MINIMAL RAYS ON CLOSED SURFACES
JAN PHILIPP SCHRÖDER
Abstract. Given an arbitrary Riemannian metric on a closed surface, we consider length-minimizing geodesics in the universal cover.
Morse and Hedlund proved that such minimal geodesics lie in bounded
distance of geodesics of a Riemannian metric of constant curvature.
Knieper asked when two minimal geodesics in bounded distance of a
single constant-curvature geodesic can intersect. In this paper we prove
an asymptotic property of minimal rays, showing in particular that intersecting minimal geodesics as above can only occur as heteroclinic
connections between pairs of homotopic closed minimal geodesics. A
further application characterizes the boundary at infinity of the universal cover defined by Busemann functions. A third application shows
that flat strips in the universal cover of a non-positively curved surface
are foliated by lifts of closed geodesics of a single homotopy class.
1. Introduction and main results
In this paper we study the behavior rays in surfaces, that is lengthminimizing geodesics in the universal cover of the surface, defined on the
half line R+ = [0, ∞). The topic is classical, starting with the works of
H. M. Morse [Mor24] and G. A. Hedlund [Hed32] and giving rise to AubryMather theory of Tonelli Lagrangian systems, cf. [Mat91] and the references
therein. We shall work with Finsler metrics, i.e. a strongly convex norm F
in each tangent space, not assuming F (−v) = F (v), cf. [BCS00] for more
information. One motivation for using general Finsler metrics rather than
Riemannian metrics is that the former can be used to describe the dynamics
of arbitrary Tonelli Lagrangian systems in high energy levels, cf. [CIPP98],
while our results are also new in the Riemannian case. The behavior of rays,
especially in dimension two, is to a large extend determined by the topology
of the surface, as we shall see shortly. In this way, rays give a framework
to the overall dynamics of geodesics of arbitrary Finsler metrics and hence
are a good starting point for studying geodesic flows (cf. e.g. the approach
in [Sch15b], [Sch15c]).
The existence of rays requires, of course, that the universal cover of the
surface is non-compact. Hence, the surface should not be elliptic, but rather
Date: July 6, 2015.
2010 Mathematics Subject Classification. Primary 37D40, Secondary 37E99, 53C22.
Key words and phrases. Finsler metric, ray, minimal geodesic, hyperbolic surface, Busemann function, horofunction boundary.
1
2
J. P. SCHRÖDER
flat or hyperbolic. We consider the following setting. Let (M, g) be either the flat torus T2 = R2 /Z2 with the euclidean metric or a hyperbolic
surface, that is, a connected, oriented, complete Riemannian surface with
constant curvature −1. Let us assume that on M we are given an arbitrary
Finsler metric F with the standing assumption that F and g are uniformly
equivalent (note that we have not yet assumed M to be compact):
p
∃cF ≥ 1 :
c−1
g(v, v) ≤ cF · F (v) ∀v ∈ T M.
F · F (v) ≤
We lift F and g to the universal cover of M , denoted by p : D → M . The
Rb
length functional lF (c; [a, b]) = a F (ċ)dt of F induces a (non-symmetric)
distance on D by
dF (x, y) := inf{lF (c; [0, 1]) | c : [0, 1] → D C 1 , c(0) = x, c(1) = y}.
Definition 1.1. An F -geodesic c : R+ → D or c : R → D is a ray, minimal
geodesic, respectively, if F (ċ) ≡ 1 and lF (c; [a, b]) = dF (c(a), c(b)) for all
a ≤ b in the domain of definition. We write
M = {ċ(0) | c : R → D is a minimal geodesic}
⊂
R+ = {ċ(0) | c : R+ → D is a ray}
⊂
{F = 1}
⊂ T D.
The set M is a closed subset of T D invariant by the geodesic flow φtF
of F and projects to a closed, φtF -invariant subset p∗ (M) of {F = 1} ⊂
T M . Interested in the behavior of the geodesic flow of F in general, one
can start by studying the dynamics inside the set p∗ (M). Estimating the
topological entropy of the restricted geodesic flow φtF |p∗ (M) , G. Knieper
posed the following question.
(A) Given an arbitrary Riemannian metric on a closed surface and a pair
of minimal geodesics c, c0 : R → D in the universal cover with bounded
Hausdorff distance
sup d(c(R), c0 (t)) + d(c0 (R), c(t)) : t ∈ R < ∞,
under which circumstances is it possible for c, c0 to intersect in D?
Note that by minimality of c, c0 there can be at most one intersection, provided c(R) 6= c0 (R). In the case where the Riemannian metric has no conjugate points, there exist no intersecting geodesics in bounded distance. We
will answer Question (A) in the general context in Corollary 1.8 below. Note
that having bounded Hausdorff distance defines an equivalence relation ∼
on M. Using the techniques in [GKOS14] and Corollaries 1.5 and 1.8 below, one can then show that the topological entropy of φtF restricted to any
equivalence class of ∼ in M vanishes. This strengthens the main result
in [GKOS14] stating that the topological entropy of φtF |p∗ (M) equals the
volume growth of the Riemannian metric in the universal cover.
We return to the general setting of being given an arbitrary Finsler metric
F on (M, g), the flat torus or a hyperbolic surface as above. The following
MINIMAL RAYS ON CLOSED SURFACES
3
theorem links the behavior of rays and minimal geodesics with respect to
F to the topology of M , the latter being encoded in the behavior of ggeodesics. It was proved by Morse [Mor24] in the hyperbolic case and by
Hedlund [Hed32] in the flat case M = T2 . Note, however, that the proofs
are quite different in the two cases.
Theorem 1.2 (Morse, Hedlund). There exists a constant C ≥ 0 depending
only on F, g with the following property: for any F -ray c : R+ → D or
minimal geodesic c : R → D, there exists a g-geodesic γ ⊂ D, such that
c(R+ ), c(R), respectively, lies within g-distance at most C from γ.
For definiteness, we let D ⊂ R2 be the open unit disc. In the case where
(M, g) is hyperbolic, we use the Poincaré metric gx = 4(1 − |x|2 )−2 h., .i on
D, such that g-geodesics are circle segments meeting S 1 orthogonally. In the
torus-case, we consider the pullback g of the euclidean metric to D under
the map D → R2 , x 7→ x/(1 − |x|2 ); here geodesics are circle segments in D
meeting S 1 in antipodal points. Using the euclidean topology in the closed
disc D ⊂ R2 , we use Theorem 1.2 to define the asymptotic direction of rays
c : R+ → D,
c(∞) = lim c(t) ∈ S 1 ,
t→∞
and set
R+ (ξ) = {ċ(0) ∈ R+ : c(∞) = ξ},
ξ ∈ S1.
It is not hard to see (cf. Section 3) that for the projection π : T D → D we
have π(R+ (ξ)) = D for any ξ ∈ S 1 , in particular R+ (ξ) 6= ∅. In Figure 1
below, we depict g-geodesics and some sets R+ (ξ).
Our aim is to characterize the structure of R+ (ξ). Consider the set
Π := {γ(∞) : γ ⊂ D g-geodesic, such that p(γ) ⊂ M is closed} ⊂ S 1 .
In the case of M = T2 the set Π consists of ξ ∈ S 1 with rational or infinite
slope; if (M, g) is hyperbolic, Π are the fixed points of hyperbolic deck
transformations with respect to the covering p : D → M . Given ξ ∈ Π, we
can consider the set of periodic minimal geodesics with endpoint ξ:
Rp+ (ξ) = {ċ(0) ∈ R+ (ξ) : p(c) ⊂ M is a closed F -geodesic}.
Note that an F -geodesic c : R → D from Rp+ (ξ) is minimal on all of R due
to periodicity; moreover, it lies within finite distance from some periodic
g-geodesic (i.e. a g-geodesic projecting to a closed curve via p : D → M ),
which is uniquely determined by ξ in the hyperbolic case. The following
characterization of R+ (ξ) for ξ ∈ Π was achieved by Morse [Mor24] in the
hyperbolic case and by Hedlund [Hed32] in the flat case.
Theorem 1.3 (Morse, Hedlund). Let ξ ∈ Π. Then we have the following.
(i) Rp+ (ξ) 6= ∅ and all minimal geodesics in Rp+ (ξ) project to a single,
prime homotopy class of free loops in M (i.e. the homotopy class is
not an iterate of any other class).
4
J. P. SCHRÖDER
(ii) If c : R+ → D is a ray from R+ (ξ), then c(t) is asymptotic to some
minimal geodesic c0 (R) from Rp+ (ξ), as t → ∞.
(iii) If c : R+ → D is a ray from R+ (ξ) and c(0) ∈ c0 (R) for some minimal
geodesic c0 from Rp+ (ξ), then c(R) = c0 (R).
(iv) If S ⊂ D is a connected component of D − π(Rp+ (ξ)) and if c0 is a
minimal geodesic from Rp+ (ξ) forming a component of the boundary
∂S, then for any x ∈ S there exists a ray c : R+ → D with c(0) = x,
which is asymptotic to c0 (R), as t → ∞.
Item (iii) shows that Rp+ (ξ) determines a lamination of D by minimal
geodesics projecting to closed geodesics in a fixed homotopy class determined
by ξ ∈ Π, which cannot be traversed by the other rays form R+ (ξ). In
particular, it makes sense to speak of pairs of neighboring minimal geodesics
in Rp+ (ξ) (i.e. in D, between the pair there are no more minimal geodesics
from Rp+ (ξ)). We write Γ for the group of deck transformations of p : D →
M . Then for M = T2 the lamination determined by Rp+ (ξ) with ξ ∈ Π is
Γ-invariant. In the hyperbolic case, it is invariant only under an infinitely
cyclic subgroup of Γ.
