# The limit set of non-orientable mapping class groups

Sayantan Khan  Department of Mathematics, University of Michigan, Ann Arbor,
MI<EMAIL_ADDRESS>https://www-personal.umich.edu/~saykhan/

###### Abstract.

We provide evidence both for and against a conjectural analogy between
geometrically finite infinite covolume Fuchsian groups and the mapping class
group of compact non-orientable surfaces. In the positive direction, we show
the complement of the limit set is open and dense. Moreover, we show that the
limit set of the mapping class group contains the set of uniquely ergodic
foliations and is contained in the set of all projective measured foliations
not containing any one-sided leaves, establishing large parts of a conjecture
of Gendulphe. In the negative direction, we show that a conjectured convex
core is not even quasi-convex, in contrast with the geometrically finite
setting.

###### Key words and phrases:

mapping class group, Teichmüller space, Thurston boundary

###### 2010 Mathematics Subject Classification:

57K20

The author was partially supported by NSF Grant DMS 1856155.

##  1\. Introduction

The moduli space $\mathcal{M}(\mathcal{N}_{g})$ of compact _non-orientable_
hyperbolic surfaces of genus $g$ is conjectured to have similarities to
infinite volume geometrically finite manifolds (in a manner similar to how
moduli spaces of compact orientable surfaces have properties similar to finite
volume hyperbolic manifolds). The main results suggesting the analogy between
moduli spaces of non-orientable surfaces and infinite volume geometrically
finite manifolds are due to Norbury and Gendulphe.

  * •

The $\mathcal{M}(\mathcal{N}_{g})$ has infinite Teichmüller volume [Gen17,
Theorem 17.1]. While the associated Teichmüller space does not have a Weil-
Peterson volume form, it has an analogous volume form with respect to which
the moduli space has infinite volume as well (see [Nor08]).

  * •

The action of the mapping class group $\mathrm{MCG}(\mathcal{N}_{g})$ on the
Thurston boundary is not minimal (Proposition 8.9 in [Gen17]).

  * •

The Teichmüller geodesic flow is not topologically transitive, and thus not
ergodic with respect to any Borel measure with full support [Gen17,
Proposition 17.5].

  * •

There exists an $\mathrm{MCG}(\mathcal{N}_{g})$-equivariant finite covolume
deformation retract of $\mathcal{T}(\mathcal{N}_{g})$.

We extend this analogy further, by showing that the limit set of
$\mathrm{MCG}(\mathcal{N}_{g})$ is contained in the complement of a full
measure dense open set.

######  Theorem 4.4.

The limit set of $\mathrm{MCG}(\mathcal{N}_{g})$ is contained in the
complement of $\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$.

Here $\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$ is the set of all
projective measured foliations that have one-sided compact leaf. The fact that
such foliations form a full measure dense open subset is classical, due to
Danthony-Nogueira (see [DN90]). This is analogous to limit sets of infinite
volume geometrically finite groups, where the complement of the limit set is a
full measure open set as well.

In [Gen17], Gendulphe constructed a retract of $\mathcal{T}(\mathcal{N}_{g})$
to $\mathcal{T}_{\varepsilon}^{-}$, the set of points in the Teichmüller space
that have no one-sided curves shorter than $\varepsilon$, and showed that it
has finite covolume. They also asked the following question about
$\mathcal{T}_{\varepsilon}^{-}$.

######  Question (Question 19.1 of [Gen17]).

Is $\mathcal{T}_{\varepsilon}^{-}$ quasi-convex with respect to the
Teichmüller metric?

We show that ${\mathcal{T}_{\varepsilon}^{-}}$ is not quasi-convex, answering
the above question.

######  Theorem 5.2.

For all $\varepsilon>0$, and all $D>0$, there exists a Teichmüller geodesic
segment whose endpoints lie in ${\mathcal{T}_{\varepsilon}^{-}}$ such that
some point in the interior of the geodesic is more than distance $D$ from
$\mathcal{T}_{\varepsilon}^{-}$.

The key idea involved in the proof of Theorem 5.2 is determining which
projective measured foliations are in the limit set. Conjecture 9.1 of [Gen17]
states that the limit set should exactly be the complement of
$\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$, the set of projective measured
foliations that do not contain any one-sided leaves (denoted
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$). We prove a result that is
slightly weaker than the conjecture.

######  Theorem 3.3.

A foliation $\lambda\in\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$ is in the
limit set of $\mathrm{MCG}(\mathcal{N}_{g})$ if all the minimal components
$\lambda_{j}$ of $\lambda$ satisfy one of the following criteria.

  1. (i)

$\lambda_{j}$ is periodic.

  2. (ii)

$\lambda_{j}$ is ergodic and orientable, i.e. all leaves exiting one side of a
transverse arc always come back from the other side.

  3. (iii)

$\lambda_{j}$ is uniquely ergodic.

Furthermore, if $\lambda_{j}$ is minimal, but not uniquely ergodic, there
exists some other foliation $\lambda_{j}^{\prime}$ supported on the same
topological foliation as $\lambda_{j}$ which is in the limit set.

We now exhibit two contexts in which one is naturally led to consider the
limit set of $\mathrm{MCG}(\mathcal{N}_{g})$.

### Counting simple closed curves

In the orientable setting, the number of simple closed geodesics of length
less than $L$ grows like a polynomial of degree $6g-6$, which is precisely the
dimension of the limit set of the mapping class group: in this case, that
happens to be all of $\mathbb{P}\mathcal{MF}$. In the non-orientable setting,
Gendulphe showed that the growth rate of the simple closed geodesics of length
less than $L$ is smaller than
$L^{\dim(\mathbb{P}\mathcal{MF}(\mathcal{N}_{g}))}$, and one might conjecture
that the growth rate is $L^{h}$, where $h$ is the Hausdorff dimension of the
limit set. Mirzakhani, in [Mir08], obtained precise asymptotics for the
counting function in the orientable case by essentially proving
equidistribution (with respect to Thurston measure) of
$\mathrm{MCG}(\mathcal{S}_{g})$-orbits in
$\mathbb{P}\mathcal{MF}(\mathcal{S}_{g})$. The same problem for non-orientable
surfaces is posed in [Wri20, Problem 9.2]: to make the above techniques work
in this setting, we need an ergodic measure supported on sets minimal with
respect to the $\mathrm{MCG}$ action, e.g.
$\overline{\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{d})}$. One way to construct
such a measure would be to replicate the construction of Patterson-Sullivan
measures for geometrically finite manifolds, which brings us back to the
analogy between $\mathcal{M}(\mathcal{N}_{d})$ and infinite volume
geometrically finite manifolds.

### Interval exchange transformations with flips

Teichmüller spaces of non-orientable surfaces also show up in the context of
_interval exchange transformations with flips_. The dynamics of interval
exchange transformations are closely related to the dynamics of
horizontal/vertical flow on an associated quadratic differential, which is
related to the geodesic flow on the Teichmüller surface via Masur’s criterion
(a version of which holds in the non-orientable setting as well). IETs with
flips do not have very good recurrence properties: in fact, almost all of them
(with respect to the Lebesgue measure) have a periodic point (see [Nog89]) and
the set of minimal IETs with flips have a lower Hausdorff dimension (see
[ST18]). To understand the IETs which are uniquely ergodic, one is naturally
led to determine which “quadratic differentials” on non-orientable surfaces
are recurrent. A necessary but not sufficient condition for recurrence of a
Teichmüller geodesic is that its forward and backward limit points lie in the
limit set. From this perspective, Theorems 3.3 and 4.4 can be seen as a
statement about the closure of the recurrent set. Constructing a measure
supported on the closure of the recurrent set can be then used to answer
questions about uniquely ergodic IETs with flips.

### Another paper on mapping class group orbit closures

A few days after the first version of this paper was posted on arXiv, the
author was notified of another recent paper by Erlandsson, Gendulphe,
Pasquinelli, and Souto [Erl+21] which proves the Conjecture 9.1 of [Gen17],
i.e. a stronger version of Theorem 3.3. The techniques they use are
significantly different, relying on careful analysis of train track charts
carrying various measured laminations. While this paper was being written,
neither the author, nor the authors of [Erl+21] were aware of each others’
work.

### Organization of the paper

Section 2.1 contains the background on non-orientable surfaces and measured
foliations, and section 2.2 contains the background on limit sets of mapping
class subgroups. These sections can be skipped and later referred to if some
notation or definition is unclear. Section 3 contains the proof of Theorem
3.3, section 4 contains the proof of Theorem 4.4, and section 5 contains the
proof of Theorem 5.2. Sections 3, 4, and 5 are independent of each other, and
can be read in any order.

### Acknowledgments

The author would like to thank Alex Wright for introducing him to the problem,
and also Jon Chaika, Christopher Zhang, and Bradley Zykoski, for several
helpful conversations throughout the course of the project. The author also
would like to thank the creators of surface-dynamics [Del+21], which helped
with many of the experiments that guided the results in this paper.

##  2\. Background

###  2.1. Non-orientable surfaces and measured foliations

For the purposes of this paper, the most convenient way to think about non-
orientable surfaces will be to attach _crosscaps_ to orientable surfaces.
Given a surface $S$, attaching a crosscap is the operation of deleting the
interior of a small embedded disc, and gluing the boundary $S^{1}$ via the
antipodal map. Attaching $k$ crosscaps to a genus $g$ surface results in a
genus $2g+k$ non-orientable surface $\mathcal{N}_{2g+k}$ (i.e. the non-
orientable surface obtained by taking the connect sum of $2g+k$ copies of
$\mathbb{RP}^{2}$). Associated to each cross cap is a one-sided curve, which
is the image of the boundary under the quotient map. We say that a curve
intersects the crosscap if it intersects the associated one-sided curve.

Consider the set $\mathscr{S}$ of simple closed curves on a non-orientable
surface $\mathcal{N}$. The elements of $\mathscr{S}$ can be classified into
two types.

Two sided curves:

Tubular neighbourhoods are cylinders.

One sided curves:

Tubular neighbourhoods are Möbius bands.

The subset of two sided curves in denoted by $\mathscr{S}^{+}$ and one sided
curves by $\mathscr{S}^{-}$. Since these two types are topologically distinct,
they form invariant subspaces with respect to the mapping class group action.
If we think of our non-orientable surface as an orientable subsurface with
crosscaps attached, a two-sided curve is one that intersects an even number of
crosscaps, and a one-sided curve is one that intersects an odd number of
crosscaps.

