# On the ergodicity of the Rel foliation

Jon Chaika  University of Utah<EMAIL_ADDRESS>and  Barak Weiss  Tel
Aviv University<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathcal{H}}$ be a stratum of translation surfaces with at least two
singularities, let $m_{{\mathcal{H}}}$ denote the Masur-Veech measure on
${\mathcal{H}}$, and let $Z_{0}$ be a flow on
$({\mathcal{H}},m_{{\mathcal{H}}})$ obtained by integrating a Rel vector
field. We prove that $Z_{0}$ is mixing of all orders, and in particular is
ergodic. We also characterize the ergodicity of flows defined by Rel vector
field, for more general spaces $({\mathcal{L}},m_{{\mathcal{L}}})$, where
${\mathcal{L}}\subset{\mathcal{H}}$ is an orbit-closure for the action of
$G=\operatorname{SL}_{2}({\mathbb{R}})$ (i.e., an affine invariant subvariety)
and $m_{{\mathcal{L}}}$ is the natural measure. Our results are conditional on
a forthcoming measure classification result of Brown, Eskin, Filip and
Rodriguez-Hertz.

##  1\. Introduction

Let ${\mathcal{H}}$ be a stratum of area-one translation surfaces and let
$G{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\operatorname{SL}_{2}({\mathbb{R}})$.
There is a $G$-invariant finite measure $m_{{\mathcal{H}}}$ on ${\mathcal{H}}$
known as the Masur-Veech measure, and the dynamics of the $G$-action on
$({\mathcal{H}},m_{{\mathcal{H}}})$ have been intensively studied in recent
years and are intimately connected to many problems in geometry and ergodic
theory, see e.g. [MaTa, Zo]. Suppose that surfaces in ${\mathcal{H}}$ have $k$
singularities, where $k\geq 2$. Then there is a $k-1$-dimensional foliation of
${\mathcal{H}}$, known as the real Rel foliation. A precise definition of the
foliation and some of its properties will be given below in §2.2. Loosely
speaking, two surfaces are in the same real Rel leaf if one can be obtained
from the other by a surgery in which singular points are moved with respect to
each other in the horizontal direction, without otherwise changing the
geometry of the surface. A natural question, which we address here, is the
ergodic properties of this foliation.

As we review in §2.2, by labeling the singularities and removing a set of
leaves of measure zero, we can think of the real Rel leaves as being the
orbits of an action of a group $Z$ on ${\mathcal{H}}$, where
$Z\cong{\mathbb{R}}^{k-1}$, and the restriction of this action to any one-
dimensional subgroup of $Z$ defines a flow. Our first main result is the
following.

######  Theorem 1.1.

Let ${\mathcal{H}}$ be a connected component of a stratum
${\mathcal{H}}(a_{1},\ldots,a_{k})$ with all $a_{i}>0$ (i.e., no marked
points). Let $m_{\mathcal{H}}$ be the Masur-Veech measure on ${\mathcal{H}}$,
let $Z\cong{\mathbb{R}}^{k-1}$ be the corresponding action given by
translation along the leaves of the real Rel foliation, and let $Z_{0}\subset
Z$ be any one-dimensional connected subgroup of $Z$. Then the $Z_{0}$-flow on
$({\mathcal{H}},m_{{\mathcal{H}}})$ is mixing of all orders (and in
particular, ergodic).

The definition of mixing of all orders is given in §3.3. For purposes of this
introduction it is enough to note that it implies ergodicity of any nontrivial
element. Note that when ${\mathcal{H}}$ has marked points, there will be
subgroups $Z_{0}$ which only move the marked points on surfaces without
otherwise changing the geometry, and the conclusion of Theorem 1.1 will not
hold. This is the only obstruction to generalizing our results to strata with
marked points, see Theorem 8.1.

The proof of Theorem 1.1, as well as the other results of this paper, relies
crucially on measure-rigidity results of Eskin, Mirzakhani and Mohammadi [EM,
EMM], and further forthcoming work extending these results, which we will
describe in §5.

Theorem 1.1 improves on the results of several authors. In those results,
ergodicity for the full Rel foliation was studied. The full Rel foliation
(also referred to as the ‘kernel foliation’, ‘isoperiodic foliation’, or
‘absolute period foliation’) will also be defined in §2.2. Its leaves are of
dimension $2(k-1)$, that is, twice the dimension of the real Rel leaves.
Loosely speaking, two surfaces are in the same leaf for this foliation if one
can be obtained from the other by moving the singularities (without otherwise
affecting the geometry of the surface). That is, we relax the hypothesis that
points can only be moved horizontally. The first ergodicity results for the
full Rel foliation were obtained by McMullen [McM], who proved ergodicity in
the two strata ${\mathcal{H}}(1,1)$ and ${\mathcal{H}}(1,1,1,1)$.
Subsequently, Calsamiglia, Deroin and Francaviglia [CDF] proved ergodicity in
all principal strata, and Hamenstädt [Ham] reproved their result by a simpler
argument. Recently, Winsor [Wi1] proved ergodicity for most of the additional
strata, and in [Wi2], showed that there are dense orbits for the $Z_{0}$-flow,
for any $Z_{0}$ as in Theorem 1.1. Note that ergodicity for a foliation is
implied by ergodicity for any of its subfoliations, and that ergodicity
implies the existence of dense leaves. and thus Theorem 1.1 generalizes all of
these results. Also note that full Rel is a foliation which is not given by a
group action, and the notions of mixing and multiple mixing do not make sense
in this case.

The papers [McM, CDF] go beyond ergodicity and obtain classifications of full
Rel leaves in their settings. We suspect that there is not a reasonable
classification of real Rel leaf-closures, indeed it is already known (see
[HW]) that there are real rel trajectories that leave every compact set never
to return.

The strata ${\mathcal{H}}$ support other interesting measures for which
similar questions could be asked. Namely, by work of Eskin, Mirzakhani and
Mohammadi [EM, EMM], for any $q\in{\mathcal{H}}$, the orbit-closure
${{\mathcal{L}}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\overline{Gq}$
is the support of a unique smooth $G$-invariant measure which we denote by
$m_{{{\mathcal{L}}}}$. Let $Z_{{{\mathcal{L}}}}$ be the subgroup of $Z$
leaving ${{\mathcal{L}}}$ invariant. Then $Z_{{{\mathcal{L}}}}$ also preserves
$m_{{{\mathcal{L}}}}$ and for many choices of ${{\mathcal{L}}}$, we have $\dim
Z_{{\mathcal{L}}}>0$. In these cases, for any closed connected $Z_{1}\subset
Z_{{{\mathcal{L}}}}$, there is a complexification $\mathfrak{R}_{1}$, which
gives a foliation of ${{\mathcal{L}}}$ whose leaves $\mathfrak{R}_{1}(q)$ have
dimension $2\dim Z_{1}$ (see §2.2). The leaves $\mathfrak{R}_{1}(q)$ have a
natural translation structure, and this induces a natural locally finite
translation-invariant measure on each leaf. With this terminology we can now
state the main result of this paper:

######  Theorem 1.2.

Let ${{\mathcal{L}}}$ be a $G$-orbit closure, and let
$m_{\mathcal{L}},\,Z_{{{\mathcal{L}}}},\mathfrak{R}_{{{\mathcal{L}}}}$ be as
above, where $\dim Z_{{{\mathcal{L}}}}>0$. Let $z_{0}$ be a nontrivial element
of $Z_{{{\mathcal{L}}}}$ and let $Z_{0}={\rm span}_{{\mathbb{R}}}(z_{0}).$
Then either

  1. (1)

$Z_{0}$ is $m_{\mathcal{L}}$ mixing of all orders (and in particular, $z_{0}$
acts ergodically); or

  2. (2)

there is an intermediate closed connected subgroup $Z_{1}$ so that
$Z_{0}\subset Z_{1}\subset Z_{{{\mathcal{L}}}}$, and the complexification
$\mathfrak{R}_{1}$ of $Z_{1}$ satisfies

     * •

for every $q\in{{\mathcal{L}}}$, the leaf $\mathfrak{R}_{1}(q)$ is closed, and

     * •

for $m_{\mathcal{L}}$-a.e. $q$, $\mathfrak{R}_{1}(q)$ is of finite volume with
respect to its translation-invariant measure, and
$\overline{Z_{0}q}=\mathfrak{R}_{1}(q)$.

Thus, in order to establish ergodicity of real Rel subfoliations on $G$-orbit-
closures, it is enough to rule out Case (2). We will prove Proposition 7.1,
which provides a simple way to achieve this, using cylinder circumferences of
surfaces in ${{\mathcal{L}}}$. Theorems 1.1 and 8.1 are deduced from Theorem
1.2 using Proposition 7.1.

The following statement is an immediate consequence of Theorem 1.2.

######  Corollary 1.3.

Let ${{\mathcal{L}}}$ be a $G$-orbit-closure, let
$m_{{{\mathcal{L}}}},Z_{{{\mathcal{L}}}}$ be as above, and let $Z_{1}\subset
Z_{{{\mathcal{L}}}}$ be one-dimensional. Suppose that the foliation induced by
the complexification $\mathfrak{R}_{1}$ has a dense leaf. Then the
$Z_{1}$-flow on $({{\mathcal{L}}},m_{{{\mathcal{L}}}})$ is mixing of all
orders (and in particular, ergodic).

The density of certain leaves of the full Rel foliation in $G$-orbit-closures
of rank one was obtained by Ygouf in [Y]. Using these results we obtain
ergodicity of one-dimensional subgroups of the real Rel foliation in many
cases. For instance, using [Y, Thm. A & Prop. 5.1] we have:

######  Corollary 1.4.

The real Rel foliation is mixing of all orders (and in particular, ergodic) in
any eigenform locus in ${\mathcal{H}}(1,1)$ with a non-square discriminant.

Recall that in [Wi2] Winsor proved the existence of dense real Rel leaves, and
dense leaves for one-dimensional flows $Z_{0}$, in all strata. Using these
results in conjunction with Corollary 1.3, one can obtain an alternative proof
of Theorem 1.1 that avoids the use of Proposition 7.1.

