# Horocycle dynamics in rank one invariant subvarieties I: weak measure
classification and equidistribution

Jon Chaika  University of Utah<EMAIL_ADDRESS>,  Barak Weiss  Dept. of
Mathematics, Tel Aviv University, Tel Aviv, Israel<EMAIL_ADDRESS>and
Florent Ygouf  Dept. of Mathematics, Tel Aviv University, Tel Aviv, Israel
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathcal{M}$ be an invariant subvariety in the moduli space of
translation surfaces. We contribute to the study of the dynamical properties
of the horocycle flow on $\mathcal{M}$. In the context of dynamics on the
moduli space of translation surfaces, we introduce the notion of a ‘weak
classification of horocycle invariant measures’ and we study its consequences.
Among them, we prove genericity of orbits and related uniform equidistribution
results, asymptotic equidistribution of sequences of pushed measures, and
counting of saddle connection holonomies. As an example, we show that
invariant varieties of rank one, Rel-dimension one and related spaces obtained
by adding marked points satisfy the ‘weak classification of horocycle
invariant measures’. Our results extend prior results obtained by Eskin-Masur-
Schmoll, Eskin-Marklof-Morris, and Bainbridge-Smillie-Weiss.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Strata of translation surfaces
    1. 2.1 Translation surfaces and developing map
    2. 2.2 $G$-action, invariant subvarieties
    3. 2.3 Rel foliation, Rel vector fields
    4. 2.4 Real Rel vector fields and real Rel flow
    5. 2.5 Real Rel pushes of ergodic $U$-invariant measures
  3. 3 Horocycle dynamics in rank-one, Rel-one invariant subvarieties and compact extension
    1. 3.1 The drift argument
    2. 3.2 Saddle connection free $U$-invariant ergodic measures
  4. 4 Adding marked points
  5. 5 Nondivergence of the horocycle flow and frequency of passage near singular sets
    1. 5.1 Singular sets and long horocycle trajectories
    2. 5.2 Singular sets and large circles
  6. 6 From measure classification to equidistribution of orbits
  7. 7 Equidistribution results for sequences of measures

##  1\. Introduction

Any stratum of translation surfaces $\mathcal{H}$ is endowed with an action of
the group
$G{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\operatorname{SL}_{2}(\mathbb{R})$,
and the restriction of this action to the subgroup

(1) $U=\\{u_{s}:s\in\mathbb{R}\\},\ \ \ \text{ where }\
u_{s}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left(\begin{matrix}1&s\\\
0&1\end{matrix}\right),$

is called the horocycle flow. It is a longstanding open problem to understand
the invariant measures for the $U$-action on $\mathcal{H}$. It has been known
for a while, that a ‘sufficiently nice’ classification of such measures would
have many interesting dynamical and geometrical consequences. See [Zor06,
HMSZ06], and see [CSW20] for a survey of recent results. This paper
contributes to the investigation of this problem in two ways. Firstly, we
obtain a partial classification of horocycle invariant ergodic measures, which
we call weak classification, in the setting of rank-one invariant subvarieties
of Rel-dimension one, or spaces obtained from them by adding marked points.
All of these notions will be defined in §2. Secondly, we explore the
implications of a weak classification result in rank-one invariant
subvarieties (of arbitrary Rel-dimension) and show that it implies genericity
of orbits, equidistribution of circle averages, and many additional
equidistribution results.

We do not know whether the weak classification result holds in general rank-
one invariant subvarieties; our approach crucially uses the Rel-dimension one
assumption — see Remark 3.4. Nevertheless, in a follow-up paper [CWY], we
extend the classification of $U$-orbit closures to general rank-one invariant
subvarietes (omitting the assumption of one-dimensional Rel). We do so without
obtaining a weak measure classification, but rather by an induction on the
dimension of Rel. Although the current paper uses completely different
techniques, it serves as the base of the induction used in [CWY].

In the remainder of this introduction we give an initial description of the
main results and discuss the proofs. Precise statements of results require
some preparations and will be given in the body of the text. Let $\mathcal{M}$
be an orbit-closure for the action of $G$. As was shown in a celebrated paper
of Eskin, Mirzakhani and Mohammadi [EMM15], such orbit-closures are (iso-area
subloci of) affine orbifolds in period coordinates, and carry a wealth of
additional structure. Following [AW], we refer to these orbit closures as
invariant subvarieties. A fundamental invariant of an invariant subvariety is
its rank, introduced in [Wri15a]. We will be interested in the rank-one case,
which is characterized by the fact that the $G$-action and the Rel foliation
(see §2.3) are complementary foliations, in the sense that a neighborhood of a
point can be described in terms of the $G$-action and the Rel-foliation. A
further invariant is the Rel dimension (the complex dimension of leaves of the
Rel foliation). Now suppose $\mathcal{N}\subset\mathcal{M}$ is an invariant
subvariety. It was shown in [EM18] that $\mathcal{N}$ is the support of a
unique $\operatorname{SL}_{2}(\mathbb{R})$-invariant Radon probability
measure, which has a geometric description in terms of the affine structure of
$\mathcal{N}$. We denote this measure by $m_{\mathcal{N}}$. It is well-known
that $m_{\mathcal{N}}$ is ergodic for the action of $U$ and, as first observed
by Calta [Cal04] and explained in §2.5, such a measure can be pushed by real
Rel to create more $U$-invariant ergodic measures. We say that $\mathcal{M}$
satisfies the weak classification of $U$-invariant measures if any ergodic
$U$-invariant probability Radon measure measure on $\mathcal{M}$, that assigns
measure zero to the set of surfaces with horizontal saddle connections, is
obtained by this construction. In Theorem 3.3 we prove that invariant
subvarieties of rank one and with Rel-dimension one satisfy weak
classification. We remark that all invariant subvarieties modeled on a complex
vector space of dimension 3 are of rank one with Rel-dimension one and thus
satisfy the hypotheses of Theorem 3.3; explicit examples are given in
Corollaries 3.5, 3.8 and 3.9.

Theorem 6.1 gives severe restrictions on the collection of weak-* limits of
measures obtained by averaging along a $U$-orbit in a rank-one invariant
subvariety. This theorem uses the rank-one structure and is false in general,
as shown in [CSW20]. We then explore the implications of weak classification
in the setting of rank-one invariant subvarieties (omitting the assumption
that the Rel-dimension is one). Under these conditions, we can derive the
following results. In Theorem 6.2 we prove that the $U$-orbit of any surface
without horizontal saddle connections is generic for some $U$-ergodic
invariant measure. Moreover, for a given test function $f$ and a given error
$\varepsilon$, genericity holds uniformly outside certain compact ‘singular
sets’, see Theorem 6.4. This result is an analogue of [DM93, Thm. 3]. We show
in Theorem 7.3 that for any $U$-invariant ergodic measure $\mu$, the
pushforwards $g_{t\ast}\mu$ converge to $m_{\mathcal{N}}$ for some invariant
subvariety $\mathcal{N}$, as $t\to\infty$ and to $m_{\mathcal{N}^{\prime}}$
for some invariant subvariety $\mathcal{N}^{\prime}\subset\mathcal{N}$, as
$t\to-\infty$ (unless $\mu$ assigns positive measure to the set of surfaces
with horizontal saddle connections, in which case they diverge in the space of
probability measures). Here