Theorem 1.3 together with Lemma 2.2 below imply a corollary on possible
intersections of rays in R+ (ξ). To be precise, we give the following definition.
Definition 1.4. Two rays intersect, if one of the rays (as a closed set in
D) intersects the relative interior of the other ray without being a subset of
the latter with the same orientation.
Corollary 1.5. If ξ ∈ Π and if two rays c, c0 : R+ → D with c(∞) =
c0 (∞) = ξ intersect, then there exists a pair c0 , c00 : R → D of disjoint,
neighboring minimal geodesics from Rp+ (ξ), such that c(t) is asymptotic to
c0 (R) and c0 (t) is asymptotic to c00 (R), as t → ∞.
In particular, if there exist no neighboring minimal geodesics in Rp+ (ξ),
then R+ (ξ) has a very simple structure: no two rays intersect, which poses
strong restrictions on the possible dynamics of rays in R+ (ξ) due to dim M =
2. This happens if and only if the minimal geodesics in Rp+ (ξ) foliate a single
closed strip in D; in the case of M = T2 this strip would have to be all of
D due to Γ-invariance of Rp+ (ξ).
The main goal of this paper is to characterize the structure of R+ (ξ) with
ξ ∈ S 1 − Π. V. Bangert [Ban88] and independently M. L. Bialy and L. V.
Polterovich [BP86] proved the following theorem for the case M = T2 (it
was generalized by the author to non-reversible Finsler metrics in [Sch15a]).
Theorem 1.6 (Bangert, Bialy, Polterovich). If M = T2 , F any Finsler
metric on M and ξ ∈ S 1 − Π, then no two rays from R+ (ξ) intersect.
The natural question is whether Theorem 1.6 has a generalization to the
hyperbolic case. This will indeed be implied by our following main result.
Main Theorem 1.7. Let (M, g) be a hyperbolic surface and F be a Finsler
metric on M , which is uniformly equivalent to g. Assume that ξ ∈ S 1 − Π
MINIMAL RAYS ON CLOSED SURFACES
5
is such that there exists a g-geodesic γ ⊂ D with γ(∞) = ξ, for which the
ω-limit set of the projected unit-speed geodesic p(γ) in T M contains a nonempty, compact set left invariant under the g-geodesic flow. Then for any
two F -rays c, c0 : R+ → D with c(∞) = c0 (∞) = ξ we have
lim inf dg (c(R+ ), c0 (t)) = 0.
t→∞
Observe that the statement of Main Theorem 1.7 is false for the case
M = T2 . Note also that the techniques in the proof differ sharply from the
arguments leading to Theorem 1.6.
Main Theorem 1.7 does not require the surface M to be compact, but
rather asks for a compactness condition on the point ξ ∈ S 1 − Π, which is
fulfilled by Lebesgue almost every point ξ ∈ S 1 in the case where (M, g) has
finite volume. As the condition on ξ ∈ S 1 in Main Theorem 1.7 is difficult
to check in general, let us from now on assume that M is compact. This
means that M is an orientable, closed surface of genus ≥ 1, while the genus
is ≥ 2 if and only if we find a Riemannian metric g on M , such that (M, g)
is hyperbolic. Uniform equivalence of F and g is fulfilled automatically.
Using Main Theorem 1.7 and Lemma 2.2 below, we obtain in the next
corollary the version of Theorem 1.6 for all closed surfaces.
Corollary 1.8. If M is a closed, orientable surface of genus ≥ 1, F any
Finsler metric on M and ξ ∈ S 1 −Π, then no two rays from R+ (ξ) intersect.
Combining Corollaries 1.8 and 1.5 it follows that, if c, c0 : R → D is a pair
of intersecting minimal geodesics with c(∞) = c0 (∞), then there exists a pair
of neighboring minimal geodesics c0 , c00 from Rp+ (c(∞)), such that c, c0 are
heteroclinic connections between c0 , c00 with opposite asymptotic behavior:
For t → ∞ we have
dg (c0 (R), c(−t)), dg (c00 (R), c(t)), dg (c00 (R), c0 (−t)), dg (c0 (R), c0 (t)) → 0.
In particular, c(−∞) = c0 (−∞) and c(−∞), c(∞) ∈ Π. This is the situation
of Corollary 1.5. Hence, we obtain a complete answer to Question (A).
Using the the main result in [Sch14] and the remarks after Corollary 1.5, if
the Finsler (or Riemannian) metric is chosen generically, then intersecting
minimal geodesics with common endpoints at infinity cannot occur at all.
Also using Corollary 1.8 it is not hard to show that given any ξ ∈ S 1 , there
exists a subset Aξ ⊂ S 1 having at most a countable complement, such that
any point in Aξ is connected to ξ by a unique minimal geodesic.
We give a second application of Main Theorem 1.7. Recall the following
fact, usually referred to as the flat strip theorem [EO’N73]: If F is Riemannian and of non-positive curvature and if c, c0 : R → D is a pair of F -geodesics
having bounded Hausdorff distance, then c, c0 bound a flat strip, that is an
isometrically embedded strip (R × [0, a], euclidean) ,→ (D, F ) with some
a > 0. The following consequence of Main Theorem 1.7 was obtained earlier
by Y. Coudéne and B. Schapira [CS14]. It plays a role in the long-standing
6
J. P. SCHRÖDER
Figure 1. Left: g-geodesics γ ⊂ D with γ(∞) = 1, the
euclidean picture above and the hyperbolic picture below.
Center (above and below): The structure of R+ (1) given
by Theorem 1.3, assuming that the g-geodesic from −1 to
1 is periodic; one can see intersecting heteroclinic geodesics
between two periodic geodesics (the latter depicted in gray);
all other rays are asymptotic to periodic geodesics. Right:
The structure of R+ (1) given by Theorem 1.6, Corollary 1.8,
respectively for 1 ∈ S 1 − Π; here no two rays intersect.
question whether the geodesic flow of the non-positively curved metric F is
ergodic.
Corollary 1.9. If M is a closed, orientable surface of genus ≥ 2, F is
Riemannian with non-positive curvature and S ⊂ X is a flat strip of positive
width, then S foliated by F -geodesics projecting via p : D → M to closed
geodesics of a common homotopy class, i.e. the strip S is periodic.
Finally, we apply our results to a compactification of the universal cover
D associated to the Finsler metric F , due to M. Gromov [Gro81], sometimes
called the horofunction compactification. Letting f ∼ f 0 in C 0 (D) if f −f 0 =
0 -topology and consider the
const., we endow C 0 (D)/∼ with the quotient Cloc
embedding
iF : D → C 0 (D)/∼ ,
iF (x) = [dF (., x)].
The closure iF (D) is a natural compactification of D induced by F . In the
case where F is Riemannian and has non-positive curvature, it is well-known
MINIMAL RAYS ON CLOSED SURFACES
7
(cf. e.g. Theorem 8.13 in [BH99]) that iF extends to a homeomorphism
IF : D → iF (D).
For general Finsler metrics F , we obtain the following characterization, recalling the set Rp+ (ξ) for ξ ∈ Π in Theorem 1.3..
Corollary 1.10. If M is a closed, orientable surface of genus ≥ 1 and F
any Finsler metric on M , then the following conditions are equivalent:
(i) iF extends to a homeomorphism IF : D → iF (D),
(ii) for all ξ ∈ Π, the lamination of D defined by set Rp+ (ξ) is connected
(possibly consisting of a single geodesic or all of D).
In the torus-case, the Γ-invariance of Rp+ (ξ), combined with a celebrated
theorem of E. Hopf [Hop48], yields the following rigidity result.
Corollary 1.11. If G is any Riemannian metric on T2 , then iG extends to
a homeomorphism IG : D → iG (D) if and only if G is a flat metric.
Remark 1.12. For general Finsler metrics F on T2 , iF extends to the
homeomorphism if and only the geodesic flow of F admits a C 0 -foliation of
the tangent bundle T T2 by flow-invariant Lipschitz graphs over T2 , which
is sometimes called C 0 -integrability. See also [MS11].
These results show that the case, where iF extends to a homeomorphism
is very exceptional and non-generic in the genus one case M = T2 . Let us
turn to the higher genus case. We already remarked that, if F is Riemannian
with non-positive curvature, or more generally has no conjugate points, then
iF extends to a homeomorphism, which ties in with the C 0 -integrability
of the geodesic flow in the universal cover in the torus case in Remark
1.12. However, these are not the only examples where iF extends to a
homeomorphism. Any Finsler metric F on M determines a conformal class
of Finsler metrics denoted by
[F ] = {(f ◦ π) · F : f ∈ C ∞ (M ), f > 0 everywhere}.
We say that a property P is conformally generic in a class [F ], if there
exists a residual subset E ⊂ [F ], such that P holds for every F 0 ∈ E. Here,
a residual subset means a countable intersection of dense, open sets in the
C ∞ -topology. Using the main result from [Sch14] and Corollary 1.10, we
obtain the following result, contrary to the flat case in Corollary 1.11.
Corollary 1.13. If M is a closed, orientable surface of genus ≥ 2, then
P :
iF extends to a homeomorphism IF : D → iF (D)
is conformally generic in every conformal class of Finsler metrics on M .
