientation induced in the boundary $c_{\tilde{w}}$ is
given by $\tilde{w}$. We denote this half-plane by $X_{\tilde{w}}^{-}$ and the
other one by $X_{\tilde{w}}^{+}$.

G. Kniper proved that the horocycles of rank $1$ $g_{t}$-recurrent vectors are
contracting and expanding. The following lemma explains what happens in
general for recurrent vectors.

######  Lemma 3.3.

[Kni98, Proposition 4.1] Let $v$ be a positively (resp. negatively) recurrent
vector in $T^{1}M$ such that, for any lift $\tilde{v}$ of $v$ to $T^{1}X$, the
geodesic $c_{\tilde{v}}$ does not bound a flat strip in $X_{\tilde{v}}^{+}$.
Then, the positive stable (resp. unstable) half-horocycle of $\tilde{v}$ is
contracting for positive (resp. negative) time, that is, for every $s\geq 0$,

$\lim_{t\to+\infty}d(g_{t}(\tilde{v}),g_{t}(h_{s}(\tilde{v})))=0.$

In particular, if $v$ has rank $1$ and is positively (resp. negatively)
recurrent, then its stable (resp. unstable) horocycle coincides with its
stable (resp. unstable) manifold and it consists of rank $1$ vectors
exclusively.

We will also need the following refinement of [Bal95, Lemma III.3.3].

######  Lemma 3.4.

Let $\gamma$ be an axial isometry and let $\tilde{v}$ be a vector tangent to
an axis of $\gamma$. If the half-plane $X_{\tilde{v}}^{+}$ is not flat, then
$\gamma$ does not fix any point in $(\tilde{v}-,\tilde{v}_{+})$. In fact, for
every $x\in(\tilde{v}-,\tilde{v}_{+})$, $\lim_{n\to+\infty}\gamma^{\pm
n}\xi=\tilde{v}_{\pm}$ and the convergence is uniform on compact subsets of
$(\tilde{v}-,\tilde{v}_{+})$.

We denote the set of vectors $v$ in $T^{1}M$ which have a lift in $T^{1}X$
with both end points in the limit set $\Lambda$ by $\Omega$. The set $\Omega$
is precisely the set of $g_{t}$-nonwandering vectors [Bal95, Section III.1].
Among these vectors, we consider the subset $\Omega_{NF}\subset\Omega$ of
those whose geodesic does not bound a flat strip.

######  Lemma 3.5.

$\Omega^{+}=\overline{h_{\operatorname{\mathbb{R}}_{+}}(\Omega_{NF})}$

###### Proof.

Let $v\in\Omega^{+}$ and take a lift $\tilde{v}$ in $T^{1}X$. By definition,
the positive end point $\tilde{v}_{+}$ is in the limit set $\Lambda$. First,
assume that the limit set $\Lambda$ intersects the interval
$(\tilde{v}_{+},\tilde{v}_{-})$. Since the action of $\Gamma$ is minimal on
the limit set $\Lambda$, there exists
$\xi\in\Lambda\cap(\tilde{v}_{+},\tilde{v}_{-})$ which is an end point of the
lift of a rank $1$ periodic geodesic. The points $\xi$ and $\tilde{v}_{+}$ can
be joined by unique geodesic, so we can choose $\tilde{w}\in
h_{\operatorname{\mathbb{R}}}\tilde{v}$ such that $\tilde{w}_{-}=\xi$. Notice
that the projection $w$ of $\tilde{w}$ to $T^{1}M$ is in the set
$\Omega_{NF}$. By the choice of $\xi$, it is clear that there exists $s\geq 0$
such that $h_{s}(w)=v$.

If $(\tilde{v}_{+},\tilde{v}_{-})$ does not intersect $\Lambda$, then $v_{+}$
is the first end point of an interval of $\partial X\setminus\Lambda$ of the
form $I=(\tilde{v}_{+},\eta)$ with $\eta\in[\tilde{v}_{-},\tilde{v}_{+})$. The
end corresponding to $I$ is either cylindrical or expanding. Since
$\tilde{v}_{+}$ is not isolated in $\Lambda$ we can choose a sequence
$\eta_{n}$ in $\Lambda\setminus\\{\tilde{v}_{+}\\}$ of end points of lifts of
closed rank $1$ geodesics converging to $\tilde{v}_{+}$. Necessarily,
$\eta_{n}\in[\tilde{v}_{-},\tilde{v}_{+})$. Let $\tilde{v}_{n}$ be the vector
with the same base point as $\tilde{v}$ that points at $\eta_{n}$. As $n$ goes
to infinity, $\tilde{v}_{n}$ tends to $\tilde{v}$. Since $\tilde{v}_{+}$ can
be joined to $\eta_{n}$ for all $n$, we can consider vectors $\tilde{w}_{n}\in
h_{\operatorname{\mathbb{R}}}\tilde{v}_{n}$ such that
$\tilde{w}_{n-}=\tilde{v}_{+}$. Since $\tilde{v}_{n-}$ tends to
$\tilde{v}_{-}$, for large $n$,
$\tilde{v}_{n-}\in(\tilde{v}_{+},\tilde{v}_{-})$. This implies that the vector
$\tilde{w}_{n}$ lies in the negative horocycle orbit of $\tilde{v}_{n}$, so
there exists $s_{n}\geq 0$ such that $\tilde{v}_{n}=h_{s_{n}}(\tilde{w}_{n})$.
The projections $w_{n}$ of $\tilde{w}_{n}$ to $T^{1}M$ belong to $\Omega_{NF}$
and satisfy $\lim_{n}h_{s_{n}}(w_{n})=v$.

Figure 2: Elements of the proof of Lemma 3.5.

∎

######  Lemma 3.6.

There exists a vector $w$ in $\Omega$ whose negative orbit
$g_{\operatorname{\mathbb{R}}_{-}}(w)$ is dense in the set $\Omega_{NF}$ of
nonwandering vectors which do not bound a flat strip.

###### Proof.

Coudène and Schapira showed that the geodesic flow in restriction to the
closure of the set of periodic hyperbolic vectors is positively (hence
negatively) transitive [CS10, CS14]. This implies the existence of a dense
negative orbit $g_{\operatorname{\mathbb{R}}_{-}}(w)$ on this set. On the
other hand, Link, Peigné and Picaud proved that the set of pairs of end points
of rank $1$ periodic vectors is dense in $\Lambda\times\Lambda$ [LPP06,
Corollaire 1.20]. This implies that the set $\Omega_{NF}$ is contained in the
closure of the set of rank $1$ periodic vectors as we show next.