(2) $\displaystyle
g_{t}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{diag}(e^{t},e^{-t})$

denotes the geodesic flow. In Theorem 7.1 we obtain a similar result for
pushing a $G$-invariant measure by a real Rel flow. Finally we show in Theorem
7.4 that in a rank one locus satisfying the weak classification of
$U$-invariant measures, pushed circle averages
$\frac{1}{2\pi}\int_{0}^{2\pi}g_{t\ast}r_{\theta}\delta_{q}$ converge as
$t\to\infty$, to the $G$-invariant measure on $\mathcal{N}=\overline{Gq}$.
Here $r_{\theta}$ is the rotation matrix by angle $\theta$. As a consequence,
every surface satisfies quadratic growth of saddle connection holonomies. Our
technique revisits and adapts arguments of Ratner [Rat91] for proving the
measure classification of unipotent flows on homogeneous spaces, and
subsequent work of Dani and Margulis [DM93] for harnessing this in order to
obtain equidistribution results. The first papers in which these ideas were
adapted to translation surfaces are [EMS03, EMWM06], in which a weak
classification of $U$-invariant measures was proved, and equidistribution for
circle averages was deduced, in the context of branched covers of Veech
surfaces. In [CW10] and [BSW16], measure classification results were obtained
in eigenform loci in genus two, which are particular cases of a rank-one
invariant subvariety with Rel-dimension one. Specifically, the paper [BSW16]
proves a complete measure classification result, which is stronger than our
‘weak classification’, as it also classifies measures for which typical
surfaces do have horizontal saddle connections. From this strong measure
classification, equidistribution results are deduced, namely a strong version
of Theorem 6.2, and analogues of Theorems 7.3, 7.1 and 7.4. Our work adapts
the techniques in [BSW16] to our more general setting and establishes that
most of these equidistribution results, and additional ones, can be proved
assuming a weaker measure classification result. A key technical input is
Proposition 5.7, which improves [BSW16, Proof of Thm. 11.1, Step 3] and
uniformly bounds the time horocycle trajectories starting in a given compact
set can spend close to certain singular sets.

###  1.1. Acknowledgements

The authors gratefully acknowledge the support of grants BSF 2016256, ISF
2019/19, ISF-NSFC 3739/21, a Warnock chair, a Simon Fellowship, and NSF grants
DMS-1452762 and DMS-2055354.

##  2\. Strata of translation surfaces

In this section we collect some standard definitions and set our notations.
See [MT02, Zor06, FM14, Wri15b, BSW16] for helpful surveys and additional
introductory material.

###  2.1. Translation surfaces and developing map

Let $S$ be a compact and oriented topological surface of genus $g$, let
$\Sigma$ be a finite subset of $S$ and let $\kappa=(k_{\xi})_{\xi\in\Sigma}$
be an integer partition of $2g-2$. That is, $\sum_{\xi\in\Sigma}k_{\xi}=2g-2$
and $k_{\xi}$ are non-negative integers. We emphasize that $\kappa_{\xi}=0$ is
allowed, in which case $\xi$ is called a marked point. A singularity-labeled
translation surface of type $\kappa$ is a triple
$q=(X_{q},\omega_{q},\mathfrak{z}_{q})$ where $X_{q}$ is a compact and
connected Riemann surface of genus $g$, $\omega_{q}$ is a non-zero holomorphic
$1$-form on $X_{q}$ whose set of zeroes is denoted by $\Sigma_{q}$ and
$\mathfrak{z}_{q}:\Sigma\to X_{q}$ is an injective map whose image contains
$\Sigma_{q}$ and such that $\mathfrak{z}_{q}(\xi)$ has order $k_{\xi}$. Where
it will not cause any confusion we will omit the words ‘singularity-labeled’
and refer to $q$ simply as a translation surface of type $\kappa$. A marking
on a translation surface $q$ is a homeomorphism $\varphi:S\to X_{q}$ such that
$\varphi(\xi)=\mathfrak{z}_{q}(\xi)$ for $\xi\in\Sigma$, and a pair
$(q,\varphi)$ is called a marked translation surface with labeled
singularities. Two marked translation surfaces $(q_{1},\varphi_{1})$ and
$(q_{2},\varphi_{2})$ are isomorphic if there is a biholomorphism
$\phi:X_{q_{1}}\to X_{q_{2}}$ such that
$\phi^{\ast}\omega_{q_{2}}=\omega_{q_{1}}$,
$\phi\circ\mathfrak{z}_{q_{1}}=\mathfrak{z}_{q_{2}}$ and
$\varphi_{2}^{-1}\circ\phi\circ\varphi_{1}$ is isotopic to the identity rel
$\Sigma$. Let $\mathcal{H}_{\mathrm{m}}$ be the space of isomorphism classes
of marked translation surfaces of type $\kappa$ with labeled singularities and
let $\mathrm{Mod}(S,\Sigma)$ be the mapping class group of $S$ that preserves
$\Sigma$ point-wise. It acts by precomposition on the space
$\mathcal{H}_{\mathrm{m}}$ and we denote by $\mathcal{H}$ the quotient and by
$\pi:\mathcal{H}_{\mathrm{m}}\to\mathcal{H}$ the quotient map. The space
$\mathcal{H}=\mathcal{H}(\kappa)$ is the moduli space of translation surfaces
of type $\kappa$ with labeled singularities. There is a structure of linear
manifold on $\mathcal{H}_{\mathrm{m}}$ that turns the following map into a
local linear isomorphism.

$\mathrm{dev}:\mathcal{H}_{\mathrm{m}}\to H^{1}(S,\Sigma;\mathbb{R}^{2}),\ \ \
\mathrm{dev}\left((q,\varphi)\right){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left(\gamma\mapsto\int_{\varphi(\gamma)}\omega_{q}\right).$

Each homeomorphism of $S$ preserving $\Sigma$ acts on homology and this
induces an action of the group $\mathrm{Mod}(S,\Sigma)$ on
$H^{1}(S,\Sigma;\mathbb{R}^{2})$. For this action, the map $\mathrm{dev}$ is
$\mathrm{Mod}(S,\Sigma)$-equivariant. The space $\mathcal{H}$ can be endowed
with the quotient topology and inherits an orbifold structure from the one on
$\mathcal{H}_{\mathrm{m}}$.

The area of a translation surface is the integral of the volume form induced
by the 1-form, i.e.

$\mathrm{Area}(q)=\frac{1}{2\mathbf{i}}\int_{S}\omega_{q}\wedge\overline{\omega}_{q}.$

We denote the subset of area-one surfaces in $\mathcal{H}_{\mathrm{m}}$ and
$\mathcal{H}$ by $\mathcal{H}_{\mathrm{m}}^{(1)}$ and $\mathcal{H}^{(1)}$
respectively.