Structure of this paper. Main Theorem 1.7 is proved in Section 2, with
the exception of the proof of one Lemma 2.15; the arguments will be selfcontained, apart from references to [Mor24]. Section 3 discusses properties
of the compactification iF (D), most of which are well-known; at the end of
Section 3 we give the proofs of Lemma 2.15 and Corollary 1.10.
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J. P. SCHRÖDER
2. The proof of Main Theorem 1.7
Throughout this Section 2, (M, g) denotes a hyperbolic surface. We denote by (D, g) the Poincaré disc model of hyperbolic geometry, i.e.
D = {x ∈ R2 : |x| < 1},
gx (v, w) = 4 · (1 − |x|2 )−2 h v, w i,
|.|, h., .i being the euclidean norm and inner product. (D, g) is (isometric
to) the Riemannian universal cover of (M, g) (hence we use the letter g for
the metric on both M and D) and we continue to write p : D → M for the
covering map. The group Γ of deck transformations τ : D → D with respect
to p acts freely on D by Möbius transformations; it extends continuously to
the closed disc D. For a subset Γ0 ⊂ Γ we write
Fix(Γ0 ) = {ξ ∈ S 1 | ∃τ ∈ Γ0 − id : τ ξ = ξ}.
As Γ consists of orientation-preserving isometries of (D, g), Γ − {id} splits
into hyperbolic transformations Γhyp (representing closed geodesics in M )
and parabolic transformations Γpar (representing cusps of M ). Then, with
the set Π of endpoints of periodic geodesics as defined in the introduction,
we find Π = Fix(Γhyp ). The set of oriented, unparametrized g-geodesics
γ ⊂ D will be denoted by G. We will say that γ ∈ G is the axis of an
element τ ∈ Γhyp , if τ γ = γ and if γ(∞) is the unstable fixed point of τ .
We let F be the Finsler metric on D lifted from some fixed Finsler metric
on M and assume that F, g are uniformly equivalent via the constant cF ≥ 1.
Hence, every τ ∈ Γ is also an F -isometry. We denote by cv : R → D the
unique F -geodesic determined by ċv (0) = v ∈ T D; note that cv is defined
on all of R due to the completeness of the metric g (Hopf-Rinow theorem).
Namely, it follows from the uniform equivalence of F, g, that
c−1
F · dg (x, y) ≤ dF (x, y) ≤ cF · dg (x, y)
∀x, y ∈ D.
We write SF D = {v ∈ T D : F (v) = 1} for the unit tangent bundle of F and
φtF : SF D → SF D for the geodesic flow of F given by φtF v = ċv (t).
Notation 2.1. We will often drop the dependence on F of objects that
are defined with respect to F ; e.g. minimal geodesics will refer to minimal
F -geodesics. Throughout, we will denote g-geodesics by γ, γn etc., while
F -geodesics will be termed c, cn etc..
Recall that rays c : R± → D or minimal geodesics c : R → D were curves
with lF (c; [a, b]) = dF (c(a), c(b)) for all a ≤ b in the domain of definition; in
particular, they are F -geodesics. We denote
R± := {v ∈ SF D | cv : R± → D is a ray},
M := {v ∈ SF D | cv : R → D is a minimal geodesic}.
The following lemma is a key property of rays; it is independent of the fact
that (M, g) is hyperbolic or dim M = 2. It excludes in particular successive
intersections of rays and shows that asymptotic rays can cross only in a
MINIMAL RAYS ON CLOSED SURFACES
9
common initial point. The idea of the proof is classical (cf. Theorem 6 in
[Mor24], another proof can be found in [Sch15a], Lemma 2.20).
Lemma 2.2. Let vn , v, wn , w ∈ R+ with vn → v, wn → w, πw = cv (a) for
some a > 0, but w 6= ċv (a). Then for all δ > 0 and sufficiently large n
inf{dg (cvn (s), cwn (t)) : s ∈ [a, ∞), t ∈ [δ, ∞)} > 0.
The analogous statement holds for R− .
Using Theorem 1.2 and the euclidean topology in R2 ⊃ D, we defined
asymptotic directions for v ∈ R± simply by cv (±∞) = limt→±∞ cv (t) ∈ S 1 .
For ξ ∈ S 1 , γ ∈ G, we set
R± (ξ) := {v ∈ R± : cv (±∞) = ξ},
M(γ) := {v ∈ M : cv (−∞) = γ(−∞) and cv (∞) = γ(∞)}.
Hence, the sets R± , M decompose:
[
R± =
R± (ξ) : ξ ∈ S 1 },
M=
[
M(γ) : γ ∈ G .
Another useful property is the continuity of asymptotic directions (cf. Theorem 7 in [Mor24]), namely
vn → v in R±
=⇒
cvn (±∞) → cv (±∞) in S 1 .
In order to motivate our approach to the proof of Main Theorem 1.7, we
will sketch how items (i) and (ii) of Morse’s Theorem 1.3 imply item (iii) of
that theorem, which can be stated as follows: If γ ∈ G is the axis of some
element τ ∈ Γhyp , c0 : R → D a minimal geodesic with τ c0 (R) = c0 (R), and
if c : R− → D is a backward ray with c(0) ∈ c0 (R) and c(−∞) = γ(−∞),
then c is a subray of c0 , i.e. c(R− ) ⊂ c0 (R).
Figure 2. The objects in the proof of Theorem 1.3 (iii).
Sketch of the proof of Theorem 1.3 (iii). (cf. Figure 2) Let τ, c, c0 as described above. By items (i) and (ii) of Theorem 1.3, both backward rays
c(t), τ −1 c(t) are asymptotic as t → −∞ to a single τ -invariant, minimal
˙ci (0) = ċi (T ) for i = 0, 1 – note that we
geodesic c1 . Let T < 0 such that τc
have the same period |T | for both c0 and c1 due to minimality of ci . Due to
the periodicity of c1 we find S −1, such that dF (τ −1 c(S + T ), c(S)) ≈ 0.
Assuming that c is not a subray of c0 , we find some ε, δ > 0, such that
dF (c(−δ), c0 (δ)) ≤ 2δ − ε
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J. P. SCHRÖDER
by shortening the vertex at c0 (0) = c(0). Hence, using the triangle inequality
and minimality of τ −1 c, we obtain a contradiction:
−T − S = dF (τ −1 c(S + T ), τ −1 c(0)) ≈ dF (c(S), c0 (−T ))
≤ dF (c(S), c(−δ)) + dF (c(−δ), c0 (δ)) + dF (c0 (δ), c0 (−T ))
≤ −S − δ + 2δ − ε − T − δ = −T − S − ε.
When proving Main Theorem 1.7, we will obtain a similar contradiction.
Assuming that the statement is false, we will make the following steps:
(A) Find a backward recurrent minimal geodesic c0 : R → D, i.e.
(1)
∃{τn } ⊂ Γ, Tn → −∞ :
˙
τd
n c0 (Tn ) → ċ0 (0)
(τn playing the role of τ −1 in the above sketch).
(B) Find a backward ray c : R− → D with
(2)
c(0) = c0 (0),
c(−∞) = c0 (−∞),
ċ(0) 6= ċ0 (0),
that also has the property (with {τn } and {Tn } given by step (A))
(3)
∃Sn → −∞ :
lim inf dF (τn c(Sn + Tn ), c(Sn )) = 0.
n→∞
Our first goal is to obtain the recurrent minimal geodesic c0 : R → D
described in step (A) above, from given recurrent g-geodesics, cf. Lemma
2.10 below. Let us see how recurrence can be seen in the universal cover.
Definition 2.3. We write γn → γ in G, if for the pairs of endpoints
γn (−∞) → γ(−∞) and γn (∞) → γ(∞) in S 1 .
A sequence {τn } ⊂ Γ is positive for γ ∈ G, if there exists a sequence
{xn } ⊂ γ with xn → γ(∞) ∈ S 1 (in the euclidean sense) and a compact
set K ⊂ D, such that τn xn ∈ K for all n. Analogously we define negative
sequences for γ ∈ G.
We will omit the proof of the following simple lemma.
˙γ is forward (backLemma 2.4. If γ ∈ G, then the projected geodesic pc
t
ward) recurrent for φg : Sg M → Sg M if and only if there exists a positive
(negative) sequence {τn } ⊂ Γ with τn γ → γ.
The following notions of “width” of R± (ξ), M(γ) will be useful.
Definition 2.5. For ξ ∈ S 1 and γ ∈ G we define
w± (ξ) := sup lim inf dg (cv (R± ), cw (t)) : v, w ∈ R± (ξ) ,
t→±∞
w0 (γ) := sup inf dg (cv (R), cw (t)) : v, w ∈ M(γ) .
t∈R
By Theorem 8 in [Mor24], in each M(γ) there are two particular minimal
geodesics, which we call the bounding geodesics of M(γ).
MINIMAL RAYS ON CLOSED SURFACES
11
Proposition 2.6 (bounding geodesics). For all γ ∈ G, there are two particular, non-intersecting, minimal geodesics c0γ , c1γ : R → D in M(γ), such
that all minimal geodesics from M(γ) lie in the closed strip in D bounded
by c0γ (R) and c1γ (R), denoted by Sγ ⊂ D.
As a rule, we will always assume that c1γ lies left of c0γ with respect to the
orientation of γ. Of course, it is possible that c0γ and c1γ coincide.
Remark 2.7. Theorem 1.2 implies that w± and w0 are finite. It is easy to
see that, if γ ∈ G and if c0γ , c1γ are the two bounding geodesics of M(γ) given
by Proposition 2.6, then
w0 (γ) = inf dg (c0γ (R), c1γ (t)).
t∈R
Next, we relate the widths w± and w0 to the action by Γ.
