Strong stable sets and asymptotic fineness

Introduction

We consider the geodesic flow $g_t$ on the unit tangent bundle $T^1S$ of a hyperbolic surface $S$. We endow $T^1S$ with the Sasaki distance $d_{Sa}$ induced by the hyperbolic metric of $S$. For $u \in T^1S$, the stable horocycle $H^{ss}_u$ is defined as the projection of the stable horocycle of any lift of $u$ to $T^1\mathbb{H}$, whereas the strong stable set $W^{ss}_u$ is the set

Following Bellis [6], we define the asymptotic fineness of the geodesic ray $u(\mathbb{R}^+)$ as

where $\operatorname{Inj}(x)$ denotes the injectivity radius of $S$ at $x \in S$. Our main result establishes a link between the asymptotic fineness and strong stable sets.

There are several ways for the asymptotic fineness of $u(\mathbb{R}^+)$ to vanish. Bellis [5] and the authors together with S. Herrero [BCDHV26] study the case where $u(\mathbb{R}^+)$ meets a sequence of arbitrarily short closed geodesics. In [2], the authors study the case where $u(\mathbb{R}^+)$ meets a sequence of arbitrarily short closed horocycles. The contribution of this article is to study a third possible case, which is not always covered by the previous works: the ray $u(\mathbb{R}^+)$ meets closed hypercycles of arbitrarily small length.

These three cases are exhaustive: if the asymptotic fineness of $u(\mathbb{R}^{+})$ vanishes and $u(\mathbb{R}^{+})$ does not diverge through a cusp, then at least one of them occurs (Proposition 3). However, they are not mutually exclusive.

The proof of Theorem 2 follows the strategy of [BCDHV26] and [2]. We adapt to closed hypercycles the notion of winding used in the previous works. One novelty is the use of the hyperbolic band model, in which the hypercycles around a given geodesic become horizontal lines, to obtain the necessary estimates.

The paper is organized as follows. In § 2 we recall the facts of hyperbolic geometry that will be used. In § 3 we prove the trichotomy (Proposition 3), we deduce Theorem 1 from Theorem 2 and the previous works, and we give an example of the third case. In § 4 we introduce the winding around a closed hypercycle and we prove the Key Proposition (Proposition 5), which controls the distance between a geodesic ray and its winding. § 5 is devoted to the proof of Theorem 2, where we construct uncountably many vectors in $W^{ssu}\setminus H^{ssu}$.

Preliminaries

Throughout the paper we use the upper half-plane model $\mathbb{H}=\{z=x+iy\in\mathbb{C}:y>0\}$ equipped with the metric

The Riemannian distance induced by this metric is denoted by $d$. We denote the unit tangent bundle of $\mathbb{H}$ by $T^{1}\mathbb{H}$, and the basepoint projection by $\pi:T^{1}\mathbb{H}\to\mathbb{H}$.

Horocycles and boundary at infinity

The boundary at infinity of $\mathbb{H}$ is $\partial\mathbb{H}=\mathbb{R}\cup\{\infty\}$. Every geodesic ray in $\mathbb{H}$ converges to a unique point of $\partial\mathbb{H}$, and two rays define the same point at infinity if and only if they remain at bounded distance from each other.

Let $\xi\in\partial\mathbb{H}$, and let $(c(t))_{t\geq0}$ be a geodesic ray converging to $\xi$. The Busemann function based at $\xi$ is defined by

This limit exists and does not depend on the choice of the geodesic ray $c$ asymptotic to $\xi$. For fixed $x\in\mathbb{H}$, the level sets of the map $y\mapsto B_{\xi}(y,x)$ are called horocycles centered at $\xi$. Geometrically, horocycles are circles tangent to the real line or horizontal lines.

For $u\in T^{1}\mathbb{H}$, we denote by $u(+\infty)\in\partial\mathbb{H}$ its forward endpoint. The stable horocycle through $u$ is the subset

We will also denote

The following lemma can be proved by elementary computations using the formula for the hyperbolic distance [4 Theorem 7.2.1].

Distance on the unit tangent bundle

We consider the distance $d_1$ on the unit tangent bundle of $T^1\mathbb{H}$ defined as follows [1]:

Fuchsian groups

Throughout the paper, $\Gamma$ denotes a torsion-free Fuchsian group acting properly discontinuously by isometries on $\mathbb{H}$. We consider the quotient surface $S = \Gamma\backslash\mathbb{H}$ and its unit tangent bundle $T^1S = \Gamma\backslash T^1\mathbb{H}$. We use a tilde to denote lifts to the universal cover: given $u \in T^1S$, we write $\widetilde{u} \in T^1\mathbb{H}$ for a unit tangent vector projecting onto $u$ under the canonical projection $T^1\mathbb{H} \to T^1S$.

The sets $H^{ssu}$ and $H^{wsu}$ are $\Gamma$-equivariant and therefore descend to well-defined subsets in $T^1S$, which we still denote by $H^{ssu}$ and $H^{wsu}$.

On the unit tangent bundle $T^1S$ of a hyperbolic surface $S = \Gamma\backslash\mathbb{H}$, the distance $d_1$ descends as follows:

where $\widetilde{u}$ and $\widetilde{v}$ are lifts of $u$ and $v$ to $T^1\mathbb{H}$. This distance is again equivalent to the Sasaki metric on $T^1S$.

The strong-stable set of a vector $u \in T^1S$ is defined as

We remark that, since the distance $d_1$ is equivalent to the Sasaki distance, they give the same notion of strong-stable set.