Let $v$ be a nonwandering vector in $\Omega_{NF}$ and take a lift
$\tilde{v}\in T^{1}X$. We know that $\tilde{v}_{-}$ and $\tilde{v}_{+}$ are in
the limit set $\Lambda$. There exists a sequence of geodesics $c_{n}$ on $X$
whose projection to $M$ is closed and such that $c_{n}(+\infty)$ converges to
$\tilde{v}_{+}$ and $c_{n}(-\infty)$ converges to $\tilde{v}_{-}$. We can
parametrize $c_{n}$ such that $c_{n}(0)$ is the point on $c_{n}$ closest to
the base point of $v$. We apply [Bal95, Lemma III.3.1]: for $R>0$, if $n$ is
big enough, any geodesic joining $c_{n}(-\infty)$ to $c_{n}(+\infty)$ is at
distance less than $R$ from $c(0)$. The vectors $\dot{c}_{n}(0)$ converge to
$v$. ∎

######  Remark 3.7.

If $v\in\Omega$ does not bound a flat half-plane, we can still apply the
previous argument by choosing $R>0$ larger than the width of the maximal strip
containing the geodesic generated by $v$. In this case, the argument shows
that there is a vector generating a geodesic biasymptotic to $v$ which is in
the closure of the set of hyperbolic periodic geodesics.

If $v\in\Omega$ and its geodesic bounds a maximal flat half-plane, then it is
also in the closure of the set of hyperbolic periodic geodesics. In fact, the
geodesic of $v$ bounds a cylindrical end. The periodic geodesics in the
interior of the cylinder are not in the closure of hyperbolic periodic
vectors. This is already studied in [CS11].

######  Proposition 3.8.

Let $v$ be a periodic vector in $T^{1}M$. Then, the half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ is dense in $\Omega^{+}$ if and only
if, for any lift $\tilde{v}$ of $v$ to $T^{1}X$, $\tilde{v}_{+}$ is not the
second end point of an interval in $\partial X\setminus\Lambda$.

###### Proof.

First, assume that $\tilde{v}_{+}$ is the second end point of an open interval
$I$ of $\partial X\setminus\Lambda$. Then the interval $I$ is bounded by the
points $\tilde{v}_{+}$ and $\tilde{v}_{-}$. So the end associated to $I$ is
bounded by a closed geodesic. For $s\geq 0$, $h_{s}(v)$ is based on the end
bounded by the closed geodesic generated by $v$. Moreover, the distance
between $h_{s}(v)$ and this geodesic tends to $+\infty$ when $s$ goes to
$+\infty$. So the half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}(v)$ is
divergent.

For the converse, we will adapt [Sch11, Proposition 3.12]. Assume that
$\tilde{v}_{+}$ is not the second end point of an interval in $\partial
X\setminus\Lambda$. Notice that the half-plane $X_{\tilde{v}}^{+}$ is not
flat. It is enough to prove the result with the additional assumption that the
geodesic generated by $\tilde{v}$ does not bound a flat strip in
$X^{+}_{\tilde{v}}$.

Step 1. We prove that
$\Omega_{NF}\subset\overline{g_{\operatorname{\mathbb{R}}}h_{\operatorname{\mathbb{R}}_{+}}(v)}$.
By Lemma 3.6, there exists a vector $w$ in $\Omega_{NF}$ whose negative orbit
$g_{\operatorname{\mathbb{R}}_{-}}(w)$ is dense in $\Omega_{NF}$. Since
$\tilde{v}_{+}$ is not the second end point of an interval in $\partial
X\setminus\Lambda$, we can lift $w$ to a vector $\tilde{w}$ in $T^{1}X$ with
$\tilde{w}_{-}$ in the interval $(\tilde{v}_{-},\tilde{v}_{+})$. Moreover,
there is a geodesic joining $\tilde{w}_{-}$ to $\tilde{v}_{+}$. We take the
vector $\tilde{u}\in h_{\operatorname{\mathbb{R}}}\tilde{v}$ tangent to this
geodesic, and we project it to a vector $u$ in $T^{1}M$. By the choice of the
point $\tilde{u}_{-}=\tilde{w}_{-}$, $u$ is in the positive horocyclic orbit
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ of $v$. By Lemma 3.3 the unstable
horocycle of $\tilde{w}$ is contracting in the past, so the negative geodesic
orbit of $u$ is also dense in $\Omega_{NF}$. Hence,
$\Omega_{NF}\subset\overline{g_{\operatorname{\mathbb{R}}}(u)}\subset\overline{g_{\operatorname{\mathbb{R}}}h_{\operatorname{\mathbb{R}}_{+}}(v)}$.

Claim. Let $\varepsilon>0$. There exists a $g_{t}$-periodic vector $u$ in
$T^{1}M$ whose geodesic does not bound a flat half-plane and such that there
exists $n,m\in\operatorname{\mathbb{N}}$ with $|ml(v)-nl(u)|<\varepsilon$.

###### Proof.

We first follow the proof of the nonarithmeticity of the length spectrum for
negatively curved surfaces by Dal’bo [Dal99]. There exist two isometries
$\gamma_{1}$ and $\gamma_{2}$ with distinct axis $c_{1}$ and $c_{2}$ that
intersect. Neither of the axis bounds a flat plane, so each has a repelling
fixed point $\gamma_{i}^{-}$ and an attracting fixed point $\gamma_{i}^{+}$ at
infinity. Up to taking the inverse of $\gamma_{1}$ we can assume that they are
ordered in counterclockwise sense as follows:
$\gamma_{1}^{+},\gamma_{2}^{-},\gamma_{1}^{-},\gamma_{2}^{+}$ (see Figure 3).
Consider the isometry $g=\gamma_{1}\gamma_{2}$. We observe that
$g[\gamma_{2}^{+},\gamma_{2}^{-}]\subset(\gamma_{2}^{+},\gamma_{2}^{-})$ and
$g^{-1}[\gamma_{2}^{-},\gamma_{2}^{+}]\subset(\gamma_{2}^{-},\gamma_{2}^{+})$.
The only type of isometry which achieves this is an axial isometry which does
not bound a flat half-plane. Moreover, the axis of $g$ intersects that of
$\gamma_{2}$ because of the position of the fixed points.

Figure 3: Order of the fixed points of $\gamma_{1}$ and $\gamma_{2}$ at
infinity.

Similarly, the isometry $g_{n}=\gamma_{1}\gamma_{2}^{n}$, $n\geq 0$, is axial,
its axis does not bound a flat half-plane and it intersects the axis of
$\gamma_{2}$. We observe that the attracting fixed point $g_{n}^{+}$ of
$g_{n}$ tends to $\gamma_{1}\gamma_{2}^{+}$ and the repelling fixed point
$g_{n}^{-}$ of $g_{n}$ tends to $\gamma_{2}^{-}$ when $n$ goes to $\infty$. We
have

$\displaystyle l(g_{n})-l(g_{n-1})$
$\displaystyle=\beta_{g_{n}^{-}}(o,\gamma_{2}^{-n}\gamma_{1}^{-1}o)+\beta_{g_{n-1}^{+}}(\gamma_{1}\gamma_{2}^{n-1}o,o)$
$\displaystyle=\beta_{g_{n}^{-}}(o,\gamma_{2}^{-1}o)+\beta_{g_{n}^{-}}(\gamma_{2}^{-1}o,\gamma_{2}^{-n}\gamma_{1}^{-1}o)+\beta_{g_{n-1}^{+}}(\gamma_{1}\gamma_{2}^{n-1}o,o)$
$\displaystyle=\beta_{g_{n}^{-}}(o,\gamma_{2}^{-1}o)+\beta_{g_{n}^{-}}(\gamma_{1}\gamma_{2}^{n-1}o,o)+\beta_{g_{n-1}^{+}}(\gamma_{1}\gamma_{2}^{n-1}o,o)$
$\displaystyle=\beta_{g_{n}^{-}}(o,\gamma_{2}^{-1}o)+2(g_{n}^{-}|g_{n-1}^{+})_{\gamma_{1}\gamma_{2}^{n-1}o}-2(g_{n}^{-}|g_{n-1}^{+})_{o}$
$\displaystyle=\beta_{g_{n}^{-}}(o,\gamma_{2}^{-1}o)+2(\gamma_{2}g_{n}^{-}|g_{n-1}^{+})_{o}-2(g_{n}^{-}|g_{n-1}^{+})_{o},$

which converges to $\beta_{\gamma_{2}^{-}}(o,\gamma_{2}^{-1}o)=l(\gamma_{2})$
when $n$ goes to infinity.