Associated with a translation surface is a translation atlas on $X_{q}$,
obtained by locally integrating $\omega_{q}$. A saddle connection on a
translation surface $q$ is a path $\gamma:[0,1]\to X_{q}$ such that
$\gamma(0),\gamma(1)\in\Sigma_{q},\gamma(t)\notin\Sigma_{q}$ for $t\in(0,1)$,
and $t\mapsto\gamma(t)$ is linear in planar charts. We refer to the vector
$\int_{\gamma}\omega_{q}\in\mathbb{C}\cong\mathbb{R}^{2}$ as the holonomy of
$\gamma$ with respect to $q$, and denote it by $\mathrm{hol}_{q}(\gamma)$.

###  2.2. $G$-action, invariant subvarieties

For the remainder of this paper we set
$G{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\operatorname{SL}_{2}(\mathbb{R})$.
The space $\mathcal{H}_{\mathrm{m}}$ is endowed with a $G$-action, for which
the developing map is equivariant with respect to the natural action of $G$ by
post-composition on $H^{1}(S,\Sigma;\mathbb{R}^{2})$. More precisely, for any
$(q,\varphi)$ and any $g\in G$, we have

$\mathrm{dev}(g(q,\varphi))=g\mathrm{dev}(q,\varphi),$

and thus, for any saddle connection $\delta$ on $q$ and any $g\in G$,

(3) $g\mathrm{hol}_{q}(\delta)=\mathrm{hol}_{gq}(\delta).$

The $G$-action preserves the area of a translation surface. Besides the group
$U=\\{u_{s}:s\in\mathbb{R}\\}$ defined in (1), we will consider

(4)
$A{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{g_{t}:t\in\mathbb{R}\\}\text{
and }B{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}AU,$

where $g_{t}$ is as in (2). The action of $A$ will be referred to as the
geodesic flow.

######  Definition 2.1 (Invariant subvariety).

An invariant subvariety is a subset $\mathcal{M}\subset\mathcal{H}$ such that
$\pi^{-1}(\mathcal{M})$ is a locally finite union of connected equilinear
manifolds, each of which projects to $\mathcal{M}$ under $\pi$.

We recall from [SSWY] that an equilinear manifold is a properly embedded
connected submanifold $X\subset\mathcal{H}_{\mathrm{m}}$ that is modeled via
the developing map on a $\mathbb{C}$-linear subspace $V\subset
H^{1}(S,\Sigma;\mathbb{C})\cong H^{1}(S,\Sigma;\mathbb{R}^{2})$ defined over
$\mathbb{R}$, and such that if $\gamma\in\mathrm{Mod}(S,\Sigma)$ preserves
$X$, then $\gamma|_{V}$ has determinant $1$. By the work of Eskin, Mirzakhani
and Mohammadi ([EM18] and [EMM15]), the orbit-closures for the
$\mathrm{GL}_{2}^{+}(\mathbb{R})$-action are exactly the invariant
subvarieties, closed invariant sets are always finite unions of invariant
subvarieties, and the orbit-closures and invariant measures for $G$ coincide
with the orbit-closures and invariant measures for its subgroup $P$. What we
call ‘invariant subvariety’ was called ‘affine invariant manifold’ in [EMM15];
our choice follows [AW]. Note also that Definition 2.1 differs slightly from
the definitions given in [EMM15] but will be more convenient for us, see
[SSWY] for a more detailed discussion of this definition.

Let $\mathcal{M}$ be an invariant subvariety. A connected equilinear manifold
$X$ contained in $\pi^{-1}(\mathcal{M})$ such that $\pi(X)=\mathcal{M}$ is
called an irreducible component of $\pi^{-1}(\mathcal{M})$. It is easy to see
that $\pi^{-1}(\mathcal{M})$ is the union of its irreducible components and
that $\mathrm{Mod}(S,\Sigma)$ acts transitively on the set of irreducible
components. In particular, since the action of $\mathrm{Mod}(S,\Sigma)$ is
smooth, the irreducible components all have the same dimension. This allows us
to define the dimension of $\mathcal{M}$, which we denote by
$\dim(\mathcal{M})$, as the dimension of any one of the irreducible
components.

Let $\mathcal{M}^{(1)}$ be the area-one locus in $\mathcal{M}$. Using Lebesgue
measures on the irreducible components of $\pi^{-1}(\mathcal{M})$ and a ‘cone
construction’, one can construct a natural smooth $G$-invariant Radon
probability measure of full support on $\mathcal{M}^{(1)}$. For more details
on this construction, using the language above, see [SSWY]. We will denote
this measure by $m_{\mathcal{M}}$. It was shown in [EM18] that all
$G$-invariant ergodic measures on strata arise in this way, and that
$m_{\mathcal{M}}$ is the unique $G$-invariant ergodic measure with
$\mathrm{supp}\,m_{\mathcal{M}}=\mathcal{M}^{(1)}$.

###  2.3. Rel foliation, Rel vector fields

We recall the following construction (see [Zor06, MW14, McM14, BSW16]). Let
${\mathrm{Res}}:H^{1}(S,\Sigma;\mathbb{R}^{2})\to H^{1}(S;\mathbb{R}^{2})$ be
the canonical cocycle restriction map and denote its kernel by $\mathfrak{R}$.
Clearly $\mathfrak{R}$ is a $\mathbb{C}$-linear subspace defined over
$\mathbb{R}$. It will be useful to have an explicit set of generators for
$\mathfrak{R}.$ For any $\xi\in\Sigma$, denote by $\xi^{*}$ the element of
$H^{1}(S,\Sigma;\mathbb{R}^{2})$ that is dual (with respect to the
intersection form) to the cycle of $H_{1}(S\smallsetminus\Sigma;\mathbb{Z})$
represented by a contractible loop on $S$ that circles counterclockwise the
point $\xi$. The elements $\\{\xi^{*}:\xi\in\Sigma\\}$ are a set of generators
for $\mathfrak{R}$, and they satisfy one relation
$\sum_{\xi\in\Sigma}\xi^{*}=0.$ In particular
$\dim_{\mathbb{C}}(\mathfrak{R})=n-1$, where $n$ is the cardinality of
$\Sigma$.

There is a foliation of $H^{1}(S,\Sigma;\mathbb{R}^{2})$ by translates of
$\mathfrak{R}$ which can be pulled back by the developing map to give a
foliation of $\mathcal{H}_{\mathrm{m}}$. Since the map
$\mathrm{dev}\circ{\mathrm{Res}}$ is $\mathrm{Mod}(S,\Sigma)$-equivariant,
there is an induced foliation of $\mathcal{H}$. The area of two translation
surfaces which are in the same leaf for this foliation are the same and thus
the foliation restricts to give a foliation of $\mathcal{H}^{(1)}$. We call
this foliation (on any one of the spaces
$\mathcal{H}_{\mathrm{m}},\mathcal{H},$ or $\mathcal{H}^{(1)}$) the Rel
foliation; it has also been called the kernel foliation, the isoperiodic
foliation, and the absolute period foliation.