###  1.1. Outline

In §2 we give background material on translation surfaces, their moduli
spaces, and the Rel foliation. In §3 we use standard facts about joinings to
build a measure $\theta$ on the product of two strata (see (3.1)), depending
on a real-Rel flow $Z_{0}$, such that if $\theta$ is the product measure, then
$Z_{0}$ is ergodic. In §3.3 we discuss a technique of Mozes that makes it
possible to upgrade ergodicity to mixing of all orders. In §4 we show that
$\theta$ is ergodic for the diagonal action of the upper triangular group
$P\subset G$ on the product of two strata. In §5 we state a far-reaching
measure rigidity result of Brown, Eskin, Filip and Rodriguez-Hertz for
$P$-ergodic measures on products of two strata. In §6 we use this measure
rigidity result, as well as prior results for the action on one stratum due to
Wright, in order to characterize the situations in which $\theta$ is not a
product measure, thus proving Theorem 1.2.

###  1.2. Acknowledgements

We are very grateful to Alex Eskin for crucial contributions to this project.
We also thank Simion Filip and Alex Wright for useful discussions, and
acknowledge the support of ISF grants 2919/19, BSF grant 2016256, NSFC-ISF
grant 3739/21, a Warnock chair, a Simons Fellowship, and NSF grants
DMS-1452762 and DMS-2055354.

##  2\. Preliminaries about translation surfaces

###  2.1. Strata, period coordinates

In this section we collect standard facts about translation surfaces, and fix
our notation. For more details, we refer to reader to [Zo, Wr1, BSW]. Below we
briefly summarize the treatment in [BSW, §2].

Let $S$ be a compact oriented surface of genus $g$,
$\Sigma=\\{\xi_{1},\ldots,\xi_{k}\\}\subset S$ a finite set,
$a_{1},\ldots,a_{k}$ non-negative integers with $\sum a_{i}=2g-2$, and
${\mathcal{H}}={\mathcal{H}}(a_{1},\ldots,a_{k})$ the corresponding stratum of
unit-area translation surfaces. We let
${\mathcal{H}}_{\mathrm{m}}={\mathcal{H}}_{\mathrm{m}}(a_{1},\ldots,a_{k})$
denote the stratum of unit-area marked translation surfaces and
$\pi:{\mathcal{H}}_{\mathrm{m}}\to{\mathcal{H}}$ the forgetful mapping. Our
convention is that singular points are labeled, or equivalently,
${\mathcal{H}}={\mathcal{H}}_{\mathrm{m}}/\operatorname{Mod}(S,\Sigma)$, where
$\operatorname{Mod}(S,\Sigma)$ is the group of isotopy classes of orientation-
preserving homeomorphisms of $S$ fixing $\Sigma$, up to an isotopy fixing
$\Sigma$.

There is an ${\mathbb{R}}_{>0}$-action that dilates the atlas of a translation
surface by $c\in{\mathbb{R}}_{>0}$. For a stratum $\mathcal{H}$ and marked
stratum $\mathcal{H}_{\mathrm{m}}$, we denote the collection of surfaces of
arbitrary area, obtained by applying such dilations, by
$\bar{\mathcal{H}},\,\bar{\mathcal{H}}_{\mathrm{m}}$. The marked stratum
$\bar{\mathcal{H}}_{\mathrm{m}}$ is a linear manifold modeled on the vector
space $H^{1}(S,\Sigma;{\mathbb{R}}^{2})$. It has a developing map
$\mathrm{dev}:\bar{\mathcal{H}}_{\mathrm{m}}\to
H^{1}(S,\Sigma;{\mathbb{R}}^{2})$, sending an element of
$\bar{\mathcal{H}}_{\mathrm{m}}$ represented by $f:S\to M$, where $M$ is a
translation surface, to $f^{*}(\mathrm{hol}(M,\cdot))$, where for an oriented
path $\gamma$ in $M$ which is either closed or has endpoints at singularities,
$\mathrm{hol}(M,\gamma)=\left(\begin{matrix}\int_{\gamma}dx\\\
\int_{\gamma}dy\end{matrix}\right)$, and $dx,dy$ are the 1-forms on $M$
inherited from the plane. Furthermore, there is an open cover
$\\{\mathcal{U}_{\tau}\\}$ of ${\mathcal{H}}_{\mathrm{m}}$, indexed by
triangulations $\tau$ of $S$ with triangles whose vertices are in $\Sigma$,
and maps $\mathrm{dev}|_{\mathcal{U}_{\tau}}:\mathcal{U}_{\tau}\to
H^{1}(S,\Sigma;{\mathbb{R}}^{2})$, which are homeomorphisms onto their image,
and such that the transition maps on overlaps for this atlas are restrictions
of linear automorphisms of $H^{1}(S,\Sigma;{\mathbb{R}}^{2})$.

This atlas of charts
$\\{(\mathcal{U}_{\tau},\mathrm{dev}|_{\mathcal{U}_{\tau}})\\}$ is known as
period coordinates. Since each $\mathcal{U}_{\tau}$ is identified via period
coordinates with an open subset of the vector space
$H^{1}(S,\Sigma;{\mathbb{R}}^{2})$, the tangent space at each
$\mathcal{U}_{\tau}$ is identified canonically with
$H^{1}(S,\Sigma;{\mathbb{R}}^{2})$, and thus the tangent bundle of
${\mathcal{H}}_{\mathrm{m}}$ is locally constant. A sub-bundle of the tangent
bundle is called locally constant or flat if it is constant in the charts
afforded by period coordinates. The $\operatorname{Mod}(S,\Sigma)$-action on
${\mathcal{H}}_{\mathrm{m}}$ is properly discontinuous, and hence
${\mathcal{H}}$ is an orbifold, and the map
$\pi:{\mathcal{H}}_{\mathrm{m}}\to{\mathcal{H}}$ is an orbifold covering map.

The group $G$ acts on translation surfaces in ${\mathcal{H}}$ by modifying
planar charts, and acts on $H^{1}(S,\Sigma;{\mathbb{R}}^{2})$ via its action
on ${\mathbb{R}}^{2}$, thus inducing a $G$-action on
${\mathcal{H}}_{\mathrm{m}}$. The $G$-action commutes with the
$\operatorname{Mod}(S,\Sigma)$-action, and thus the map $\pi$ is
$G$-equivariant for these actions. The $G$-action on
${\mathcal{H}}_{\mathrm{m}}$ is free, since
$\mathrm{dev}(g\mathbf{q})\neq\mathrm{dev}(\mathbf{q})$ for any nontrivial
$g\in G$. We will use the following subgroups of $G$:

$\begin{split}g_{t}=\left(\begin{matrix}e^{t}&0\\\
0&e^{-t}\end{matrix}\right),\ \ \ u_{s}=&\left(\begin{matrix}1&s\\\
0&1\end{matrix}\right)\\\ U=\\{u_{s}:s\in{\mathbb{R}}\\},\ \ \
P=&\left\\{\left(\begin{matrix}a&b\\\ 0&a^{-1}\end{matrix}\right):a>0,\
b\in{\mathbb{R}}\right\\}.\end{split}$

###  2.2. Rel foliation and real Rel foliation

We define and list some important properties of the Rel foliation, the real-
Rel foliation, and the corresponding action on the space of surfaces without
horizontal saddle connections. See [MW2, BSW] for more details. See also [Zo,
McM], and references therein.

We have a canonical splitting
${\mathbb{R}}^{2}={\mathbb{R}}\oplus{\mathbb{R}}$ and we write
${\mathbb{R}}^{2}={\mathbb{R}}_{\mathrm{x}}\oplus{\mathbb{R}}_{\mathrm{y}}$ to
distinguish the two summands in this splitting. There is a corresponding
splitting

(2.1)
$H^{1}(S,\Sigma;{\mathbb{R}}^{2})=H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{x}})\oplus
H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{y}}).$

We also have a canonical restriction map
${\mathrm{Res}}:H^{1}(S,\Sigma;{\mathbb{R}}^{2})\to H^{1}(S;{\mathbb{R}}^{2})$
(given by restricting a cochain to absolute periods). Since ${\mathrm{Res}}$
is topologically defined, its kernel $\ker({\mathrm{Res}})$ is
$\operatorname{Mod}(S,\Sigma)$-invariant. Moreover, from our convention that
singular points are marked, the $\operatorname{Mod}(S,\Sigma)$-action on
$\ker({\mathrm{Res}})$ is trivial.

Let

(2.2)
$\mathfrak{R}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\ker({\mathrm{Res}})\
\ \text{ and }\ \
Z{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathfrak{R}\cap
H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{x}}).$

Since $H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{x}})$ and
$H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{y}})$ are naturally identified with each
other via their identification with $H^{1}(S,\Sigma;{\mathbb{R}})$, for each
$Z_{1}\subset Z$ we can define the space $\mathfrak{R}_{1}$ spanned by the two
copies of $Z_{1}$ in $H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{x}})$ and
$H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{y}})$ respectively. The space
$\mathfrak{R}_{1}$ is the complexification of $Z_{1}$. This terminology arises
from viewing $H^{1}(S,\Sigma;{\mathbb{R}}^{2})$ as
$H^{1}(S,\Sigma;\mathbb{C})$, a vector space over $\mathbb{C}$, viewing
$H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{x}})$ and
$H^{1}(S,\Sigma;{\mathbb{R}}_{\mathrm{y}})$ as the real and imaginary subspace
of this complex vector space. With this viewpoint, $\mathfrak{R}_{1}$ is the
$\mathbb{C}$-span of $Z_{1}$.

For any subspace $Z_{1}\subset\mathfrak{R}$, we can foliate the vector space
$H^{1}(S,\Sigma;{\mathbb{R}}^{2})$ by affine subspaces parallel to $Z_{1}$.
Pulling back this foliation using the period coordinate charts gives rise to a
foliation of $\bar{\mathcal{H}}_{\mathrm{m}}$. Since monodromy acts trivially
on $\mathfrak{R}$, this foliation descends to a well-defined foliation on
$\bar{\mathcal{H}}$. It is known (see e.g. [BSW, Prop. 4.1]) that the area of
a surface is constant on leaves of the Rel foliation, and thus the Rel
foliation and any of its subfoliations descends to a foliation of
$\mathcal{H}$. The foliation corresponding to $\mathfrak{R}$ (respectively, to
$Z$) is known as the Rel foliation (respectively, the real Rel foliation).