If $z_{n-1}$ is a point in the intersection of $c_{2}$ with the axis of
$g_{n-1}$, we have

$\displaystyle l(g_{n})$ $\displaystyle\leq d(g_{n}z_{n-1},z_{n-1})$
$\displaystyle<d(\gamma_{1}\gamma_{2}^{n}z_{n-1},\gamma_{1}\gamma_{2}^{n-1}z_{n-1})+d(\gamma_{1}\gamma_{2}^{n-1}z_{n-1},z_{n-1})=l(\gamma_{2})+l(g_{n-1}),$

where the last inequality is strict because the axis of $g_{n-1}$ is distinct
from that of $\gamma_{2}$.

Now, we apply Lemma 3.9, to see that, for every $\varepsilon^{\prime}>0$,
there exist $k,n^{\prime},m^{\prime}\in\operatorname{\mathbb{N}}$ such that

$0<|m^{\prime}l(\gamma_{2})-n^{\prime}l(g_{k})|<\varepsilon^{\prime}.$

Finally, we compare $l(v)$ with $l(\gamma_{2})$. If the ratio
$l(v)/l(\gamma_{2})$ is irrational, we can find
$n,m\in\operatorname{\mathbb{N}}$ such that

$0<|ml(v)-nl(\gamma_{2})|<\varepsilon$

by Dirichlet’s approximation theorem. If it is equal to a rational number
$p/q$, $p,q\in\operatorname{\mathbb{N}}^{*}$, then putting
$\varepsilon^{\prime}=\varepsilon/p$, $m=m^{\prime}q$ and $n=n^{\prime}$, we
obtain

$0<|ml(v)-nl(g_{k})|<\varepsilon.$

The vector $u$ in the claim is tangent to an axis of $\gamma_{2}$ or $g_{n}$.
∎

Step 2. We prove that
$g_{\operatorname{\mathbb{R}}}(v)\subset\overline{h_{\operatorname{\mathbb{R}}_{+}}(v)}$.

Let $\varepsilon>0$. We write $\varepsilon_{0}:=ml(v)-nl(u)$, where $u$, $n$
and $m$ are given by the claim. Since $v_{+}$ is not the second enpoint of an
interval in $\partial X\setminus\Lambda$, it is possible to take a lift
$\tilde{u}$ of $u$ to $T^{1}X$ such that $\tilde{u}_{-}$ is between
$\tilde{v}_{-}$ and $\tilde{v}_{+}$. We denote by $\gamma$ and $\gamma_{0}$
the isometries that translate the vectors $\tilde{v}$ and $\tilde{u}$
respectively.

Since $\tilde{v}_{+}$ is not the second end point of an interval in $\partial
X\setminus\Lambda$, for any point in $(\tilde{v}_{-},\tilde{v}_{+})$ there is
a geodesic joining it to $\tilde{v}_{+}$. geodesic. For
$j\in\operatorname{\mathbb{N}}$ big enough, $\gamma_{0}^{-j}\tilde{v}_{-}$ is
close to $\tilde{u}_{-}$. So we can consider a vector $\tilde{v}_{j}$ which
points positively to $\tilde{v}_{+}$, negatively to
$\gamma_{0}^{-j}\tilde{v}_{-}$ and $\tilde{v}_{j}\in
h_{\operatorname{\mathbb{R}}}\tilde{v}$. In fact $\tilde{v}_{j}$ is in the
positive half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}$ of
$\tilde{v}$. We also consider the vector $\tilde{w}\in
h_{\operatorname{\mathbb{R}}}\tilde{v}$ with $\tilde{w}_{-}=\tilde{u}_{-}$,
which is the limit of the vectors $\tilde{v}_{j}$ when $j$ goes to $+\infty$.

Figure 4: Elements of the proof of Step 2.

For $i\in\operatorname{\mathbb{N}}$, we define a new vector
$\tilde{u}_{i,j}=\gamma^{i}g_{-il(v)}\tilde{v}_{j}$ obtained by pushing
$\tilde{v}_{j}$ backwards by the geodesic flow a time $il(v)$ and then
bringing it back to $h_{\operatorname{\mathbb{R}}}\tilde{v}$ using the
isometry $\gamma$. The vectors $\tilde{u}_{i,j}$ are again in the positive
half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}(\tilde{v})$ of $\tilde{v}$.
We observe that, for $j\in\operatorname{\mathbb{N}}$ large enough and for all
$i\in\operatorname{\mathbb{N}}$, the vector
$\gamma_{0}^{j}\gamma^{-i}\tilde{u}_{i,j}$ is in the weak unstable manifold of
$\tilde{v}$. Wisely choosing $i$ and $j$, we will be able to control which
horocycle this vector belongs to.

We set the following notations:

$\displaystyle t_{0}:=$
$\displaystyle\beta_{\tilde{v}_{-}}(\pi(\tilde{v}),\pi(\tilde{u})),$
$\displaystyle t_{1,j}:=$
$\displaystyle\beta_{\gamma_{0}^{-j}\tilde{v}_{-}}(\pi(\tilde{u}),\pi(\tilde{v}_{j})),$
$\displaystyle t_{1}:=$
$\displaystyle\beta_{\tilde{u}_{-}}(\pi(\tilde{u}),\pi(\tilde{w})),$
$\displaystyle t_{2}:=$ $\displaystyle
2(\tilde{v}_{-}|\tilde{u}_{+})_{\pi(\tilde{u})},$ $\displaystyle t_{3,j}:=$
$\displaystyle-2(\gamma_{0}^{-j}\tilde{v}_{-}|\tilde{u}_{+})_{\pi(\tilde{u})},$
$\displaystyle t_{3}:=$
$\displaystyle-2(\tilde{u}_{-}|\tilde{u}_{+})_{\pi(\tilde{u})}.$

By continuity, $\lim_{j}t_{1,j}=t_{1}$ and $\lim_{j}t_{3,j}=t_{3}$. The
quantities $t_{2}$ and $t_{3}$ are finite, because there are geodesic joining
$\tilde{v}_{-}$ to $\tilde{u}_{+}$ and $\tilde{u}_{-}$ to $\tilde{u}_{+}$.
Finally, we let $T:=t_{0}+t_{1}+t_{2}+t_{3}$ and we choose $j_{0}$ large such
that $t_{0}+t_{1,j}+t_{2}+t_{3,j}$ is $\varepsilon$-close to $T$ for $j\geq
j_{0}$.