Hypercycles and hyperbolic isometries

Let $\gamma$ be a hyperbolic isometry of $\mathbb{H}$. It fixes exactly two points in $\partial\mathbb{H}$, the repelling fixed point $\gamma^-$ and the attracting fixed point $\gamma^+$. The geodesic $c$ joining $\gamma^-$ to $\gamma^+$ is the axis of $\gamma$, and $\gamma$ acts on $c$ as a translation of length

The displacement of a point $z \in\mathbb{H}$ by $\gamma$ only depends on its distance to the axis [4 Theorem 7.35.1]:

In particular, the displacement is minimal along the axis and increases with the distance to the axis.

For $r > 0$, the set of points of $\mathbb{H}$ at distance $r$ from $c$ consists of two curves, one on each side of $c$, called hypercycles of $\gamma$. Each of them is invariant under $\gamma$. In the upper half-plane model, they are arcs of Euclidean circles, or Euclidean half-lines, joining $\gamma^-$ to $\gamma^+$. Given a point $q \in\mathbb{H}$, we denote by $\mathcal{H}(\gamma,q)$ the hypercycle of $\gamma$ through $q$.

If $\gamma$ belongs to $\Gamma$, the hypercycle $\mathcal{H}(\gamma,q)$ projects to a closed curve in $S = \Gamma\backslash\mathbb{H}$, which we call a closed hypercycle. A direct computation shows that the arc of $\mathcal{H}(\gamma,q)$ between $q$ and $\gamma q$ has length

If $\gamma$ is primitive in $\Gamma$, this is the length of the closed hypercycle of $\gamma$.

The band model

For some computations it is convenient to use the band model

The map $z \mapsto ie^z$ is an isometry from $\mathbb{B}$ onto $\mathbb{H}$, which sends the real line $\{y=0\}$ onto the imaginary axis. The boundary at infinity of $\mathbb{B}$ consists of the two lines $\{y=\pm\pi/2\}$ together with the two points $\pm\infty$. We will use the following facts.

1. The real line $\{y=0\}$ is a geodesic. For $\lambda>0$, the translation $z\mapsto z+\lambda$ is a hyperbolic isometry with axis $\{y=0\}$ and translation length $\lambda$.

2. The vertical segments $\{x=k\}$, $k\in\mathbb{R}$, are geodesics orthogonal to $\{y=0\}$.

3. The horizontal lines $\{y=k\}$, $k\in(-\pi/2,\pi/2)\setminus\{0\}$, are hypercycles. More precisely, for $x\in\mathbb{R}$ and $y\in(-\pi/2,\pi/2)$,

4. Equations of the geodesics relative to $\{y=0\}$ (see Figure 1). Every geodesic other than $\{y=0\}$ is of one of the following three types.

(a) Ultraparallel geodesics. A geodesic at positive distance from $\{y=0\}$ has an equation of the form

Its endpoints are $x_0\pm r+i\epsilon\pi/2$, where $\epsilon$ is the sign of $\alpha$ and $\cosh r=|\alpha|$, and its point closest to $\{y=0\}$ is $x_0+iy_0$, where $\sin y_0=1/\alpha$.

(b) Intersecting geodesics. A geodesic intersecting $\{y=0\}$ has an equation of the form

Its endpoints are $x_0+s+i\pi/2$ and $x_0-s-i\pi/2$, where $\sinh s=\beta$. It meets $\{y=0\}$ at $x_0$, and the angle $\theta\in(0,\pi)$ from the positive direction of $\{y=0\}$ to the geodesic satisfies $\cot\theta=\beta$. The case $\beta=0$ gives the geodesics of item (2).

(c) Asymptotic geodesics. A geodesic asymptotic to $\{y=0\}$ has an equation of the form

Its endpoints are $x_0+i\epsilon\pi/2$ and $+\infty$ if $\delta=1$, or $-\infty$ if $\delta=-1$.

Vanishing asymptotic fineness

Given $x \in S$, we denote by $\operatorname{Inj}(x)$ the injectivity radius of $S$ at $x$. The next proposition reduces the study of the vectors whose geodesic ray has vanishing injectivity radius, without diverging through a cusp, to three cases. Each case is governed by the geometry of the thin parts visited by the ray: short closed geodesics, cusps, or short geodesic loops enclosed by hypercycles.

Proof. Since $\inf_{t \ge0} \operatorname{Inj}(u(t)) = 0$, there exists a sequence of times $t_n \ge0$ with $\operatorname{Inj}(u(t_n)) \to0$.

Fix a lift $\widetilde{u} \in T^1\mathbb{H}$ of $u$. The injectivity radius at $u(t_n)$ equals half the length of the shortest geodesic loop based at $u(t_n)$, so there exists $\gamma_n \in\Gamma\setminus\{\operatorname{id}\}$ such that

Since $\Gamma$ is torsion free, each $\gamma_n$ is either parabolic or hyperbolic. After passing to a subsequence, either all the $\gamma_n$ are parabolic, or all of them are hyperbolic.