The orientable double cover of $\mathcal{N}_{g}$ is the orientable surface
$\mathcal{S}_{g-1}$, and comes with an orientation reversing involution
$\iota$. Since this is an orientation double cover, the subgroup of
$\pi_{1}(\mathcal{N}_{g})$ corresponding to this cover is characteristic, i.e.
left invariant by every homeomorphism induced automorphism of the fundamental
group. A useful consequence of this fact is that one can lift mapping classes
uniquely.

######  Fact.

Any self homeomorphism of $\mathcal{N}_{g}$ lifts to a unique orientation
preserving self homeomorphism of $\mathcal{S}_{g-1}$, and as a consequence,
one has the injective homomorphism induced by the covering map $p$.

$\displaystyle
p^{\ast}:\mathrm{MCG}(\mathcal{N}_{d})\hookrightarrow\mathrm{MCG}^{+}(\mathcal{S}_{d-1})$

Furthermore, this inclusion preserves the mapping class type, i.e. finite
order, reducible and pseudo-Anosov maps in $\mathrm{MCG}(\mathcal{N}_{g})$
stay finite order, reducible, and pseudo-Anosov in
$\mathrm{MCG}(\mathcal{S}_{g-1})$.

One also obtains a map from $\mathcal{T}(\mathcal{N}_{g})$ to
$\mathcal{T}(\mathcal{S}_{g-1})$ using the fact that mapping classes can be
lifted canonically. Given a point $(p,\varphi)$ in
$\mathcal{T}(\mathcal{N}_{g})$, where $p$ is a hyperbolic surface homeomorphic
to $\mathcal{N}_{g}$, and $\varphi$ is an isotopy class of homeomorphism from
$\mathcal{N}_{g}$ to $p$, we define the image of $(p,\varphi)$ in
$\mathcal{T}(\mathcal{S}_{g-1})$ to be $(\widetilde{p},\widetilde{\varphi})$,
where $\widetilde{p}$ is the orientation double cover of $p$, and
$\widetilde{\varphi}$ is the orientation preserving lift of the homeomorphism
$\varphi$. One can also explicitly describe the image of this map. To do so,
we consider the extended Teichmüller space of $\mathcal{S}_{g-1}$, i.e. also
allowing orientation reversing markings. This space has two connected
components, one for each orientation, and there is a canonical involution,
given by reversing the orientation, that exchanges the two connected
components. We denote this conjugation map by $\overline{\cdot}$. There is
another involution, induced by the orientation reversing deck transformation
of $\mathcal{S}_{g-1}$, which we denote by $\iota^{\ast}$. This map also
exchanges the two components of the extended Teichmüller space. The image of
$\mathcal{T}(\mathcal{N}_{g})$ is precisely the set of points fixed by the
composition of these two maps, i.e. $\overline{\iota^{\ast}}$. We skip the
proof of these two facts, since they follow by relatively elementary covering
space arguments, and summarize the result in the following theorem.

######  Theorem 2.1 (Embedding Teichmüller spaces).

Given a point $(p,\varphi)$ in $\mathcal{T}(\mathcal{N}_{g})$, there is a
unique point $(\widetilde{p},\widetilde{\varphi})$ in
$\mathcal{T}(\mathcal{S}_{g-1})$, where $\widetilde{p}$ is the pullback of the
metric, and $\widetilde{\varphi}$ is the unique orientation preserving lift of
the marking. The image of the inclusion map is the intersection of the
invariant set of $\overline{\iota^{\ast}}$ with the connected component of the
extended Teichmüller space corresponding to orientation preserving maps.

It turns out that the image of $\mathcal{T}(\mathcal{N}_{g})$ in
$\mathcal{T}(\mathcal{S}_{g-1})$ is an isometrically embedded submanifold, and
the geodesic flow can be represented by the action of the diagonal subgroup of
$SL(2,\mathbb{R})$.

To understand the Teichmüller geodesic flow on $\mathcal{T}(\mathcal{N}_{g})$,
we need to determine what the cotangent vectors look like: let $X$ be a point
in $\mathcal{T}(\mathcal{N}_{g})$ and let $\widetilde{X}$ be the corresponding
point in $\mathcal{T}(\mathcal{S}_{g-1})$. Then the map on the extended
Teichmüller space induced by the orientation reversing deck transformation
maps $\widetilde{X}$ to $\overline{\widetilde{X}}$, i.e. the conjugate Riemann
surface. Following that with the canonical conjugation map brings us back to
$\widetilde{X}$. Let $q$ be a cotangent vector at $\overline{\widetilde{X}}$,
i.e. an anti-holomorphic quadratic differential on the Riemann surface
$\overline{\widetilde{X}}$. Pulling back $q$ along the canonical conjugation
map gives a holomorphic quadratic differential on $X$. In local coordinate
chart on $\widetilde{X}$, this looks like $q(z)dz^{2}$ if on the corresponding
chart on $\overline{\widetilde{X}}$ it looked like $q(\overline{z})dz^{2}$. We
want this to equal $\iota^{\ast}q$, which will also be a holomorphic quadratic
differential on $\widetilde{X}$. If that happens, then $\iota^{\ast}q$ is a
cotangent vector to the point $X$ in $\mathcal{T}(\mathcal{N}_{g})$.

######  Example 2.2 (A cotangent vector to a point in
$\mathcal{T}(\mathcal{N}_{3})$).

Consider the quadratic differential $q$ on a genus two Riemann surface
pictured in Figure 1.

$a$
---
$a$
---
$b$
---
$b$
---
$c$
---
$c$
---
$c^{\prime}$
---
$c^{\prime}$
---
$a^{\prime}$
---
$a^{\prime}$
---
$b^{\prime}$
---
$b^{\prime}$
---
Figure 1. A quadratic differential $q$ on $\mathcal{S}_{2}$ given by the slit
torus construction.

Observe that this particular quadratic differential is the global square of an
abelian differential, so it makes sense to talk about the pairing between
$\sqrt{q}$ and the homology classes
$\\{a,a^{\prime},b,b^{\prime},c,c^{\prime}\\}$. Recall that the action of a
mapping class like $\iota$ is merely relabelling homology classes: in this
case $\iota$ swaps $a$ with $-a^{\prime}$, $b$ with $b^{\prime}$, and $c$ with
$-c^{\prime}$. That gives us the following expressions involving $\sqrt{q}$.

$\displaystyle\langle\iota^{\ast}\sqrt{q},a\rangle$
$\displaystyle=\langle\sqrt{q},-a^{\prime}\rangle$ (1)
$\displaystyle\langle\iota^{\ast}\sqrt{q},b\rangle$
$\displaystyle=\langle\sqrt{q},b^{\prime}\rangle$ (2)
$\displaystyle\langle\iota^{\ast}\sqrt{q},c\rangle$
$\displaystyle=\langle\sqrt{q},-c^{\prime}\rangle$ (3)

On the other hand, the conjugation action conjugates the complex value of each
pairing.

$\displaystyle\langle\overline{\sqrt{q}},a\rangle$
$\displaystyle=\overline{\langle\sqrt{q},a\rangle}$ (4)
$\displaystyle\langle\overline{\sqrt{q}},b\rangle$
$\displaystyle=\overline{\langle\sqrt{q},b\rangle}$ (5)
$\displaystyle\langle\overline{\sqrt{q}},c\rangle$
$\displaystyle=\overline{\langle\sqrt{q},c\rangle}$ (6)

For $q$ to be invariant under $\overline{\iota}$, both of the above set of
equations must be satisfied, which imposes certain conditions on $q$. For
instance, if the complex lengths of $a$ and $a^{\prime}$ to be conjugates of
each other, the complex lengths of $b$ and $b^{\prime}$ to be negative
conjugates of each other, and forces the complex length of $c$ and
$c^{\prime}$ to be real. Only the quadratic differentials satisfying these
constraints will be the cotangent vectors to points in the image of
$\mathcal{T}(\mathcal{N}_{3})$.

To realize the quadratic differential directly as an object on
$\mathcal{N}_{3}$, we can quotient out the flat surface given by $q$ by the
orientation reversing deck transformation. Doing that for our example gives
the non-orientable flat surface gives the picture seen in Figure 2.

$a$
---
$a$
---
$b$
---
$b$
---
$c$
---
$c$
---
Figure 2. A quadratic differential on $\mathcal{N}_{3}$.

This example suggests what the right definition of a quadratic differential on
a non-orientable surface ought to be: in the flat picture, rather than
allowing gluing via just the maps $z\mapsto\pm z+c$, we also allow
$z\mapsto\pm\overline{z}+c$. This leads to the definition of _dianalytic
quadratic differentials_ (which we’ll abbreviate to DQDs).

######  Definition 2.3 (Dianalytic quadratic differential (adapted from
[Wri15])).

A dianalytic quadratic differential is the quotient of a collection of
polygons in $\mathbb{C}$, modulo certain equivalences. The quotienting
satisfies the following conditions.

  1. (1)

The interiors of the polygons are disjoint.

  2. (2)

Each edge is identified with exactly one other edge, and the mapping must be
of one of the following four forms: $z\mapsto z+c$, $z\mapsto-z+c$,
$z\mapsto\overline{z}+c$, or $z\mapsto-\overline{z}+c$.

  3. (3)

Extending the edge identification map to a small enough open neighbourhood of
a point on the edge should not map it to an open neighbourhood of the image of
the point: in other words, it should get mapped to the “other side” of the
edge.

Two such quotiented collections of polygons are considered the same if they
differ by a composition of the following moves.

  1. (1)

A polygon may be translated, rotated by $\pi$ radians, or reflected across the
real or imaginary axis.

  2. (2)

A polygon may be cut along a straight line to form two polygons, or two
polygons sharing an edge may be glued together to form a single polygon.

Given a DQD, we can pull it back to the orientation double cover, getting an
actual quadratic differential: this operation corresponds to identifying a
cotangent vector to a point in $\mathcal{T}(\mathcal{N}_{g})$ to the
corresponding cotangent vector in $\mathcal{T}(\mathcal{S}_{g-1})$.

To verify that $\mathcal{T}(\mathcal{N}_{g})$ is isometrically embedded, all
we need to do is verify that the Teichmüller geodesic flow takes the quadratic
differentials satisfying the symmetry condition $\iota^{\ast}(q)=\overline{q}$
to quadratic differentials that satisfy the symmetry conditions.