Because the $\operatorname{Mod}(S,\Sigma)$-monodromy action fixes all points
of $\mathfrak{R}$, the leaves of the Rel foliation, and any of its sub-
foliations, acquire a translation structure. In particular, they are equipped
with a natural measure.

For any $v\in Z$ we have a constant vector field, well-defined on
${\mathcal{H}}_{\mathrm{m}}$ and on ${\mathcal{H}}$, everywhere equal to $v$.
Integrating this vector field we get a partially defined real REL flow
(corresponding to $v$) $(t,q)\mapsto\mathrm{Rel}_{tv}(q)$; the flow may not be
defined for all time due to possible ‘collide of zeroes’. For every
$q\in{\mathcal{H}}$ it is defined for $t\in I_{q}$, where the domain of
definition $I_{q}=I_{q}(v)$ is an open subset of ${\mathbb{R}}$ which contains
$0$. The sets $I_{q}(v)$, are explicitly described in [BSW, Thm. 6.1]. Let
$\hat{\mathcal{H}}$ denote the set of surfaces in $\mathcal{H}$ with no
horizontal saddle connections. Then $I_{q}={\mathbb{R}}$ for all
$q\in\hat{\mathcal{H}}$.

If $q\in{\mathcal{H}},\,s\in{\mathbb{R}}$ and $\tau\in I_{q}$ then $\tau\in
I_{u_{s}q}$, and $\mathrm{Rel}_{\tau v}(u_{s}q)=u_{s}\mathrm{Rel}_{\tau
v}(q)$. Similarly, if $q\in{\mathcal{H}},\,t\in{\mathbb{R}}$ and $\tau\in
I_{q}$ then
$\tau^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}e^{t}\tau\in
I_{g_{t}q}$ and $\mathrm{Rel}_{\tau^{\prime}v}(g_{t}q)=g_{t}\mathrm{Rel}_{\tau
v}(q)$. In particular, since $P$ preserves $\hat{\mathcal{H}}$ and
$P=\\{g_{t}u_{s}:t,s\in{\mathbb{R}}\\}$, there is an action of $P\ltimes Z$ on
$\hat{\mathcal{H}}$, given by $(p,z).q=p\mathrm{Rel}_{z}(q)$.

##  3\. Preliminaries from ergodic theory

###  3.1. Ergodic decomposition

We will use the notation $\mathbb{G}\circlearrowright(X,\mu)$ to indicate that
$\mathbb{G}$ is a locally compact second countable group, $(X,\mathcal{B})$ is
a standard Borel space, and $\mu$ is a probability measure on $\mathcal{B}$
preserved by the $\mathbb{G}$-action. We say that
$\mathbb{G}\circlearrowright(Y,\nu)$ is a factor of $(X,\mu)$ if there is a
measurable $\mathbb{G}$-invariant conull subset $X_{0}\subset X$, and a
measurable map $T:X_{0}\to Y$ such that $T\circ g=g\circ T$ for all
$g\in\mathbb{G}$, and $\nu=T_{*}\mu$. In this situation we refer to $T$ as the
factor map. Given a factor map, there is a (unique up to nullsets) measure
disintegration $\mu=\int\mu_{y}\,d\nu(y)$, for a Borel mapping
$y\mapsto\mu_{y}$ from $Y$ to the space of Borel probability measures on $X$,
such that $\mu_{y}(T^{-1}(y))=1$ for $\nu$-a.e. $y$. Equivalently we can write
$\mu=\int_{x}\mu^{\prime}_{x}\,d\mu(x)$, where
$\mu^{\prime}_{x}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mu_{T(x)}$.
For a closed subgroup $H\subset\mathbb{G}$, we say that $\mu$ is $H$-ergodic
if any invariant set is null or conull. We have the following well-known
ergodic decomposition theorem:

######  Proposition 3.1.

Suppose $\mathbb{G}\circlearrowright(X,\mu)$, and $H$ is a closed subgroup of
$\mathbb{G}$. Then there is a factor of $H\circlearrowright(X,\mu)$, called
the space of ergodic components and denoted by $X/\\!\\!/H$, with the
following properties:

  * (i)

For $\nu$-a.e. $y\in X/\\!\\!/H$, $\mu_{y}$ is $H$-invariant and $H$-ergodic.

  * (ii)

$H$ acts trivially on $X/\\!\\!/H$.

  * (iii)

$H\circlearrowright(X,\mu)$ is ergodic if and only if $X/\\!\\!/H=\\{pt.\\}.$

  * (iv)

The properties (i)–(iii) uniquely determine the factor $X/\\!\\!/H$ up to
measurable isomorphism.

  * (v)

If $H\lhd\mathbb{G}$ then $\mathbb{G}\circlearrowright(X/\\!\\!/H,\nu)$.

###### Proof.

For (i) and (ii) see [Va, Thm. 4.4] (in the notation of [Va], these assertions
follow from the fact that $\beta$ is a map on points and is $H$-invariant).
Assertion (iii) is immediate from definitions and (iv) follows from [Va, Lemma
4.4]. For (v), one can argue using the uniqueness property (iv), and the fact
that the image of an $H$-invariant ergodic measures under any element
$g\in\mathbb{G}$ is also $H$-invariant and ergodic. ∎

######  Remark 3.2.

An action is called prime if it has no factors (besides the action itself, and
the trivial action on a point). The construction above shows that if
$H\lhd\mathbb{G}$, $\mathbb{G}^{\prime}$ is a subgroup of $\mathbb{G}$ so that
$\mathbb{G}^{\prime}\circlearrowright(X,\mu)$ is prime and
$H\circlearrowright(X,\mu)$ is not isomorphic to the trivial action, then
$H\circlearrowright(X,\mu)$ is ergodic. This is not the approach we will take
for proving Theorem 1.1.

###  3.2. Joinings

We recall some well-known facts about joinings, see [dlR] and references
therein. Let $\mathbb{G}\circlearrowright(X_{i},\mu_{i})$ for $i=1,2$. A
joining is a measure $\theta$ on $X_{1}\times X_{2}$, invariant under the
diagonal action of $\mathbb{G}$, such that $\pi_{i*}\theta=\mu_{i}$. A self-
joining is a joining in case $X_{1}=X_{2}$. If $(X_{i},\mu_{i})\to(Z,\nu)$ is
a joint factor then the relatively independent joining over $Z$ is the joining
$\int_{Z}(\mu_{1})_{z}\times(\mu_{2})_{z}\,d\nu(z)$, where
$\mu_{i}=\int_{Z}(\mu_{i})_{z}\,d\nu(z)$ is the disintegration of $\mu_{i}$.
In case $X_{1}=X_{2}=X$, and $Z=X/\\!\\!/H$ is the space of ergodic components
of the action of $H$ on $(X,\mu)$ as in Proposition 3.1, we obtain the
relatively independent self-joining over $X/\\!\\!/H$. This joining satisfies:

######  Proposition 3.3.

The following are equivalent:

  * •

$H\circlearrowright(X,\mu)$ is ergodic.

  * •

The relatively independent self-joining over $X/\\!\\!/H$ is $\mu\times\mu$.

We note two properties of this self-joining. We fix a topology on $X$ which
generates the $\sigma$-algebra, and denote by ${\rm supp}\,\mu$ the
topological support of $\mu$, i.e., the smallest closed set of full measure.

######  Proposition 3.4.

Let $\theta$ be the measure on $X\times X$ which is the relatively independent
self-joining over $X/\\!\\!/H$, for some $H$, and let $T:X\to X/\\!\\!/H$ be
the factor map. Then the following hold:

  * •

We have

(3.1) $\theta=\int_{X}\mu_{T(x)}\times\mu_{T(x)}\,d\mu(x).$

  * •

If $X={\rm supp}\,\mu$ then ${\rm supp}\,\theta$ contains the diagonal
$\Delta_{X}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{(x,x):x\in
X\\}$.

###### Proof.

Formula (3.1) is immediate from the definition of the relatively independent
self-joining over $X/\\!\\!/H$. Since each $\mu^{\prime}_{x}=\mu_{T(x)}$ is
$H$-invariant and ergodic, and $\mu^{\prime}_{x}(T^{-1}(T(x)))=1$, the set
$\\{x\in X:x\notin{\rm supp}\,\mu^{\prime}_{x}\\}$ is a nullset. From this,
and from (3.1) we obtain the second assertion. ∎

###  3.3. Ergodicity, mixing, and mixing of all orders

For $\mathbb{G}\circlearrowright(X,\mu)$, let $L^{2}_{0}(\mu)$ denote the
Hilbert space of $L^{2}$-functions on $(X,\mu)$ of integral zero, and let
$k\geq 2$. The action is called $k$-mixing if for any $f_{1},\ldots,f_{k}\in
L^{2}_{0}(\mu)$ and for any $k-1$ sequences
$\left(g^{(i)}_{n}\right)_{n\in{\mathbb{N}}}\in\mathbb{G},\ i=1,\ldots,k-1$,
for which all of the sequences

$\left(g^{(i)}_{n}\right)_{n\in{\mathbb{N}}}\ \ (1\leq i\leq k-1)\ \ \ \text{
and }\left(g^{(i)}_{n}(g^{(j)}_{n})^{-1}\right)_{n\in{\mathbb{N}}}\ \ (1\leq
i<j\leq k-1)$

eventually leave every compact subset of $\mathbb{G}$, we have

$\int_{X}f_{1}\left(g^{(1)}_{n}x\right)\cdots
f_{k-1}\left(g_{n}^{(k-1)}x\right)f_{k}(x)\,d\mu(x)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\prod_{i=1}^{k}\int_{X}f_{i}\,d\mu.$

We say that the action is mixing if it is 2-mixing, and mixing of all orders
if it is mixing of order $k$ for any $k\geq 2$. It is easy to check that
mixing implies ergodicity of any unbounded subgroup of $\mathbb{G}$. We have
the following:

######  Proposition 3.5.

Let $Z_{0}\cong{\mathbb{R}}$ and let $\\{g_{t}\\}$ be a one-parameter group
acting on $Z_{0}$ by dilations, i.e., for all $v\in Z_{0}$ and
$t\in{\mathbb{R}}$ we have $g_{t}v=e^{\lambda t}v$ for some $\lambda\neq 0$.
Let $F=\\{g_{t}\\}\ltimes Z_{0}$ and let $F\circlearrowright(X,\mu)$ be a
probability space. The following are equivalent:

  1. (a)

the restricted flow $Z_{0}\circlearrowright(X,\mu)$ is ergodic;

  2. (b)

the restricted flow $Z_{0}\circlearrowright(X,\mu)$ is mixing of all orders;

  3. (c)

the restricted flow $Z_{0}\circlearrowright(X,\mu)$ is mixing;

  4. (d)

any nontrivial element of $Z_{0}$ acts ergodically.