Recall our convention that the singularities of surfaces in $\mathcal{H}$ are
labeled. This implies that elements of $\mathrm{Mod}(S,\Sigma)$ do not permute
the singularities and act trivially on $\mathfrak{R}.$ For any
$v\in\mathfrak{R}$, the constant vector field $v$ can be lifted to a vector
field on $\mathcal{H}_{\mathrm{m}}$ by $\mathrm{dev}$ and since singularities
are not permuted by $\mathrm{Mod}(S,\Sigma)$, this vector field descends to a
well-defined vector field on $\mathcal{H}$, which is called the Rel vector
field associated to $v$ and denoted $\vec{v}$.

######  Definition 2.2.

Let $q\in\mathcal{H}$ and let $v\in\mathfrak{R}$. We say that
$\mathrm{Rel}_{v}(q)$ is well-defined and equal to $q^{\prime}$ if there is a
smooth path $\gamma:I\to\mathcal{H}$, where $I$ is an interval that contains
$0$ and $1$, and such that $\gamma(0)=q$, $\gamma(1)=q^{\prime}$ and $\forall
t\in I,\ \frac{d}{dt}\gamma(t)=\vec{v}$. We denote by $\Delta$ the subset of
$\mathcal{H}\times\mathfrak{R}$ consisting of pairs $(q,v)$ for which
$\mathrm{Rel}_{v}(q)$ is well-defined.

We summarize some properties of this partially defined map, see [BSW16, §4]
for proofs.

The set $\Delta$ is open and the map

$\displaystyle\Delta\to\mathcal{H},\ \ \ \ \ (q,v)\mapsto\mathrm{Rel}_{v}(q)$

is continuous. The group $G$ acts on $\mathfrak{R}$ by restriction of its
action by post-composition on $H^{1}(S,\Sigma;\mathbb{C})$, and for any $g\in
G$,

(5) $(q,v)\in\Delta\iff(gq,gv)\in\Delta,\ \ \text{ and }\
\mathrm{Rel}_{gv}(gq)=g\mathrm{Rel}_{v}(q).$

The set $\Delta$ is completely described in [MW14, BSW16] in terms of saddle
connections. In particular, if $n=2$ and $q$ has no saddle connection whose
holonomy is parallel to $v$, then $(q,v)\in\Delta$.

Let $\mathcal{M}\subset\mathcal{H}$ be an invariant subvariety, let $X$ be an
irreducible component of $\pi^{-1}(\mathcal{M})$ and let $V$ be the
$\mathbb{C}$-linear subspace of $H^{1}(S,\Sigma;\mathbb{C})$ defined over
$\mathbb{R}$, on which $X$ is modeled. We define

$\mathfrak{R}_{X}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathfrak{R}\cap
V.$

Since $\mathrm{Mod}(S,\Sigma)$ acts transitively on the set of irreducible
components of $\pi^{-1}(\mathcal{M})$ and $\mathrm{Mod}(S,\Sigma)$ acts
trivially on $\mathfrak{R}$, it is easy to see that if $Y$ is another
irreducible component then $\mathfrak{R}_{X}=\mathfrak{R}_{Y}$. We thus denote
by $\mathfrak{R}_{\mathcal{M}}$ this common subspace. By construction, for any
$q\in\mathcal{M}$ and $v\in\mathfrak{R}_{\mathcal{M}}$ we have
$\mathrm{Rel}_{v}(q)\in\mathcal{M}$ whenever the latter is well-defined, and
$\mathfrak{R}_{\mathcal{M}}$ is the biggest linear subspace of $\mathfrak{R}$
with this property. Whenever $\mathcal{N}\subset\mathcal{M}$ is an inclusion
of invariant subvarieties, we have
$\mathfrak{R}_{\mathcal{N}}\subset\mathfrak{R}_{\mathcal{M}}.$ We refer to the
number $\dim_{\mathbb{C}}(\mathfrak{R}_{\mathcal{M}})$ as the Rel dimension of
$\mathcal{M}$.

Let $V\subset H^{1}(S,\Sigma;\mathbb{C})$ be the local model of an irreducible
component $X$ of $\pi^{-1}(\mathcal{M})$. It follows from [AEM17, Theorem 1.4]
that ${\mathrm{Res}}(V)$ is a symplectic subspace of $H^{1}(S;\mathbb{C})$
with respect to the symplectic structure induced by the cup product, and in
particular, that $\dim_{\mathbb{C}}({\mathrm{Res}}(V))$ is even. In particular
this implies that for any affine subvariety $\mathcal{M}$,
$\dim_{\mathbb{C}}(\mathcal{M})-\dim_{\mathbb{C}}(\mathfrak{R}_{\mathcal{M}})$
is an even number.

######  Definition 2.3 (Rank).

The rank of $\mathcal{M}$ is defined to be
$\frac{1}{2}\left(\dim_{\mathbb{C}}(\mathcal{M})-\dim_{\mathbb{C}}(\mathfrak{R}_{\mathcal{M}})\right)$.

The rank is a fundamental invariant of an invariant subvariety. See [Wri15a]
for additional characterizations. In this paper we will be particularly
interested in the case $\mathrm{rank}(\mathcal{M})=1$. A simple dimension
computation implies that in this case, the Rel leaves and the foliation into
$G$-orbits are complementary foliations on $\mathcal{M}^{(1)}$; that is, in
the rank one case, for any $q\in\mathcal{M}$ there are neighborhoods
$\mathcal{U}$ in $\mathcal{M}^{(1)}$ and neighborhoods
$\mathcal{V}_{1},\mathcal{V}_{2}$ of the identity in $G$ and of $0$ in
$\mathfrak{R}$, such that the map

$\mathcal{V}_{1}\times\mathcal{V}_{2}\to\mathcal{U},\ \ \ \ \
(g,v)\mapsto\mathrm{Rel}_{v}(gq)$

is a homeomorphism. It follows from the discussion preceding Definition 2.3
that if $\dim_{\mathbb{C}}(\mathcal{M})=2$ then necessarily
$\mathcal{M}^{(1)}$ is a closed $G$-orbit, and that being a rank-one invariant
subvariety of Rel-dimension one is equivalent to the condition
$\dim_{\mathbb{C}}(\mathcal{M})=3$.

For fixed $v\in\mathfrak{R}$, we denote by $\mathcal{H}^{\prime}_{v}$ the set
of surfaces $q\in\mathcal{H}$ such that $(q,v)\in\Delta$; and for any
$v\in\mathfrak{R}_{\mathcal{M}}$, we denote by $\mathcal{M}^{\prime}_{v}$ the
set of surfaces $q\in\mathcal{M}$ such that $(q,v)\in\Delta$.