######  Lemma 2.4.

If $q$ satisfies $\iota^{\ast}(q)=\overline{q}$, then for any $t$,
$\iota^{\ast}(g_{t}q)=\overline{g_{t}q}$.

###### Proof.

Recall that if $q$ satisfies the given condition, we must have the following
hold for any homology class $a$.

$\langle\sqrt{q},\iota(a)\rangle=\overline{\langle\sqrt{q},a\rangle}$ (7)

If $q$ is not the global square of an abelian differential, we may have to
pass to the holonomy double cover. Observe now what $g_{t}$ does to $q$.

$\langle\sqrt{g_{t}q},\iota(a)\rangle=e^{t}\mathrm{Re}\langle\sqrt{q},\iota(a)\rangle+ie^{-t}\mathrm{Im}\langle\sqrt{q},\iota(a)\rangle$
(8)

Using (7), we simplify (8) to the following.

$\displaystyle\langle\sqrt{g_{t}q},\iota(a)\rangle$
$\displaystyle=e^{t}\mathrm{Re}\langle\sqrt{q},a\rangle-
ie^{-t}\mathrm{Im}\langle\sqrt{q},a\rangle$ (9)
$\displaystyle=\overline{\langle\sqrt{g_{t}q},a\rangle}$ (10)

This proves the lemma. ∎

######  Remark.

The key idea that diagonal matrices commute: the conjugation action is really
multiplication by $\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ which happens to
commute with the diagonal matrices of determinant $1$, which are exactly the
matrices corresponding to geodesic flow. On the other hand, the conjugation
matrix does not commute with the horocycle flow matrices, and that shows that
the horocycle flow is not well defined on the cotangent bundle of
$\mathcal{T}(\mathcal{N}_{g})$.

Lemma 2.4 shows that the Teichmüller geodesic flow for the cotangent bundle of
$\mathcal{T}(\mathcal{N}_{g})$ is the restriction of the geodesic flow for the
ambient space $\mathcal{T}(\mathcal{S}_{g-1})$.

Theorem 2.1 gives us an alternative perspective into the action of
$\mathrm{MCG}(\mathcal{N}_{g})$ on $\mathcal{T}(\mathcal{N}_{g})$.
$\mathrm{MCG}(\mathcal{N}_{g})$ can be thought of as the subgroup of
$\mathrm{MCG}(\mathcal{S}_{g-1})$ that stabilizes a totally real isometrically
embedded submanifold $\mathcal{T}(\mathcal{N}_{g})$. With this perspective,
$\mathrm{MCG}(\mathcal{N}_{g})$ can be thought of as the higher dimensional
generalization of the subgroups obtained by stabilizing Teichmüller discs,
i.e. Veech groups.

We now state a few classical results about measured foliations on non-
orientable surfaces that show why the theory diverges significantly from the
orientable case.

A measured foliation on a non-orientable surface $\mathcal{N}_{g}$ is singular
foliation along with an associated transverse measure, up to equivalence by
Whitehead moves. Any leaf of a measured foliation can either be non-compact or
compact: in the former case, the closure of the non-compact leaf fills out a
subsurface. Restricted to the subsurface given by the closure of a non-compact
leaf, the foliation is minimal, i.e. the orbit of every point under the flow
given by the foliation is dense. For a compact leaf, there are two
possibilities for the topology of the subsurface containing it: if the closed
leaf is the core curve or the boundary curve of an embedded Möbius strip, then
the subsurface is the maximal neighbourhood of the periodic leaf that is
foliated by periodic leaves as well, and this turns out to be an embedded
Möbius strip. If the compact leaf is not the core curve or the boundary curve
of an embedded Möbius strip, then it is the core curve of an embedded
cylinder, and the maximal neighbourhood of the periodic leaf foliated by
periodic leaf is an embedded cylinder. The identification of leaves with
associated subsurfaces lets us decompose a measured foliation into its minimal
components. Note the slightly confusing terminology: when the minimal
component is a Möbius strip or a cylinder, then the foliation restricted to
the component is not minimal, but when the minimal component has higher genus,
then the foliation restricted to that component indeed is minimal.

We denote the set of measured foliations on $\mathcal{N}_{g}$ by
$\mathcal{MF}(\mathcal{N}_{g})$, the set of foliations whose minimal
components do not contain a Möbius strip by
$\mathcal{MF}^{+}(\mathcal{N}_{g})$, and the set of foliations whose minimal
components contain at least one Möbius strip by
$\mathcal{MF}^{-}(\mathcal{N}_{g})$. Via the standard identification between
simple closed curves and measured foliations, we can associate
$\mathbb{Q}$-weighted two-sided multicurves on $\mathcal{N}_{g}$ to a subset
of $\mathcal{MF}^{+}(\mathcal{N}_{g})$, denoted by
$\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$.

Quotienting out $\mathcal{MF}(\mathcal{N}_{g})$ by the $\mathbb{R_{+}}$-action
given by scaling the transverse measure gives us the set of projective
measured foliations $\mathbb{P}\mathcal{MF}(\mathcal{N}_{g})$. The subsets
$\mathcal{MF}^{-}(\mathcal{N}_{g})$, $\mathcal{MF}^{+}(\mathcal{N}_{g})$, and
$\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$ are $\mathbb{R}$-invariant, and
thus descend to their projective versions
$\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$,
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$, and
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$. The set
$\mathbb{P}\mathcal{MF}(\mathcal{N}_{g})$ is the boundary of the Teichmüller
space of $\mathcal{N}_{g}$, and admits a continuous mapping class group
action. It is when considering the mapping class group action that we see
differences between the orientable and the non-orientable case.

######  Theorem 2.5 (Proposition 8.9 of [Gen17]).

The action of $\mathrm{MCG}(\mathcal{N}_{g})$ (for $g\geq 2$) on
$\mathcal{MF}(\mathcal{N}_{g})$ is not minimal. In fact, the action is not
even topologically transitive.

######  Remark.

The proof of non minimality and topological non-transitivity follow from the
fact that one can construct a $\mathrm{MCG}(\mathcal{N}_{g})$-invariant
continuous function on $\mathcal{MF}(\mathcal{N}_{g})$. That is because
starting with a foliation in $\mathcal{MF}^{+}(\mathcal{N}_{g})$, it is
impossible to approximate an element of $\mathcal{MF}^{-}(\mathcal{N}_{g})$
since one does not have Dehn twists about one-sided curves.

One can now consider subspaces of $\mathcal{MF}(\mathcal{N}_{g})$ where the
$\mathrm{MCG}(\mathcal{N}_{g})$ action might be nicer. There are two natural
subspaces: $\mathcal{MF}^{+}(\mathcal{N}_{g})$, and
$\mathcal{MF}^{-}(\mathcal{N}_{g})$. Danthony-Nogueira proved the following
theorem about $\mathcal{MF}^{-}(\mathcal{N}_{g})$ in [DN90].

######  Theorem 2.6 (Theorem II of [DN90]).

$\mathcal{MF}^{-}(\mathcal{N}_{g})$ is an open dense subset of
$\mathcal{MF}(\mathcal{N}_{g})$ of full Thurston measure.

Theorem 2.5 and Theorem 2.6 suggest that studying the
$\mathrm{MCG}(\mathcal{N}_{g})$ dynamics restricted to
$\mathcal{MF}^{-}(\mathcal{N}_{g})$ will be hard since one will not have
minimality, or ergodicity with respect to any measure with full support. In
Section 3, we get a lower bound for the set on which
$\mathrm{MCG}(\mathcal{N}_{g})$ acts minimally.

###  2.2. Limit sets of mapping class subgroups

The first results on limit sets of subgroups of mapping class groups were
obtained by Masur for handlebody subgroups [Mas86], and McCarthy-Papadopoulos
for general mapping class subgroups [MP89]. They defined two distinct notions
of limit sets; while they did not give distinct names to the two different
definitions, we will do so for the sake of clarity.

######  Definition 2.7 (Dynamical limit set).

Given a subgroup $\Gamma$ of the mapping class group, the dynamical limit set
$\Lambda_{\mathrm{dyn}}(\Gamma)$ is the minimal closed invariant subset of
$\mathbb{P}\mathcal{MF}$ under the action of $\Gamma$.

######  Definition 2.8 (Geometric limit set).

Given a subgroup $\Gamma$ of the mapping class group, and a point $x$ in the
Teichmüller space, its boundary orbit closure
$\Lambda_{\mathrm{geo},x}(\Gamma)$ is intersection of its orbit closure with
the Thurston boundary, i.e. $\overline{\Gamma x}\cap\mathbb{P}\mathcal{MF}$.
The geometric limit set is the union of all boundary orbit closures, as we
vary $x$ in the Teichmüller space, i.e.
$\Lambda_{\mathrm{geo}}(\Gamma)=\bigcup_{x\in\mathcal{T}}\Lambda_{\mathrm{geo},x}(\Gamma)$.

######  Remark.

The specific family of subgroups considered by McCarthy-Papadopoulos were
subgroups containing at least two non-commuting pseudo-Anosov mapping classes,
in which case the dynamical limit set is unique. The mapping class groups
$\mathrm{MCG}(\mathcal{N}_{g})$ considered as a subgroup of
$\mathrm{MCG}(\mathcal{S}_{g-1})$ certainly satisfies this property, letting
us talk about _the_ dynamical limit set.

Both of these definitions are natural generalizations of the limit sets of
Fuchsian groups acting on $\mathbb{H}^{2}$. In the hyperbolic setting, the two
notions coincide, but for mapping class subgroups, the dynamical limit set may
be a proper subset of the geometric limit set.

For simple enough subgroups, one can explicitly work out
$\Lambda_{\mathrm{dyn}}(\Gamma)$ and $\Lambda_{\mathrm{geo}}(\Gamma)$: for
instance, when $\Gamma$ is the stabilizer of the Teichmüller disc associated
to a Veech surface, $\Lambda_{\mathrm{dyn}}(\Gamma)$ is the visual boundary of
the Teichmüller disc, which by Veech dichotomy, only consists of either
uniquely ergodic directions on the Veech surface, or the cylinder directions,
where the coefficients on the cylinders are their moduli in the surface. On
the other hand, $\Lambda_{\mathrm{geo}}(\Gamma)$ consists of all the points in
$\Lambda_{\mathrm{dyn}}(\Gamma)$, but it additionally contains all possible
convex combinations of the cylinders appearing in
$\Lambda_{\mathrm{dyn}}(\Gamma)$ (see Section 2.1 of [KL07]).