Using standard properties of Busemann functions, we get

$\displaystyle\beta_{\tilde{v}_{-}}(\pi(\tilde{v}),\gamma_{0}^{j}\gamma^{-i}\pi(\tilde{u}_{i,j}))=$
$\displaystyle\beta_{\tilde{v}_{-}}(\pi(\tilde{v}),\pi(g_{-il(v)}\gamma_{0}^{j}\tilde{v}_{j}))=\beta_{\tilde{v}_{-}}(\pi(\tilde{v}),\gamma_{0}^{j}\pi(\tilde{v}_{j}))+il(v)$
$\displaystyle=$ $\displaystyle
t_{0}+t_{1,j}+il(v)+\beta_{\tilde{v}_{-}}(\pi(\tilde{u}),\gamma_{0}^{j}\pi(\tilde{u}))$
$\displaystyle=$ $\displaystyle
t_{0}+t_{1,j}+il(v)+2(\tilde{v}_{-}|\tilde{u}_{+})_{\pi(\tilde{u})}-2(\tilde{v}_{-}|\tilde{u}_{+})_{\gamma_{0}^{j}\pi(\tilde{u})}$
$\displaystyle-\beta_{\tilde{u}_{+}}(\pi(\tilde{u}),\gamma_{0}^{j}\pi(\tilde{u}))$
$\displaystyle=$ $\displaystyle t_{0}+t_{1,j}+t_{2}+t_{3,j}+il(v)-jl(u).$

For $k\in\operatorname{\mathbb{N}}$, we put $j_{k}=j_{0}+km$ and $i_{k}=kn$.
We write $T^{\prime}=T-j_{0}l(u)$. There exists
$-\varepsilon<\alpha_{k}<\varepsilon$ such that
$g_{\alpha_{k}}\gamma_{0}^{j_{k}}\gamma^{-j_{k}}\tilde{u}_{i_{k},j_{k}}$ is in
the horocycle $H^{u}(g_{T^{\prime}+k\varepsilon_{0}}\tilde{v})$.

We project $\tilde{u}_{i_{k},j_{k}}$ to vectors $u_{i_{k},j_{k}}$ in the
quotient $T^{1}M$. We know that they are in the the positive half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ of $v$ and also in the weak unstable
leaf of $v$. Recall that we chose $\tilde{v}$ so that its geodesic does not
bound a flat strip in $X_{\tilde{v}}^{+}$. By Lemma 3.3, the positive unstable
half-horocycle of $v$ contracts in the past, so we can find
$r\in\operatorname{\mathbb{N}}$, depending on $k$, such that the distance
between $g_{-rl(v)+\alpha_{k}}u_{i_{k},j_{k}}\in
H^{u}(g_{T^{\prime}+k\varepsilon_{0}}v)$ and
$g_{T^{\prime}+k\varepsilon_{0}}v$ is less than $\varepsilon$. We deduce that
$g_{-rl(v)}u_{i_{k},j_{k}}\in h_{\operatorname{\mathbb{R}}}v$ is at distance
from $g_{T^{\prime}+k\varepsilon_{0}}v$ less than $2\varepsilon$. Since this
holds for all $k\in\operatorname{\mathbb{N}}$, we have shown that any vector
in the $g_{t}$-orbit of $v$ is $3\varepsilon$-close to a vector in
$h_{\operatorname{\mathbb{R}}_{+}}(v)$.

Conclusion. Putting together the previous steps, we have

$h_{\operatorname{\mathbb{R}}_{+}}(\Omega_{NF})\subset
h_{\operatorname{\mathbb{R}}_{+}}\overline{g_{\operatorname{\mathbb{R}}}h_{\operatorname{\mathbb{R}}_{+}}(v)}=\overline{h_{\operatorname{\mathbb{R}}_{+}}g_{\operatorname{\mathbb{R}}}(v)}\subset\overline{h_{\operatorname{\mathbb{R}}_{+}}(v)},$

which combined with Lemma 3.5 gives the conclusion.

∎

######  Lemma 3.9.

Let $\alpha>0$ and $(\beta_{k})$ be a sequence of real numbers such that, for
all $k$, $\beta_{k}-\beta_{k-1}<\alpha$ and
$\lim_{k}\beta_{k}-\beta_{k-1}=\alpha$. Then, for all $\varepsilon>0$, there
exist $k,n,m\in\operatorname{\mathbb{N}}$ such that

$0<|m\alpha-n\beta_{k}|<\varepsilon.$

###### Proof.

If $\beta_{k}/\alpha$ is irrational for some $k$, the inequality is just an
application of the Dirichlet’s approximation theorem.

Assume that all the quotients $\beta_{k}/\alpha$ are rational. We can find
coprime natural numbers $p_{k}$ and $q_{k}$ such that
$\beta_{k}/\alpha=p_{k}/q_{k}$. First, we observe that $q_{k}$ must be
unbounded. Indeed, if they were bounded by some $q$, the numbers
$\beta_{k}/\alpha$ would be in $1/Q\operatorname{\mathbb{N}}$ with
$Q=\prod_{i=1}^{q}i$. But this would imply that $\beta_{k}-\beta_{k-1}=\alpha$
for large $k$, contradicting the hypothesis.

We fix $k$ such that $\alpha/q_{k}<\varepsilon$. By Bézout’s identity, there
exist $n,m\in\operatorname{\mathbb{N}}$ such that $|mq_{k}-np_{k}|=1$.
Therefore $|m\alpha-n\beta_{k}|=\alpha/q_{k}<\varepsilon$. ∎

Using the terminology of [Sch11], we next prove that a radial point which is
not the second end point of an interval is right horocyclic. Given
$\xi\in\partial X$, $p\in X$, and $R>0$, we consider the set of points in the
horoball $B_{\xi}(p)$ which are at distance greater that $R$ from the geodesic
joining $p$ to $\xi$. This set has to connected component. We denote the
connected component which is in the positive half-plane bounded by the
geodesic from $p$ to $\xi$ by $B_{\xi}^{+}(p,R)$.

######  Lemma 3.10.

Let $\xi$ be a limit point which and is not the second end point of an
interval of $\partial X\setminus\Lambda$. Assume that $\xi$ is not the center
of a parabolic isometry. Fix $o\in X$. For every $p\in X$ and every $R>0$,
there exists $\gamma\in\Gamma$ such that $\gamma o\in B_{\xi}^{+}(p,R)$.

###### Proof.

The proof goes by contraposition. Suppose that, for some $p\in X$ and some
$R>0$, $\Gamma o$ does not intersect $B_{\xi}^{+}(p,R)$. Let $d$ be the
diameter of the compact part $K$ of $M$. Then $\Pi^{-1}(K)$ does not intersect
the set $B_{\xi}^{+}(c_{p,\xi}(d),R+d)$. So $B_{\xi}^{+}(c_{p,\xi}(d),R+d)$
must be contained in a connected component $Y_{0}$ of the complement in $X$ of
the set $\Pi^{-1}(K)$ which is a lift of a neighborhood $E_{i}$ of an end.
This implies that the end point $\xi$ is in $\overline{Y_{0}}$. If $E_{i}$ is
simple parabolic, then $\xi$ is a simple parabolic limit point.