Parabolic case. Assume that every $\gamma_n$ is parabolic. Let $x_n \in\partial\mathbb{H}$ be its fixed point and let $\widetilde{H}_n$ be the horocycle centered at $x_n$ passing through $\widetilde{u}(t_n)$. Denote by $\ell(\widetilde{H}_n,\gamma_n)$ the length of the horocyclic arc of $\widetilde{H}_n$ between $\widetilde{u}(t_n)$ and $\gamma_n\widetilde{u}(t_n)$. A direct computation in the upper half-plane model shows that

hence $\ell(\widetilde{H}_n,\gamma_n) \to0$. The horocycle $\widetilde{H}_n$ projects to a closed horocycle of $S$ of length at most $\ell(\widetilde{H}_n,\gamma_n)$ passing through $u(t_n)$. So $u(\mathbb{R}^{+})$ intersects a sequence of closed horocycles whose lengths tend to $0$. As $u(\mathbb{R}^{+})$ does not diverge through a cusp, this shows that $u$ is cusp-recurrent in the sense of [2], and we are in case (2).

Hyperbolic case. Assume that every $\gamma_n$ is hyperbolic. Let $c_n$ be its axis and $\ell(\gamma_n)$ its translation length. Clearly, $\ell(\gamma_n) \le d\bigl(\widetilde{u}(t_n),\gamma_n\widetilde{u}(t_n)\bigr) = \varepsilon_n$ tends to $0$ when $n \to\infty$.

We distinguish two subcases.

Subcase (i): $u(\mathbb{R}^{+})$ meets the closed geodesic obtained by projecting $c_n$ onto $S$ for infinitely many $n$. Passing to a subsequence, this holds for every $n$. These closed geodesics have length at most $\ell(\gamma_n)$. Since $\ell(\gamma_n) \to0$, we are in case (1).

Subcase (ii): $u(\mathbb{R}^{+})$ meets the closed geodesic obtained by projecting $c_n$ onto $S$ for at most finitely many $n$. Passing to a subsequence, $u(\mathbb{R}^{+})$ does not meet these closed geodesics for any $n$. The hypercycle $\mathcal{H}(\gamma_n,\widetilde{u}(t_n))$ projects to a closed hypercycle in $S$ passing through $u(t_n)$. Let $\gamma'_n$ be a generator of the stabilizer of $c_n$ in $\Gamma$, so that $\ell(\gamma'_n) \le\ell(\gamma_n)$. By (2.2), the length of the projection of $\mathcal{H}(\gamma_n,\widetilde{u}(t_n))$ is

Using the inequality $x \le2\sinh(x/2)$ and (1), we obtain

which tends to $0$ when $n \to+\infty$. Hence we are in case (3).

We now deduce Theorem 1 from Proposition 3, Theorem 2 and the previous works.

Proof of Theorem 1. Let $S$ be a hyperbolic surface and $u \in T^1S$. Since $\operatorname{Inj}$ is continuous and positive on $S$, the asymptotic fineness of $u(\mathbb{R}^{+})$ vanishes if and only if $\inf_{t\ge0}\operatorname{Inj}(u(t))=0$.

Assume first that the asymptotic fineness of $u(\mathbb{R}^{+})$ is positive. Then Bellis proved that $H^{ss}_u=W^{ss}_u$ [5 Théorème D]. If the geodesic ray $u(\mathbb{R}^{+})$ diverges through a cusp, then $H^{ss}_u$ is a closed horocycle around that cusp and the equality $H^{ss}_u=W^{ss}_u$ is clear.

Assume now that the asymptotic fineness of $u(\mathbb{R}^{+})$ vanishes and that $u(\mathbb{R}^{+})$ does not diverge through a cusp. By Proposition 3, at least one of the following cases occurs.

1. The ray $u(\mathbb{R}^{+})$ crosses closed geodesics whose lengths tend to $0$. The conclusion follows from [BCDHV26, Main Theorem].

2. The ray $u(\mathbb{R}^{+})$ intersects closed horocycles whose lengths tend to $0$, so $u$ is cusp-recurrent and the conclusion follows from [2 Theorem 2].

3. The ray $u(\mathbb{R}^{+})$ intersects closed hypercycles whose lengths tend to $0$. The conclusion follows from Theorem 2.

In all cases, $H^{ss}_u \subsetneq W^{ss}_u$ and $W^{ss}_u$ is an uncountable union of stable horocycles.

The following example shows that case (3) of Proposition 3 can occur while cases (1) and (2) do not.

Winding around a closed hypercycle

Proof. As $q$ belongs to $\tilde{u}(\mathbb{R}^{+})$, we have

Thus

For the other inequality, we observe that $\gamma\tilde u(+\infty)$ is between $\tilde u(+\infty)$ and the attracting fixed point $x_\gamma^+$ of $\gamma$. Then the geodesic ray $[\tilde u(0),\gamma\tilde u(+\infty))$ is closer than $\tilde u(\mathbb{R}^+)$ to the axis $c$ of $\gamma$. Let $q'$ be the point of $[\tilde u(0),\gamma\tilde u(+\infty))$ closest to $c$. We have $d(q',c)\le d(q,c)$. Then $d(q',\gamma q')\le d(q,\gamma q)$.

Since $q'$ is in the geodesic ray $[\tilde u(0),\gamma\tilde u(+\infty))$, we have

Therefore

Proof. For simplicity, we denote $\tilde v=\operatorname{Wind}_{\gamma}(\tilde u)$ and $\tau=\tau_{\gamma,\tilde u}$. Let $q=\tilde u(t_1)$, $t_1\ge0$, be the point that realizes the distance of $\tilde u(\mathbb{R})$ to the axis of $\gamma$. We observe that $\tilde u(0)$ and $\gamma\tilde u(+\infty)$ lie in the same half-plane in $\mathbb{H}\setminus(\gamma^-,\gamma^+)$. Hence, the geodesic generated by $\tilde v$ does not intersect the axis of $\gamma$. Let $q'=\tilde v(t'_1)$, $t'_1\in\mathbb{R}$, be the point that realizes the distance of $\tilde v(\mathbb{R})$ to the axis of $\gamma$. The first step is to control the distance between $q$ and $q'$.