The gap between $\Lambda_{\mathrm{geo}}$ and $\Lambda_{\mathrm{dyn}}$ suggests
the following operation on subsets of $\mathbb{P}\mathcal{MF}$, which we will
call _saturation_.

######  Definition 2.9 (Saturation).

Given a projective measured foliation $\lambda$, we define its saturation
$\mathrm{Sat}(\lambda)$ to be the image in $\mathbb{P}\mathcal{MF}$ of set of
all non-zero measures invariant measures on the topological foliation
associated to $\lambda$. Given a subset $\Lambda$, we define its saturation
$\mathrm{Sat}(\Lambda)$ to be the union of saturations of the projective
measured laminations contained in $\Lambda$.

Observe that for a uniquely ergodic foliation $\lambda$,
$\mathrm{Sat}(\lambda)=\\{\lambda\\}$, for a minimal but not uniquely ergodic
$\lambda$, $\mathrm{Sat}(\lambda)$ is the convex hull of all the ergodic
measures supported on the topological lamination associated to $\lambda$, and
for a foliation with all periodic leaves, $\mathrm{Sat}(\lambda)$ consists of
all foliations that can be obtained by assigning various weights to the core
curves of the cylinders.

Going back to the example of the stabilizer of the Teichmüller disc of a Veech
surface, we see that
$\Lambda_{\mathrm{geo}}(\Gamma)=\mathrm{Sat}(\Lambda_{\mathrm{dyn}}(\Gamma))$.
One may ask if this is always the case.

######  Question 2.10.

Is
$\Lambda_{\mathrm{geo}}(\Gamma)=\mathrm{Sat}(\Lambda_{\mathrm{dyn}}(\Gamma))$
for all $\Gamma$?

We know from Theorem 3.3 that $\Lambda_{\mathrm{geo}}(\Gamma)$ is contained in
$\mathrm{Sat}(\Lambda_{\mathrm{dyn}}(\Gamma))$ when
$\Gamma=\mathrm{MCG}(\mathcal{N}_{g})$.

McCarthy-Papadopoulos also formulated an equivalent definition of
$\Lambda_{\mathrm{dyn}}(\Gamma)$, which is easier to work with in practice.

######  Theorem (Theorem 4.1 of [MP89]).

$\Lambda_{\mathrm{dyn}}(\Gamma)$ is the closure in $\mathbb{P}\mathcal{MF}$ of
the stable and unstable foliations of all the pseudo-Anosov mapping classes in
$\Gamma$.

### List of notation

Here we describe some of the more commonly used symbols in the paper.

  * $\mathcal{S}_{g}$: The compact orientable surface of genus $g$.

  * $\mathcal{N}_{g}$: The compact non-orientable surface of genus $g$.

  * $\iota$: The deck transformation of the orientation double cover of a non-orientable surface.

  * $\mathcal{T}(S)$: The Teichmüller space of $S$.

  * $\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{d})$: The set of points in $\mathcal{T}(\mathcal{N}_{d})$ where no one-sided curve is shorter than $\varepsilon$.

  * $\mathrm{MCG}(S)$: The mapping class group of $S$.

  * $\mathcal{MF}(S)$: The space of measured foliations on $S$.

  * $\mathbb{P}\mathcal{MF}(S)$: The space of projective measured foliations on $S$.

  * $\mathcal{MF}^{+}(\mathcal{N}_{d})$, $\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{d})$: The set of (projective) measured foliations on $\mathcal{N}_{d}$ containing no one-sided leaves.

  * $\mathcal{MF}^{-}(\mathcal{N}_{d})$, $\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{d})$: The set of (projective) measured foliations on $\mathcal{N}_{d}$ containing some one-sided leaf.

  * $\mathcal{MF}(S;\mathbb{Q})$, $\mathbb{P}\mathcal{MF}(S;\mathbb{Q})$: The set of all (projective) weighted rational multicurves on $S$.

  * $\Lambda_{\mathrm{geo}}(\Lambda)$: The geometric limit set of the discrete group $\Lambda$.

  * $\Lambda_{\mathrm{dyn}}(\Lambda)$: The dynamical limit set of the discrete group $\Lambda$.

  * $\ell_{\mathrm{hyp}}(M,\gamma)$: The hyperbolic length of $\gamma$ with respect to the hyperbolic structure on $M\in\mathcal{T}(S)$. We will suppress $M$ when it is clear from context.

  * $\ell_{\mathrm{flat}}(q,\gamma)$: The flat length of $\gamma$ with respect to the flat structure given by the DQD $q$. We will suppress $q$ when it is clear from context.

  * $\mu_{c}$: The probability measure on a transverse arc given by the closed curve $c$.

##  3\. Lower bound for the limit set

A natural lower bound for $\Lambda_{\mathrm{dyn}}(\mathcal{N}_{g})$ is the
closure of the set of rational two-sided multicurves
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$. For any
$\lambda\in\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$, and any
psuedo-Anosov $\gamma$, conjugating $\gamma$ with large enough powers of the
Dehn multi-twist given by $\lambda$ gives us a sequence of pseudo-Anosov maps
whose stable foliation approaches $\lambda$, which shows that
$\Lambda_{\mathrm{dyn}}(\mathcal{N}_{g})$ must contain $\lambda$. Note that
the same argument does not work if
$\lambda\in\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g},\mathbb{Q})$, since one
cannot Dehn twist about one-sided curves. In Section 4, we show that the
geometric limit set is indeed contained in the complement of
$\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$.

In [Gen17], Gendulphe made the following conjecture about
$\overline{\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})}$.

######  Conjecture 3.1 (Conjecture 9.1 of [Gen17]).

For $g\geq 4$,
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})=\overline{\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})}$.

We prove a slightly weaker version of the above conjecture, by describing a
subset of the foliations that can be approximated by multicurves in
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$. To state the
theorem, we need to define what it means for a minimal foliation to be
orientable.

######  Definition 3.2 (Orientable foliation).

A local orientation on a foliation is the choice of a locally constant tangent
direction on the leaves in a small open set. If the local orientation can be
extended to an entire minimal foliation, the foliation is said to be
orientable.

In the setting of orientable surfaces, the vertical foliations of translation
surfaces are orientable, while there are some directions in half-translation
surfaces where the foliation is non-orientable. There exist similar examples
of orientable and non-orientable foliations on non-orientable surfaces.

Having defined the notion of orientable foliations, we can state the main
theorem of this section.

######  Theorem 3.3.

A foliation $\lambda\in\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$ can be
approximated by foliations in
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$ if all the minimal
components $\lambda_{j}$ of $\lambda$ satisfy one of the following criteria.

  1. (i)

$\lambda_{j}$ is periodic.

  2. (ii)

$\lambda_{j}$ is ergodic and orientable.

  3. (iii)

$\lambda_{j}$ is uniquely ergodic.

Furthermore, if $\lambda_{j}$ is minimal, but not uniquely ergodic, there
exists some other foliation $\lambda_{j}^{\prime}$ supported on the same
topological foliation as $\lambda_{j}$ that can be approximated by elements of
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g},\mathbb{Q})$.

Before we prove this result, we need to define the _orbit measure_ associated
to simple curve, and define what it means for an orbit measure to be _almost
invariant_. Consider an arc $\eta$ transverse to a measured foliation
$\lambda$. We assign one of the sides of $\eta$ to be the “up” direction, and
the other side to be the “down” direction. This lets us define the first
return map to $T$.

######  Definition 3.4 (First return map).

The first return map $T$ maps a point $p\in\eta$ to the point obtained by
flowing along the foliation in the “up” direction until the flow intersects
$\eta$ again. The point of intersection is defined to be $T(p)$. If the flow
terminates at a singularity, $T(p)$ is left undefined: there are only
countable many points in $\eta$ such that this happens.

Since $\lambda$ is a measured foliation, it defines a measure on $\eta$: we
can scale it so that it is a probability measure. It follows from the
definition of transverse measures that the measure is $T$-invariant. It is a
classical result of Katok [Kat73] and Veech [Vee78] that the set of
$T$-invariant probability measures is a finite dimensional simplex contained
in the Banach space of bounded signed measures on $\eta$. Given an orbit of a
point $p$ under the $T$-action of length $L$, we construct a probability
measure on $\eta$, called the orbit measure of $p$.

######  Definition 3.5 (Orbit measure).

The orbit measure of length $L$ associated to the point $p$ is the following
probability measure on $\eta$.

$\displaystyle\mu_{p,L}\coloneqq\frac{1}{L}\sum_{i=0}^{L-1}\delta_{T^{i}(p)}$

Here, $\delta_{x}$ is the Dirac delta measure at the point $x$.

One might expect that if a point $p$ equidistributes, then a long orbit
measure starting at $p$ will be “close” to an invariant measure. We formalize
this notion by metrizing the Banach space of signed finite measures on $\eta$.

######  Definition 3.6 (Lèvy-Prokhorov metric).

Define $\left\lVert\cdot\right\rVert_{\mathrm{BL}}$ denote the bounded
Lipschitz norm on the space of continuous functions on $\eta$.

$\displaystyle\left\lVert f\right\rVert_{\mathrm{BL}}\coloneqq\left\lVert
f\right\rVert_{\infty}+\sup_{x\neq
y}\frac{\left|f(x)-f(y)\right|}{\left|x-y\right|}$

Then the Lèvy-Prokhorov distance $d_{\mathrm{LP}}$ between the probability
measures $\mu_{1}$ and $\mu_{2}$ is defined to be the following.

$\displaystyle d_{\mathrm{LP}}(\mu_{1},\mu_{2})\coloneqq\sup_{\left\lVert
f\right\rVert_{\mathrm{BL}}\leq 1}\int
f(\mathrm{\,d}\mu_{1}-\mathrm{\,d}\mu_{2})$

Using the Lèvy-Prokhorov metric, we can define what it means for a probability
measure to be $\varepsilon$-almost $T$-invariant.

######  Definition 3.7 ($\varepsilon$-almost $T$-invariance).

A measure $\mu$ is $\varepsilon$-almost $T$-invariant if
$d_{\mathrm{LP}}(\mu,T\mu)\leq\varepsilon$. Here $T\mu$ is the pushforward of
$\mu$ under $T$.

We state the following easy fact about orbit measures without proof.

######  Fact.

An orbit measure of length $L$ is $\frac{2}{L}$-almost $T$-invariant.