If $E_{i}$ is cylindrical, expanding or exceptional parabolic, then $\xi$ is
necessarily the first end point of the interval
$I=\overline{Y_{0}}\cap\partial X$. The first two cases are impossible, since
the set $B_{\xi}^{+}(c_{p,\xi}(d),R+d)$ would not be contained in $Y_{0}$. The
case that $E_{i}$ is exceptional parabolic and $\xi$ is not the center of the
horoball $Y_{0}$ is not possible for the same reason. The only remaining case
is that $E_{i}$ is exceptional parabolic and $\xi$ is the center of an
exceptional parabolic isometry. ∎

######  Proposition 3.11.

Let $v\in\Omega_{+}$. Assume that the horocycle of $v$ is not periodic. Then,
the half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}(v)$ is dense in
$\Omega^{+}$ if and only if, for any lift $\tilde{v}$ of $v$ to $T^{1}X$,
$\tilde{v}_{+}$ is not the second end point of an interval in $\partial
X\setminus\Lambda$.

###### Proof.

First, we prove that if $\tilde{v}_{+}$ is the second end point of an interval
of $\partial X\setminus\Lambda$, the half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}v$ converges to an end of the surface. We
consider the interval $I$ of $\partial X\setminus\Lambda$ bounded by
$\tilde{v}_{+}$. This interval corresponds to an end of the surface
$\varepsilon$. We take take a standard neighborhood $E$ of that end, and
consider the connected component $\tilde{E}$ of the lift $\Pi^{-1}(E)$
corresponding to the interval $I$. Recall that the boundary of $\tilde{E}$
consists of a geodesic or a horocycle joining the two end points of $I$.

Let us treat the case that $\varepsilon$ is expanding. The horocycle
$\pi(h_{\operatorname{\mathbb{R}}}\tilde{v})$ cuts the geodesic $\tilde{E}$
orthogonally at a unique point $x$. Let $\tilde{v}^{\prime}$ be the vector in
$h_{\operatorname{\mathbb{R}}}\tilde{v}$ with basepoint $x$. The half-
horocycle $h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime}$ stays in
$\tilde{E}$ and diverges from the geodesic boundary $\partial\tilde{E}$. On
the surface $M$, this means that the half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}v^{\prime}$ diverges through the end
$\varepsilon$.

Now we deal with the case that $\varepsilon$ is exceptional parabolic or
cylindrical. The boundary of $\tilde{E}$ is a horocycle which projects to a
closed one, so its center is different from $\tilde{v}_{+}$. Let $x$ be any
point in $\partial\tilde{E}$ and consider the vectors $\tilde{v}^{\prime}$ and
$\tilde{w}$ based at $x$ pointing $\tilde{v}_{+}$ and the center of
$\partial\tilde{E}$, respectively. Since there exists no geodesic joining
$\tilde{v}_{+}$ and the center of $\partial\tilde{E}$, which is in $\bar{I}$,
the two horocycles $\pi(h_{\operatorname{\mathbb{R}}}\tilde{v}^{\prime})$ and
$\partial\tilde{E}=\pi(h_{\operatorname{\mathbb{R}}}\tilde{w})$ intersect just
once. Therefore, $\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime})$ is
contained in $\tilde{E}$. By Proposition 2.26, it diverges from
$\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{w})$. Moreover,
$\pi(g_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime})$ stays in the
horoball $\tilde{E}$ and diverges from
$\pi(h_{\operatorname{\mathbb{R}}_{-}}\tilde{w})$. Since
$\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime})$ is on the half-
plane $X^{+}_{\tilde{v}^{\prime}}$,
$\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime})$ also diverges from
$\pi(h_{\operatorname{\mathbb{R}}_{-}}\tilde{w})$. The horocycle
$\pi(h_{\operatorname{\mathbb{R}}}\tilde{v})$ is equidistant from
$\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}^{\prime})$, so we also know
that $\pi(h_{\operatorname{\mathbb{R}}_{+}}\tilde{v})$ eventually enters the
region $\tilde{E}$ and diverges from $\partial\tilde{E}$. This proves that
$h_{\operatorname{\mathbb{R}}_{+}v}$ converges to $\varepsilon$.

Now, we assume that $\tilde{v}_{+}$ is not the second end point of an interval
in $\partial X\setminus\Lambda$ and we want to show that
$h_{\operatorname{\mathbb{R}}_{+}}v$ is dense in $\Omega^{+}$. We can choose a
reference point $o$ on a lift of the interior of the compact part of $M$. No
geodesic passing through $o$ bounds an end of the manifold. If $c$ is a
geodesic starting at $o$, the two intervals $(c(-\infty),c(+\infty))$ and
$(c(+\infty),c(-\infty))$ necessarily intersect the limit set $\Lambda$.

By Lemma 3.10, there exists a sequence $\gamma_{n}$ in $\Gamma$ such that
$\gamma_{n}o$ stays in the half-plane $X^{+}_{\tilde{v}}$ and converges to the
end point $\tilde{v}_{+}$, and $t_{n}:=\beta_{\tilde{v}_{+}}(o,\gamma_{n}o)$
tends to $+\infty$. For $n\in\operatorname{\mathbb{N}}$, we consider the
vector $\tilde{w}_{n}$ based on $\gamma_{n}o$ pointing at $\tilde{v}_{+}$.
Notice that $\tilde{v}_{n}:=g_{-t_{n}}\tilde{w}_{n}$ is in the positive stable
horocycle of $v$. We write $\tilde{v}_{n}=h_{s_{n}}(\tilde{v})$ for some
$s_{n}\geq 0$.

The sequence of vectors $\gamma_{n}^{-1}\tilde{w}_{n}$ belongs to
$T^{1}_{o}X$. Up to taking a subsequence we can assume that it converges to a
vector $u\in T^{1}_{o}X$. The sequence $\gamma_{n}^{-1}\tilde{v}_{+}$
converges to $\tilde{u}_{+}$, which is also in the limit set $\Lambda$.

By the observation made above, the limit set intersects the interval
$(\tilde{u}_{-},\tilde{u}_{+})$. Let $p$ be a $g_{t}$-periodic rank $1$ vector
on $T^{1}M$ whose geodesic does not bound an end of the surface $M$. We choose
a lift $\tilde{p}$ of $p$ to $T^{1}X$ such that $\tilde{p}_{-}$ is in the
interval $(\tilde{u}_{-},\tilde{u}_{+})$. By Proposition 3.8, the half-
horocycle $h_{\operatorname{\mathbb{R}}_{+}}p$ is dense in $\Omega^{+}$. The
negative end point $\tilde{p}_{-}$ can be joined to any other point at
infinity. We consider the vector $\tilde{z}$ in
$h_{\operatorname{\mathbb{R}}}\tilde{u}$ with $\tilde{z}_{-}=\tilde{p}_{-}$
and the vectors $\tilde{w}_{n}^{\prime}\in
h_{\operatorname{\mathbb{R}}}(\gamma_{n}^{-1}w_{n})$ with
$\tilde{w}_{n-}^{\prime}=\tilde{p}_{-}$. Since the geodesic of $\tilde{z}$
does not bound a flat strip, the vectors $\tilde{w}_{n}^{\prime}$ tend to
$\tilde{z}$ when $n$ goes to infinity. There exist real numbers
$s>0,s_{n}^{\prime}\in\operatorname{\mathbb{R}}$ such that
$\tilde{z}=h_{s}(\tilde{u})$,
$\tilde{w}_{n}^{\prime}=h_{s_{n}^{\prime}}(\gamma_{n}^{-1}\tilde{w}_{n})$ and
$\lim_{n}s_{n}^{\prime}=s$.