######  Remark 3.6.

The group $F$ appearing in Proposition 3.5 is isomorphic as a Lie group to the
subgroup $P$ of upper triangular matrices in $G$, but in our application we
will use it for the group generated by a one-parameter real Rel flow $Z_{0}$
and the diagonal flow $\\{g_{t}\\}$.

###### Proof.

Clearly b $\implies$ c $\implies$ d $\implies$ a. We assume that the
$Z_{0}$-flow is ergodic. To see that it is mixing, it is enough by [P, Chap.
2, Prop. 5.9] to prove that it has countable Lebesgue spectrum, and for this,
use [KT, Prop. 1.23 & Prop. 2.2]. The proof of mixing of all orders follows
verbatim from an argument of Mozes [Mo], for mixing actions of Lie groups
which are ‘Ad-proper’. Since our group $F$ is not Ad-proper, we cannot cite
[Mo] directly, so we sketch the proof. For notational convenience we deduce
3-fold mixing from mixing (the proof that ‘$k$-fold mixing $\implies k+1$-fold
mixing’, for $k\geq 3$, is identical but requires more cumbersome notation).

We use additive notation in the group $Z_{0}$, and denote the action of
$Z_{0}$ on $X$ by $(z,x)\mapsto z.x$. Let
$\left(b_{n}\right)_{n\in{\mathbb{N}}}$ and
$\left(c_{n}\right)_{n\in{\mathbb{N}}}$ be sequences in $Z_{0}$ such that each
of the sequences
$\left(b_{n}\right)_{n\in{\mathbb{N}}},\,\left(c_{n}\right)_{n\in{\mathbb{N}}},\,\left(b_{n}+c_{n}\right)_{n\in{\mathbb{N}}}$
eventually leaves every compact set, and let $f_{1},f_{2},f_{3}$ be in
$L^{2}_{0}(\mu)$. We need to prove that

$\int_{X}f_{1}(x)f_{2}(b_{n}.x)f_{3}((b_{n}+c_{n}).x)\,d\mu(x)\stackrel{{\scriptstyle
n\to\infty}}{{\longrightarrow}}\int_{X}f_{1}\,d\mu\,\int_{X}f_{2}\,d\mu\,\int_{X}f_{3}\,d\mu.$

For each $n$, define a measure $\mu_{n}$ on
$X^{3}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}X\times X\times X$ by

$\int_{X^{3}}f\,d\mu_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\int_{X}f(x,b_{n}.x,(b_{n}+c_{n}).x)\,d\mu(x),\
\ \forall f\in C_{c}(X^{3}).$

That is, $\mu_{n}$ is the pushforward of the diagonal measure on $X^{3}$ by
the sequence $(0,b_{n},b_{n}+c_{n})$. It is easy to see that 3-mixing is
equivalent to the fact that the weak-* limit of $\mu_{n}$ is the measure
$\mu^{3}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mu\times\mu\times\mu$.
The group $F^{3}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}F\times
F\times F$ acts on $X^{3}$ by acting separately on each component, and as in
[Mo], since $Z_{0}$ is mixing, it suffices to show that any measure $\nu$ on
$X^{3}$ which is a weak-* limit of a subsequence of
$\left(\mu_{n}\right)_{n\in{\mathbb{N}}}$, is invariant under
$(0,u,v)\in{\mathbb{R}}^{3}\subset F^{3},$ for some
$(u,v)\in{\mathbb{R}}^{2}\smallsetminus(0,0)$. We claim that for any
$s\in{\mathbb{R}}$ the measure $\mu_{n}$ is invariant under

$h_{n}(s){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left(g_{s},b_{n}\cdot
g_{s}\cdot(-b_{n}),(b_{n}+c_{n})\cdot g_{s}\cdot(-b_{n}-c_{n})\right),$

where the multiplication is in the group $F^{3}$. Indeed, since $\mu$ is
$\\{g_{s}\\}$-invariant,

$\int_{X^{3}}f\,d\mu_{n}=\int_{X}f\left(g_{s}x,b_{n}.(g_{s}x),(b_{n}+c_{n}).(g_{s}x)\right)\,d\mu(x),$

and

$h_{n}(s)\cdot(\mathrm{id}_{F},b_{n},b_{n}+c_{n})=(g_{s},b_{n}\cdot
g_{s},(b_{n}+c_{n})\cdot g_{s}).$

That is, applying $h_{n}(s)$ changes one description of $\mu_{n}$ to another.

We embed $F$ as a multiplicative group of matrices in
$\operatorname{GL}_{2}({\mathbb{R}})$ and let $d_{F}$ be the metric on $F$
induced by some norm on $\operatorname{GL}_{2}({\mathbb{R}})$. By a
straightforward computation we have

$h_{n}(s)=\left(g_{s},\ (1-e^{\lambda s})b_{n}\cdot g_{s},\ (1-e^{\lambda
s})(b_{n}+c_{n})\cdot g_{s}\right),$

and $d_{F}(\mathrm{id}_{F},h_{n}(s_{n}))$ is a continuous function of $s$
which goes to $0$ as $s\to 0$ and for any fixed $s>0$, increases to infinity
as $n\to\infty$. Therefore we can choose $s_{n}\to 0$ so that
$d_{F}(\mathrm{id}_{F},h_{n}(s_{n}))=1$ for all large enough $n$. As in [Mo],
$\nu$ is invariant under some subsequential limit of $h_{n}(s_{n})$ which is
of the form $(0,u,v)$ for some $(u,v)\in{\mathbb{R}}^{2}\smallsetminus(0,0)$.
This establishes our sufficient condition. ∎

##  4\. The relatively independent self-joining for a Rel flow

Recall that $\hat{{\mathcal{L}}}\subset{{\mathcal{L}}}$ is the set of surfaces
without horizontal saddle connections, and this is a $P$-invariant set of full
measure with respect to $m_{{{\mathcal{L}}}}$. We can combine the product
action of $Z_{{\mathcal{L}}}\times Z_{{\mathcal{L}}}$ on
$\hat{{\mathcal{L}}}\times\hat{{\mathcal{L}}}$ with the diagonal action of $P$
to obtain an action of the semi-direct product
$P\ltimes(Z_{{\mathcal{L}}}\times Z_{{\mathcal{L}}})$ on
$\hat{{\mathcal{L}}}\times\hat{{\mathcal{L}}}$. Since
$\hat{{\mathcal{L}}}\subset{{\mathcal{L}}}$ is of full measure, and the
arguments of this section involve passing to sets of full measure, in the
remainder of this section we will ignore the distinction between
${{\mathcal{L}}}$ and $\hat{{\mathcal{L}}}$.

######  Proposition 4.1.

Let $Z\subset Z_{{\mathcal{L}}}$ be a closed connected subgroup. If $\theta$
is an invariant probability measure for an action of the semidirect product
$P\ltimes(Z\times Z)$ on ${{\mathcal{L}}}\times{{\mathcal{L}}}$, then any
$f\in L^{2}(\theta)$ which is $\\{g_{t}\\}$-invariant is also $Z\times
Z$-invariant.

###### Proof.

For any $z\in Z\times Z$, $g_{t}zg_{-t}\to_{t\to-\infty}0$. So the claim
follows from the Mautner phenomenon, see e.g. [EW, Prop 11.18]. ∎

######  Proposition 4.2.

Let $({{\mathcal{L}}},m_{{\mathcal{L}}})$ be a $G$-orbit-closure with a fully
supported $P$-invariant ergodic measure, let $Z\subset Z_{{\mathcal{L}}}$ be a
connected closed subgroup, and let $\theta$ on
${{\mathcal{L}}}\times{{\mathcal{L}}}$ be the relatively independent joining
over ${{\mathcal{L}}}/\\!\\!/Z$. Then $\theta$ is $P$-invariant and
$\\{g_{t}\\}$-ergodic (and hence $P$-ergodic). Also
$\Delta_{{{\mathcal{L}}}}\subset{\rm supp}\,\theta$.

As we will see in §5, under the conditions of the Proposition,
$m_{{\mathcal{L}}}$ is the so-called ‘flat measure’ on ${{\mathcal{L}}}$.

###### Proof.

Let $\pi:{\mathcal{L}}\times{\mathcal{L}}\to{{\mathcal{L}}}$ be the projection
onto the first factor, and let $\nu=\pi_{*}\theta.$ For each
$x\in{{\mathcal{L}}}$, let
$\Omega_{x}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\pi^{-1}(x)=\\{x\\}\times{{\mathcal{L}}}$
be the fiber, and let $\theta_{x}$ be the fiber measure appearing in the
disintegration $\theta=\int_{{{\mathcal{L}}}}\theta_{x}\,d\nu(x)$. Then $Z$
acts on $\Omega_{x}$ via the second factor in $Z\times Z$, and $\theta_{x}$ is
$Z$-invariant and ergodic by the definition of the ergodic decomposition.