The following lemma shows that a long orbit measure is close to an invariant
measure.

######  Lemma 3.8.

Let $\left\\{n_{j}\right\\}$ be a sequence of positive integers and let
$\\{\mu_{ij}\\}$ be orbit measures such that $1\leq i\leq n_{j}$ and
$d(\mu_{ij},T{\mu}_{ij})\leq l_{j}$, where $\lim_{j\to\infty}l_{j}=0$. For any
$\varepsilon>0$, there exists an $J$ large enough such that for all $j>J$,
$\mu_{ij}$ is within distance $\varepsilon$ of an invariant measure.

###### Proof.

Let $A_{k}$ be the closure of all the $\mu_{ij}$ such that $j\geq k$. The set
$A_{k}$ is compact, because it is a closed subset of a compact set, and we
have that $\bigcap_{k=1}^{\infty}A_{k}$ is contained in the set of invariant
measures. By compactness, we have that for some large enough $J$, $A_{J}$ must
be in a $\varepsilon$-neighbourhood of the set of invariant measures, and
therefore every $\mu_{ij}$ for $j>J$ must distance at most $\varepsilon$ away
from an invariant measure. ∎

We now sketch a proof of the following lemma about simplices in finite
dimensional normed spaces.

######  Lemma 3.9.

Let $V$ be a finite dimensional normed vector space, and $S$ be a simplex in
$V$. Let $\\{p_{1},\ldots,p_{n}\\}$ be points in $S$ such that they are all at
least distance $\varepsilon$ from a vertex $v$. Then there exists a positive
constant $k$ such that any convex combination of $\\{p_{i}\\}$ is distance at
least $\dfrac{\varepsilon}{k}$ from $v$.

###### Sketch of proof.

We shift the simplex so that the vertex $v$ is at the origin. It will also
suffice to let $\\{p_{1},\ldots,p_{n}\\}$ be the vectors joining $0$ to the
other vertices scaled to have norm $\varepsilon$. The convex combinations of
$\\{p_{1},\ldots,p_{n}\\}$ will form a compact set not containing $0$. Since
the norm is a continuous function, the norm will achieve a minimum
$\varepsilon^{\prime}$ on that compact set, and the minimum will not be $0$.
Then $k=\dfrac{\varepsilon^{\prime}}{\varepsilon}$ is the required value of
$k$. ∎

We now prove a lemma that gives us a criterion for deducing when a long orbit
measure is close to an ergodic measure.

######  Lemma 3.10.

Let $\\{n_{i}\\}$ be a sequence of positive integers, and let $\\{p_{ij}\\}$
and $\\{L_{ij}\\}$ be points in $\eta$ and positive integers respectively,
where $1\leq j\leq n_{i}$ and $\min_{j}L_{ij}$ goes to $\infty$ as $i$ goes to
$\infty$. Consider the following sequence of probability measures, indexed by
$i$.

$\displaystyle\mu_{i}\coloneqq\frac{\sum_{j=1}^{n_{i}}L_{ij}\cdot\mu_{p_{ij},L_{ij}}}{\sum_{j=1}^{n_{i}}L_{ij}}$

If the sequence $\\{\mu_{i}\\}$ converges to an ergodic measure $\nu$, then
there exists a subsequence of the orbit measures $\mu_{p_{ij},L_{ij}}$ also
converging to $\nu$.

###### Proof.

Suppose for the sake of a contradiction that no subsequence of
$\mu_{p_{ij},L_{ij}}$ converged to $\nu$. That would mean there exists a small
enough $\varepsilon>0$ and a large enough $i_{0}$ such that for all $i>i_{0}$,
the measures $\mu_{p_{ij},L_{ij}}$ are more than distance $\varepsilon$ from
$\nu$. Since $\min_{j}L_{ij}$ goes to $\infty$, there exists some other large
enough $i_{1}>i_{0}$ such that for all $i>i_{1}$, $\mu_{p_{ij},L_{ij}}$ is
within distance $\frac{\varepsilon}{k}$ of the simplex of invariant
probability measures, where $k$ is a large integer we will pick later: this is
a consequence of 3.8. Using this, we decompose $\mu_{p_{ij},L_{ij}}$ as the
sum of an invariant measure $\iota_{ij}$ and a signed measure $e_{ij}$, such
that $d_{\mathrm{LP}}(0,e_{ij})\leq\frac{\varepsilon}{k}$.

$\displaystyle\mu_{p_{ij},L_{ij}}=\iota_{ij}+e_{ij}$

Observe that the weighted average of $\mu_{p_{ij},L_{ij}}$ will differ from
the weighted average of $\iota_{ij}$ by at most $\frac{\varepsilon}{k}$. Also
note that all the invariant measures $\iota_{ij}$ are distance at least
$\varepsilon-\frac{\varepsilon}{k}$ from $\nu$. Since $\nu$ is the vertex of a
finite-dimensional convex set, we know from 3.9 that any weighted average of
the $\iota_{ij}$ must be at least distance
$\frac{\varepsilon-\frac{\varepsilon}{k}}{k^{\prime}}$ from $\nu$, where the
multiplicative factor $k^{\prime}$ only depends on the geometry of the convex
set of invariant probability measures, and not $\varepsilon$ or $k$. By
picking $k>2k^{\prime}$ we can ensure that any weighted average of the
$\mu_{p_{ij},L_{ij}}$ must be at least distance
$\frac{\varepsilon}{2k^{\prime}}$ from $\nu$. But this would contradict our
hypothesis that the measures $\mu_{i}$ converge to $\nu$. That would that
there exists some subsequence of $\mu_{p_{ij},L_{ij}}$ that converges to
$\nu$, which proves the lemma. ∎

We now have everything we need to prove Theorem 3.3.

###### Proof of Theorem 3.3.

If a minimal component $\lambda_{j}$ is periodic, then the proof is
straightforward. Since $\lambda$ contains no one-sided component, the core
curve of $\lambda_{j}$ must be two-sided, possibly with an irrational
coefficient. Approximating the core curve with rational coefficients proves
the result in case (i).

In case (ii), we have that $\lambda_{j}$ is not periodic, but an ergodic
orientable foliation. Pick an arc $\eta_{0}$ transverse to $\lambda_{j}$ such
that the leaf passing through the left endpoint $p_{0}$ of $\eta_{0}$
equidistributes with respect to the ergodic transverse measure of
$\lambda_{j}$. We can find such a leaf because almost every leaf
equidistributes with respect to the ergodic measure. We now inductively define
a sequence of points $\\{p_{i}\\}$, sequence of sub-intervals $\eta_{i}$, and
a sequence of segments $\\{a_{i}\\}$ of the leaf passing through $p_{0}$. Let
$p_{1}$ be the first return of the leaf going up through $p_{0}$ to the
interval $\eta_{0}$. Define the sub-interval $\eta_{1}$ to be the sub-interval
whose left endpoint is $p_{0}$ and right endpoint is $p_{1}$. Let $a_{1}$ be
the segment of the leaf starting at $p_{0}$ and ending at $p_{1}$. Given a
point $p_{i}$, define $p_{i+1}$ to be the first return to the interval
$\eta_{i}$, $\eta_{i+1}$ to be the interval whose left endpoint is $p_{0}$ and
right endpoint is $p_{i+1}$, and $a_{i+1}$ to be the segment of the leaf
starting at $p_{i}$ and ending at $p_{i+1}$.

Since we have assumed $\lambda_{j}$ is an orientable foliation, we have that
the leaf we are working with always enters $\eta_{0}$ from the bottom, and
exits from the top. If we pick $\eta_{0}$ to be small enough, we can pick a
local orientation, and keep track of how a positively oriented frame returns
to each $p_{i}$, i.e. with or without the orientation flipped (see Figure 3).

$\eta_{0}$
---
$\eta_{i}$
---
$p_{i}$
---
$p_{0}$
---
$a_{i}$
---
$\eta_{0}$
---
$\eta_{i}$
---
$p_{i}$
---
$p_{0}$
---
$a_{i}$
---
Figure 3. Two possibilities for first return to $\eta_{i}$: on the left, the
arc returns without the local orientation flipping, and on the right, the arc
returns with the local orientation flipped.

If the flow returns infinitely often without the orientation flipped, we join
the endpoint $p_{i}$ to $p_{0}$ by going left along $\eta_{i}$ to get a simple
closed curve that is two-sided. Furthermore, the geodesic tightening of the
resulting curve is very close to the original curve, because the initial and
final tangent vectors can be made arbitrarily close since they both face the
“up” direction: the Anosov closing lemma then tells us that an orbit of the
geodesic flow that approximately closes up can be perturbed by a small amount
to exactly close up. This gives us a long geodesic that equidistributes with
respect to the ergodic measure, and therefore an approximation by two-sided
curves.

If the flow does not return without the orientation flipped infinitely often,
it must always return with the orientation flipped after some large enough
$i_{0}$. In that case, consider the simple two-sided curves $c_{i}$ obtained
by concatenating $a_{i}$ with the arc on $\eta_{i-1}$ joining $p_{i-1}$ and
$p_{i}$ (see Figure 4).

$\eta_{0}$
---
$\eta_{i}$
---
$p_{i}$
---
$p_{0}$
---
$a_{i}$
---
$p_{i-1}$
---
Figure 4. The curve $c_{i}$ is colored blue. Since the leaf from $p_{0}$
returns with the local orientation flipped to both $p_{i-1}$ and $p_{i}$, the
curve $c_{i}$ is two-sided.

We have that as $i$ goes to $\infty$, the length of $c_{i}$ must go to
$\infty$ as well, otherwise a subsequence would converge to a closed vertical
curve starting at $p_{0}$, which cannot happen since the leaf through $p_{0}$
equidistributes. Also, note that the average of the curves $c_{i}$ weighted by
their lengths for $i^{\prime}<i<i^{\prime\prime}$ where $i^{\prime\prime}\gg
i^{\prime}$ is close to the ergodic measure, since we assumed that the leaf
through $p_{0}$ equidistributes. This lets us invoke 3.10 to claim that there
is a subsequence of $c_{i}$ whose orbit measures converge to the ergodic
measure. Consequently, the geodesic representatives of $c_{i}$ converge to
$\lambda_{j}$, since the geodesic tightening is close to the original curve,
by the virtue of the initial and final tangent vectors being arbitrarily
close. This resolves the two cases that can appear in the case of an
orientable foliation, proving the result for case (ii).