Proposition 2.8 (monotonicity of width). Let γ, γ 0 ∈ G and {τn } ⊂ Γ be
a positive sequence for γ, such that τn γ → γ 0 . Then w+ (γ(∞)) ≤ w0 (γ 0 ).
Analogously, if {τn } is negative for γ, then w− (γ(−∞)) ≤ w0 (γ 0 )
We will write cn → c for minimal geodesics, if ċn (tn ) → ċ(0) for some
sequence tn ∈ R.
Proof. Let v, w ∈ R+ (γ(∞)). We let γv , γw ∈ G be the g-geodesics connecting πv, πw to γ(∞), respectively. As τn is positive for γ, both γv , γw
will converge under the τn to γ 0 (γv , γw are asymptotic to γ in +∞ with
respect to dg , which is invariant under the isometries τn ). This shows, using
Theorem 1.2, that the rays τn cv , τn cw have (after taking subsequences) limit
minimal geodesics c0v , c0w from M(γ 0 ) and by definition of w0 we find
inf dg (c0v (R), c0w (t)) ≤ w0 (γ 0 ).
t∈R
Let now t ∈ R be fixed, then by τn cw → c0w , we find tn ∈ R with τn cw (tn ) →
c0w (t), and since {τn } is positive, we have tn → ∞. Since τn are isometries
with respect to dg , it follows from the definitions, that
lim inf dg (cv (R), cw (s)) ≤ lim dg (cv (R), cw (tn ))
s→∞
n→∞
= lim dg (τn cv (R), τn cw (tn ))
n→∞
= dg (c0v (R), c0w (t)).
The claim follows, as t, v, w were arbitrary.
Corollary 2.9. If γ ∈ G is forward recurrent, then w+ (γ(∞)) = w0 (γ);
if γ is backward recurrent, then w− (γ(−∞)) = w0 (γ). If G ⊂ G is such
that the projection p∗ (G) ⊂ Sg M is φtg -invariant, compact and minimal (i.e.
contains no non-trivial, closed, invariant subsets), then
w± (·(±∞))|G = w0 |G =: w(G) = const. .
12
J. P. SCHRÖDER
Proof. Letting γ, γ 0 ∈ G and {τn }, {τn0 } ⊂ Γ positive for γ, γ 0 , respectively
with τn γ → γ 0 and τn0 γ 0 → γ, we find by Proposition 2.8 and the definition
of w+ , w0
w+ (γ(∞)) ≤ w0 (γ 0 ) ≤ w+ (γ 0 (∞)) ≤ w0 (γ) ≤ w+ (γ(∞)),
i.e. all inequalities are equalities. The same holds for w− using negative
sequences. If γ is recurrent, we can take γ 0 = γ. If G ⊂ G is minimal in the
above sense, then for any γ ∈ G there exists a positive (negative) sequence
{τn } ⊂ Γ, such that {τn γ} = G. The claim follows.
By Theorem 1.3, if γ ∈ G is an axis, i.e. invariant under some element
of τ ∈ Γhyp , then the set of τ -invariant minimal geodesics Rp+ (γ(∞)) is
non-empty. The following lemma is a generalization of this fact to recurrent
γ ∈ G: given “homotopy information” {τn } leading to recurrence of γ, we
can find recurrent motions in M(γ) with the same “homotopy information”.
Recall the definition of the bounding geodesics ciγ in Proposition 2.6.
Lemma 2.10 (step (A, 1)). Let γ ∈ G be forward recurrent and {τn } ⊂ Γ
be a positive sequence for γ with τn γ → γ. Then for i = 0, 1 there exist
minimal geodesics ci : R → D in M(γ) and sequences Tni → ∞, such that
i
i
˙
ċi (0) is a limit point of {τd
n ci (Tn )} and lim inf t→∞ dg (cγ (R), ci (t)) = 0.
An analogous statement holds for backward recurrent γ ∈ G.
Proof. We will prove the lemma for i = 0. Let us say that a subset A ⊂
M(γ) is {τn }-invariant, if for all v ∈ A any limit geodesic of {τn cv } belongs
again to A. Then consider the set
A := v ∈ M(γ) : inf dg (c1γ (R), cv (t)) ≥ w0 (γ) .
t∈R
We claim that the set A has the following properties:
(i)
(ii)
(iii)
(iv)
A is closed, non-empty and φtF -invariant,
A is {τn }-invariant,
for all v ∈ A, we have lim inf t→∞ dg (c0γ (R), cv (t)) = 0,
if v, w ∈ A, then cv (R) and cw (R) are equal or disjoint.
Item (i) is easy to see, observing ċ0γ ∈ A. To show item (iii), suppose
inf t≥T dg (c0γ (R), cv (t)) ≥ ε for some v ∈ A, T ∈ R, ε > 0. Since v ∈ A and
cv lies between c0γ and c1γ , we find for t ≥ T
dg (c1γ (R), c0γ (t)) = inf dg (c1γ (R), cv (s)) + dg (cv (s), c0γ (t)) ≥ w0 (γ) + ε.
s∈R
Using w+ (γ(∞)) = w0 (γ) in Corollary 2.9, we find a contradiction. Item
(iv) follows from Lemma 2.2 together with item (iii). Let us prove item (ii).
Let v ∈ A and assume that after taking a subsequence
˙
w := lim τd
n cv (tn ) ∈ M(γ)
n→∞
MINIMAL RAYS ON CLOSED SURFACES
13
for some sequence tn → ∞ (note that M(γ) is {τn }-invariant). Suppose also
that τn c1γ → c1 . By v ∈ A we have for t ∈ R
w0 (γ) ≤ lim inf dg (c1γ (R), cv (tn + t)) = lim inf dg (τn c1γ (R), τn cv (tn + t))
n→∞
n→∞
1
= dg (c1 (R), cw (t)) ≤ inf dg (cγ (R), c1 (s)) + dg (c1 (s), cw (t))
s∈R
=
dg (c1γ (R), cw (t)).
For the last equality observe, since cv lies right of c1γ , that under the application of {τn } the limit geodesic cw will lie right of c1 , so c1 lies in the closed
strip between c1γ and cw . As t was arbitrary, this shows w ∈ A.
Let A be the collection of subsets of A that have properties (i) and (ii)
above. Then A 6= ∅ by A ∈ A and A is partially ordered by ⊂. Moreover,
any linearly ordered subset of A has the intersection of its members as a
⊂-smallest element (the intersection is non-empty by Cantor’s Intersection
Theorem, if using item (iv) we think of the lamination A as a set of points
in some compact, transverse interval). Hence, Zorn’s Lemma shows the
existence of a minimal element A0 ∈ A with respect to inclusion.
Since A0 defines a lamination of D and A0 is closed, there exists a leftmost
minimal geodesic c0 in A0 and we claim that c0 is {τn }-recurrent (in the sense
of the statement of the lemma). For this, let c1 be the leftmost limit geodesic
of {τn c0 } and A1 ⊂ A0 be the initial vectors of minimal geodesics in A0
lying right of or on c1 . Since the limiting progress under {τn } preserves the
ordering in the lamination A, any {τn }-limit of A1 lies right of the leftmost
limit of c0 , that is, right of c1 . This shows that A1 is {τn }-invariant and by
minimality of A0 we find A1 = A0 and hence c0 = c1 , i.e. c0 has the desired
forward recurrence property.
Our next goal is to obtain the ray c : R− → D described in step (B) at
the beginning of this section, cf. Lemma 2.13 below. For this, we use the
concept of ω-compactness.
Definition 2.11. A g-geodesic γ ∈ G is ω-compact, if the ω-limit set of the
geodesic p(γ) in Sg M contains a compact, φtg -invariant set. We define
Ω := {ξ ∈ S 1 | ∃ ω-compact γ ∈ G : ξ = γ(∞)}.
Remark 2.12. Note that if γ ∈ G is ω-compact, then so are all other γ 0 ∈ G
with γ 0 (∞) = γ(∞), since all such γ 0 are asymptotic to γ with respect to
the metric g. Hence, ω-compactness is really a property of the point γ(∞).
In terms of Definitions 2.5 and 2.11, our Main Theorem 1.7 claims
w+ (ξ) = 0
∀ξ ∈ Ω − Fix(Γhyp ).
Lemma 2.13 (step (B, 2)). Assume w+ (ξ) > 0 for some ξ ∈ Ω − Fix(Γhyp ).
Then there exist a backward recurrent γ ∈ G, i ∈ {0, 1} and a backward ray
c : R− → D with c(−∞) = γ(−∞), c(0) ∈ ciγ (R), ċ(0) ∈
/ ċiγ (R) and the
14
J. P. SCHRÖDER
following property: If {τn } ⊂ Γ is a negative sequence for γ with τn γ → γ,
such that {τn c} converges to a minimal geodesic c∗ in M(γ), then
lim inf dg (c∗ (R), c(t)) = 0.
t→−∞
The proof of Lemma 2.13 needs some understanding of the way that
sequences τn γ can approximate other γ 0 under the assumption w+ (γ(∞)) >
0. Recall the notation Sγ ⊂ D for the closed strip between the bounding
geodesics c0γ , c1γ of M(γ) in Proposition 2.6.