It follows from Proposition 3.1(v) that $\theta$ is $P$-invariant. To prove
ergodicity, let $f\in L^{2}({{\mathcal{L}}}\times{{\mathcal{L}}},\theta)$ be a
$P$-invariant function. By Proposition 4.1, $f$ is $Z\times Z$-invariant. For
each $x\in{{\mathcal{L}}}$, let
$f_{x}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}f|_{\Omega_{x}}$. There
is ${{\mathcal{L}}}_{0}\subset{{\mathcal{L}}}$ such that
$m_{{{\mathcal{L}}}}({{\mathcal{L}}}_{0})=1$ and for every
$x\in{{\mathcal{L}}}_{0}$, $f_{x}$ belongs to $L^{2}(\Omega_{x},\theta_{x})$
and is $Z$-invariant. Hence, by ergodicity, there is
$\bar{f}:{{\mathcal{L}}}_{0}\to{\mathbb{R}}$ such that for every
$x\in{{\mathcal{L}}}_{0}$, $\bar{f}(x)$ is the $\theta_{x}$-almost-sure value
of $f_{x}$. Since $f$ is $P$-invariant for the diagonal action of $P$,
$\bar{f}$ is $P$-invariant for the action of $P$ on ${{\mathcal{L}}}$. By
ergodicity of $P\circlearrowright({{\mathcal{L}}},m_{{\mathcal{L}}})$,
$\bar{f}$ is $\nu$-a.e. constant, and thus $f$ is $\theta$-a.e. constant.

The last assertion follows from Proposition 3.4. ∎

##  5\. An upgraded magic wand theorem

The celebrated ‘magic wand’ Theorem of Eskin and Mirzakhani [EM], and ensuing
work of Eskin, Mirzakhani and Mohammadi [EMM], classified $P$\- and
$G$-invariant probability measures and orbit-closures on strata of translation
surfaces. These results can be summarized as follows (see [EM, Defs. 1.1 &
1.2, Thms. 1.4 & 1.5]):

######  Theorem 5.1.

Let
$\mathcal{H},\,\mathcal{H}_{\mathrm{m}},\,\bar{\mathcal{H}},\,\bar{\mathcal{H}}_{\mathrm{m}}$
be as in §2.1. Any $P$-invariant ergodic probability measure $m$ has the
following properties:

  * (i)

It is $G$-invariant.

  * (ii)

There is a complex-affine manifold $\mathcal{N}$ and a proper immersion
$\varphi:\mathcal{N}\to\bar{\mathcal{H}}$ such that

${{\mathcal{L}}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}{\rm
supp}\,m={\mathcal{H}}\cap\varphi(\mathcal{N}).$

  * (iii)

There is an open $G$-invariant subset $U\subset\bar{\mathcal{H}}$ satisfying
$m(U)=1$, and for any $x\in U\cap{{\mathcal{L}}}$ there is an open set $V$
containing $x$ such that $V$ is evenly covered by
$\mathcal{V}\subset{\mathcal{H}}_{\mathrm{m}}$ under the map
$\pi:\bar{{\mathcal{H}}}_{\mathrm{m}}\to\bar{{\mathcal{H}}}$, and
$\psi{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{dev}\circ(\pi|_{\mathcal{V}})^{-1}\circ\varphi$
coincides on its domain with a $\mathbb{C}$-linear map, with real
coefficients.

  * (iv)

The subspace
$W{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Im}(\psi)$ is
symplectic, and the measure $m$ is obtained via the cone construction from the
Lebesgue measure on $W$.

  * (v)

The complement ${{\mathcal{L}}}\smallsetminus U$ is a finite union of supports
of measures satisfying properties (i)–(iv), for which the manifolds
$\mathcal{N}^{\prime}$ appearing in (ii) satisfy
$\dim\mathcal{N}^{\prime}<\dim\mathcal{N}.$

Any orbit-closure for the $P$-action is a set ${{\mathcal{L}}}$ as above.

We will refer to ${{\mathcal{L}}}$ as an orbit-closure and to
$m=m_{{\mathcal{L}}}$ as a flat measure on ${{\mathcal{L}}}$. Orbit-closures
are referred to as affine invariant manifolds and also as invariant
subvarieties. The use of an evenly covered neighborhood in item (iii) is a
standard approach for defining period coordinates (see e.g. [MS]). We refer to
[Wr1] for a survey containing more information on orbit-closures.

In a forthcoming work of Brown, Eskin, Filip and Rodriguez-Hertz, the same
conclusion is obtained for the diagonal actions of $P$ and $G$ on a product of
strata ${\mathcal{H}}\times{\mathcal{H}}^{\prime}$. Namely, the following is
shown:

######  Theorem 5.2.

Let ${\mathcal{H}},{\mathcal{H}}^{\prime}$ be strata of translation surfaces,
and let $P$ and $G$ act on ${\mathcal{H}}\times{\mathcal{H}}^{\prime}$ via
their diagonal embeddings in $G\times G$. Then all of the conclusions of
Theorem 5.1 hold for this action (with
$\bar{\mathcal{H}}\times\bar{\mathcal{H}}^{\prime}$ replacing
$\bar{\mathcal{H}}$).

##  6\. Proof of main result

Using Theorem 5.2 and further work of Wright [Wr2], we can prove our main
result.

###### Proof of Theorem 1.2.

Let $Z_{0}={\rm span}_{{\mathbb{R}}}(z_{0})$ be a one-dimensional connected
real Rel subgroup. Assume that (1) fails, so that the action of $Z_{0}$ on
$\left({{\mathcal{L}}},m_{\mathcal{L}}\right)$ is not mixing of all orders.
Then, by Proposition 3.5 it is not ergodic. Let $\theta$ be the relatively
independent self-joining over ${{\mathcal{L}}}/\\!\\!/Z_{0}$. Applying
Propositions 3.3 and 3.4 we have that $\theta\neq m_{{{\mathcal{L}}}}\times
m_{{{\mathcal{L}}}}$ and $\Delta_{{{\mathcal{L}}}}\subset{\rm supp}\,\theta$.
Applying Proposition 4.2 and Theorem 5.2, we have that there is a
$G$-invariant open subset $U$ of full $\theta$-measure such that $U\cap{\rm
supp}\,\theta$ is the isomorphic image of an affine complex-linear manifold
whose dimension is strictly smaller than $2\dim\bar{\mathcal{H}}$, and
$\theta$ is obtained from Lebesgue measure on this complex-linear manifold by
the cone construction.

We claim that the set

$U_{1}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{q\in\mathcal{H}:(q,q)\in
U\\}$

is of full measure for $(\pi_{1})_{*}\theta$, where
$\pi_{1}:{{\mathcal{L}}}\times{{\mathcal{L}}}\to{{\mathcal{L}}}$ is the
projection onto the first factor. Indeed, the measure $\theta$ is invariant
under $Z_{0}\times\\{\mathrm{Id}\\}$, and hence so is its support. Since
$Z_{0}$ acts by homeomorphisms where defined, and using property (v) in
Theorems 5.1 and 5.2, we have that the set $U$ is also
$Z_{0}\times\\{\mathrm{Id}\\}$-invariant. Thus for any $Z_{0}$-ergodic
measure, it is either null or conull. Thus if $q\notin U_{1}$ and $q$ is
generic for the measure $\mu_{T(q)}$ appearing in (3.1), $\mu_{T(q)}$ assigns
measure zero to $U$. If this were to happen for a positive measure of $q$ it
would follow from (3.1) that $U$ does not have full measure for $\theta$.

For $q\in U_{1}$, let $N_{q}$ denote the connected component of
$U\cap\pi_{1}^{-1}(q)\cap{\rm supp}\,\theta$ containing $(q,q)$. Since the
fibers $\pi_{1}^{-1}(q)$ are also affine submanifolds of
${{\mathcal{L}}}\times{{\mathcal{L}}}$, we have that the $N_{q}$ are affine
submanifolds contained in $\pi_{1}^{-1}(q)\cong{{\mathcal{L}}}$, so we can
identify them with invariant submanifolds in ${{\mathcal{L}}}$ (which we
continue to denote by $N_{q})$. With this notation we have $q\in N_{q}$.

The mapping $q\mapsto T(N_{q})$ is locally constant; that is, letting
$V\subset\bar{\mathcal{H}}$ and
$\mathcal{V}\subset\bar{\mathcal{H}}_{\mathrm{m}}$ be open sets such that
$\pi|_{\mathcal{V}}:\mathcal{V}\to V$ is a homeomorphism and $q\in V$, the map
$q\mapsto\mathrm{dev}\circ\pi|_{\mathcal{V}}^{-1}(q)$ sends a neighborhood of
$q$ in $N_{q}$ to an affine subspace $W_{q}$ of
$H^{1}(S,\Sigma;{\mathbb{R}}^{2})$, and the corresponding linear spaces
$W_{q}-W_{q}$ are the same for all $q\in V$. Since $m_{{{\mathcal{L}}}}\times
m_{{{\mathcal{L}}}}$ is the unique $P$-invariant ergodic measure on
${{\mathcal{L}}}\times{{\mathcal{L}}}$ of full support, we have $\dim
N_{q}<\dim{{\mathcal{L}}}$ for every $q\in U_{1}$.

Let $\bar{N}_{q}$ denote the set of surfaces (not necessarily of area one)
which are obtained by rescaling surfaces in $N_{q}$, and let

$\mathfrak{N}_{q}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}T_{q}(\bar{N}_{q})$

(the tangent space to $\bar{N}_{q}$ at $q$, thought of as a subset of the
tangent space $T_{q}(\bar{{\mathcal{L}}})$). The assignment
$q\mapsto\mathfrak{N}_{q}$ defines a proper flat sub-bundle of the tangent
bundle $T(\bar{{\mathcal{L}}})$. Flat sub-bundles of $T(\bar{{\mathcal{L}}})$
were classified in [Wr2]. According to [Wr2, Thm. 5.1],
$\mathfrak{N}_{q}\subset\mathfrak{R}_{{{\mathcal{L}}}}$ for each $q$, and
$\mathfrak{N}_{q}$ is a complex linear subspace which is locally constant.
Since $\mathfrak{R}_{{{\mathcal{L}}}}$ is acted on trivially by monodromy, we
in fact have that $\mathfrak{N}_{q}$ is independent of $q$, and we denote it
by $\mathfrak{R}$. The leaves $\mathfrak{R}(q)$ are contained in $\bar{N}_{q}$
for each $q$, and of the same dimension. That is, $\mathfrak{R}(q)$ is the
connected component of $\bar{N}_{q}$ containing $q$. Since Rel deformations do
not affect the area of the surface, we see that $\bar{N}_{q}=N_{q}$. In
particular $\mathfrak{R}(q)$ is closed for each $q$.