###  2.4. Real Rel vector fields and real Rel flow

Following [BSW16], we write
$\mathbb{R}^{2}=\mathbb{R}_{x}\oplus\mathbb{R}_{y}$ for the canonical
splitting into $x$ and $y$ coordinates. This induces the splitting

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

where $H^{1}(S,\Sigma;\mathbb{R}_{x})\cong H^{1}(S,\Sigma;\mathbb{R})\cong
H^{1}(S,\Sigma;\mathbb{R}_{y})$. The subspace $H^{1}(S,\Sigma;\mathbb{R}_{x})$
shall be referred to as the horizontal space. If $v\in
H^{1}(S,\Sigma;\mathbb{R}_{x})$, it will be convenient to denote by
$\mathbf{i}\cdot v$ the corresponding vector in
$H^{1}(S,\Sigma;\mathbb{R}_{y})$; that is, with respect to the splitting (6),
$(x,0)$ projects to $x\in H^{1}(S,\Sigma;\mathbb{R}_{x})$ and $(0,x)$ projects
to $\mathbf{i}\cdot x\in H^{1}(S,\Sigma;\mathbb{R}_{y})$. We denote

$Z{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathfrak{R}\cap
H^{1}(S,\Sigma;\mathbb{R}_{x})\ \ \ \text{ and
}Z_{\mathcal{M}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathfrak{R}_{\mathcal{M}}\cap
H^{1}(S,\Sigma;\mathbb{R}_{x})$

where $\mathcal{M}\subset\mathcal{H}$ is an invariant subvariety. If $v\in Z$,
the vector field $\vec{v}$ is called a real Rel vector field. For any $r>0$,
we denote by $\mathcal{M}_{r}$ the set of surfaces in $\mathcal{M}$ which do
not have a horizontal saddle connection strictly shorter than $r$, and by
$\mathcal{M}_{\infty}$ the set of surfaces in $\mathcal{M}$ that do not have
horizontal saddle connections. With respect to the action of $G$ by post-
composition on $\mathfrak{R}$, the space $\mathfrak{R}_{\mathcal{M}}$ is
invariant, and the subgroup $B\subset G$ defined in (4) leaves the subspace
$Z_{\mathcal{M}}$ invariant. We can use this action to define semi-direct
products, as follows

######  Definition 2.4.

Let $\mathcal{M}\subset\mathcal{H}$ be an invariant subvariety. We define

$\displaystyle
N_{\mathcal{M}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}B\ltimes
Z_{\mathcal{M}}\ \text{ and }\
L_{\mathcal{M}}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}G\ltimes\mathfrak{R}_{\mathcal{M}}.$

The set $\mathcal{M}_{\infty}$ is a dense $G_{\delta}$ subset of
$\mathcal{M}$. Since $B$ preserves horizontal saddle connections and real Rel
is defined on surfaces without horizontal saddle connections, the map

$N_{\mathcal{M}}\times\mathcal{M}_{\infty}\ni(b,z,q)\mapsto\mathrm{Rel}_{z}(bq)$

is well-defined. Moreover, it satisfies the axioms of a group action (see
[MW14]). In particular, for $z\in Z_{\mathcal{M}}$, the map
$(t,q)\mapsto\mathrm{Rel}_{tz}(q)$ defines a real Rel flow on
$\mathcal{M}_{\infty}$, and this flow commutes with the horocycle flow. On the
other hand, the same map defined on the larger set
$L_{\mathcal{M}}\times\mathcal{M}$ need not satisfy the axioms of a group
action, even if well-defined. In order to stress this point, for general
$q\in\mathcal{M}$ and $(g,v)\in L_{\mathcal{M}}$ for which
$\mathrm{Rel}_{v}(gq)$ is well-defined, we will write $gq\pluscirc
v{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Rel}_{v}(gq).$

###  2.5. Real Rel pushes of ergodic $U$-invariant measures

For fixed $q\in\mathcal{H}$, let $Z^{(q)}$ denote the set of $z\in Z$ for
which $\mathrm{Rel}_{z}(q)$ is well-defined. By (5), $Z^{(q)}=Z^{(uq)}$ for
any $u\in U$ and $q\in\mathcal{H}$, and hence, for any ergodic $U$-invariant
probability measure $\mu$ on $\mathcal{H}$, there is a set $Z^{(\mu)}\subset
Z$ such that $Z^{(q)}=Z^{(\mu)}$ for $\mu$-a.e. $q\in\mathcal{H}$. Let $z\in
Z^{(\mu)}$. Denoting by $\mathcal{B}(\mathcal{M})$ the Borel $\sigma$-algebra
on $\mathcal{M}$, we define a measure by

(7) $\displaystyle\forall X\in\mathcal{B}(\mathcal{M}),\ \ \
{\mathrm{Rel}_{z}}_{\ast}\mu(X)=\mu(\mathrm{Rel}_{-z}(\mathcal{H}_{-z}^{\prime}\cap
X)).$

The measure ${\mathrm{Rel}_{z}}_{\ast}\mu$ is also $U$-invariant and ergodic.
In particular, let $\mathcal{M}\subset\mathcal{H}$ be an invariant subvariety.
If $\mu$ is an ergodic $U$-invariant measure supported on $\mathcal{M}$ and
$z\in Z^{(\mu)}\cap Z_{\mathcal{M}}$, then ${\mathrm{Rel}_{z}}_{\ast}\mu$ is
also an ergodic $U$-invariant measure supported on $\mathcal{M}$. We finally
denote

(8) $\displaystyle
N_{\mu}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left\\{(b,z)\in
N_{\mathcal{M}}:z\in Z^{(b_{\ast}\mu)}\ \text{and}\
{\mathrm{Rel}_{z}}_{\ast}(b_{\ast}\mu)=\mu\right\\}.$

It follows from [BSW16, Cor. 4.20] that $N_{\mu}$ is a closed subgroup of
$N_{\mathcal{M}}$, and thus $N^{\circ}_{\mu}$, the connected component of the
identity in $N_{\mu}$, is a connected Lie group. We recall that a point
$q\in\mathcal{H}$ is generic for an ergodic $U$-invariant measure $\mu$ if for
any compactly supported continuous function
$\varphi:\mathcal{M}\to\mathbb{R}$,

$\displaystyle\frac{1}{T}\int_{0}^{T}f(u_{s}q)\,ds\underset{T\to\infty}{\longrightarrow}\int_{\mathcal{H}}f\
d\mu.$

The set of generic points of $\mu$, which we denote by $\Omega_{\mu}$, has
full measure for $\mu$. We will need the following:

######  Proposition 2.5 ([BSW16], Cor. 4.24, Prop. 4.27 & Cor. 4.28).

Let $\mu$ be an ergodic $U$-invariant measure. Then:

  * •

$Z^{(\mu)}\subset Z^{(q)}$ for any $q\in\Omega_{\mu}$.

  * •

If $q,q^{\prime}\in\Omega_{\mu}$ and $n=(b,z)\in N_{\mathcal{M}}$ satisfy that
$z\in Z^{(b_{\ast}\mu)}$ and $q^{\prime}=\mathrm{Rel}_{z}(bq)$, then $n\in
N_{\mu}$.

  * •

If $n=(b,z)\in N_{\mathcal{M}}$ with $z\in Z^{(\mu)}$, and $q\in\Omega_{\mu}$,
then $\mathrm{Rel}_{z}(bq)\in\Omega_{\mu^{\prime}}$ for
$\mu^{\prime}=\mathrm{Rel}_{z\ast}(b_{\ast}\mu)$.