For case (iii), we define the points $p_{i}$, the nested intervals $\eta_{i}$,
and the arcs $a_{i}$ in a similar manner as to case (ii). The key difference
is that we no longer have that the foliation is orientable, which means the
leaf can approach $p_{i}$ in one of four possible ways: from the “up” or the
“down” direction, and with or without the orientation flipped.

In case that the leaf approaches $p_{i}$ from the “down” direction without the
orientation flipped infinitely often, the same closing argument as case (ii)
works. Suppose now that the leaf approaches $p_{i}$ from the “up” direction,
but without the orientation flipping, infinitely often. We then construct
simple two-sided curves by concatenating the flow with the arc joining $p_{i}$
to $p_{0}$. While this curve does equidistribute with respect to the ergodic
measure, it is not necessary that its geodesic tightening will do so. Denote
the geodesic tightening by $c_{i}^{\prime}$: we have that its intersection
number with $\lambda_{j}$ goes to $0$ as $i$ goes to $\infty$. By the
compactness of the space of transverse probability measures, we must have that
$\mu_{c_{i}^{\prime}}$ converges to some projective measured foliation
$\gamma$ which has $0$ intersection number with $\lambda_{j}$, but is still
supported on a subset of the support of $\lambda_{j}$. This means $\gamma$
must be another projective measured foliation in the topological conjugacy
class of $\lambda_{j}$, i.e. is supported on the same underlying foliation.
This proves the furthermore case of theorem. If $\lambda_{j}$ is actually
uniquely ergodic, there is only one measure in the simplex of invariant
probability measures, namely the uniquely ergodic one, and therefore
$\mu_{g_{i}}$ is forced to converge to it.

Suppose now that neither of the first two scenarios occur, i.e. the leaf
returns to $p_{i}$ from the “up” or “down” direction, but with the orientation
always flipped. We deal with this case like we did with the second subcase of
case (ii). See Figure 5 for the construction of the two-sided curves $c_{i}$.

$\eta_{0}$
---
$\eta_{i}$
---
$p_{0}$
---
Figure 5. Construction of the blue curve $c_{i}$ when the leaf always returns
with orientation flipped from the “up” or “down” direction.

We have that the geodesic tightenings of the curves $c_{i}$ are close to the
original curve by the Anosov closing lemma, and that the weighted averages of
the $c_{i}$ converge the ergodic measure, which means by 3.10 we have a
subsequence $\mu_{c_{i}}$ that converges to the ergodic measure. This proves
the result for case (iii), and therefore the theorem. ∎

##  4\. Upper bound for the limit set

In this section, we prove that
$\Lambda_{\mathrm{geo}}(\mathrm{MCG}(\mathcal{N}_{g}))$ is contained in
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$. We do so by defining an
$\mathrm{MCG}(\mathcal{N}_{g})$-invariant subset
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$, and showing that the
intersection of its closure with $\mathbb{P}\mathcal{MF}(\mathcal{N}_{g})$ is
contained in $\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$.

######  Definition 4.1 (One-sided systole superlevel set).

For any $\varepsilon>0$, the set
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$ is the set of all points in
$\mathcal{T}(\mathcal{N}_{g})$ where the length of the shortest one-sided
curve is greater than or equal to $\varepsilon$.

We can state the main theorem of this section.

######  Theorem 4.2.

For any $\varepsilon>0$,
$\overline{\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})}\cap\mathbb{P}\mathcal{MF}(\mathcal{N}_{g})$
is contained in $\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$.

The key idea of the proof is proving a quantitative estimate on the Fenchel-
Nielsen coordinates of points converging to points in
$\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$.

######  Proposition 4.3.

Let $\\{m_{i}\\}$ be a sequence of points in $\mathcal{T}(\mathcal{N}_{g})$
converging to a projective measured foliation $[\lambda]$. If $p$ is a one-
sided atom of $\lambda$, for any Fenchel-Nielsen coordinate chart containing
$p$ as a cuff, the length coordinate of $p$ goes to $0$.

###### Proof.

Consider the following decomposition of the measured foliation $\lambda$.

$\displaystyle\lambda=1\cdot p+\lambda_{\mathrm{at}}+\lambda_{\mathrm{Leb}}$

Here, $\lambda_{\mathrm{at}}$ are the minimal components on periodic
components other than $p$, i.e. cylinders and Möbius strips, and
$\lambda_{\mathrm{Leb}}$ are non-periodic minimal components. In the above
expression, $p$ is the one-sided curve considered as a measured foliation
(since we’re picking a representative of $[\lambda]$, we can pick one such
that $p$ has weight $1$)

Pick simple closed curves $p_{0}$, $p_{1}$, and $p_{2}$, where $p_{0}$ is the
curve $p$, and $\\{p_{0},p_{1},p_{2}\\}$ bound a pair of pants. Furthermore,
we impose the following conditions on $p_{1}$ and $p_{2}$.

$\displaystyle i(p_{1},\lambda_{\mathrm{at}})$ $\displaystyle=0$
$\displaystyle i(p_{2},\lambda_{\mathrm{at}})$ $\displaystyle=0$

Note that this can always be done, by deleting the support of
$\lambda_{\mathrm{at}}$, and looking at the resulting subsurfaces. Neither
$p_{1}$ nor $p_{2}$ can be the same as $p_{0}$, since $p_{0}$ is one-sided. We
focus our attention to this pair of pants, and consider the curves on it,
labelled in Figure 6.

$p_{1}$
---
$p_{2}$
---
$p_{0}$
---
$p_{3}$
---
$p_{4}$
---
Figure 6. The curves restricted to a pair of pants.

The curve labelled $p_{3}$ has intersection number at least $1$ with $\lambda$
which means its length should be going to $\infty$. We can bound the length of
$p_{3}$ above and below via the lengths of the orthogeodesic $p_{4}$ and the
length of $p_{0}$.

$\ell_{\mathrm{hyp}}(p_{0})\leq\ell_{\mathrm{hyp}}(p_{3})\leq\ell_{\mathrm{hyp}}(p_{4})+\ell_{\mathrm{hyp}}(p_{0})$
(11)

Observe that we get the upper bound by observing that the red and cyan arcs
are isotopic to $p_{3}$ relative to their endpoints being fixed. The cyan arcs
have length at most $\ell_{\mathrm{hyp}}(p_{0})$ in this setting; if one
allowed a twist parameter, the length of the cyan arcs would be proportional
to the twist parameters. The point of this inequality is that we can estimate
$\ell_{\mathrm{hyp}}(p_{4})$ using $\ell_{\mathrm{hyp}}(p_{0})$,
$\ell_{\mathrm{hyp}}(p_{1})$ and $\ell_{\mathrm{hyp}}(p_{2})$ via hyperbolic
trigonometry. Cut the pair of pants along the seams, to get a hyperbolic
right-angled hexagon, pictured in Figure 7.

$\frac{l(p_{4})}{2}$
---
$\frac{l(p_{1})}{2}$
---
$\frac{l(p_{2})}{2}$
---
$f\cdot l(p_{0})$
---
$(1-f)\cdot l(p_{0})$
---
Figure 7. The right angled hexagon obtained by cutting the pants along the
seams.

To get good estimates on $\ell_{\mathrm{hyp}}(p_{4})$, we need a universal
lower bound on the fraction $f$ as we move in the Teichmüller space. The
analysis splits up into two cases, but it’s not a priori clear that these two
cases are exhaustive. We will deal with the two cases, and then show that any
other case can be reduced to the second case by changing $p_{1}$ and $p_{2}$.

### Case I

We’re in this case if $p_{1}$ and $p_{2}$ don’t intersect the foliation
$\lambda$ at all.

$\displaystyle i(p_{1},\lambda)$ $\displaystyle=0$ $\displaystyle
i(p_{2},\lambda)$ $\displaystyle=0$

In this case, we can pass to a subsequence of $\\{m_{i}\\}$ such that the
corresponding values of $f$ are always greater than $\frac{1}{2}$ or less than
$\frac{1}{2}$. In the former case, we focus on $p_{1}$, and in the latter
case, we focus on $p_{2}$. Without loss of generality, we’ll suppose
$f\geq\frac{1}{2}$. In that case, we cut along the orthogeodesic $p_{4}$, and
get a hyperbolic right-angled pentagon, which is the top half of Figure 7. We
can relate the lengths of $\ell_{\mathrm{hyp}}(p_{0})$,
$\ell_{\mathrm{hyp}}(p_{1})$, and $\ell_{\mathrm{hyp}}(p_{4})$ using the
following identity for hyperbolic right-angled pentagons (see [Thu79] for the
details).

$\displaystyle\sinh\left(f\cdot\ell_{\mathrm{hyp}}(p_{0})\right)\cdot\sinh\left(\frac{\ell_{\mathrm{hyp}}(p_{4})}{2}\right)=\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)$
(12)

Now suppose that $\ell_{\mathrm{hyp}}(p_{0})$ does not go to $0$. Then we must
have that for all $m_{i}$, $\ell_{\mathrm{hyp}}(p_{0})\geq 2\varepsilon$ for
some $\varepsilon>0$. By the lower bound on $f$, we have that the first term
on the left hand side of the above expression is bounded below by
$\varepsilon$. Rearranging the terms gives us the following upper bound on
$\ell_{\mathrm{hyp}}(p_{4})$.

$\ell_{\mathrm{hyp}}(p_{4})\leq
2\cdot\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{\varepsilon}\right)$
(13)

Using (11) and (13), we get an upper bound for $\ell_{\mathrm{hyp}}(p_{3})$.

$\ell_{\mathrm{hyp}}(p_{3})\leq\ell_{\mathrm{hyp}}(p_{0})+2\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{\varepsilon}\right)$
(14)

Since $\frac{i(p_{0},\lambda)}{i(p_{3},\lambda)}=0$, as $\\{m_{i}\\}$
approaches $\lambda$, the ratio of lengths of $p_{0}$ and $p_{3}$ approach
$0$.

$\lim_{i\to\infty}\frac{\ell_{\mathrm{hyp}}(p_{0})}{\ell_{\mathrm{hyp}}(p_{3})}=0$
(15)

Using (14), we can find a lower bound for
$\frac{\ell_{\mathrm{hyp}}(p_{0})}{\ell_{\mathrm{hyp}}(p_{3})}$, which goes to
$0$ by (15).