To do the next computations we use the band model $\mathbb{B}$ (see § 2.5). We can assume that the axis of $\gamma$ is the geodesic $y=0$ and that $\gamma$ has the form $\gamma(z)=z+\lambda$, where $\lambda>0$ is the translation length of $\gamma$.

The geodesics at positive distance from $y=0$ are those of item (4a). Without loss of generality, we can assume that the equation of $\tilde u(\mathbb{R})$ has $x_0=0$ and $\alpha>1$. Hence,

which lies in the upper half-strip $y>0$. The endpoints at infinity of $\tilde u(\mathbb{R})$ are

where $r>0$ satisfies $\cosh(r)=\alpha$. The point $q$ realizing the distance to the axis $y=0$ is $q=y_0i$ with $\alpha=1/\sin(y_0)$. The initial point $\tilde u(0)$ lies between $\tilde u(-\infty)$ and $q$, so it has negative real part.

The geodesic $\tilde v(\mathbb{R})$ is also at positive distance from the axis $y=0$ and in the upper half-strip. Let

be the equation of $\tilde{v}(\mathbb{R})$. The endpoints of $\tilde{v}(\mathbb{R})$ are

where $r'>0$ satisfies $\cosh(r')=\alpha'$. The point $q'$ realizing the distance to the axis is $q'=x'_0+y'_0i$ with $\alpha'=1/\sin(y'_0)$.

By definition, $\tilde{v}(+\infty)=\gamma\tilde{u}(+\infty)$, which implies $r+\lambda=x'_0+r'$. The point $\tilde{v}(-\infty)$ must lie between $-r+\pi/2i$ and $\pi/2i$.

From the position of the endpoints of $\tilde{u}(\mathbb{R})$ and $\tilde{v}(\mathbb{R})$, we obtain the following estimates

(1) $0<y'_0<y_0$,

(2) $0<r'-r<\lambda/2$,

(3) $0<x'_0<\lambda$.

Consider the function $f(t)=\arcsin(1/\cosh(t))$ defined for $t\in\mathbb{R}$. We have $f(r)=y_0$ and $f(r')=y'_0$. An easy computation shows that $f$ is 1-Lipschitz, hence

Introducing $q''=x'_0+y_0i$, we have

First, we notice that $\gamma q=\lambda+y_0i$. Since $0<x'_0<\lambda$, the point $q''=x'_0+y_0i$ belongs to the hypercycle segment between $q$ and $\gamma q$. This implies that $d(q,q'')\le d(q,\gamma q)$.

The points $q''$ and $q'$ lie on the vertical geodesic $\{x=x'_0\}$ (item (2)). Hence $d(q'',q')$ is the length of the vertical segment joining them and, since $0<y'_0<y_0$,

By item (3), $\cosh(d(0,q))=1/\cos(y_0)$. Since $d(0,q)$ is the distance from $q$ to the axis of $\gamma$, (1) gives

These together imply

If $0\le x\le1$ then $\sinh(x/2)\le x$. Since $d(q,\gamma q)\le1$, we obtain

The rest of the proof follows from the bound of the distance between $q$ and $q'$. Recall that $q=\tilde{u}(t_1)$ and $q'=\tilde{v}(t'_1)$. By the triangular inequality,

We now bound the distance between the basepoints of the geodesics of $\tilde{u}$ and $\tilde{v}$.

Case 1. If $0\le t\le t_1$, then

Since the distance in $\mathbb{H}$ between two geodesic rays starting at the same point is increasing, it follows

Using the bounds established above, one obtains

which implies the statement.

Case 2: If $t \ge t_1$, then

Since $\tilde{v}$ and $\gamma\tilde{u}$ have the same point at infinity, the distance $d(\tilde{v}(t),\gamma\tilde{u}(t))$ is decreasing (Lemma 1), and hence for $t \ge t_1$,

Moreover,

By the bounds proved above, the first two terms on the right hand side are bounded by $2d(q,\gamma q)$. We conclude that, for $t \ge t_1$, $d(\tilde{v}(t),\gamma\tilde{u}(t)) \le5d(q,\gamma q)$.

Let us now compute the bounds for the distance $d_1$. Recall that

We can distinguish three scenarios:

(a) If $t \le t_1-1$, by the first case, we have that both

are bounded by $4d(q,\gamma q)$. Then

(b) If $t \ge t_1$, by the second case, we can bound both

by $5d(q,\gamma q)$. Then

(c) Let $t_1-1<t<t_1$. We have

The first and the second terms in the last line are bounded by $4d(q,\gamma q)$ and $5d(q,\gamma q)$ by Cases 1 and 2 above, respectively.

In order to bound the third term we apply (1). Writing $z=\tilde{u}(t+1)$ and $c$ for the axis of $\gamma$, we obtain

We notice that

We obtain

Since $d(q,\gamma q)\le1$, we have

We get

Finally, we observe that

thanks to Proposition 4. Similarly we have

Together with the previous bounds, these imply the statement. $\square$

Proof of Theorem 2

Rays intersecting arbitrarily short hypercycles

Let $S=\Gamma\backslash\mathbb{H}$ be a hyperbolic surface and let $u\in T^1S$ be such that the geodesic ray $u(\mathbb{R}^+)$ meets closed hypercycles of arbitrarily small length. Notice that each closed hypercycle has a closed geodesic in its free homotopy class, which is shorter. The case where $u(\mathbb{R}^+)$ meets infinitely many of these closed geodesics falls in the first case of Proposition 3 and the conclusion of Theorem 2 follows from [BCDHV26]. Therefore, we may assume that $u(\mathbb{R}^+)$ does not meet the closed geodesics associated to these hypercycles.