Figure 5: Elements of the proof of Proposition 3.11

Now, since

$\gamma_{n}\tilde{w}_{n}^{\prime}=h_{s_{n}^{\prime}}(\tilde{w}_{n})=h_{s_{n}^{\prime}}g_{t_{n}}\tilde{v}_{n}=h_{s_{n}^{\prime}}g_{t_{n}}h_{s_{n}}\tilde{v},$

and $s_{n}^{\prime}$ is positive for $n$ large enough, we have

$\tilde{v}_{n}^{\prime}:=g_{-t_{n}}\gamma_{n}\tilde{w}_{n}^{\prime}=g_{-t_{n}}h_{s_{n}^{\prime}}g_{t_{n}}h_{s_{n}}\tilde{v}\in
h_{\operatorname{\mathbb{R}}_{+}}(\tilde{v}).$

On the other hand,

$\gamma_{n}^{-1}\tilde{v}_{n}^{\prime}\in
W^{wu}(\tilde{w}_{n}^{\prime})=W^{wu}(\tilde{p}).$

Since $\tilde{p}$ has rank $1$, the weak unstable leaf of $\tilde{p}$
contracts in the past. Putting
$\bar{t}:=\beta_{\tilde{p}_{-}}(\pi(\tilde{z}),\pi(\tilde{p}))$, we have

$d(\gamma_{n}^{-1}\tilde{v}_{n}^{\prime},g_{-t_{n}+\bar{t}}\,\tilde{p})=d(g_{-t_{n}}\tilde{w}_{n}^{\prime},g_{-t_{n}+\bar{t}}\,\tilde{p})\xrightarrow[n\to+\infty]{}0.$

In the quotient $T^{1}M$, this means that the projections $v_{n}^{\prime}\in
h_{\operatorname{\mathbb{R}}_{+}}(v)$ of $\tilde{v}_{n}^{\prime}$ approach the
periodic orbit of $p$. So they accumulate to a $g_{t}$-periodic vector with
dense positive half-horocycle. This finishes the proof.

∎

####  3.2.2 Half-horocycles and collared ends

Consider an interval $I$ in the complement of the limit set $\Lambda$ that
corresponds to a cylindrical or exceptional parabolic. Recall that there
exists an horocycle with center in the closure of $I$ whose projection to the
surface $M$ is a closed curve that bounds a neighborhood of the end associated
to $I$. In the next proposition we study horocyles centered at points in $I$
different from this distinguished point.

######  Proposition 3.12.

Let $I=(\eta_{1},\eta_{2})$ be a interval corresponding to a cylindrical or
exceptional parabolic end. Let $\eta\in[\eta_{1},\eta_{2}]$ be the center of
the isometry associated to the end. Let $v\in T^{1}M$ such that, for any lift
$\tilde{v}$ of $v$ to $T^{1}X$, $\tilde{v}_{+}\in[\eta_{1},\eta_{2}]$.

  * •

If $\tilde{v}_{+}\in[\eta_{1},\eta)$, then the the positive half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ is dense in $\Omega^{+}$.

  * •

If $\tilde{v}_{+}\in(\eta,\eta_{2}]$, then the the positive half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}(v)$ is divergent.

###### Proof.

Let $H$ be a horocycle in $X$ centered at $\eta$ whose projection to $M$
bounds a neighborhood $E_{i}$ of the end associated to $I$. The fact that a
half-horocycle be dense in $\Omega^{+}$ or divergent does only depend on its
center. Without loss of generality, we can assume that the basepoint $x$ of
$\tilde{v}$ is in $H$. Let $H^{\prime}$ be the horocycle centered at
$\tilde{v}_{+}$ passing through $x$. The angle at $x$ from $H$ to $H^{\prime}$
is positive if $\tilde{v}_{+}\in(\eta,\eta_{2}]$ and negative if
$\tilde{v}_{+}\in[\eta_{1},\eta)$. Since $H$ and $H^{\prime}$ only intersect
at $x$ by Proposition 2.5, the half-horocycle
$\pi(h_{\operatorname{\mathbb{R}}_{+}}v)$ stays inside the horoball bounded by
$H$ if $\tilde{v}_{+}\in(\eta,\eta_{2}]$ and stays outside of it if
$\tilde{v}_{+}\in[\eta_{1},\eta)$. Moreover, by Proposition 2.26, the distance
between $\pi(h_{s}(v))$ and $H$ goes to infinity when $s\to+\infty$. This
shows that the half-horocycle $h_{\operatorname{\mathbb{R}}_{+}}v$ is
divergent when $\tilde{v}_{+}\in(\eta,\eta_{2}]$.

We assume that $\tilde{v}_{+}\in(\eta,\eta_{2}]$ and show that the half-
horocycle is dense. Let $\gamma$ be the exceptional parabolic or axial
isometry associated to $I$. This isometry fixes $H$, and up to taking the
inverse, we can assume that $\gamma^{n}x$ tends to $\eta_{1}$ when $n$ goes to
infinity. For $n\in\operatorname{\mathbb{N}}$, consider the vector $w_{n}$
with basepoint $\gamma^{n}x$ pointing to $\tilde{v}_{+}$. Observe that
$\tilde{w}_{n}=\gamma^{n}\tilde{v}$. We put
$t_{n}:=\beta_{\tilde{v}_{+}}(x,\gamma^{n}x)$ and consider the vectors
$\tilde{v}_{n}:=g_{-t_{n}}\tilde{w}_{n}\in
h_{\operatorname{\mathbb{R}}}\tilde{v}$. Moreover, since $\gamma$ fixes
$\tilde{v}_{+}$,

$t_{n}=n\beta_{\tilde{v}_{+}}(x,\gamma x),$

where $\beta_{\tilde{v}_{+}}(x,\gamma x)$ is strictly positive. Also,
$\gamma^{n}x$ is in the half plane $X^{+}_{\tilde{v}}$, so $\tilde{v}_{n}$ is
in the positive half-orbit $h_{\operatorname{\mathbb{R}}_{+}}\tilde{v}$. We
remark that $\gamma^{-n}\tilde{v}_{n}=g_{-t_{n}}(\tilde{v})$.