We will also need the following result, which is proved in [CW10, §6] for
eigenform loci in genus 2, and is a special case of a general result proved in
[SSWY]:

######  Proposition 2.6.

Let $\mathcal{M}$ be a rank-one invariant subvariety and let $\mu$ be a
$U$-invariant ergodic probability measure on $\mathcal{M}$. Suppose that $\mu$
assigns zero measure to surfaces with horizontal saddle connections, and that
$\mathrm{Rel}_{x\ast}\mu=\mu$ for every $x\in Z_{\mathcal{M}}$. Then
$\mu=m_{\mathcal{M}}$.

Notational conventions. We will be interested in the horocycle dynamics on
invariant subvarieties of rank one. Since the horocycle flow preserves the
area of a surface, it is sufficient to consider the area one locus in an
invariant subvariety. From now on, to simplify notation, we will restrict
attention to area-one surfaces and we will write $\mathcal{H}$ for
$\mathcal{H}^{(1)}$, $\mathcal{M}$ for $\mathcal{M}^{(1)}$, etc. Also,
although some statements are valid in greater generality, from now until the
end of the paper, $\mathcal{M}$ will denote a rank one invariant subvariety in
$\mathcal{H}$.

##  3\. Horocycle dynamics in rank-one, Rel-one invariant subvarieties and
compact extension

This section is concerned with the classification of ergodic $U$-invariant
measures supported on invariant subvarieties of rank one and Rel-dimension one
and related compact extensions where marked points are added. Throughout this
section we assume that, in addition to being of rank one, $\mathcal{M}$ is of
Rel-dimension one.

###  3.1. The drift argument

The following key argument is modeled on Ratner’s work and was adapted to the
present context in [EMM06, BSW16].

######  Proposition 3.1.

Let $\mu$ be an ergodic $U$-invariant probability measure on $\mathcal{M}$. If
$\mu$ is not supported on a closed $U$-orbit then $\dim(N^{\circ}_{\mu})\geq
2$.

######  Remark 3.2.

Contrary to [BSW16, Prop 9.4], we do not need to assume that $\mu$ does not
assign positive measure to the set of surfaces with horizontal saddle
connection.

###### Proof.

Let $\varepsilon>0$ and let $K\subset\Omega_{\mu}$ be a compact subset such
that $\mu(K)>1-\varepsilon$. Such a compact set exists by inner regularity of
the measure $\mu$. By combining Birkhoff’s ergodic theorem and Egorov’s
theorem, one obtains that there is another compact set
$K_{0}\subset\mathcal{M}$ and $T_{0}$ large enough so that for any $T>T_{0}$
and any $q\in K_{0}$,

(9) $\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q\in K\\}|>1-2\varepsilon.$

Let $\mathfrak{g}$ denote the Lie algebra of $G$, which we identify with the
set of $2\times 2$ matrices with vanishing trace, endowed with the Lie bracket
$[A,B]=AB-BA$, and let $\exp:\mathfrak{g}\to G$ be the exponential map. We
also choose a $G$-equivariant linear isomorphism
$\mathfrak{R}_{\mathcal{M}}\simeq\mathbb{R}^{2}$ where the action of $G$ on
$\mathbb{R}^{2}$ is the usual linear action. Under this isomorphism, we have
$Z_{\mathcal{M}}\simeq\mathbb{R}\times\\{0\\}$. In the remainder of this
proof, we shall abuse slightly notations and write elements of
$\mathfrak{R}_{\mathcal{M}}$ as elements of $\mathbb{R}^{2}$.

If $\mu$ is not supported on a closed orbit, there is a point $q\in K_{0}$ and
a sequence of points $q_{n}\in K_{0}\smallsetminus Uq$ that converges to $q$.
For $n$ large enough, we can find $V_{n}\in\mathfrak{g}$ and
$W_{n}\in\mathfrak{R}_{\mathcal{M}}$ such that

$(\exp(V_{n}),W_{n})\in L_{\mathcal{M}}\smallsetminus U,\ \ \
(V_{n},W_{n})\to(0,0)\ \ \ \text{ and }\ q_{n}=\exp(V_{n})q\pluscirc W_{n}.$

Write

$\displaystyle V_{n}=\begin{pmatrix}a_{n}&&u_{n}\\\
v_{n}&&-a_{n}\end{pmatrix}\ \ \text{ and }\ W_{n}=(x_{y},y_{n}).$

Let $P_{n}(s)=-v_{n}s^{2}+(1-2a_{n})s+u_{n}$. A computation using (5) shows
that for any $s>0$,

$\displaystyle u_{s}q_{n}=\exp\left(\begin{smallmatrix}a_{n}+sv_{n}&&0\\\
v_{n}&&-(a_{n}+sv_{n})\end{smallmatrix}\right)u_{P_{n}(s)}q\pluscirc(x_{n}+sy_{n},y_{n}).$

Let $0<\eta<1$ and define

$\displaystyle
I_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{s\in\mathbb{R}:\max(|a_{n}+sv_{n}|,|x_{n}+sy_{n}|)<\eta\varepsilon\\}.$

If $n$ is large enough, the set $I_{n}$ contains $0$ and we denote by $J_{n}$
the connected component of $I_{n}$ that contains $0$. For any $s>0$, we have
$P_{n}^{\prime}(s)=-2(sv_{n}+a_{n})+1$ and thus if $s\in I_{n}$ we get
$|P_{n}^{\prime}(s)|<1+2\eta\varepsilon<1+2\varepsilon$. We want to find $s\in
J_{n}$ such that

(10) $u_{s}q_{n}\text{ and }u_{P_{n}(s)}q\text{ are both in }K$

and

(11)
$\frac{\eta\varepsilon}{4}\leq\max(|a_{n}+sv_{n}|,|x_{n}+sy_{n}|)\leq\eta\varepsilon.$

Notice that if $n$ is large enough then
$S_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\sup(J_{n})$ is larger
than $T_{0}$ and

$\displaystyle\frac{1}{S_{n}}\left|\left\\{s\in[0,S_{n}]:\max(|a_{n}+sv_{n}|,|x_{n}+sy_{n}|)<\frac{\eta\varepsilon}{4}\right\\}\right|<\frac{1}{2}.$

Finally, the set of times $s\in[0,S_{n}]$ for which $u_{s}q_{n}$ is in $K$ has
measure at least $(1-2\varepsilon)S_{n}$ and likewise for $u_{s}q$. Since
$P_{n}^{\prime}(s)$ is bounded by $1+2\varepsilon$ on $J_{n}$, the set of
times $s\in[0,S_{n}]$ for which $u_{P_{n}(s)}q$ belongs to $K$ has measure at
least $(1-4\varepsilon)S_{n}$. Consequently, whenever $\varepsilon$ is small
enough, for all $n$ large enough (depending on $T_{0}$ and hence on
$\varepsilon$), $s=s_{n}$ satisfies the desired properties (10) and (11). We
choose $\eta$ small enough to that $\exp(v)p\pluscirc w$ is well defined for
any $v$, $w$ with coefficients smaller that $\eta\varepsilon$ and $p\in K$.
Passing to a converging subsequence in $u_{s_{n}}q$ and $u_{p_{n}(s)}q$ and
invoking Proposition 4.6 in [BSW16], we obtain a pair of generic points $p$
and $p^{\prime}$ in $K$ such that $p^{\prime}=\exp(V)p\pluscirc W$ with