Lemma 2.14. Let γ, γ 0 ∈ G, {τn } ⊂ Γ be a positive sequence for γ with
τn γ → γ 0 and v, w ∈ R+ (γ(∞)). If for all n both sets τn cv (R+ ), τn cw (R+ )
lie in the same connected component of the open set D − Sγ 0 , then
lim inf dg (cv (R+ ), cw (t)) = 0.
t→∞
Proof of Lemma 2.14. Suppose that τn cv → c0v and τn cw → c0w , then ċ0v , ċ0w ∈
M(γ 0 ) by τn γ → γ 0 and by the assumption that both τn cv , τn cw lie on the
same side of Sγ 0 we find that both c0v and c0w coincide with one bounding
geodesic of M(γ 0 ), i.e. c0v (R) = c0w (R). The claim follows as in the proof of
Proposition 2.8.
Figure 3. The objects in the proof of Lemma 2.13.
Proof of Lemma 2.13. Let ξ ∈ Ω − Fix(Γhyp ) and w+ (ξ) > 0. We fix an ωcompact γ 0 ∈ G with γ 0 (∞) = ξ and write G ⊂ G for the closed, Γ-invariant
set of g-geodesics lifted from a compact, φtg -invariant, minimal subset of
Sg M found in the ω-limit set of p(γ 0 ) due to ω-compactness of γ 0 . In the
case where G does not consist of the Γ-translates of a single axis, we replace
γ 0 by some g-geodesic in G. In this way we can assume by Proposition 2.8
and Corollary 2.9, that w+ (ξ) = w0 (γ 0 ) = w(G) > 0 in the case where G is
non-periodic; here we set c0 := c0γ 0 , c1 := c1γ 0 . In the case where G is periodic,
we choose v, w ∈ R+ (ξ) with lim inf t→∞ dg (cv (R), cw (t)) > 0 and such that
cv lies right of cw ; here we set c0 := cv , c1 := cw .
Since G projects to the ω-limit set of p(γ 0 ), we can find γ 00 ∈ G and a
positive sequence {τn } ⊂ Γ for γ 0 , such that τn γ 0 → γ 00 . Under the action of
MINIMAL RAYS ON CLOSED SURFACES
15
{τn }, the rays c0 , c1 will converge to minimal geodesics in M(γ 00 ). By the
assumption of ω-compactness we have ξ ∈
/ Fix(Γpar ) and by ξ ∈
/ Fix(Γhyp ),
we find ξ ∈
/ Fix(Γ). Hence, without loss we can assume τn ξ 6= γ 00 (∞) for
all n. This together with Lemma 2.14 shows that one of the rays τn ci will
cross one of the bounding geodesics ciγ 00 of M(γ 00 ). We will assume that
τn c0 crosses c1γ 00 from the right to the left and set i = 1; the other cases are
analogous. The situation is displayed in Figure 3.
Let xn ∈ τn c0 (R+ ) ∩ c1γ 00 (R) and choose a new sequence {τn0 } ⊂ Γ, such
that (after passing to subsequences) τn0 γ 00 → γ and τn0 xn → x for some
γ ∈ G, x ∈ D (here we use the compactness of p∗ (G) ⊂ Sg M ). Letting
either yn ∈ τn c0 (R+ ) ∩ c0γ 00 (R) or yn = τn c0 (0) if the intersection is empty,
we have dg (xn , yn ) → ∞ (in the first case use that any limit of {τn c0 } lies
in the strip Sγ 00 ; in the second case assume that dg (xn , yn ) stays bounded
and apply Lemma 2.14 to contradict lim inf dg (c0 (R+ ), c1 (t)) > 0). We take
subsequences and find minimal geodesics c00 , c01 : R → D with τn0 τn ci → c0i
for i = 0, 1. Note that c0i (−∞) = γ(−∞) by yn → γ 00 (−∞).
We claim that the minimal geodesic c00 (t) has to leave the strip Sγ to
the left, as t → +∞. To see this, observe that at each xn , left of τn c0 ,
there is a half strip Sn+ of uniform width w := lim inf dg (c0 (R+ ), c1 (t)) > 0
between τn c0 , τn c1 , which will under {τn0 } converge to a strip of width w left
of the limit geodesic c00 , starting at x. On the other hand, there is a similar
half strip right of any limit geodesic of {τn0 c1γ 00 } in M(γ), obtained from the
sequence {τn0 Sγ 00 }. Assuming c00 (∞) = γ(∞), this would mean that in the
positive end of Sγ , there are two separate strips, one of width w > 0, the
other of width w(G), i.e. w(G) = w+ (γ(∞)) ≥ w(G) + w, a contradiction.
Hence, we can assume that c00 (0) ∈ c1γ (R) and ċ00 (0) ∈
/ ċ1γ (R).
0
We set c := c0 . Consider a negative sequence {τn00 } ⊂ Γ for γ with
00
τn γ → γ (γ belongs to a minimal set G and hence is backward recurrent)
and τn00 c00 → c∗0 for some minimal geodesic c∗0 : R → D in M(γ). To finish the
proof of the lemma, we need to show lim inf t→−∞ dg (c∗0 (R), c00 (t)) = 0. Let
us first assume that γ is an axis. In this case, τn00 = τ kn for some τ ∈ Γhyp
with τ γ = γ and a sequence {kn } ⊂ Z. Using Theorem 1.3 we find that
c∗0 is τ -invariant and c0 is asymptotic to c∗0 . The claim follows, so we now
consider the case where G is non-periodic. Assume that after passing to a
subsequence we have also τn00 c01 → c∗1 and note that then ċ∗i ∈ M(γ). Arguing
by contradiction, we suppose
ε := lim inf dg (c∗0 (R), c0 (t)) > 0.
t→−∞
We consider first the case where c∗0 lies right of c0 , asymptotically in −∞.
Using that the c0i were obtained from the bounding geodesics ci = ciγ 0 we
find by γ 0 ∈ G
inf dg (c00 (R), c01 (t)) = w0 (γ 0 ) = w(G),
t∈R
16
J. P. SCHRÖDER
hence
w(G) = w− (γ(−∞)) ≥ lim inf dg (c∗0 (R), c01 (t))
≥ lim inf
t→−∞
t→−∞
∗
dg (c0 (R), c00 (t))
+ inf dg (c00 (R), c1 (t)) = ε + w(G),
t∈R
a contradiction. In the remaining case, where c∗0 lies left of c00 , observe that
also inf t∈R dg (c∗0 (R), c∗1 (t)) = w(G), hence as before
w(G) = w− (γ(−∞)) ≥ lim inf dg (c00 (R), c∗1 (t))
≥ lim inf
t→−∞
t→−∞
0
dg (c0 (R), c∗0 (t))
+ inf dg (c∗0 (R), c∗1 (t)) = ε + w(G),
t∈R
again a contradiction.
In Lemma 2.16 below, we finish the second part of step (B) from the
beginning of this section. To prove it, we need the following lemma, whose
prove is delayed to Section 3 below.
Lemma 2.15. Let ξ ∈ S 1 − Fix(Γpar ) and {τn } ⊂ Γ, such that τn ξ → ξ.
Then there exists a function u : D → R with u(0) = 0 and the following
properties:
(i) u is globally Lipschitz continuous with respect to dg ,
(ii) τn u → u uniformly in bounded sets, where τn u := u ◦ τn−1 − u ◦ τn−1 (0),
(iii) for any v ∈ R− (ξ), the function t ∈ R− 7→ t − u ◦ cv (t) is bounded and
non-decreasing.
Note how the function u : D → R in Lemma 2.15 relates topological
information of M in item (ii) with variational and dynamical information
of the metric F in item (iii). Indeed, we will see in the proof of the next
Lemma 2.16, that minimal geodesics in R− (ξ), which are recurrent under
{τn } have e.g. the property, that t − u ◦ cv (t) ≡ −u ◦ u(πv).
Lemma 2.16 (step (B, 3)). Let ξ ∈ Ω − Fix(Γhyp ) and assume w+ (ξ) > 0.
We let γ ∈ G be the backward recurrent g-geodesic, c : R− → D be the
backward ray and i ∈ {0, 1} given by Lemma 2.13, such that c(0) ∈ ciγ (R).
If ci : R → D is the backward recurrent minimal geodesic associated to γ, i
by Lemma 2.10, {τn } ⊂ Γ a negative sequence for γ and Tni → −∞, such
i
˙
that ċi (0) is a limit point of {τd
n ci (Tn )}, then c intersects ci (transversely)
and
∃Sn → −∞ :
lim inf dg (c(Sn ), τn c(Tni + Sn )) = 0.
n→∞
Proof. We will assume i = 0, the other case being analogous. The intersection of c and c0 follows from the intersection of c with c0γ , using Lemma 2.2
and lim inf t→−∞ dg (c0γ (R), c0 (t)) = 0.
0
∗
˙
We take a subsequence to obtain τd
n c0 (Tn ) → ċ0 (0) and τn c → c for some
∗
minimal geodesic c in M(γ). By Lemma 2.13 we have
lim inf dg (c∗ (R), c(S)) = 0.
S→−∞
MINIMAL RAYS ON CLOSED SURFACES
17
We fix ε > 0 and let S ∈ R− , such that dg (c∗ (R), c(S)) ≤ ε. Let u : D → R
be the function given by Lemma 2.15 for the case τn γ(−∞) → γ(−∞)
(note that in the proof of Lemma 2.13, p(γ) lies in a compact set in M , so
γ(−∞) ∈
/ Fix(Γpar )). We find some T ∈ R− and n0 ∈ N, such that
(4)

dg (τn0 c(T + S), c(S)) ≤ 2ε,



 d (τ c (T 0 ), c (0)) ≤ ε,
0
g n0 0 n0

u)
◦
c(S)
−
u ◦ c(S)| ≤ ε,
|(τ
n0



|(τn0 u) ◦ c0 (0) − u ◦ c0 (0)| ≤ ε.