Let $p$ be a $g_{t}$-periodic rank $1$ vector on $T^{1}M$ whose geodesic does
not bound an end of the surface $M$. We choose a lift $\tilde{p}$ of $p$ to
$T^{1}X$ such that $\tilde{p}_{-}$ is in the interval
$(\tilde{v}_{-},\tilde{v}_{+})$. By Proposition 3.8, the half-horocycle
$h_{\operatorname{\mathbb{R}}_{+}}p$ is dense in $\Omega^{+}$. The negative
end point $\tilde{p}_{-}$ can be joined to any other point at infinity. We
consider the vector $\tilde{z}$ in $h_{\operatorname{\mathbb{R}}}\tilde{v}$
with $\tilde{z}_{-}=\tilde{p}_{-}$. For every $n\in\operatorname{\mathbb{N}}$,
there exists $s_{n}\geq 0$, such that
$h_{s}(\gamma^{-n}\tilde{v}_{n})=g_{-t_{n}}(\tilde{z})$. As in the proof of
Proposition 3.11, the projection $v_{n}$ of $\tilde{v}_{n}$ to $T^{1}M$ tends
to a vector in the periodic orbit of $p$. Since the positive half-orbit of $p$
is dense in $\Omega^{+}$, we get the desired result.

∎

###  3.3 Closure of horocyclic orbits

######  Proposition 3.13.

Any accumulation point of an orbit $\Gamma\xi$, $\xi\in\partial X$, is in the
limit set $\Lambda$.

###### Proof.

We reason by contradiction. Assume that there exists a sequence
$(\gamma_{n})_{n}$ of pairwise distinct isometries in $\Gamma$ such that
$\gamma_{n}\xi$ converges to $\eta\in\partial X\setminus\Lambda$. The point
$\eta$ belongs to an open interval $I$ of $\partial X\setminus\Lambda$. There
exists an $n_{0}$ such that, for $n\geq n_{0}$, $\gamma_{n}\xi$ belongs to
$I$. Since $g_{n}=\gamma_{n}\gamma_{n_{0}}^{-1}$ sends an element of $I$ to
another, it leaves invariant $I$, but is different from the identity by
assumption. We observe that $g_{n}$, $n\geq n_{0}$, can never be simple
parabolic.

If $g_{n_{0}}$ is axial, the only way that it leave $I$ invariant is that the
extremities of $I$ are the attracting and the repelling fixed points of
$g_{n_{0}}$. So $I$ is associated to an expanding end of the surface. This
implies that all the $g_{n}$ for $n\geq n_{0}$ are powers of a common axial
isometry $\gamma$. The only possible accumulation points of
$\langle\gamma\rangle\gamma_{n_{0}}\xi$ are the extremities of $I$, which are
in the limit set.

If $g_{n_{0}}$ is exceptional parabolic, the only way that it leave $I$
invariant is that $I$ is equal to the set of fixed points of $g_{n_{0}}$. So
$I$ is associated to an exceptional parabolic end of the surface. This implies
that all the $g_{n}$ for $n\geq n_{0}$ are powers of a common exceptional
parabolic isometry $\gamma$ which fixes $I$. Therefore, the sequence
$\gamma_{n}\xi=g_{n}\gamma_{n_{0}}\xi$ is constant for $n\geq n_{0}$ and equal
to $\eta$. Thus $\eta$ is not an accumulation point.

∎

As a consequence of Proposition 3.13, there is a unique minimal subset for the
action of $\Gamma$ on $\partial X$.

######  Proposition 3.14.

Any accumulation point of an orbit $h_{\operatorname{\mathbb{R}}}v$ is in
$\Omega^{+}$.

###### Proof.

It is an application of the duality between the action of the horocyclic flow
on $T^{1}M$ and the action of $\Gamma$ on $\partial X$. ∎

######  Definition 3.15.

A geodesic ray $c:\operatorname{\mathbb{R}}_{+}\to M$ is eventually minimizing
if there exists $t_{0}$ such that for $t,t^{\prime}\geq t_{0}$,

$d(c(t^{\prime}),c(t))=|t^{\prime}-t|.$

We notice that an eventually minimizing geodesic ray converges to an end of
the manifold. If the end is parabolic or cylindrical, the geodesic is
perpendicular to the periodic horocycles that bound the end.

Now we can finally prove the main result of the article.

_Proof of Theorem 1.1._ Let $g_{\operatorname{\mathbb{R}}_{+}}v$ be a geodesic
ray that converges to an end $\varepsilon$. Consider a parametrization in
coordinates $(r,\theta)$ of a neighborhood of $\varepsilon$ given by Theorems
2.15 and 2.16. First, we assume that the end is parabolic or cylindrical. The
only eventually minimizing geodesic rays that converge to $\varepsilon$ are
those that when they enter the neighborhood of $\varepsilon$ are $r$-curves.
Any horocycle associated to a vector tangent to such curves is periodic.

The other geodesic rays that converge to $\varepsilon$ are not eventually
minimizing. By Proposition 3.12, one half-horocycle associated to vectors
tangent to these rays is dense in $\Omega^{+}$. Thus, we have the inclusion
$h_{\operatorname{\mathbb{R}}}v\cup\Omega^{+}\subset\overline{h_{\operatorname{\mathbb{R}}}v}$.
Proposition 3.14 gives the other inclusion.

If the end is expanding, then by Proposition 2.22 the horocycle of $v$
converges to the end $\varepsilon$.

Finally, if $g_{\operatorname{\mathbb{R}}_{+}}v$ does not converge to any end,
then any lift of $v$ to $T^{1}X$ points to a non parabolic limit point. In
particular $v$ belongs to $\Omega^{+}$. By Proposition 2.28, the limit point
is radial and by Proposition 3.11, at least one half-horocycle of $v$ is dense
in $\Omega^{+}$. $\square$

###  3.4 Minimal subsets of the horocyclic flow

We now seek $h_{s}$-invariant closed nonempty subsets of $T^{1}M$ which are
minimal with this property. Periodic noncompact closed orbits of $h_{s}$ are
the trivial examples of such sets. We obtain Corollary 1.2, which asserts that
there is at most one nontrivial minimal subset and gives a necessary and
sufficient condition for its existence.

_Proof of Corollary 1.2._ The set $\Omega^{+}$ is always $h_{s}$-invariant,
closed and nonempty. According to Proposition 2.28 and Lemma 2.11, saying that
all the points $\Lambda$ are horocyclic is equivalent to asking that $\Lambda$
contains no center of a parabolic isometry. If $\Lambda$ does not contain
centers of parabolic isometries, then no $v$ in $\Omega_{+}$ has a periodic
horocycle. By Proposition 3.11, every $v$ in $\Omega^{+}$ has dense orbit in
$\Omega^{+}$, so $\Omega^{+}$ is minimal.

For the converse, a minimal subset is equal to the closure of the orbit of any
of its elements. Assume that $\overline{h_{\operatorname{\mathbb{R}}}v}$ is
minimal but $h_{\operatorname{\mathbb{R}}}v$ is not closed. Then
$h_{\operatorname{\mathbb{R}}}v$ accumulates somewhere in $\Omega^{+}$, so in
fact $\Omega^{+}$ is contained in $\overline{h_{\operatorname{\mathbb{R}}}v}$.
Since the latter is minimal, they must be equal. But for $\Omega^{+}$ to be
minimal, there cannot be periodic horocycles in $\Omega^{+}$, put otherwise,
$\Lambda$ has no centers of parabolic isometries. $\square$

The condition of the corollary amounts to ask that $M$ has neither simple
parabolic ends nor exceptional parabolic ends with center in the limit set. We
do not know if the latter can occur.