$\displaystyle V=\begin{pmatrix}a&&0\\\ 0&&-a\end{pmatrix}\ \mathrm{and}\
W=(x,0),$

and $\eta\varepsilon/4\leq\max(a,x)\leq\varepsilon$. From Proposition 2.5 that
$\exp(V,W)\in N_{\mu}$. Since $\varepsilon$ was arbitrary, this shows that
$N_{\mu}\smallsetminus U$ contains elements arbitrarily close to the identity.
Since $N_{\mu}$ is a Lie subgroup of $N_{\mathcal{N}}$, this implies $\dim
N_{\mu}^{\circ}>1$. ∎

###  3.2. Saddle connection free $U$-invariant ergodic measures

The set of surfaces with horizontal saddle connections is invariant under $U$.
In particular, if $\mu$ is an ergodic $U$-invariant measure, the measure of
this set is either 1 or 0. In the second case, we say $\mu$ is a saddle
connection free measure.

######  Theorem 3.3.

Let $\mu$ be a saddle connection free ergodic $U$-invariant probability
measure on $\mathcal{M}$. Then there is some invariant subvariety
$\mathcal{N}\subset\mathcal{M}$ and some $x\in Z_{\mathcal{M}}$ such that
$\mu=\mathrm{Rel}_{x\ast}m_{\mathcal{N}}$.

###### Proof.

It is well-known that the normalized length measure on a periodic $U$-orbit is
not saddle connection free. Thus, by Proposition 3.1 there are $a\geq 0$,
$w\in Z_{M}$ and a homomorphism $\rho:\mathbb{R}\to
N^{\circ}_{\mu}\smallsetminus U$ such that

(12)
$\left.\frac{d}{dt}\right|_{t=0}\rho(t)=\Bigl{(}\left(\begin{smallmatrix}a&&0\\\
0&&-a\end{smallmatrix}\right),w\Bigr{)},\ \ \text{ where }(a,w)\neq(0,0).$

If $a=0$, we have that $\rho(t)=(\mathrm{Id},tw)$ and thus
$\rho(t)_{\ast}\mu={\mathrm{Rel}_{tw}}_{\ast}\mu=\mu$. Therefore $\mu$ is
invariant under both $U$ and $\mathrm{Rel}_{tw}$ for all $t$. Since
$\mathcal{M}$ is of Rel-dimension 1, we deduce via Proposition 2.6 that
$\mu=m_{\mathcal{M}}$. If $a\neq 0$ we have

$\displaystyle\rho(t)=\Bigl{(}\left(\begin{smallmatrix}e^{at}&&0\\\
0&&e^{-at}\end{smallmatrix}\right),e^{at}w-w\Bigr{)}$

and it follows via (5) that the measure
$\mu_{0}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}{\mathrm{Rel}_{-w}}_{\ast}\mu$
is invariant under the upper triangular subgroup $B$. Here we have used the
assumption that $\mu$ assigns zero measure to surfaces without horizontal
saddle connections, since it implies $Z_{\mathcal{M}}\subset Z^{(\mu)}$, that
is $\mathrm{Rel}_{-w}(q)$ is well-defined for every $w\in Z_{\mathcal{M}}$,
for $\mu$-a.e. $q$. It follows from [EM18] that there is an invariant
subvariety $\mathcal{N}\subset\mathcal{M}$ such that $\mu_{0}=m_{\mathcal{N}}$
and thus $\mu=\mathrm{Rel}_{w\ast}m_{\mathcal{N}}$. ∎

######  Remark 3.4.

The drift argument relies on a fine analysis of the displacement between two
horocycle trajectories of nearby points. In order to remove the assumption
that $\mathcal{M}$ is of Rel-dimension one, it is tempting to argue
inductively, replacing the horocycle flow by a well-chosen flow in the group
$N_{\mu}$ in order to pick up additional invariance of the measure. However,
to compute the displacement between two nearby trajectories under this new
flow, one would need the ‘group-action property’ (5), with $G$ replaced by
larger groups $G\ltimes\mathfrak{R}_{0},$ for
$\mathfrak{R}_{0}\subset\mathfrak{R}$ (even the weaker property
$\mathrm{Rel}_{x_{1}+x_{2}}(q)=\mathrm{Rel}_{x_{1}}(\mathrm{Rel}_{x_{2}}(q))$
for suitable $x_{1},x_{2}\in\mathfrak{R}$ might suffice). These relations are
not true in general.

We now list some examples to which our results apply. For relevant
definitions, see [McM06]. Let $D\equiv 0\mod 4$ or $D\equiv 1\mod 4$, let
$\mathcal{O}_{D}$ be the quadratic order of discriminant $D$ and let $\Omega
E_{D}$ be the space of Prym eigenforms for real multiplication by
$\mathcal{O}_{D}$. Let $\Omega E_{D}^{odd}(2,2)$ and $\Omega E_{D}(2,1,1)$ be
the intersection of $\Omega E_{D}$ with the strata $\mathcal{H}^{odd}(2,2)$
and $\mathcal{H}(2,1,1)$. It follows from [LN18] that these spaces are not
empty whenever $D\not\equiv 5\ \mathrm{mod}\ 8$, in which case there are at
most two connected components. Any such connected component is a rank 1
invariant subvariety of Rel-dimension 1, see [LN16, Lemma 3.1]. Thus, Theorem
3.3 has the following corollary:

######  Corollary 3.5.

Let $D\equiv 0,1\ \mathrm{mod}\ 4$, $D\not\equiv 5\ \mathrm{mod}\ 8$ be a
positive integer that is not a square and let $\mathcal{M}$ be a connected
component of $\Omega E_{D}^{odd}(2,2)$ or $\Omega E_{D}(2,1,1)$. Then
$\mathcal{M}$ satisfies the weak classification of $U$-invariant measures.