Moreover, by Lemma 2.15 (iii) we may assume that |S| is large enough, such
that
(5)
(S + t) − u ◦ c(S + t) − S − u ◦ c(S) ≤ ε
∀t ≤ 0.
(4), the minimality of c and the triangle inequality show for S and n0 as
chosen above, that
dg (c(S), τn0 c(Tn00 + S))
≤ dg (c(S), τn0 c(T + S)) + dg (τn0 c(T + S), τn0 c(Tn00 + S))
≤ 2ε + |T − Tn00 |.
We estimate |T − Tn00 |. First, let us see that for the minimal geodesic c0
(6)
t = u ◦ c0 (t) − u ◦ c0 (0)
∀t ≤ 0.
0 ) we find by definition
Indeed, by τn u → u and τn c0 (.+Tn0 ) → c0 (both in Cloc
of τn u = u ◦ τn−1 − u ◦ τn−1 (0), that
t − u ◦ c0 |t0 = lim t − (τn u) ◦ τn c0 (. + Tn0 )|t0
n→∞
= lim t − u ◦ τn−1 ◦ τn c0 (. + Tn0 )|t0
n→∞
= lim t − u ◦ c0 (. + Tn0 )|t0
n→∞
= 0,
using again Lemma 2.15 (iii) for the last equality.
Next, observe that by (5)
(7)
|T − u ◦ c(S + T ) + u ◦ c(S)| ≤ ε.
18
J. P. SCHRÖDER
Letting L ≥ 0 be a Lipschitz constant for u, we find
|T − Tn00 |
(6)
= |T − u ◦ c0 (Tn00 ) + u ◦ c0 (0)|
≤ T − u ◦ c(S + T ) + u ◦ c(S) + u ◦ c(S + T ) − u ◦ τn−1
c(S)
0
+ u ◦ τn−1
c(S) − u ◦ τn−1
(0) − u ◦ c(S) + u ◦ τn−1
c (0) − u ◦ c0 (Tn00 )
0
0
0 0
c (0) + u ◦ τn−1
(0)
+ u ◦ c0 (0) − u ◦ τn−1
0 0
0
(7)
≤ ε + L · dg (c(S + T ), τn−1
c(S))
0
c (0), c0 (Tn00 ))
+ (τn0 u) ◦ c(S) − u ◦ c(S) + L · dg (τn−1
0 0
+ u ◦ c0 (0) − (τn u) ◦ c0 (0)
0
(4)
≤ 3ε + 3Lε.
This concludes the proof.
We are now ready to prove Main Theorem 1.7 along the lines sketched at
the beginning of this section. Recalling Definitions 2.5 and 2.11, we need to
show
w+ (ξ) = 0
∀ξ ∈ Ω − Fix(Γhyp ).
Figure 4. The argument in the proof of Main Theorem 1.7.
Proof of Main Theorem 1.7. Let ξ ∈ Ω − Fix(Γhyp ). Arguing by contradiction, assume w+ (ξ) > 0. Then by Lemmata 2.10, 2.13 and 2.16, we are
given a minimal geodesic c0 : R → D, a backward ray c : R− → D with
c(0) = c0 (0), c(−∞) = c0 (−∞), ċ(0) 6= ċ0 (0) and sequences {τn } ⊂ Γ and
Sn , Tn → −∞, such that after taking subsequences
˙
(8)
τd
dg (c(Sn ), τn c(Tn + Sn )) → 0.
n c0 (Tn ) → ċ0 (0),
We find a small δ > 0 and, depending on the angle between ċ0 (0) and ċ(0),
some ε > 0 with
dF (c(−δ), c0 (δ)) ≤ 2δ − ε.
˙
By τd
n c0 (Tn ) → ċ0 (0) we then obtain for large n, that
(9)
dF (c(−δ), τn c0 (Tn + δ)) ≤ 2δ − ε/2.
MINIMAL RAYS ON CLOSED SURFACES
19
The minimality of τn c and c0 (0) = c(0) show for sufficiently large n
|Tn + Sn | = dF (τn c(Sn + Tn ), τn c(0))
≤ dF (τn c(Sn + Tn ), c(Sn )) + dF (c(Sn ), c(−δ))
+ dF (c(−δ), τn c0 (Tn + δ)) + dF (τn c0 (Tn + δ), τn c(0))
(8),(9)
≤ ε/4 + |Sn + δ| + (2δ − ε/2) + |Tn + δ|
= |Tn + Sn | − ε/4.
This is a contradiction.
3. Gromov’s compactification
We discuss in this section the compactification of (D, F ) due to Gromov [Gro81], that we already saw in the introduction; see also [Bal95] or
[BH99], while many aspects of the theory are also treated in A. Fathi’s weak
KAM theory [Fat08]. Most of the statements in this section are well-known
throughout the literature. In this section we will also prove Lemma 2.15,
which we used in the proof of Main Theorem 1.7 in Section 2.
We start by letting (X, g) be a complete Riemannian manifold, carrying
also a Finsler metric F , which is uniformly equivalent to g via a constant
cF ≥ 1. Thus, we have the (non-symmetric) Finsler distance dF and the
Riemannian distance dg on X. Letting f ∼ f 0 in C 0 (X), if f − f 0 = const.,
we consider the map
iF : X → C 0 (X)/∼ ,
iF (x) = [dF (., x)]
0 -topology in C 0 (X)/ . Let us start with a simple,
and use the quotient Cloc
∼
well-known proposition, whose proof will be omitted; it relies on the triangle
inequality, while for the compactness of iF (X) one uses the Arzela-Ascoli
theorem.
Proposition 3.1. The map iF is an embedding and the closure iF (X) is
compact. If η = lim iF (xn ) ∈ iF (X), then η ∈ iF (X) if and only if {xn } is
bounded. Moreover, the image iF (X) is open in iF (X).
The following terminology stems from the fact that the functions u ∈ η ∈
ig (X) − ig (X) have the horocycles as their level sets.
Definition 3.2. We shall call iF (X) the horofunction compactification and
∂F X := iF (X) − iF (X) the horofunction boundary of (X, F ).
The triangle inequality shows for [u] = lim iF (xn ) ∈ iF (X) and x, y ∈ X
(10)
u(x) − u(y) = lim dF (x, xn ) − dF (y, xn ) ≤ dF (x, y).
In particular, by uniform equivalence of F, g, any representative u of any
element in iF (X) is globally Lipschitz with respect to dg with Lipschitz
20
J. P. SCHRÖDER
constant cF , hence differentiable almost everywhere. Observe moreover that,
if [u] ∈ iF (X), v ∈ SF X and a > 0, then
u ◦ cv (a) − u ◦ cv (0) ≥ −dF (cv (0), cv (a)) ≥ −lF (cv ; [0, a]) = −a,
while equality implies, using (10), that the same holds true for any a0 ∈ [0, a]
and cv |[0,a] is a minimizing segment. This motivates the following definition.
Definition 3.3. Given an element η = [u] ∈ iF (X), the set
Cal(η) := {v ∈ SF X | ∃a > 0 : u ◦ cv (a) − u ◦ cv (0) = −a}
is called the calibrating set of η. The curve cv |[0,a] in the definition of
Cal(η) = Cal([u]) is said to calibrate η or u.
Obviously, the set Cal(η) is independent of the choice u ∈ η. The next
lemma is well-known, cf. e.g. Theorem 4.3.8 in [Fat08]. In the proof we can
see how the set Cal(η) relates to the differential of the functions u ∈ η.
Lemma 3.4. Let [u] ∈ iF (X).
(i) If u is differentiable in x ∈ X, then card(Cal([u]) ∩ Tx X) ≤ 1.
(ii) If cv |[0,a] calibrates u, then u is differentiable in the set cv (0, a) ⊂ X.
Remark 3.5. One can show (Lipschitz) regularity results on the set Cal([u])
strengthening Lemma 3.4 (i), cf. Sections 4.11 and 4.13 in [Fat08].
Proof. (i). For any v ∈ Tx X, we have by (10), that
−du(x)v = limt&0
1
t
· u(cv (0)) − u(cv (t)) ≤ limt&0
1
t
Rt
0
F (φsF v)ds = F (v),
which becomes an equality, if v ∈ Cal([u]). Hence, for v ∈ Cal([u]) we find
−du(x)v = max{−du(x)w : w ∈ Tx X, F (w) = 1}.
Due to the strict convexity of F 2 |Tx X , the point v where this maximum is
attained is unique.
(ii). For any x ∈ X we find by (10)
ψ− (x) := −dF (cv (0), x) + u ◦ cv (0) ≤ u(x)
≤ dF (x, cv (a)) + u ◦ cv (a) =: ψ+ (x).
As cv |[0,a] calibrates u, we find ψ− = u = ψ+ in cv (0, a), while ψ± are both
smooth in cv (0, a) by minimality of the segment cv |[0,a] . The differentiability
of u in cv (0, a) follows.
For a sequence of subsets An ⊂ SF X we shall write
lim An := {v ∈ SF X : v is a limit point of some sequence vn ∈ An }.
Lemma 3.6.
(i) If x ∈ X and η = iF (x), then
Cal(η) = {v ∈ SF X : πv 6= x, cv (dF (πv, x)) = x}.