###  3.5 Nonwandering set of the horocyclic flow

In this section, we prove that the nonwandering set of the horocyclic flow is

$\operatorname{NW}(h_{s})=\Omega^{+}\cup\operatorname{Per}(h_{s}).$

_Proof of Theorem 1.3._ First, we prove the inclusion to the left. Periodic
vectors are always nonwandering and nonperiodic vectors in $\Omega_{+}$ have
dense orbits in $\Omega^{+}$, which implies that they are also nonwandering
(in fact, recurrent) because the $h_{s}$ orbits of $\Omega^{+}$ are not
isolated. We can see this as follows. Let $v\in\Omega^{+}$ be nonperiodic and
let $U$ be a neighborhood of $v$ in $\Omega^{+}$. For any $R>0$, the set
$U\setminus h_{[-R,R]}(v)$ is always nonempty, for example, it contains
$g_{t}(v)$ for $t$ small. By the density of $h_{\operatorname{\mathbb{R}}}v$,
there exists $s\in\operatorname{\mathbb{R}}$ such that $h_{s}(v)$ is in
$U\setminus h_{[-R,R]}(v)$, so $|s|>R$. So $h_{\operatorname{\mathbb{R}}}v$
intersects every neighborhood of $v$ with arbitrarily large times.

Now, let us show that a vector $v$ which is not in
$\Omega^{+}\cup\operatorname{Per}(h_{s})$ is wandering. We take a lift
$\tilde{v}$ to $T^{1}X$. Its end point $\tilde{v}_{+}$ is outside the limit
set and is not fixed by an exceptional parabolic isometry. Let $I$ be the
interval of $\partial X\setminus\Lambda$ that contains $\tilde{v}_{+}$ and let
$\varepsilon$ be the end associated to it. Let $\tilde{E}$ be the lift of a
standard neighborhood of $\varepsilon$, which consists of a horoball or a
half-plane. Its boundary $\partial\tilde{E}$ is a geodesic or a horocycle.

We distinguish two cases. If the end $\varepsilon$ is expanding, we consider
any open interval $J$ containing $\tilde{v}_{+}$ relatively compact in $I$. If
the end is exceptional parabolic or cylindrical, we also consider an open
interval $J$ containing $\tilde{v}_{+}$ relatively compact in $I$, but we ask
in addition that $\overline{J}$ does not contain the center of the horoball
$\tilde{E}$.

Given a small $\varepsilon>0$, we consider an open neighborhood of $\tilde{v}$
of the form

$V=\\{\tilde{w}\in T^{1}X|\pi(\tilde{w})\in
B(\pi(v),\varepsilon),\,\tilde{w}_{+}\in J\\}.$

First we deal with the parabolic or cylindrical case. By Proposition 2.26, the
distance between the horocycle $\partial\tilde{E}$ and $\pi(h_{s}(v)))$ goes
to infinity for both $s\to\pm\infty$. Since $\tilde{v}_{+}\in I$, for positive
or negative large time $s$, $\pi(h_{s}(\tilde{v}))$ is the horoball
$\tilde{E}$ and stays inside for larger times. We denote the sign of the time
$s$ by $\delta\in\\{-1,1\\}$. In fact, the same result implies the following
property: for every $\tilde{w}\in\overline{V}$, there exists $s_{0}>0$ such
that for every $s\in\operatorname{\mathbb{R}}$ such that $\delta s\geq s_{0}$,
$h_{s}(\tilde{w})$ is based on $\tilde{E}$ and

$\lim_{s\to\delta\cdot\infty}d(\partial\tilde{E},\pi(h_{s}(\tilde{w})))=+\infty.$

This property is also true when $\varepsilon$ is an expanding end for both
choices of the sign $\delta$ by Proposition 2.22. Since $\tilde{V}$ is
compact, we can find $s_{0}$ such that for $s$ such that $\delta s\geq s_{0}$,
$\pi(h_{s}(\overline{V}))\subset\tilde{E}$ and

$\inf_{\tilde{w}\in
h_{s}(\overline{V})}d(\partial\tilde{E},\tilde{w})>\sup_{\tilde{w}\in\overline{V}}d(\Gamma\partial\tilde{E},\tilde{w}).$

This guarantees that if we take the projection $U$ of $V$ to the quotient
$T^{1}M$, for $\delta s\geq s_{0}$, $h_{s}(U)$ does not intersect $U$. This
says that the vector $v$ is wandering.

$\square$

## References

  * [Bal82] Werner Ballmann.  Axial isometries of manifolds of nonpositive curvature.  Math. Ann., 259(1):131–144, 1982.
  * [Bal95] Werner Ballmann.  Lectures on spaces of nonpositive curvature, volume 25 of DMV Seminar.  Birkhäuser Verlag, Basel, 1995.  With an appendix by Misha Brin.
  * [BGS85] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder.  Manifolds of nonpositive curvature, volume 61 of Progress in Mathematics.  Birkhäuser Boston, Inc., Boston, MA, 1985.
  * [BH99] Martin R. Bridson and André Haefliger.  Metric spaces of non-positive curvature, volume 319 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [CS10] Yves Coudene and Barbara Schapira.  Generic measures for hyperbolic flows on non-compact spaces.  Israel J. Math., 179:157–172, 2010.
  * [CS11] Yves Coudène and Barbara Schapira.  Counterexamples in non-positive curvature.  Discrete Contin. Dyn. Syst., 30(4):1095–1106, 2011.
  * [CS14] Yves Coudène and Barbara Schapira.  Generic measures for geodesic flows on nonpositively curved manifolds.  J. Éc. polytech. Math., 1:387–408, 2014.
  * [Dal99] Françoise Dal’bo.  Remarques sur le spectre des longueurs d’une surface et comptages.  Bol. Soc. Brasil. Mat. (N.S.), 30(2):199–221, 1999.
  * [Dal00] Françoise Dal’bo.  Topologie du feuilletage fortement stable.  Ann. Inst. Fourier (Grenoble), 50(3):981–993, 2000.
  * [Ebe79] Patrick Eberlein.  Surfaces of nonpositive curvature.  Mem. Amer. Math. Soc., 20(218):x+90, 1979.
  * [EO73] P. Eberlein and B. O’Neill.  Visibility manifolds.  Pacific J. Math., 46:45–109, 1973.
  * [HIH77] Ernst Heintze and Hans-Christoph Im Hof.  Geometry of horospheres.  J. Differential Geometry, 12(4):481–491, 1977.
  * [Ker23] B. Kerékjártó.  Vorlesungen über Topologie, volume 8 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer Berlin, Heidelberg, 1923.
  * [Kni98] Gerhard Knieper.  The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds.  Ann. of Math. (2), 148(1):291–314, 1998.
  * [LPP06] Gabriele Link, Marc Peigné, and Jean-Claude Picaud.  Sur les surfaces non-compactes de rang un.  Enseign. Math. (2), 52(1-2):3–36, 2006.
  * [Sch11] Barbara Schapira.  Density and equidistribution of half-horocycles on a geometrically finite hyperbolic surface.  J. Lond. Math. Soc. (2), 84(3):785–806, 2011.