By Proposition 3.4, for a.e. $q$, $N_{q}$ is the support of the ergodic
component $(m_{{{\mathcal{L}}}})_{q}$, and in particular

$(m_{{{\mathcal{L}}}})_{q}(N_{q})<\infty,\ \ \text{ for a.e. }q.$

Since $Z_{0}$ acts ergodically with respect to $(m_{{{\mathcal{L}}}})_{q}$, we
have that almost surely $N_{q}=\mathfrak{R}(q)$. Since the measure
$(m_{{{\mathcal{L}}}})_{q}$ is affine in charts, it is a scalar multiple of
the translation-invariant measure on $\mathfrak{R}(q)$, and thus the volume
$V_{q}$ of $\mathfrak{R}(q)$ (with respect to its translation-invariant
measure) is almost surely finite. It is clear that the function $q\mapsto
V_{q}$ is $U$-invariant, and by ergodicity, it is constant almost surely. ∎

######  Remark 6.1.

We note that the above argument works under much weaker conclusions than those
given in Theorem 5.2. Indeed, in the first step of the argument, Theorem 5.2
was used simply to extract a $G$-invariant assignment $q\mapsto N_{q}$, where
$N_{q}$ is a subspace of $T_{q}({{\mathcal{L}}})$, which is proper if $\theta$
is not the product joining. A fundamental fact about such $G$-invariant
assignments is that they are very restricted – besides [Wr2], see [EFW] and
[Fi]. In particular, [Fi] gives strong restrictions on assignments that are
only assumed to be defined almost everywhere and measurable.

##  7\. A topological condition for Rel ergodicity

Let $Z_{0}\subset Z$ be a subspace. We say that a translation surface $x$ is
$Z_{0}$-stably periodic if it can be presented as a finite union of horizontal
cylinders and the $Z_{0}$-orbit of $x$ is well defined. Recall that a
horizontal separatrix is a horizontal leaf whose closure contains at least one
singularity, and it is a horizontal saddle connection if its closure contains
two singularities. Then the condition of being $Z_{0}$-stably periodic is
equivalent to requiring that all horizontal separatrices starting at singular
points are on horizontal saddle connections, and $Z_{0}$ preserves the
holonomy of every horizontal saddle connection on $x$. In case $Z=Z_{0}$ is
the full real rel group, we say that $x$ is fully stably periodic. This is
equivalent to saying that all horizontal separatrices starting at singular
points are on saddle connections, and all horizontal saddle connections start
and end at the same singularity. In particular, for any cylinder $C$ on a
fully stably periodic surface, each boundary component of $C$ is made of
saddle connections starting and ending at the same singular point $\xi$; we
say that the boundary component only sees singularity $\xi$. For more
information on the real Rel action on surfaces which are horizontally
completely periodic, see [HW, §6.1].

######  Proposition 7.1.

Suppose $x$ is a surface which is $Z_{0}$-stably periodic, and $v\in Z_{0}$
moves two singularities $p$ and $q$ with respect to each other. Suppose that
$x$ contains two cylinders $C_{1}$ and $C_{2}$ that both only see singularity
$p$ on one boundary component and only see singularity $q$ on another boundary
component. Finally suppose the circumferences $c_{1},c_{2}$ of these cylinders
satisfy $\frac{c_{1}}{c_{2}}\notin{\mathbb{Q}}$. Then Case (2) of Theorem 1.1
does not hold for $x$.

###### Proof.

Since $\frac{c_{1}}{c_{2}}\notin{\mathbb{Q}}$, the trajectory
$\\{{\mathrm{Rel}}_{tv}(x):t\in{\mathbb{R}}\\}$ is not closed, let
$\mathcal{L}$ denote its closure. We claim that the tangent space to
$\mathcal{L}$ is not contained in $Z$. Let $\sigma_{1}$ denote a saddle
connection from $p$ to $q$ in $C_{1}$ and let $\sigma_{2}$ denote a saddle
connection from $q$ to $p$ in $C_{2}$. Let $\sigma$ be the concatenation. Then
$\sigma$ represents an absolute homology class because it goes from $p$ back
to $p$, and it is nontrivial because the vertical component of its holonomy on
$x$ is nonzero. If we consider the restriction of the rel-action to $C_{1}\cup
C_{2}$ then it only affects the twist parameters, which is a 2-dimensional
space. This space can be generated by the horizontal holonomy of $\sigma_{1}$
and the horizontal holonomy of $\sigma_{2}$. Since
$\frac{c_{1}}{c_{2}}\notin{\mathbb{Q}}$, this restricted action does not give
a closed orbit. So the tangent space to $\mathcal{L}$ contains directions,
which continuously affect the holonomy of $\sigma$. Since $\sigma$ is an
absolute period, we see that the tangent space to $\mathcal{L}$ is not
contained in $Z$. ∎

##  8\. Checking the condition for strata

Let $\mathcal{H}=\mathcal{H}(a_{1},\ldots,a_{k})$ and for
$i,j\in\\{1,\ldots,k\\}$, let $\xi_{i},\xi_{j}$ be the corresponding singular
points of a surface in $\mathcal{H}$. Let $z\in\mathfrak{R}$ be a Rel
cohomology class. We say that $z$ moves $\xi_{i},\xi_{j}$ with respect to each
other if for some (equivalently, every) $\alpha\in H_{1}(S,\Sigma)$
represented by a path starting at $\xi_{i}$ and ending at $\xi_{j}$, we have
$z(\alpha)\neq 0$. Below when we discuss a stratum
$\mathcal{H}(a_{1},\ldots,a_{k})$ we allow $a_{i}=0$, that is we allow marked
points. We call points with cone angle $2\pi$ (that is, with $a=0$) removable
singularities, and otherwise we call them non-removable. The following result,
which clearly implies Theorem 1.1, allows strata with removable singularities.

######  Theorem 8.1.

Let ${\mathcal{H}}$ be a connected component of a stratum
${\mathcal{H}}(a_{1},\ldots,a_{k})$. Let $m_{\mathcal{H}}$ be the Masur-Veech
measure on ${\mathcal{H}}$, let $Z$ be the corresponding real Rel foliation,
and let $Z_{0}\subset Z$ be a one-dimensional connected subgroup of $Z$.
Suppose that there are $1\leq i<j\leq k$ with corresponding singular points
$\xi_{i},\xi_{j}$, such that $a_{i}>0,\,a_{j}>0$ and such that some element of
$Z_{0}$ moves $\xi_{i},\xi_{j}$ with respect to each other. Then the
$Z_{0}$-flow on $({\mathcal{H}},m_{{\mathcal{H}}})$ is mixing of all orders
(and in particular, ergodic).

Clearly, Theorem 8.1 follows from Theorem 1.2, Proposition 7.1, and the
following result.

######  Proposition 8.2.

Let $\mathcal{H}\subset\mathcal{H}(a_{1},\ldots,a_{k})$ be a connected
component of a stratum of translation surfaces with at least two non-removable
singular points. If $p\neq q$ is any pair of non-removable singularities then
there exists $M\in\mathcal{H}$, which has cylinders $C_{1},C_{2}$ with
circumferences $c_{1},c_{2}$ so that

  1. (1)

$M$ is fully stably periodic.

  2. (2)

$\frac{c_{1}}{c_{2}}\notin\mathbb{Q}$.

  3. (3)

Both $C_{1}$ and $C_{2}$ only see singularity $p$ on one boundary component
and only see singularity $q$ on the other boundary component.

For the proof of Proposition 8.2 we will also need the following:

######  Proposition 8.3.

Let $\mathcal{H}=\mathcal{H}(a_{1},\ldots,a_{k})$ be a stratum of translation
surfaces with at least two singular points (that is $k\geq 2$). If $p\neq q$
is any pair of distinct singularities (possibly removable), then there exists
$M\in\mathcal{H}$, so that $M$ is fully stably periodic and there exists a
cylinder on $M$ that only sees singularity $p$ on one boundary component, and
only sees singularity $q$ on the other boundary component.

Propositions 8.2 and 8.3 will both be proved by induction, after some
preparations.

######  Lemma 8.4 (The basic surgery – gluing in a torus).

Let $\mathcal{H}=\mathcal{H}(b_{1},\ldots,b_{\ell})$ be a stratum of
translation surfaces, and let $M\in\mathcal{H}$, with singularities labeled by
$\xi_{1},\ldots,\xi_{\ell}$, so that the order of $\xi_{i}$ is $b_{i}$.
Suppose $M$ has a horizontal cylinder $C$, with circumference $c$, where one
boundary component is made of saddle connections that begin and end at
$\xi_{i}$, and the other is made of saddle connections that begin and end at
$\xi_{j}$, where $b_{i}\geq 0$ and $b_{j}\geq 0$ (so that $\xi_{i},\xi_{j}$
might be removable). Then for all $w>0$ there exists
$M^{\prime}\in\mathcal{H}(b_{1},\ldots,b_{i}+1,\ldots,b_{j}+1,\ldots,b_{\ell})$,
with singularities labeled $\xi^{\prime}_{1},\ldots,\xi^{\prime}_{\ell}$,
which has two horizontal cylinders $C^{\prime}_{1}$, $C^{\prime}_{2}$, where
$C^{\prime}_{1}$ has circumference $c+w$ and $C^{\prime}_{2}$ has
circumference $w$. The complements $M\smallsetminus C$ and
$M^{\prime}\smallsetminus(C_{1}\cup C_{2})$ are isometric, by an isometry
mapping $\xi^{\prime}_{i}$ to $\xi_{i}$ for all $i$. The cylinders $C_{1}$ and
$C_{2}$ only see singularity $\xi^{\prime}_{i}$ on one boundary component, and
$\xi^{\prime}_{j}$ on another. Moreover, if $M$ is fully stably periodic then
so is $M^{\prime}$.

###### Proof.

It will be easier to follow the proof while consulting Figures 1 (before) and
2 (after).