In particular, π(Cal(η)) = X − {x} = X − {i−1
F (η)}.
(ii) If η ∈ ∂F X, then π(Cal(η)) = X and for any v ∈ Cal(η), the curve
cv : R+ → X calibrates η. In particular, Cal(η) ⊂ R+ .
MINIMAL RAYS ON CLOSED SURFACES
21
(iii) If η, ηn ∈ iF (X) with ηn → η, then
lim Cal(ηn ) − π −1 (i−1
F (η)) = Cal(η).
Note that i−1
F (η) = ∅ if η ∈ ∂F X.
Proof. (i). For any y ∈ X − {x}, consider an arc-length minimizing segment
cy : [0, dF (y, x)] → X from y to x. Then for [u] = iF (x), t ∈ [0, dF (y, x)]
u ◦ cy (t) − u(y) = dF (cy (t), x) − dF (y, x) = −dF (y, cy (t)) = −t,
i.e. ċy (0) ∈ Cal(iF (x)). Conversely, let v ∈ Cal(iF (x)), such that cv |[0,a]
calibrates iF (x) for some a > 0. Arguing as before with y := cv (a), we find
that the concatenation cv |[0,a] ∗ cy calibrates iF (x), hence cv |[0,dF (πv,x)] is a
minimizing segment from πv to x.
(ii). Let η = lim iF (xn ). Since η ∈
/ iF (X), Proposition 3.1 shows that
{xn } tends to infinity. If x ∈ X, we obtain a vector v ∈ SF X with πv = x
by taking a limit of ċn (0), where cn is a minimizing segment from x to xn .
By the unboundedness of {xn } and item (i), cvn calibrate iF (xn ) on [0, a]
for any given a ∈ R+ and large n. The continuity of
(11)
([u], v) ∈ iF (X) × SF X 7→ u ◦ cv (a) − u ◦ cv (0)
shows that cv |[0,a] calibrates η. Hence, π(Cal(η)) = X and with a → ∞,
from any x ∈ X there initiates a ray calibrating η on all of R+ . Let now
v ∈ Cal([u]). We find then a ray cε : R+ → X with cε (0) = cv (ε) calibrating
η, and Lemma 3.4 shows ċε (0) = ċv (ε) (for small ε). Taking the limit ε & 0
we find that cv : R+ → X calibrates η.
(iii). Let first vn ∈ Cal(ηn ), such that vn → v ∈
/ π −1 (i−1
F (η)) (if η ∈
iF (X)) and suppose cvn calibrate ηn on an interval [0, an ]. By πv 6= i−1
F (η)
and items (i) and (ii), we can suppose that a := inf an > 0, such that again
by the continuity of the map (11), cv calibrates η on [0, a]. Conversely, if
v ∈ Cal(η), let t > 0 and vn ∈ Cal(ηn ), such that πvn = cv (t) (note that by
v ∈ Cal(η), we have inf dg (πv, i−1
F (ηn )) > 0, so vn exist for small t). Lemma
3.4 and lim vn ∈ Cal(η) show lim vn = ċv (t) and since lim Cal(η) is closed,
we have v ∈ lim Cal(η) by letting t & 0. πv 6= i−1
F (η) is trivial.
The elements in the horofunction compactification are uniquely determined by their calibrating sets, as we shall prove now. The idea is the same
as in the proof of Lemma 3.4 (i).
Corollary 3.7. If η, η 0 ∈ iF (X) with Cal(η) = Cal(η 0 ), then η = η 0 .
Proof. Given v ∈ SF X with πv = x, by smoothness and convexity of F 2 |Tx X
we find a unique half line R+ φ in the dual space Tx∗ X, such that
φ(v) = max{φ(w) : w ∈ SF X, πw = x}.
Let η = [u] and consider a point x ∈ X, where du(x) exists. The arguments
for Lemma 3.4 (i) show that, if v ∈ Cal(η) ∩ Tx X, then −du(x) ∈ R+ φ, i.e.
Cal(η) determines du(x) uniquely up to a positive multiple. The condition
22
J. P. SCHRÖDER
u ◦ cv (t) = u(x) − t for small t then determines du(x) uniquely. Hence, the
differential of u is determined by Cal(η) in the full measure set where the
Lipschitz function u is differentiable, intersected with the set full measure
set π(Cal(η)), cf. Lemma 3.6.
Our next goal is to link the horofunction compactification iF (X) to the
geometry of (X, g). For this, we come back to our original setting from
the introduction, i.e. X = D is the open unit disc in R2 and g is either
the Poincaré metric gx = 4(1 − |x|2 )−2 h., .i or the pullback of the euclidean
metric under the map f : D → R2 , f (x) = x/(1 − |x|2 ). We assume that D
is the universal cover of the surface M = D/Γ, which in the latter case is
assumed to be the 2-torus. F is a lift to D of a Finsler metric on M .
We shall need a refinement of Theorem 1.2, cf. [Mor24], [Hed32].
Lemma 3.8 (Morse, Hedlund). There exists a constant C ≥ 0 depending
only on F, g with the following property: If x, y ∈ D and if c : [0, dF (x, y)] →
D is a minimizing F -geodesic segment from x to y, then for the (unique)
g-geodesic segment γx,y from x to y we find
max{dg (γx,y , c(t)) : 0 ≤ t ≤ dF (x, y)} ≤ C.
Recall that in the introduction, we defined for F -rays c : R+ → D the
asymptotic direction by c(∞) = limt→∞ c(t) ∈ S 1 , using the euclidean topology in R2 ⊃ D. With Lemma 3.8, we can associate a unique asymptotic
direction to the sets Cal(η) ⊂ R+ for η ∈ ∂F D.
Corollary 3.9. If η ∈ ∂F D, then there exists a unique asymptotic direction
H(η) ∈ S 1 with Cal(η) ⊂ R+ (H(η)). Moreover,
[
R+ (ξ) =
Cal(η) : H(η) = ξ .
Proof. Let η = lim iF (xn ), v, v 0 ∈ Cal(η) and t > 0. By the arguments
in the proof of Lemma 3.6 (ii), we find rays c, c0 : R+ → D calibrating
η with c(0) = cv (t), c0 (0) = cv0 (t) as a limit of minimal segments from
cv (t), cv0 (t), respectively to xn (after possibly passing to a subsequence). By
Lemma 3.4 we find ċ(0) = ċv (t) and ċ0 (0) = ċv0 (t). Letting γn , γn0 ⊂ D
be the g-geodesic segments from c(0), c0 (0) to xn , we can find limit g-rays
γ = lim γn , γ 0 = lim γn0 . By the non-positive curvature of g, there exists a
constant B = dg (c(0), c0 (0)) ≥ 0, such that the curves γn , γn0 have distance
bounded by B for all n and the same is true in the limit. Hence, using
Lemma 3.8 for c(∞) = γ(∞) and c0 (∞) = γ 0 (∞), we find
cv (∞) = c(∞) = γ(∞) = γ 0 (∞) = c0 (∞) = cv0 (∞).
This shows that H(η) is well-defined. For the second claim, let v ∈ R+ (ξ)
and η = lim iF (cv (tn )) for some sequence tn → ∞, such that the limit exists.
Each subsegment cv |[0,tn ] calibrates iF (cv (tn )) by minimality of cv and hence
cv : R+ → D calibrates the limit η by Lemma 3.6 (iii), i.e. v ∈ Cal(η). By
assumption on v and definition of H, H(η) = cv (∞) = ξ.
MINIMAL RAYS ON CLOSED SURFACES
23
Using Corollary 3.9, we can define a “projection” to the closed disc D,
(
i−1 (η) : η ∈ iF (D)
hF : iF (D) → D,
hF (η) = F
.
H(η) : η ∈ ∂F D
By definition, hF (η) ∈ D is the unique endpoint of all curves calibrating η.
Lemma 3.10. hF : iF (D) → D is continuous and surjective. In particular,
if hF is also injective, then hF is a homeomorphism.
Proof. Let ηn → η in iF (D) and x ∈ D with iF (x) 6= η. If cn : [0, an ) → D
is a maximal curve calibrating ηn with cn (0) = x (note that iF (x) 6= ηn for
large n by the continuity of i−1
F ), then hF (ηn ) = cn (an ). We can by the
−1
continuity of iF assume that η ∈ ∂F D, and hence an → ∞ or an = ∞.
Take a subsequence, such that the limits ξ := lim cn (an ) ∈ S 1 and v :=
lim ċn (0) ∈ R+ exist. Using Lemma 3.8, we find cv (∞) = ξ. On the other
hand, by Lemma 3.6 (iii), v ∈ Cal(lim ηn ) = Cal(η). Hence,
hF (η) = cv (∞) = ξ = lim cn (an ) = lim hF (ηn ),
independent of the chosen subsequence.
To show that hF is a surjection, let w.l.o.g. ξ ∈ S 1 . If {xn } ⊂ D is any
sequence with xn → ξ in the euclidean sense and such that lim iF (xn ) exists,
then by continuity
ξ = lim xn = lim hF (iF (xn )) = hF (lim iF (xn )) ∈ hF (iF (D)).
The last statement is a standard fact in topology, using that iF (D) is compact and Hausdorff.
Our next goal is to prove Lemma 2.15 from Section 2, finishing the proof
of Main Theorem 1.7. In the following proposition, we show how to find
special elements in h−1
F (ξ), using dim D = 2.