Recall that $\mathcal{H}(\gamma,q)$ denotes the hypercycle of $\gamma$ through $q$ (Section 2.4). If $\gamma$ is in good position with respect to $\tilde{u}$ and $q$ is the point of $\tilde{u}(\mathbb{R})$ closest to the axis of $\gamma$, then $\tilde{u}(\mathbb{R})$ is tangent to $\mathcal{H}(\gamma,q)$ at $q$.

exist a lift $\tilde{u}$ of $u$ with $\tilde{u}(0)=i$ and $\tilde{u}(+\infty)=\infty$, and a sequence $(\gamma_n)_{n\geq0}$ of hyperbolic isometries in $\Gamma$ such that

1. $\tilde{u}$ is in good position with respect to each $\gamma_n$,

2. the point $q_n=\tilde{u}(t_n)$ of $\tilde{u}(\mathbb{R})$ closest to the axis of $\gamma_n$ is such that $(t_n)_{n\geq0}$ is increasing and tends to $+\infty$,

3. $0<\gamma_n^-<\gamma_n^+<\gamma_{n+1}^-$ for all $n\geq0$,

4. the hypercycles $\mathcal{H}(\gamma_n,q_n)$ are pairwise disjoint,

5. $d(q_n,\gamma_nq_n)$ tends to $0$.

Proof. By hypothesis, there are times $s_n\geq0$ such that $u(s_n)$ belong to a hypercycle whose length tend to $0$ when $n$ tends to $+\infty$.

Up to conjugating the group $\Gamma$ by an isometry of $\mathbb{H}$, we can assume that $u$ lifts to $\tilde{u}\in T^1\mathbb{H}$ such that $\tilde{u}(0)=i$ and $\tilde{u}(+\infty)=\infty$. For every $n$, there exists a primitive hyperbolic isometry $\gamma_n$ such that the hypercycle $\mathcal{H}(\gamma_n,\tilde{u}(s_n))$ of $\gamma_n$ through $\tilde{u}(s_n)$ project to the closed hypercycles whose length tend to $0$. In particular, $\varepsilon_n:=d(\tilde{u}(s_n),\gamma_n\tilde{u}(s_n))$ is smaller than the length of the hypercycle, so it also tends to $0$.

Since $\ell(\gamma_n)\leq\varepsilon_n\to0$, up to passing to a subsequence, the isometries $\gamma_n$ can be taken to be pairwise distinct. It can also be assumed that their fixed points are different from $0$ and $\infty$.

Let $c_n$ be the axis of $\gamma_n$. By hypothesis $\tilde{u}(\mathbb{R}^+)$ does not intersect $c_n$. Since the fixed points of $\gamma_n$ are different from $0$ and $\infty$, the geodesics $\tilde{u}(\mathbb{R})$ and $c_n$ are not asymptotic. Let $q_n=\tilde{u}(t_n)$ be the point of $\tilde{u}(\mathbb{R})$ closest to $c_n$. By (2.1), the displacement $d(z,\gamma_nz)$ is smaller the closer $z$ lies to $c_n$. In particular,

We claim that $t_n>0$ for $n$ large enough. Otherwise, if $t_n\leq0$, then $t_n\leq0\leq s_n$, so $\tilde{u}(0)=i$ is closer to $c_n$ than $\tilde{u}(s_n)$. By the property of the displacement function, we would get

which if it happened for infinitely many $n$ would contradict the discreteness of $\Gamma$. Up to taking a subsequence, we assume $t_n>0$. Moreover, $q_n$ does not lie on $c_n$, thus $\tilde{u}(\mathbb{R})$ stays at positive distance from $c_n$.

Observe that $\tilde{u}(\mathbb{R})$ is tangent to the hypercycle of $\gamma_n$ through $q_n$. Replacing $\gamma_n$ by $\gamma_n^{-1}$ if necessary, the orientation of $\tilde{u}(\mathbb{R})$ agrees with the one induced by $\gamma_n$ on the hypercycle through $q_n$. Together with $t_n>0$, this shows that $\tilde{u}$ is in good position with respect to $\gamma_n$.

Moreover, $t_n\to+\infty$. Otherwise, up to passing to a subsequence, the points $q_n$ stay in a compact subset of $\mathbb{H}$, and $d(q_n,\gamma_nq_n)\to0$ with pairwise distinct $\gamma_n$ would contradict the discreteness of $\Gamma$. Up to passing to a subsequence, we may assume that $(t_n)_{n\geq0}$ is increasing.

The axis $c_n$ does not meet $\tilde{u}(\mathbb{R})$, hence it is contained in one of the two half-planes $\{x>0\}$ or $\{x<0\}$. Up to taking a subsequence and conjugating again the group $\Gamma$ by the reflection $z\mapsto-\overline{z}$, which fixes $\tilde{u}(\mathbb{R})$ pointwise and preserves good position, we may assume that they all lie in $\{x>0\}$. Then $\gamma_n^->0$ and $\gamma_n^+>0$, and good position forces the attracting fixed point to be on the right of the repelling one, so that $0<\gamma_n^-<\gamma_n^+$.