$\triangle$$\triangle$$c$ Figure 1. The surface $M$ has a cylinder of
circumference $c$, and its boundary components see only the singularities
$\xi_{i}$ and $\xi_{j}$ (denoted by $\circ$ and $\bullet$). The edges not
labeled by $\triangle$ are connected to $M\smallsetminus C$.
$c$$w$//$\triangle$$\triangle$$\square$$\square$ Figure 2. To obtain
$M^{\prime}$ from $M$, glue in a torus (rectangle on the right). This
transforms $C$ into a cylinder $C^{\prime}_{1}$ of circumference $c+w$, and
adds a horizontal cylinder $C^{\prime}_{2}$ of circumference $w$. Edges not
labeled by $\triangle$, $\square$, / or the color green are attached to
$M^{\prime}\smallsetminus(C^{\prime}_{1}\cup C^{\prime}_{2})$.

Given a polygonal presentation for $M$, we give a polygonal presentation for
$M^{\prime}$. Let $M$ be a polygon representation for $M$ in which the
cylinder $C$ is represented by a parallelogram $P$ (in Figure 1, the large
rectangle in the center of the presentation), with two horizontal sides of
length $c$, non-horizontal sides identified to each other, and the singular
points $\xi_{i}$, $\xi_{j}$ on adjacent corners of $P$. Thus the non-
horizontal sides of $P$ represent a saddle connection $\sigma$ on $M$
connecting $\xi_{i}$ to $\xi_{j}$. We consider the two non-horizontal sides of
$P$ as distinct and label them by $\sigma_{1},\sigma_{2}$. Let $P^{\prime}$ be
a parallelogram with sides parallel to those of $P$, where the horizontal
sides have length $w$ and the nonhorizontal sides are longer than the ones on
$P$ (in Figure 2, $P^{\prime}$ is to the right of $P$).

Label the two horizontal sides of $P^{\prime}$ by $h^{\prime}_{1}$ and
$h^{\prime}_{2}$, and identify them by a translation. Partition the non-
horizontal sides of $P^{\prime}$ into two segments. The segments
$\sigma^{\prime}_{1},\sigma^{\prime}_{2}$ are parallel to each other and have
the same length as $\sigma_{1},\sigma_{2}$, and start at a corner of $P$. The
segments $\gamma^{\prime}_{1},\gamma^{\prime}_{2}$ comprise the remainder of
the non-horizontal sides of $P^{\prime}$ (and in particular, have the same
length). Identify $\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ by a
translation, and identify $\sigma^{\prime}_{1},\sigma^{\prime}_{2}$ to
$\sigma_{1},\sigma_{2}$ by a translation so that each $\sigma^{\prime}_{i}$ is
attached to the $\sigma_{j}$ with the opposite orientation. Let $M^{\prime}$
be the translation surface corresponding to this presentation. It is clear
that $M^{\prime}$ has the required properties.

∎

###### Proof of Proposition 8.3.

The proof is by induction on $\sum a_{i}$.

Base of induction: The base case is the stratum ${\mathcal{H}}(a_{1},0^{s})$,
that is, one singular point (removable or non-removable) of order $a_{1}$, and
some number $s\geq 1$ of removable singular points. In this case we take a
surface in ${\mathcal{H}}(a_{1})$ which is made of one horizontal cylinder. We
label the singular point by $\xi_{1}$ and place additional removable singular
points $\xi_{2},\ldots,\xi_{s+1}$ in the interior of the cylinder, at
different heights (so that the resulting surface has no horizontal saddle
connections between distinct singularities) and so that $\xi_{i}$ and
$\xi_{j}$ are on opposite sides of a cylinder.

Inductive step: Suppose
${\mathcal{H}}^{\prime}={\mathcal{H}}(a_{1},\ldots,a_{k})$ is our stratum,
where at least two of the singularities are non-removable. Let
$p^{\prime},q^{\prime}$ be labels of singular points for surfaces in
${\mathcal{H}}^{\prime}$, corresponding to indices $i\neq j$. To simplify
notation assume $i=1,j=2$. There are three cases to consider: $a_{i}=a_{j}=0$,
or one of $a_{i},a_{j}$ are positive, or both are positive.

If $a_{i}=a_{j}=0$ then by assumption $k\geq 4$. We take a cylinder $C$ on a
fully stably completely periodic surface $M$ in
${\mathcal{H}}={\mathcal{H}}(a_{1},\ldots,\hat{a}_{i},\ldots,\hat{a}_{j},\ldots,a_{k})$.
The notation $\hat{a}_{i}$ means that the symbol should be ignored; that is on
a stratum of the same genus with $k-2\geq 2$ singular points obtained by
removing two removable singular points. We place two singular points marked
$p^{\prime},q^{\prime}$ in the interior of $C$ at different heights. If
$a_{i}>0$ and $a_{j}=0$ is zero we take a fully stably periodic surface $M$ in
${\mathcal{H}}(a_{1},\ldots,a_{i}-1,\ldots,\hat{a}_{j},\ldots,a_{k})$, find a
cylinder $C$ on $M$ whose boundary component is made of saddle connections
starting and ending at $\xi_{i}$, place a marked point labeled $\xi_{j}$ in
the interior of $C$. If $a_{i}$ and $a_{j}$ are both positive we use the
induction hypothesis to find a surface
$M\in{\mathcal{H}}(a_{1},\ldots,a_{i}-1,\ldots,a_{j}-1,\ldots,a_{k})$ with a
cylinder whose boundary components see $\xi_{i}$ and $\xi_{j}$, and we perform
the surgery in Lemma 8.4 to this cylinder. ∎

######  Lemma 8.5 (Two surgeries involving genus two surfaces).

Let ${\mathcal{H}}=\mathcal{H}(b_{1},\ldots,b_{k})$ be a stratum of
translation surfaces and let $M\in{\mathcal{H}}$ have a horizontal cylinder
$C$, with circumference $c$. Let $p$ and $q$ be singular points with order
$b_{i},b_{j}$ respectively, such that one boundary component of $C$ only sees
singularity $p$ and the other only sees singularity $q$. Then for any
$w_{1},w_{2}>0$ there exists
$M^{\prime}\in{\mathcal{H}}^{\prime}=\mathcal{H}(b_{1},\ldots,b_{i}+2,\ldots,b_{j}+2,\ldots,b_{k})$
which has three cylinders $C_{1},C_{2},C_{3}$ with circumferences
$c+w_{1}+w_{2},w_{1}$ and $w_{2}$ respectively. The complements
$M\smallsetminus C$ and $M^{\prime}\smallsetminus(C_{1}\cup C_{2}\cup C_{3})$
are isometric by an isometry preserving the labels of singular points, and
$C_{1},C_{2},C_{3}$ all have one boundary component that sees only $p$, and
another that sees only $q$. Thus, if $M$ is fully stably periodic so is
$M^{\prime}$. Moreover, if the $b_{i}$ are all even, so that
${\mathcal{H}}^{\prime}$ has even and odd spin components, we can choose
$M^{\prime}$ to be in either the even or odd connected component.

###### Proof.

Once again we encourage the reader to consult Figures 3 and 4.

$c$$w^{\prime}$$w$//////$\triangle$$\triangle$$\square$$\square$$\nabla$$\nabla$
Figure 3. First option for $M^{\prime}$ in Lemma 8.5. Attaching the subsurface
on the right increases the genus by 2. Unlabeled edges are attached to
$M^{\prime}\smallsetminus(C_{1}\cup C_{2}\cup C_{3})$.
$c$$w^{\prime}$$w$//////$\triangle$$\triangle$$\nabla$$\nabla$$\square$$\square$
Figure 4. Second option for $M^{\prime}$, with a different spin.

In Lemma 8.4 we made a slit in $M$, running through $P$ from top to bottom,
and glued in a torus with a slit. In this case we make an identical slit, this
time gluing in a genus two surface with a slit. This surface is presented in
Figures 3 and 4 as made up of three rectangles. It is straightforward to check
that $M^{\prime}\in{\mathcal{H}}^{\prime}$ and that it has cylinders
satisfying the desired properties. It remains to check the final assertion
about the parity of the spin structure.

Recall from [KZ, eqn. (4)] that where defined, the spin structure of a surface
$M$ of genus $g$ can be computed as follows. Let $\alpha_{i},\beta_{j}$ (where
$1\leq i,j\leq g$) be a symplectic basis for $H_{1}(M)$, realized explicitly
as smooth curves on $M$. This means that all of these curves are disjoint,
except for $\alpha_{i}$ and $\beta_{i}$ which intersect once. For each curve
$\gamma$, let $\mathrm{ind}(\gamma)$ be the turning index, that is the total
number of circles made by the tangent vector to $\gamma$, as one goes around
$\gamma$. The parity of $M$ is then the parity of the integer
$\sum_{i=1}^{g}(1+\mathrm{ind}(\alpha_{i}))(1+\mathrm{ind}(\beta_{i}))$. It is
shown in [KZ] that this number is well-defined (independent of the choice of
the symplectic basis) when all the singular points have even order.

Suppose $M$ has genus $g$ and is equipped with a symplectic basis. Since any
non-separating simple closed curve can be completed to a symplectic basis, we
can assume that $\alpha_{1}$ is the core curve of $C$, and the other curves in
the basis do not intersect the saddle connection from $p$ to $q$ passing
through $C$. We construct a symplectic basis for $M^{\prime}$ in both cases,
by modifying $\alpha_{1}$, keeping
$\alpha_{2},\ldots,\alpha_{g},\beta_{1},\ldots,\beta_{g}$, and adding new
curves $\alpha_{g+1},\alpha_{g+2},\beta_{g+1},\beta_{g+2}$. The modified
curves are shown in Figures 5, 6, and the reader can easily check that these
new curves still form a symplectic basis, and that these two choices add two
numbers of different parities to the spin structure. ∎

$\alpha_{g+1}$$\beta_{g+1}$$\beta_{g+2}$$\alpha_{g+2}$$\alpha_{1}$ Figure 5.
Modifying the symplectic basis. Gluings as in Figure 3.
$\alpha_{1}$$\alpha_{g+1}$$\beta_{g+1}$$\beta_{g+2}$$\alpha_{g+2}$ Figure 6.
Modifying the symplectic basis, second case. Gluings as in Figure 4. Note the
change in the rotation number of $\beta_{g+2}$.