1
Proposition 3.11 (boundary of h−1
F (ξ)). For ξ ∈ S , there exist two unique
−1
elements η0 , η1 ∈ hF (ξ) with the following property: for any sequence ξn →
ξ in S 1 with ξn 6= ξ and any sequence ηn ∈ h−1
F (ξn ), any limit point of {ηn }
lies in {η0 , η1 }. More precisely, assuming the counterclockwise orientation
of S 1 , we have
(
η0 : if ξn < ξ ∀n
lim ηn =
.
n→∞
η1 : if ξn > ξ ∀n
Proof. (cf. Figure 5) Let ξn , ξn0 → ξ and assume that in the counterclockwise
0
orientation of S 1 , we have ξn , ξn0 > ξ. Choose a pair ηn ∈ h−1
F (ξn ), ηn ∈
−1
0
0
0
h−1
F (ξn ) and let η, η ∈ hF (ξ) be limits of {ηn }, {ηn }, respectively (after
passing to a subsequence). We claim that Cal(η) = Cal(η 0 ), which also shows
η = η 0 by Corollary 3.7. By symmetry, we show only Cal(η) ⊂ Cal(η 0 ).
Let v ∈ Cal(η) and t > 0. We claim ċv (t) ∈ Cal(η 0 ), which shows v ∈
Cal(η 0 ), since t > 0 is arbitrary and Cal(η 0 ) is closed. By Lemma 3.6 (iii),
24
J. P. SCHRÖDER
Figure 5. The argument in the proof of Proposition 3.11.
The gray circles indicate the (almost) intersections that contradict Lemma 2.2.
0 )) = D,
there exist vn ∈ Cal(ηn ) ⊂ R+ (ξn ) with vn → v. By π(Cal(ηm
0
0
0
0
we find vm ∈ Cal(ηm ) ⊂ R+ (ξm ) with πvm = cv (t), converging to a vector
v 0 ∈ Cal(η 0 ). Let us assume v 0 6= ċv (t). The points πv, ξ, ξn form almost a
triangle in R2 , bounded by a segment in S 1 and two rays cv , cvn , while the
rays cv , cvn run almost parallel past the point πv 0 = cv (t). For large m, we
0 lies on the segment in S 1 bounding the triangle. By
can assume that ξm
0
v 0 6= ċv (t), the ray cvm
now has to leave the triangle (as it is thin at πv 0 )
0 , contradicting Lemma 2.2.
and then reenter it to reach ξm
In the next lemma, we shall use the reversed Finsler metric F (v) = F (−v),
as Lemma 2.15 deals with backward rays c : R− → D.
Lemma 3.12. Let ξ ∈ S 1 , [u] ∈ ∂F D with hF ([u]) = ξ and v ∈ R− (ξ).
Then the function
t ∈ R− 7→ t − u ◦ cv (t)
is bounded and non-decreasing.
Proof. We let [u] ∈ hF−1 (ξ) and c, c0 : R− → D be two backward rays with
ċ, ċ0 ∈ R− (ξ). By (10), dF (x, y) = dF (y, x) and F (ċ) = 1 we have for a ≤ b
a − u ◦ c(a) ≤ b − u ◦ c(b),
hence the upper bound by −u ◦ c(0) and monotonicity follow. Let us find a
lower bound. By Theorem 1.2, we have
B := sup dg (c0 (R− ), c(t)) < ∞
t∈R−
and hence for all t ∈ R− we find some s ∈ R− with dg (c0 (s), c(t)) ≤ B.
We first assume that t ≤ s, then by minimality of c, c0 and the triangle
inequality
−t = dF (c(t), c(0))
≤ dF (c(t), c0 (s)) + dF (c0 (s), c0 (0)) + dF (c0 (0), c(0))
≤ cF · B − s + cF · dg (c0 (0), c(0)).
MINIMAL RAYS ON CLOSED SURFACES
25
If s ≤ t, then similarly −s ≤ cF · B − t + cF · dg (c(0), c0 (0)). In any case,
|t − s| is bounded from above by cF (dg (c(0), c0 (0)) + B) and hence
c−1
F · dg (c0 (t), c(t)) ≤ dF (c0 (t), c(t))
≤ dF (c0 (t), c0 (s)) + dF (c0 (s), c(t))
≤ c2F · |t − s| + cF B
≤ c3F · (dg (c(0), c0 (0)) + B) + cF B =: C.
As u is Lipschitz with respect to dg with Lipschitz constant cF ,
|u ◦ c(t) − u ◦ c0 (t)| ≤ cF · dg (c0 (t), c(t)) ≤ c2F · C.
By Lemma 3.6 (ii) we find some ray c0 : R− → D calibrating u, meaning
u ◦ c0 (0) − u ◦ c0 (t) = −t for t ≤ 0. Hence,
|u ◦ c(t) − t| ≤ |u ◦ c(t) − u ◦ c0 (t)| + |u ◦ c0 (t) − t| ≤ c2F · C + |u ◦ c0 (0)|.
We can now prove Lemma 2.15. It deals with the hyperbolic setting, so
the group Γ of deck transformations of the covering D → M decomposes:
Γ = {id} ∪ Γhyp ∪ Γpar , as described at the beginning of Section 2. Given
ξ ∈ S 1 − Fix(Γpar ) and {τn } ⊂ Γ with τn ξ → ξ, we have to find a function
u : D → R with u(0) = 0 and the following properties:
(i) u is globally Lipschitz continuous with respect to dg ,
0 , where τ u := u ◦ τ −1 − u ◦ τ −1 (0),
(ii) τn u → u in Cloc
n
n
n
(iii) for any v ∈ R− (ξ), the function t ∈ R− 7→ t − u ◦ cv (t) is bounded and
non-decreasing.
Proof of Lemma 2.15. Any function u with [u] ∈ hF−1 (ξ) and u(0) = 0 automatically satisfies properties (i) and (iii) by Lemma 3.12. Property (ii) is
fulfilled by u, if τn [u] → [u], where τ [u] := [u ◦ τ −1 ]. Hence, we have to show
the existence of η ∈ h−1
(ξ) with τn η → η.
F
Let ξ, τn as in the statement of the lemma. If τn ξ = ξ for all but finitely
many n, then ξ ∈ Fix(Γhyp ). Let τ ∈ Γhyp be an element fixing ξ, such that
ξ is the stable fixed point of τ , then τn = τ kn for some sequence {kn } ⊂ Z.
Consider some τ -invariant F -minimal geodesic c0 given by Theorem 1.3, set
x0 := c0 (0) and let
η := lim iF (τ n x0 ) = lim [dF (τ n x0 , .)] ∈ h−1
(ξ),
F
n→∞
n→∞
while it is well-known that the limit η exists (it is known as the backward
Busemann function of c0 with respect to F ). Then, for u ∈ η, we find
τm η = [u ◦ τ −km ] = lim [dF (τ n x0 , τ −km (.))] = lim [dF (τ km +n x0 , .)] = η.
n→∞
n→∞
If τn ξ 6= ξ for infinity many n, we can assume e.g. ξ < τn ξ for all n
in the counterclockwise orientation of S 1 , the other case being analogous.
Proposition 3.11 then shows τn η1 → η1 ∈ hF−1 (ξ).
26
J. P. SCHRÖDER
Finally, we prove Corollary 1.10 from the introduction. It rests on the
following observation.
Lemma 3.13. For ξ ∈ S 1 , the set R+ (ξ) admits no (transverse) intersections of rays if and only if card h−1
F (ξ) = 1.
Proof. If card h−1
F (ξ) = 1, then Corollary 3.9 shows R+ (ξ) = Cal(η) for the
unique element η ∈ h−1
F (ξ), while the set Cal(η) admits no intersections by
Lemma 3.4. Conversely, if R+ (ξ) admits no intersections and η ∈ h−1
F (ξ),
then R+ (ξ) = Cal(η): if v ∈ R+ (ξ) and t > 0, then from cv (t) there
initiates a ray calibrating η. If R+ (ξ) admits no intersections, then this
ray is a subray of cv , so with t & 0, v ∈ Cal(η). Corollary 3.7 shows the
uniqueness of η ∈ h−1
F (ξ).
Recall that Π ⊂ S 1 denotes the set of endpoints of axes with respect to
the background metric g. We shall from now on assume that the surface
M is compact. Corollary 1.10 states that iF : D → iF (D) extends to
a homeomorphism IF : D → iF (D) if and only if for each ξ ∈ Π, the
set of periodic rays from Rp+ (ξ) foliate a connected strip in D. The map
hF : iF (D) → D defined above is a continuous extension of i−1
F , so by
Lemma 3.10, iF extends to a homeomorphism if and only if hF is injective.
1
Proof of Corollary 1.10. As hF |iF (D) = i−1
F is injective and hF (η) ∈ S if
and only if η ∈ ∂F D, the map iF extends to a homeomorphism if and only if
1
1
card h−1
F (ξ) = 1 for all ξ ∈ S , i.e. if for all ξ ∈ S the set R+ (ξ) admits no
1
intersecting rays, using Lemma 3.13. If ξ ∈ S − Π, this condition is always
true by combining Theorem 1.6 and Corollary 1.8. For ξ ∈ Π the condition
is true if and only if there exists no pair of neighboring periodic minimal
geodesics in Rp+ (ξ) (cf. Theorem 1.3 and Corollary 1.5), i.e. if and only if
the lamination of D induced by Rp+ (ξ) is connected.
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of surfaces. Preprint, to appear in Journal of Modern Dynamics, Vol. 9 (2015).
Faculty of Mathematics, Ruhr University, 44780 Bochum, Germany
E-mail address: [email protected]