$\lim_{i\to\infty}\frac{\ell_{\mathrm{hyp}}(p_{0})}{\ell_{\mathrm{hyp}}(p_{0})+2\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{\varepsilon}\right)}=0$
(16)

Since we assumed that the length coordinate $\ell_{\mathrm{hyp}}(p_{0})$ is
bounded away from $0$, the only way the above expression can go to $0$ if
$\ell_{\mathrm{hyp}}(p_{1})$ goes to $\infty$. This is where the hypotheses of
the Case I come in. Since $i(p_{1},\lambda)$ is $0$, the following equality
must hold.

$\lim_{i\to\infty}\frac{\ell_{\mathrm{hyp}}(p_{1})}{\ell_{\mathrm{hyp}}(p_{3})}=0$
(17)

This means the lower bound for
$\frac{\ell_{\mathrm{hyp}}(p_{1})}{\ell_{\mathrm{hyp}}(p_{3})}$ must go to
$0$.

$\lim_{i\to\infty}\frac{\ell_{\mathrm{hyp}}(p_{1})}{\ell_{\mathrm{hyp}}(p_{0})+2\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{\varepsilon}\right)}=0$
(18)

But this can’t happen if $\ell_{\mathrm{hyp}}(p_{1})$ approaches $\pm\infty$.
This contradiction means our assumption that $\ell_{\mathrm{hyp}}(p_{0})$ was
bounded away from $0$ must be wrong, and thus proves the result in Case I.

### Case II

We’re in this case if the following inequality holds.

$0<i(p_{1},\lambda)<1$ (19)

The picture in this case looks similar to Figure 7. However, we can’t
necessarily pass to a subsequence where $f\geq\frac{1}{2}$ (and the trick of
working with $1-f$ won’t work, since we know nothing about $p_{2}$). This is
one of the points where the hypothesis on $p_{1}$ comes in. Since
$\frac{i(p_{2},\lambda)}{i(p_{1},\lambda)}$ is finite, we must have that the
ratio of lengths
$\frac{\ell_{\mathrm{hyp}}(p_{2})}{\ell_{\mathrm{hyp}}(p_{1})}$ approaches
some finite value as well. The fraction $f$ is a continuous function of
$\frac{\ell_{\mathrm{hyp}}(p_{2})}{\ell_{\mathrm{hyp}}(p_{1})}$, approaching
$0$ only as the ratio approaches $\infty$ (this follows from the same identity
as (12)). Since the ratio approaches a finite value, we have a positive lower
bound $f_{0}$ for $f$.

Assuming as before that $\ell_{\mathrm{hyp}}(p_{0})$ is bounded away from $0$,
and $\tau(p_{0})$ bounded away from $\pm\infty$, and repeating the
calculations of the previous case, we get the following two inequalities.

$\frac{\ell_{\mathrm{hyp}}(p_{1})}{\ell_{\mathrm{hyp}}(p_{3})}\geq\frac{\ell_{\mathrm{hyp}}(p_{1})}{\ell_{\mathrm{hyp}}(p_{0})+2\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{f_{0}\varepsilon}\right)}$
(20)
$\frac{\ell_{\mathrm{hyp}}(p_{0})}{\ell_{\mathrm{hyp}}(p_{3})}\geq\frac{\ell_{\mathrm{hyp}}(p_{0})}{\ell_{\mathrm{hyp}}(p_{0})+2\sinh^{-1}\left(\frac{\cosh\left(\frac{\ell_{\mathrm{hyp}}(p_{1})}{2}\right)}{f_{0}\varepsilon}\right)}$
(21)

The right hand side of (21) must approach $0$, and that forces either
$\ell_{\mathrm{hyp}}(p_{1})$ or $\ell_{\mathrm{hyp}}(p_{0})$ to approach
$\infty$. But that means the right hand term of (20) must approach $1$, which
cannot happen, by the hypothesis of case II. This means
$\ell_{\mathrm{hyp}}(p_{0})$ goes to $0$, proving the result in case II.

### Reducing to case II

Suppose now that both $p_{1}$ and $p_{2}$ have an intersection number larger
than $1$ with $\lambda$. We can modify one of them to have a small
intersection number with $\lambda$. First, we assume that
$\lambda_{\mathrm{Leb}}$ is supported on a single minimal component, i.e.
every leaf of $\lambda_{\mathrm{Leb}}$ is dense in the support. We now perform
a local surgery on $p_{1}$: starting at a point on $p_{1}$ not contained in
the support of $\lambda_{\mathrm{Leb}}$, we follow along until we intersect
$\lambda_{\mathrm{Leb}}$ for the first time. We denote this point by $\alpha$.
We now go along $p_{1}$ in the opposite direction, until we hit the support of
$\lambda_{\mathrm{Leb}}$ again, but rather than stopping, we keep going until
the arc has intersection number $0<\delta<1$ with $\lambda_{\mathrm{Leb}}$. We
then go back to $\alpha$, and follow along a leaf of $\lambda_{\mathrm{Leb}}$
rather than $p_{1}$, until we hit the arc. This is guaranteed to happen by the
minimality of $\lambda_{\mathrm{Leb}}$. Once we hit the arc, we continue along
the arc, and close up the curve. This gives a new simple closed curve which
intersection number with $\lambda$ is at most $\delta$. This curve is our
replacement for $p_{1}$. If $\lambda_{\mathrm{Leb}}$ is not minimal, we repeat
this process for each minimal component. We pick $p_{2}$ in a manner such that
$p_{0}$, $p_{1}$, and $p_{2}$ bound a pair of pants. Since $\delta<1$, we have
reduced to case II. This concludes the proof of the theorem. ∎

######  Remark (On the orientable version of 4.3).

The same idea also works in the orientable setting, although the analysis of
the various cases gets a little more delicate. The first change one needs to
make is in the statement of the proposition: we no longer need to require $p$
to be a one-sided atom, and correspondingly, either the length coordinate
$\ell_{\mathrm{hyp}}(p)$ can go to $0$, or the twist coordinate $\tau(p_{0})$
can go to $\pm\infty$. To see how the twist coordinate enters the picture,
observe that (11), which was the main inequality of the proof, turns into the
following in the orientable version.

$\ell_{\mathrm{hyp}}(p_{4})\leq\ell_{\mathrm{hyp}}(p_{3})\leq\tau(p_{0})+\ell_{\mathrm{hyp}}(p_{4})$
(22)

Here, $\tau(p_{0})$ is the twist parameter about $p_{0}$, and $p_{4}$ is the
orthogeodesic multi-arc (there may be one or two orthogeodesics, depending on
the two cases described below).

The proof splits up into two cases, depending on whether both sides of $p$ are
the same pair of pants, or distinct pairs of pants. This was not an issue in
the non-orientable setting, since $p$ was one-sided. If both sides of $p$ are
the same pair of pants, then the analysis is similar to what we just did,
since the curve $p_{3}$ stays within a single pair of pants. In the other,
$p_{3}$ goes through two pair of pants, and its length is a function of the
twist parameters, as well the cuff lengths of four curves, rather than two
curves, the four curves being the two remaining cuffs of each pair of pants.
The analysis again splits up into two cases, depending on the intersection
number of the cuffs with $\lambda$, but reducing all the other cases to case
II becomes tricky because we need to simultaneously reduce the intersection
number of two curves, rather than one, as in the non-orientable setting. This
added complication obscures the main idea of the proof, which is why we chose
to only prove the non-orientable version.

This quantitative estimate of 4.3 gives us a proof for Theorem 4.2.

###### Proof of Theorem 4.2.

Suppose that the theorem were false, and there was a foliation
$[\lambda]\in\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$ in the closure of
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$. Suppose $p$ is a one-sided
atom in $\lambda$. Then 4.3 tells us that the hyperbolic length of $p$ goes to
$0$, but the length of $p$ must be greater than $\varepsilon$ in
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$. This contradicts our initial
assumption, and the closure of
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$ can only intersect
$\mathbb{P}\mathcal{MF}(\mathcal{N}_{g})$ in the complement of
$\mathbb{P}\mathcal{MF}^{-}(\mathcal{N}_{g})$. ∎

######  Corollary 4.4.

The geometric limit set
$\Lambda_{\mathrm{geo}}(\mathrm{MCG}(\mathcal{N}_{g}))$ is contained in
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$.

###### Proof.

Every point $p\in\mathcal{T}(\mathcal{N}_{g})$ is contained in
$\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$ for some small enough
$\varepsilon$. This means
$\Lambda_{\mathrm{geo},p}(\mathrm{MCG}(\mathcal{N}_{g}))$ is contained in
$\mathbb{P}\mathcal{MF}^{+}(\mathcal{N}_{g})$ by Theorem 4.2. Taking the union
over all $p$ proves the result. ∎

##  5\. Failure of quasi-convexity for ${\mathcal{T}_{\varepsilon}^{-}}$

In the setting of Teichmüller geometry, convexity is usually too strong of a
requirement. For instance, metric balls in Teichmüller space are not convex,
but merely quasi-convex (see [LR11]).

######  Definition 5.1 (Quasi-convexity).

A subset $S$ of ${\mathcal{T}(S)}$ is said to be quasi-convex if there is some
uniform constant $D>0$ such that the geodesic segment joining any pair of
points in $S$ stays within distance $D$ of $S$.

Our goal for this section will be to prove the following theorem.

######  Theorem 5.2.

For all $\varepsilon>0$, and all $D>0$, there exists a Teichmüller geodesic
segment whose endpoints lie in ${\mathcal{T}_{\varepsilon}^{-}}$ such that
some point in the interior of the geodesic is more than distance $D$ from
$\mathcal{T}_{\varepsilon}^{-}$.

We begin by finding Teichmüller geodesic segments whose endpoints lie in
$\mathcal{T}_{\varepsilon}^{-}$ such that at a point in the interior, some
one-sided curve gets very short. Once we have arbitrarily short one-sided
curves in the interior of the geodesic segments, estimates relating
Teichmüller distance and ratios of hyperbolic lengths of curves will give us
the result.

######  Proposition 5.3.

For all $g\geq 8$ and any $\delta>0$, there exists a Teichmüller geodesic
segment $l$ whose endpoints lie in $\mathcal{T}_{\varepsilon}^{-}$, and a
point $p$ in $l$ such that some one-sided curve has length less than $\delta$
with respect to the hyperbolic metric on $p$.

To prove this result, we will need two lemmas relating hyperbolic and flat
lengths.