Note that in Proposition 8.2 we care about all connected components of strata.
We need to record some information about the classification of connected
components of strata, due to Kontsevich and Zorich. A translation surface is
hyperelliptic if it admits an involution which acts on absolute homology as
$-\mathrm{Id}$ (see [FM] or [KZ, §2.1] for more details). A connected
component of a stratum is hyperelliptic if all surfaces in the component are
hyperelliptic.

######  Proposition 8.6 ([KZ], Theorems 1 & 5 and Corollary 5 of Appendix B).

Let ${\mathcal{H}}(a_{1},\ldots,a_{k})$ be a stratum with $a_{i}.>0$ for all
$i$. The following holds:

  * •

${\mathcal{H}}$ has three connected components in the following cases:

    * –

$k=1,a_{1}=2g-2,g\geq 4.$

    * –

$k=2$, $a_{1}=a_{2}=g-1$, $g\geq 5$ is odd. One is hyperelliptic, and the two
non-hyperelliptic strata are distinguished by the spin invariant.

  * •

${\mathcal{H}}$ has two connected components in the following cases:

    * –

All of the $a_{i}$ are even, $g\geq 4$, and either $k\geq 3$ or $a_{1}>a_{2}$.
The components are distinguished by their spin.

    * –

$a_{1}=a_{2}$ and $g$ is either $3$ or is even. One of the components is
hyperelliptic and the other is not. When $g=3$ the hyperelliptic component is
even, and the other one is odd.

  * •

${\mathcal{H}}$ is connected in all other cases.

###### Proof of Proposition 8.2.

The proof will be case-by-case. Here are the cases:

  1. (i)

${\mathcal{H}}(1,1)$.

  2. (ii)

All the $a_{i}$ are nonzero and ${\mathcal{H}}$ is connected.

  3. (iii)

All the $a_{i}$ are nonzero and ${\mathcal{H}}$ has two connected components
distinguished by spin.

  4. (iv)

All the $a_{i}$ are nonzero and ${\mathcal{H}}$ has two connected components
distinguished by hyperellipticity.

  5. (v)

All the $a_{i}$ are nonzero and ${\mathcal{H}}$ has three connected
components.

  6. (vi)

Some of the $a_{i}$ are zero.

Case i. There is just one connected component and the desired surface is a
$Z$-shaped surface, with three horizontal cylinders $C_{1},C_{2},C_{3}$ of
circumferences $c_{1},c_{1}+c_{3},c_{3}$, where $C_{1},C_{3}$ are simple. We
put all of the removable singular points in the interior of $C_{3}$, and
choose choose $c_{1},c_{3}$ so that $c_{1}/(c_{1}+c_{3})\notin{\mathbb{Q}}$.
It is clear that with these choices the conditions are satisfied.

Case ii. The stratum ${\mathcal{H}}$ is connected, and we have at least two
singularities of positive order. So with no loss of generality that they are
labelled 1 and 2\. The result follows from Lemma 8.4, applied to a surface in
${\mathcal{H}}(a_{1}-1,a_{2}-1,a_{3},\ldots,a_{k})$, and taking $w\notin
c\mathbb{Q}$, so that $w/(c+w)\notin{\mathbb{Q}}$.

Case iii. We apply the surgery in Lemma 8.5, with
$w_{1}/w_{2}\notin{\mathbb{Q}}$. Namely if $p$ and $q$ are labelled $i,j$, we
let $b_{i}=a_{i}-2$, $b_{j}=a_{j}-2$ and $b_{\ell}=a_{\ell}$ for $\ell\neq
i,j$.

Case iv. There are two connected components. One is hyperelliptic, one is not.
This means that $a_{1}=a_{2}$ and either $g=3$ (in which case $a_{1}=a_{2}=2$)
or $g\geq 4$ is even (in which case $a_{1}=a_{2}=g-1$). In this case we give
explicit surfaces, one in each connected component. The first surface (the
${\mathcal{H}}(2,2)$ case is shown in Figure 7) is a ‘staircase’ surface made
of gluing $2g$ rectangles to each other. The rectangles are labelled $(k,B)$
and $(k,T)$ for $k=1,\ldots,g$. The top (respectively, bottom) of $(k,B)$ is
glued to the bottom (resp., top) of $(k,T)$ for $k=1,\ldots,g$, and the left
(resp., right) of $(k,T)$ is glued to the right (resp., left) of $(k+1,B)$ for
$k=1,\ldots,g-1$. The horizontal sides of $(1,B)$ are glued to each other, as
are the horizontal sides of $(g,T)$. This surface is hyperelliptic since it
has a hyperelliptic involution rotating each rectangle around its midpoint,
and this involution swaps the singularities (see [KZ, Remark 3]). The second
surface is obtained as follows. We first construct a hyperelliptic surface in
${\mathcal{H}}(a_{1}-2,a_{2}-2)$ as in the previous paragraph. Then we perform
the surgery described in Lemma 8.5. The resulting surface has a horizontal
cylinder intersecting three vertical cylinders, and thus, by [Li, Lemma 2.1],
is not hyperelliptic. See Figure 8 for an example in ${\mathcal{H}}(2,2)$. In
both of these constructions there are no restrictions on the sidelengths of
the rectangles, and we can easily arrange that two of the circumferences are
incommensurable.

Figure 7. A surface in $\mathcal{H}^{hyp}(2,2)$. Figure 8. A surface in
$\mathcal{H}^{nonhyp}(2,2)$.

Case v. In this case $a_{1}=a_{2}=g-1$ for $g\geq 5$ odd. Applying the
argument in Case iii, we obtain the required surfaces in the odd and even
connected components. To obtain the required surface in the hyperelliptic
component we use the ‘staircase surface’ describe in Case iv.

Case vi.  Assume with no loss of generality that the removable singularities
are labelled $k^{\prime}+1,\ldots,k$ for some $k^{\prime}\geq 2$, and let
${\mathcal{H}}^{\prime}={\mathcal{H}}(a_{1},\ldots,a_{k^{\prime}})$. Note that
the singularities $p$ and $q$ have label in $\\{1,\ldots,k^{\prime}\\}$. Apply
the preceding considerations to obtain a surface in ${\mathcal{H}}^{\prime}$
with the required cylinders. By examining the proof in all preceding case one
sees that the number of horizontal cylinders on this surface is at least
three, that is there is at least one cylinder $C_{3}$ which is distinct from
the cylinders $C_{1},C_{2}$, and we modify $M$ by adding $k-k^{\prime}$ in
general position in the interior of $C_{3}$, to obtain the desired surface.

∎

## References

  * [BSW] M. Bainbridge, J. Smillie and B. Weiss, Horocycle dynamics: new invariants and eigenform loci in the stratum ${\mathcal{H}}(1,1)$,  preprint (2016) to appear in Mem. AMS.
  * [CDF] G. Calsamiglia, B. Deroin and S. Francaviglia, A transfer principle: from periods to isoperiodic foliations, preprint (2015) https://arxiv.org/pdf/1511.07635.pdf
  * [EW] M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, Grad. Texts in Math. 259, Springer (2010).
  * [EFW] A. Eskin, S. Filip and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, Ann. of Math. (2) 188 (2018), no. 1, 281–-313.
  * [EM] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\operatorname{SL}_{2}({\mathbb{R}})$-action on moduli space, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 95-–324.
  * [EMM] A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\operatorname{SL}_{2}({\mathbb{R}})$-action on moduli space,  Ann. Math. (2) 182 (2015), no. 2, 673–721.
  * [Ham] U. Hamenstädt, Ergodicity of the absolute period foliation, Israel J. Math. 225 (2018), no. 2, 661–680.
  * [FM] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, Vol. 49 (2012), Princeton University Press.
  * [Fi] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich Cocycle, Invent. Math. 205 (2016), no. 3, 617-–670
  * [HW] P. Hooper and B. Weiss, Rel leaves of the Arnoux-Yoccoz surfaces, with an appendix by L. Bary-Soroker, M. Shusterman and U. Zannier, Selecta Math., (2018) 24, no. 2, 875–934.
  * [KT] A. B. Katok and J.-P., Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in Handbook of Dynamical Systems, Vol. 1B, 649–753, Elsevier (2006).
  * [KZ] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631–678.
  * [Li] K. Lindsey, Counting invariant components of hyperelliptic translation surfaces, Israel J. Math. 210 (2015), p. 125-146.
  * [MS] H. Masur and J. Smillie, Hausdorff dimension of sets of nonergodic measured foliations, Ann. Math. 134 (1991) 455–543.
  * [MaTa] H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of dynamical systems, Enc. Math. Sci. Ser. (2001).
  * [McM] C. McMullen, Moduli spaces of isoperiodic forms on Riemann surfaces, Duke Math. J. 163 (2014), no. 12, 2271?2323.
  * [MW2] Y. Minsky and B. Weiss, Cohomology classes represented by measured foliations, and Mahler’s question for interval exchanges, Annales Sci. de L’ENS 2 (2014) 245–284.
  * [Mo] S. Mozes, Mixing of all orders of Lie groups actions, Invent. Math. 107 (1992), no. 2, 235-–241.
  * [P] K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, 2. Cambridge University Press (1983).
  * [dlR] T. de la Rue, An introduction to joinings in ergodic theory,  Disc. Cont. Dyn. Sys. 15 1 121–142 (2006).
  * [Va] V. S. Varadarajan, Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191-–220.
  * [Wi1] K. Winsor, Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms,  preprint (2021) https://arxiv.org/abs/2109.12669
  * [Wi2] K. Winsor, Dense real Rel flow orbits and absolute period leaves, preprint (2022), https://people.math.harvard.edu/~kwinsor/preprints/Winsor_DenseLeaves_060622.pdf
  * [Wr1] A. Wright, Translation surfaces and their orbit closures: an introduction for a broad audience,  EMS Surv. Math. Sci. 2 (2015), no. 1, 63–108.
  * [Wr2] A. Wright, The field of definition of affine invariant submanifolds of the moduli space of abelian differentials, Geom. Topol. 18 (2014), no. 3, 1323-–1341.
  * [Y] F. Ygouf, A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds, J. Top. 16 (2023) 1–19.
  * [Zo] A. Zorich, Flat surfaces, in Frontiers in number theory, physics and geometry, P. Cartier, B. Julia, P. Moussa and P. Vanhove (eds), Springer (2006).