The geodesic realizing the distance between $\tilde{u}(\mathbb{R})$ and $c_n$ is orthogonal to both, and it passes through the point $q_n$. Hence it is a Euclidean half-circle centered at $0$. One of the end points $x_n$ of this geodesic is between $\gamma_n^-$ and $\gamma_n^+$, so $\gamma_n^-<x_n<\gamma_n^+$. By definition, $q_n=\tilde{u}(t_n)=ie^{t_n}$, then $x_n=e^{t_n}$ tends to $+\infty$, and so does $\gamma_n^+$. We claim that $\gamma_n^-$ also tends to $+\infty$. Otherwise, up to passing to a subsequence, we may assume that $\gamma_n^-$ converges to some $\xi\in\mathbb{R}$. Then the axes $c_n$ converge to the geodesic with endpoints $\xi$ and $+\infty$. Picking a point $z$ on this geodesic, it is accumulated by points $z_n$ on $c_n$. Since $\ell(\gamma_n)$ tends to $0$, we have $d(z,\gamma_nz) \to0$, which contradicts the fact that $\Gamma$ is discrete. Hence, we must have $\lim_{n}\gamma_n=+\infty$.

The hypercycle $\mathcal{H}(\gamma_n,q_n)$ is an arc of the Euclidean circle through $\gamma_n^-$, $\gamma_n^+$ and $q_n$, which is tangent to the imaginary axis at $q_n$ (see Figure 3). Then the Euclidean distance between $0$ and $\mathcal{H}(\gamma_n,q_n)$ tends to $+\infty$. For this we can observe that this distance is greater than the distance from $0$ to the Euclidean circle of radius $\gamma_n^-$ centered at $\gamma_n^-+\gamma_n^-i$. This allows us to extract a subsequence such that the hypercycles $\mathcal{H}(\gamma_n,q_n)$ are pairwise disjoint and $\gamma_n^+<\gamma_{n+1}^-$ for all $n\geq0$. □

From now on we fix the vector $u\in T^1S$, its lift $\widetilde{u}$ with $\widetilde{u}(0)=i$ and $\widetilde{u}(+\infty)=\infty$, and the sequence $(\gamma_n)_{n\geq0}$ of hyperbolic isometries given by Proposition 6.

Construction of the vector $w_\alpha$ in $\overline{H^{ss}u}$

Let $(\alpha_n)_{n\geq0}$ be a subsequence of $(\gamma_n)_{n\geq0}$. We write $\beta_n=\alpha_0\cdots\alpha_n\in\Gamma$, and we denote by $\alpha_n^-$, $\alpha_n^+$ the repelling and attracting fixed points of $\alpha_n$, so that $0<\alpha_n^-<\alpha_n^+<\alpha_{n+1}^-$. For each $n\geq0$, let $q_n^\alpha$ be the point of $\widetilde{u}(\mathbb{R})$ closest to the axis of $\alpha_n$, and set

Since $(\alpha_n)$ is a subsequence of $(\gamma_n)$, we have $\delta_n^\alpha\to0$. We introduce the vectors $\widetilde{v}_n\in T^1\mathbb{H}$ defined by

Lemma 2. The sequence $(\widetilde{v}_n)_{n\geq0}$ converges to a vector $\widetilde{v}_\alpha\in T^1\mathbb{H}$ defined by $\widetilde{v}_\alpha(0)=i$ and $\widetilde{v}_\alpha(+\infty)=\lim_{n\to+\infty}\beta_n(\infty)$.

Proof. Since every $\widetilde{v}_n$ is based at $i$, it suffices to prove that the sequence $(\beta_n(\infty))_{n\geq0}\subset\partial\mathbb{H}$ converges. Each $\alpha_i$ is a hyperbolic isometry with repelling and attracting fixed points $0<\alpha_i^-<\alpha_i^+$. Since $\infty$ lies outside $[\alpha_i^-,\alpha_i^+]$, it is attracted by $\alpha_i$ toward $\alpha_i^+$, so that $\alpha_i^+<\alpha_i\infty<+\infty$. Moreover, $\alpha_i$ preserves the order on the interval $(\alpha_i^+,+\infty)$, hence $\alpha_i^+<x<y$ implies $\alpha_i^+<\alpha_ix<\alpha_iy$. Using the nesting $\alpha_i^+<\alpha_{i+1}^-$, we obtain for every $n\geq 1

Applying $\alpha_0,\ldots,\alpha_{n-2}$ successively, which preserve the order on the relevant intervals, yields

Hence $(\beta_n(\infty))_{n\geq0}$ is a decreasing sequence of positive numbers, and therefore converges. □

The ray $\beta_n^{-1}\widetilde{v}_n(\mathbb{R}^+)$ issues from $\beta_n^{-1}i$ and points to $\beta_n^{-1}(\beta_n\infty)=\infty$; it is therefore the vertical geodesic ray starting at $\beta_n^{-1}i$. The next lemma places this ray in good position with respect to $\alpha_{n+1}$ and compares its displacement with $\delta_{n+1}^\alpha$. This is the geometric point where the present case departs from [BCDHV26]: the ray does not cross the axis of $\alpha_{n+1}$ but remains tangent to one of its hypercycles.