######  Lemma 5.4.

Let $q$ be any area $1$ DQD on $\mathcal{N}_{g}$, and let $\gamma$ be a simple
closed curve of $q$. Suppose that $\ell_{\mathrm{hyp}}(\gamma)\leq\delta$
(with respect to the unique hyperbolic metric coming from the flat structure
$q$). Then $\ell_{\mathrm{flat}}(\gamma)\leq k\sqrt{\delta}$, where $k$ is
some absolute constant.

###### Sketch of proof.

If $\ell_{\mathrm{hyp}}(\gamma)\leq\delta$, then there exists an annulus
around $\gamma$ of modulus proportional to $\frac{1}{\delta}$. By the results
in [Min92], this annulus can be homotoped to be a primitive annulus, i.e. an
annulus that does not pass through a singularity of the flat metric. Such
annuli are either expanding, i.e. concentric circles in the flat metric, or
flat, and in either case, we have an upper bound on the flat length of the
core curve in terms of the modulus. This proves the result. ∎

######  Lemma 5.5.

Let $q$ be an area $1$ DQD on $\mathcal{N}_{g}$, and consider the unique
hyperbolic metric with the same conformal structure. Let $A$ be a primitive
annulus in $q$, i.e. an annulus whose interior does not pass through a
singularity of the flat metric. Let the modulus of $A$ be $m$. Then the
hyperbolic length of the isotopy class of the core curve of the annulus is at
most $\frac{\pi}{m}$.

###### Sketch of proof.

Without loss of generality, we can pass to the orientable double cover. This
changes the hyperbolic lengths by at most a factor of two. Consider the
interior of the annulus as a Riemann surface, and put the unique hyperbolic
metric on that surface. With respect to this hyperbolic metric, the length of
the core curve is $\frac{\pi}{m}$. Since the interior doesn’t contain any
singularities, the inclusion map is holomorphic, and holomorphic maps are
distance reducing with respect to the hyperbolic metric. This proves the
result. ∎

To find a geodesic segment whose endpoints lie in
$\mathcal{T}_{\varepsilon}^{-}$, we will construct a DQD $q$, and use 5.4 to
find large enough $t$ such that both $g_{t}(q)$ and $g_{-t}(q)$ are in
$\mathcal{T}_{\varepsilon}^{-}$. We will then show that some one sided curve
on $q$ is very short using 5.5, which will prove 5.3.

###### Proof of 5.3.

We will prove the result by constructing explicit examples in genus $4$ and
$9$, and then connect summing orientable surfaces of genus $j$ to get examples
in genus $4+2j$ and $9+2j$.

We first list the two properties we require from the DQD $q$ we want to
construct, and show that having those properties proves the result.

  1. (a)

There exists an embedded annulus in $q$ with a very large modulus whose core
curve is the square of a one-sided curve in $\pi_{1}(\mathcal{N}_{g})$.

  2. (b)

The vertical and horizontal foliations decompose as a union of cylinders, i.e.
the vertical and horizontal flow is periodic, and no closed orbit is a one-
sided curve. Furthermore, deleting the core curves of the cylinders in the
horizontal or vertical direction result in a disjoint union of _orientable_
subsurfaces.

We now show why having these two properties proves the result. Suppose we have
a DQD $q$ satisfying (a) and (b). 5.5 tells us that satisfying (a) means that
the one-sided curve whose square is the core curve of the annulus will be very
short. To find a large enough $t$ such that $g_{t}(q)$ has no one-sided curves
shorter than $\varepsilon$, pick a $t$ enough such that each vertical cylinder
in $g_{t}(q)$ is at least $2k\sqrt{\varepsilon}$ wide. Consider now any closed
curve who flat length is less than $k\sqrt{\varepsilon}$. It must either be
homotopic to one of the core curves of the vertical cylinders, or can be
homotoped to be completely contained in one of the subsurfaces obtained by
deleting all the core curves. That is because it was neither of these cases,
it would cross at least one of these cylinders, and since the cylinders are at
least $2k\sqrt{\varepsilon}$ wide, the flat length of the curve would exceed
$k\sqrt{\varepsilon}$. If the curve is the core curve of a cylinder, or
completely contained in one of the subsurfaces, it must be two-sided, by
condition (b).

This proves that all one-sided curves have flat length exceeding
$k\sqrt{\varepsilon}$, and therefore hyperbolic length exceeding
$\varepsilon$. The same argument also works for $g_{-t}(q)$, proving the
result.

We now construct explicitly the DQDs satisfying conditions (a) and (b) in
genus $4$, $9$, and above.

### The $g=4$ case

Consider the area $1$ DQD on $\mathcal{N}_{4}$ depicted in Figure 8.

$a$
---
$a$
---
$b$
---
$b$
---
$b^{\prime}$
---
$b^{\prime}$
---
$c$
---
$c$
---
$c^{\prime}$
---
$c^{\prime}$
---
$d$
---
$d$
---
$d^{\prime}$
---
$d^{\prime}$
---
Figure 8. A DQD on $\mathcal{N}_{4}$.

We impose the following constraint on the depicted DQD: the edges
$\\{c,c^{\prime},d,d^{\prime}\\}$ are all oriented at an angle of
$\pm\frac{\pi}{4}$, and have the same length.

Observe that by making the length of $c$ (and correspondingly $c^{\prime}$,
$d$, and $d^{\prime}$) go to $0$, while keeping the area $1$ lets us embed an
annulus of high modulus (pictured as dotted semi circle in Figure 8) around
any curve in $\\{c,c^{\prime},d,d^{\prime}\\}$. This shows that the DQD we
constructed satisfies condition (a).

Checking condition (b) is easy, but tedious. For convenience, we have labelled
the core curves of the vertical cylinders in red, blue, and green: the reader
can check that they are all two-sided, and deleting them results in orientable
subsurfaces. In fact, deleting the core curves results in $2$ pairs of pants.

### The $g=9$ case

Consider the area $1$ DQD on $\mathcal{N}_{9}$ depicted in Figure 9. To keep
the picture from getting cluttered, we describe the edge gluing maps in words:
the edges labelled $c$ are glued via the map $z\mapsto-\overline{z}+k$, the
edges labelled $b$ and $e$ are glued via $z\mapsto-z+k$, where $k$ is some
constant. All the other gluings are translation gluings.

$a$
---
$b$
---
$a$
---
$b$
---
$c$
---
$c$
---
$x_{v}$
---
$x_{v}$
---
$y_{v}$
---
$y_{v}$
---
$x_{h}$
---
$y_{h}$
---
$y_{h}$
---
$x_{h}$
---
$d$
---
$e$
---
$d$
---
$e$
---
Figure 9. A DQD on $\mathcal{N}_{9}$. To display the gluing maps on the small
slits, we have a zoomed in picture in the ellipses.

We impose the following constraints on the DQD.

  1. (i)

The edges labelled $c$ are oriented at an angle of $\pm\frac{\pi}{4}$, and the
lengths of $\\{x_{h},y_{h},x_{v},y_{v}\\}$ are
$\frac{\ell_{\mathrm{flat}}(c)}{4\sqrt{2}}$.

  2. (ii)

The left edge of $x_{v}$ is aligned with the left edge of $c$, the left edge
of $y_{v}$ is aligned with the midpoint of $c$, the top edge of $x_{h}$ is
aligned with the top edge of $c$, and the top edge of $y_{h}$ is aligned with
the midpoint of $c$.

By making $c$ smaller, while keeping the area equal to $1$, one can embed an
annulus of high modulus in the DQD, pictured in dotted olive green in Figure
9. This shows that our construction satisfies condition (a).

To see that deleting the core curves of the horizontal cylinders results in
orientable subsurfaces, note that deleting the core curves passing through $c$
results in $2$ pairs of pants, and a genus $3$ orientable surface with one
boundary component. This is again easy, but tedious to verify, so we leave the
verification to the reader. This shows that the example satisfies condition
(b).

### The induction step

To get higher genus DQDs satisfying conditions (a) and (b), we start with the
$g=4$ and $g=9$ examples and connect-sum an orientable surface using the slit
construction. To ensure that the new surfaces still satisfy conditions (a) and
(b), we need to ensure that the slit we construct if far away from the annulus
of condition (a), as well as all the vertical and horizontal leaves passing
through $\\{c,c^{\prime},d,d^{\prime}\\}$ in the $g=4$ example, and the
vertical and horizontal leaves passing through $c$ in the $g=9$ example. This
will ensure that the resulting higher genus surface still satisfies conditions
(a) and (b). ∎

To relate Teichmüller distance to hyperbolic lengths, we need Wolpert’s lemma
([Wol79])

######  Lemma 5.6 (Wolpert’s Lemma).

Let $M$ and $M^{\prime}$ be two points in $\mathcal{T}(\mathcal{S}_{g})$, and
let $\gamma$ be a simple closed curve on $\mathcal{S}_{g}$. Let $R$ be the
Teichmüller distance between $M$ and $M^{\prime}$. Then the ratio of the
hyperbolic length of $\gamma$ and $R$ are related by the following
inequalities.

$\displaystyle\exp(-2R)\leq\frac{\ell_{\mathrm{hyp}}(M,\gamma)}{\ell_{\mathrm{hyp}}(M^{\prime},\gamma)}\leq\exp(2R)$

Using Proposition 5.3 and Wolpert’s lemma, we can prove Theorem 5.2.

###### Proof of Theorem 5.2.

Suppose that $\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$ was indeed
quasi-convex. That would mean that there exists some $R>0$, depending on
$\varepsilon$ such that every point in the interior of any geodesic segment
with endpoints in $\mathcal{T}_{\varepsilon}^{-}$ was within $R$ distance of
some point in $\mathcal{T}_{\varepsilon}^{-}(\mathcal{N}_{g})$. Proposition
5.3 lets us construct a sequence of Teichmüller geodesic segments such that
for on some interior point, the length of a given one-sided curve $\gamma$
goes to $0$. If those points were within distance $R$ of
$\mathcal{T}_{\varepsilon}^{-}$, there would be some point in
$\mathcal{T}_{\varepsilon}^{-}$ where the length of $\gamma$ was at most
$\exp(2R)$ times the length of $\gamma$ in the geodesic, by Wolpert’s lemma.
But since the length of $\gamma$ in the geodesic goes to $0$, the length in
the corresponding closest point in $\mathcal{T}_{\varepsilon}^{-}$ must also
go to $0$. This violates the definition of $\mathcal{T}_{\varepsilon}^{-}$,
giving us a contradiction, and proving the result. ∎

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