Lemma 3. For every $n\geq0$, the vector $\beta_n^{-1}\widetilde{v}_n$ is in good position with respect to $\alpha_{n+1}$. Moreover, if $q_{n+1}'$ denotes the point of $\beta_n^{-1}\widetilde{v}_n(\mathbb{R})$ closest to the axis of $\alpha_{n+1}$, then

Proof. For $n\geq0$, let $R_n$ be the closed region of the first quadrant bounded by the segment of the imaginary axis between $0$ and $q_n^\alpha$, the arc of the hypercycle $\mathcal{H}(\alpha_n,q_n^\alpha)$ between $q_n^\alpha$ and $\alpha_n^-$, and the segment $[0,\alpha_n^-]$ of $\partial\mathbb{H}$. Since the hypercycles $\mathcal{H}(\alpha_n,q_n^\alpha)$ are pairwise disjoint and $q_n^\alpha=\widetilde{u}(t_n^\alpha)$ with $(t_n^\alpha)$ increasing, we have $R_n\subset R_{n+1}$ for all $n \ge0$. Moreover, since $\mathcal{H}(\alpha_n,q_n^\alpha)$ is invariant under $\alpha_n$ and $\bar{\alpha}_n^{-1}$ moves the points of $\partial\mathcal{H}$ toward its attracting fixed point $\bar{\alpha}_n$, one checks that $\bar{\alpha}_n^{-1}(R_n) \subset R_n$.

We prove by induction that $\beta_n^{-1}i \in R_n$ for all $n \ge0$. For $n=0$ we have $\alpha_0^{-1}(i) \in\alpha_0^{-1}(R_0) \subset R_0$, because $i \in R_0$. Assume $\beta_n^{-1}i \in R_n$. Then

Since $\beta_n^{-1}i \in R_n \subset R_{n+1}$, the vertical ray $\beta_n^{-1}\widetilde v_n(\mathbb{R}^+)$ lies on the left of $\bar\alpha_n$, hence it stays at positive distance from the axis of $\alpha_{n+1}$, and is tangent to a hypercycle of $\alpha_{n+1}$ enclosed by $\mathcal{H}(\alpha_{n+1},q_{n+1}^\alpha)$. The orientation of $\beta_n^{-1}\widetilde v_n(\mathbb{R}^+)$ agrees with the one induced by $\alpha_{n+1}$, so $\beta_n^{-1}\widetilde v_n$ is in good position with respect to $\alpha_{n+1}$. Finally, the point $q'_{n+1}$ realizing the distance to the axis lies on a hypercycle enclosed by $\mathcal{H}(\alpha_{n+1},q_{n+1}^\alpha)$, hence closer to the axis than $q_{n+1}^\alpha$. As the displacement function decreases with the distance to the axis, we conclude

By Lemma 3, the winding $\operatorname{Wind}_{\alpha_{n+1}}(\beta_n^{-1}\widetilde v_n)$ is defined, and it is the vector based at $\beta_n^{-1}i$ pointing at $\alpha_{n+1}\infty$. On the other hand, $\beta_n^{-1}\widetilde v_{n+1}$ is based at $\beta_n^{-1}i$ and points at $\beta_n^{-1}(\beta_{n+1}\infty)=\alpha_{n+1}\infty$. Hence, for all $n\ge0$,

By the definition of the winding time (Definition 7) applied to $\alpha_{n+1}$ and $\beta_n^{-1}\widetilde v_n$, together with Proposition 4 and Lemma 3,

We define

By construction $g_{r_n}\beta_n^{-1}\widetilde v_n\in H^{ssu}\widetilde{\ },$ so its projection satisfies $g_{r_n}v_n\in H^{ssu}$.

Proof. By definition of $r_n$ and (5.1),

Hence $|r_m-r_n|\le\sum_{i=n+1}^{m}\delta_i^\alpha$, so $(r_n)$ is a Cauchy sequence and converges. Moreover

using $|r_0|=|\tau_{\alpha_0,\widetilde u}|\le\delta_0^\alpha$ (Proposition 4). Letting $n\to+\infty$ gives the last inequality.

Construction of the vector $w_\alpha$ in $W^{ssu}$

Throughout this subsection we fix a subsequence $(\alpha_n)_{n\ge0}$ of $(\gamma_n)_{n\ge0}$ such that $\delta_n^\alpha\le1$ for all $n$ and $\sum_{n\ge0}\delta_n^\alpha<+\infty$, and we write $\Delta=\sum_{n\ge0}\delta_n^\alpha$.

By Proposition 7 the winding times $r_n$ converge to $r_\alpha$, and by Lemma 2 the vectors $\widetilde v_n$ converge to $\widetilde v_\alpha$. We set, for $n\ge0$,

We denote by $v_n$, $v_\alpha$, $w_n$, $w_\alpha$ the projections to $T^1S$ of $\widetilde v_n$, $\widetilde v_\alpha$, $\widetilde w_n$, $\widetilde w_\alpha$, respectively. We remark that, since $\beta_n^{-1}\widetilde w_n=g_{r_n}\beta_n^{-1}\widetilde v_n\in H^{ssu}\widetilde{\ },$ the vectors $w_n$ lie in $H^{ssu}$ and $w_n$ converges to $w_\alpha$.

Our goal is to show that in fact $w_\alpha\in W^{ssu}$. For this we show that the distance between $g_t w_n$ and $g_t u$ tends to zero uniformly in $n$ and then we take the limit in $n$.

Proof. Fix $\varepsilon> 0$. Since $\sum_k \delta_k^\alpha< +\infty$, we may fix $N$ with $17\sum_{k>N}\delta_k^\alpha< \varepsilon/2$. We will use that, if $n > N$, then

Recall that $\beta_k^{-1}\widetilde{v}_{k+1}=\operatorname{Wind}_{\alpha_{k+1}}(\beta_k^{-1}\widetilde{v}_k)$ and $\tau_{\alpha_{k+1},\beta_k^{-1}}=r_{k+1}-r_k$, and that, by Lemma 3, the point of $\beta_k^{-1}\widetilde{v}_k(\mathbb{R})$ closest to the axis of $\alpha_{k+1}$ has displacement at most $\delta_{k+1}^\alpha\le1$. Applying the Key Proposition 5 to $\beta_k^{-1}\widetilde{v}_k$ and $\alpha_{k+1}$ and then projecting to $T^1S$, we obtain

Substituting $s=t+r_k$ gives

for all $t\ge-r_k$. Since $|r_k|\le\Delta$, this holds for all $t\ge\Delta$ in particular. Therefore the second term in (5.2) is bounded by $17\sum_{k=N}^{n-1}\delta_{k+1}^\alpha<\varepsilon/2$ for all $t\ge\Delta$.

To bound the first term in (5.2) or to treat the case $n\le N$, we may observe again that $w_i\in H^{ss}u$, so $d_1(gtw_i,gtu)\to0$ as $t\to+\infty$. Hence there exists $T\ge\Delta$ such that $d_1(gtw_i,gtu)<\varepsilon/2$ for all $t>T$. This concludes the proof. □

Since $w_\alpha=\lim_{n\to+\infty}w_n$, letting $n\to+\infty$ in Proposition 8 gives, for every $\varepsilon>0$ and every $t>T$,

Hence $\lim_{t\to+\infty}d_1(gtw_\alpha,gtu)=0$, that is, $w_\alpha\in W^{ss}u$.

Construction of the vector $w_\alpha$ in $W^{ss}u\setminus H^{ss}u$

We denote by $\Sigma$ the set of subsequences $\alpha=(\alpha_n)_{n\ge0}$ of $(\gamma_n)_{n\ge0}$ such that

By Proposition 8, every $\alpha\in\Sigma$ satisfies $w_\alpha\in W^{ss}u$. Since $\delta_n=d(q_n,\gamma_nq_n)\to0$, the set $\Sigma$ is nonempty. For $\alpha\in\Sigma$ we set

which is a positive number by Lemma 2.

Proof. We use the following property, which follows from the dynamics of the $\alpha_n$ as in the proof of Lemma 2: for every $\alpha\in\Sigma$ and every $n\ge0$,

hence

Assume, for contradiction, that $\xi(\Sigma)$ is countable and enumerate it as $\{x_0,x_1,\ldots\}$. We construct $\alpha'=(\alpha'_n)\in\Sigma$ with $\xi(\alpha')\ne x_i$ for all $i$.

Choose $\alpha'_0=\gamma_{k_0}$ with $\delta'_0<1$ and $\alpha'_0{}^+>x_0$. Inductively, having chosen $\alpha'_0,\ldots,\alpha'_n$, pick $\alpha'_{n+1}=\gamma_{k_{n+1}}$ with $k_{n+1}>k_n$ such that

This is possible since $\gamma_k^+ \to+\infty$ and $\delta_k \to0$. Since $\sum_{n\ge0}\delta_n^{\alpha'} < \sum_{n\ge0}2^{-n} < +\infty$ and $\delta_n^{\alpha'} \le1$, the resulting sequence $\alpha'$ belongs to $\Sigma$.

By (5.3) and the choice of $\alpha'_n$, we have for $n \ge1$

whence $\xi(\alpha') \ne x_n$. Finally $\xi(\alpha') > \alpha_0^{\prime+} > x_0$, so $\xi(\alpha') \ne x_0$. This contradicts $\xi(\alpha') \in\xi(\Sigma)=\{x_i\}$. □

Corollary 1. There exists a subsequence $\alpha\in\Sigma$ such that $w_\alpha\notin H^{ws}_u$. Moreover, $W^{ss}_u$ contains an uncountable union of pairwise distinct stable horocycles.

Proof. If $w \in T^1S$ belongs to $H^{ws}_u$, then $\widetilde w(+\infty) \in\Gamma\widetilde u(+\infty)=\Gamma_\infty$ for any lifts. Since $\Gamma$ is countable and $\xi(\Sigma)$ is uncountable by Proposition 9, the set $\xi(\Sigma)$ meets uncountably many $\Gamma$-orbits in $\partial\mathbb{H}$. Hence there exists $\alpha\in\Sigma$ with $\xi(\alpha)=\widetilde w_\alpha(+\infty) \notin\Gamma_\infty$, that is, $w_\alpha\notin H^{ws}_u$. By Proposition 8, all these $w_\alpha$ lie in $W^{ss}_u$, and their forward endpoints lie in distinct $\Gamma$-orbits, so the stable horocycles $H^{ss}_{w_\alpha} \subset W^{ss}_u$ are pairwise distinct and uncountable in number. □

Declaration on the use of AI

During the preparation of this work the authors used Claude (Anthropic) to assist with the writing and the proofreading of the manuscript. The mathematical content and the arguments are entirely the authors’ own. The authors have reviewed all the text and take full responsibility for the content of the article.

References

IMERL, CMAT, Universidad de la República, IFUMI, IRL CNRS 2030, Uruguay

Email address: [email protected]

IRMAR, Université de Rennes, France and IFUMI, IRL CNRS 2030

Email address: [email protected]