Let $\mathcal{M}$ be a connected component of $\Omega E_{D}$. In the case
where $D$ is a square, $\mathcal{M}$ consists of branched covers of flat tori
and the fact that $\mathcal{M}$ satisfies the weak classification of
$U$-invariant measures follows from the work of Eskin, Markloff and Morris
[EMWM06, Theorem 2.1]. When $D$ is not a square, $\mathcal{M}$ is a
nonarithmetic invariant subvariety, which means that its field of definition
is a non trivial algebraic extension of $\mathbb{Q}$. The examples mentioned
in Corollary 3.5 do not arise via a covering construction and thus provide the
first non-trivial examples in genus 3 of invariant subvarieties in which a
classification of $U$-invariant measures is known. It is worth mentioning that
any nonarithmetic rank 1 invariant subvariety of Rel-dimension 1 in
$\mathcal{H}^{odd}(2,2)$ arises in this way, see [Ygo20]. The other Prym
eigenform loci in genus 3 are
$\Omega(\kappa){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\Omega
E_{D}\cap\mathcal{H}(\kappa)$ for
$\kappa\in\\{4^{odd},4^{hyp},(2,2)^{hyp},(1,1,1,1)\\}$. The first three are a
finite union of Teichmüller curves (i.e., satisfy $dim(\mathcal{M})=2$), see
[LN18, Prop 2.3] for a proof in the case $(2,2)^{hyp}$, and the latter has
Rel-dimension 2. In genus 4 and 5, no explicit description of the connected
components of the Prym eigenform loci is known besides the case
$\mathcal{H}(6)$ that is addressed in [LN20]. Nevertheless, we can still use
Prym eigenforms to produce more examples of rank one invariant subvarieties of
Rel-dimension 1. We recall the following that lists all the potential rank one
invariant subvarieties of Rel-dimension 1 arising from Prym eigenforms:

######  Lemma 3.6 ([LN16, Lemma 3.1]).

Let $D\equiv 0,1\ \mathrm{mod}\ 4$, let $g=4,5$, let $\kappa$ be an integer
partition of $2g-2$ such that $\Omega E_{D}(\kappa)\neq\emptyset$ and let
$\mathcal{M}$ be a connected component of $\Omega E_{D}(\kappa)$. Then
$\mathcal{M}$ is a rank one invariant subvariety and its Rel-dimension is 1
if, and only if, $\kappa$ belongs to the following list:

$(3,3),(2,2,2)^{even},(1,1,4),(1,1,2,2),(4,4)^{even}.$

######  Proposition 3.7.

For any $\kappa$ as in Lemma 3.6, there is $D\equiv 0,1\ \mathrm{mod}\ 4$ such
that $D$ is not a square and $\Omega E_{D}(\kappa)\neq\emptyset$.

###### Proof.

It is explained in [LN16] that each $\kappa$ as in Lemma 3.6 is associated
with a rank 2 invariant subvariety denoted by $Prym(\kappa)$, and the latter
can easily be verified to be arithmetic. It thus follows from [EFW18, Theorem
1.7] that this affine variety itself contains a nonarithmetic rank one
invariant subvariety $\mathcal{M}$. It is not hard to see that $\mathcal{M}$
must contain a surface $q$ that contains a hyperbolic matrix $A$ in its Veech
group. See for instance [Wri14, Remark 4.2]. It follows from [McM06, Theorem
3.5] that $q\in\Omega E_{D}(\kappa)$ with
$\mathbb{Q}(\sqrt{D})=\mathbb{Q}(tr(A))$. If $tr(A)$ is rational then it
follows from [McM03, Theorem 9.8] that $q$ covers a torus, in contradiction
with the fact that $\mathcal{M}$ is nonarithmetic. ∎

Using Theorem 3.3, we get:

######  Corollary 3.8.

Let $\mathcal{M}$ be a connected component of $\Omega E_{D}(\kappa)$ where
$\kappa$ and $D$ are as in Proposition 3.7. Then $\mathcal{M}$ satisfies the
weak classification of $U$-invariant measures.

The Pentagon variety described in [EMMW20] yields another infinite family of
quadratic rank one invariant subvarieties of Rel-dimension 1 contained in
$\mathcal{H}(2,2,1,1)$. It is explained in [EMMW20, Appendix B] that the
intersection of the Pentagon variety with $\mathcal{H}(2,2,1,1)$ is a non-
empty arithmetic rank two invariant subvariety of Rel-dimension 1. Using once
more [EFW18, Theorem 1.7], we deduce the existence of infinitely many
quadratic rank one invariant subvarieties of Rel-dimension 1.

######  Corollary 3.9.

Let $\mathcal{M}$ be one of the infinitely many quadratic rank one invariant
subvarieties contained in the intersection of the Pentagon variety with
$\mathcal{H}(2,2,1,1)$. Then $\mathcal{M}$ satisfies the weak classification
of $U$-invariant measures.

##  4\. Adding marked points

Let $\Sigma$ be a collection of singularities and let
$\kappa=(k_{\xi})_{\xi\in\Sigma}$ be a partition of $2g-2$. We will be
interested in the case that some of the $k_{\xi}$ are zero (and thus
correspond to marked points). Let

(13)
$\displaystyle\Sigma_{0}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{\xi\in\Sigma:k_{\sigma}\neq
0\\}$

be the subset of ‘non-removable’ singularities, let
$\mathfrak{S}\subset\Sigma$ be a subset containing $\Sigma_{0}$, and
$\kappa_{\SS}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}(k_{\xi})_{\xi\in\SS}$.
By definition of $\Sigma_{0}$, $\kappa_{\SS}$ is also an integer partition of
$2g-2$. Let
$\mathcal{H}_{\SS}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathcal{H}(\kappa_{\SS})$
be the moduli space of singularity-labeled translation surfaces of type
$\kappa_{\SS}$. We also consider the space
$\mathcal{H}_{\mathrm{m}}(\kappa_{\SS})$ of marked surfaces of type
$\kappa_{\SS}$ and we denote by
$\pi_{\SS}:\mathcal{H}_{\mathrm{m}}(\kappa_{\SS})\to\mathcal{H}_{\SS}$ the
canonical projection that forgets the marking. We define the forgetful map

(14) $\displaystyle p_{\SS}:\mathcal{H}\to\mathcal{H}_{\mathfrak{S}},\ \
q\mapsto(X_{q},\omega_{q},{\mathfrak{z}_{q}}|_{\SS}),$

and the corresponding map at the level of marked surfaces

(15)
$\displaystyle\tilde{p}_{\SS}:\mathcal{H}_{\mathrm{m}}\to\mathcal{H}_{\mathrm{m}}(\kappa_{\mathfrak{S}}),\
\ (q,f)\mapsto((X_{q},\omega_{q},{\mathfrak{z}_{q}}|_{\SS}),f).$

These two maps satisfy

(16) $\displaystyle\pi_{\SS}\circ\tilde{p}_{\SS}=p_{\SS}\circ\pi,$

and we have:

######  Proposition 4.1.

The fibers of $\tilde{p}_{\SS}$ are connected.

###### Proof.

Let $(q_{1},f_{1})$ and $(q_{2},f_{2})$ be two marked surfaces in
$\mathcal{H}_{\mathrm{m}}$ that have the same image under $\tilde{p}_{\SS}$.
By definition, this means that there is biholomorphism $\varphi:X_{q_{1}}\to
X_{q_{2}}$ such that $\varphi^{*}\omega_{q_{2}}=\omega_{q_{1}}$ and
$\varphi\circ{\mathfrak{z}_{q_{1}}}|_{\SS}={\mathfrak{z}_{q_{2}}}|_{\SS}$, as
well as an isotopy $(\psi_{t}:S\to S)_{t\in[0,1]}$ that connects the identify
map of $S$ to $f_{2}\circ\varphi\circ f_{1}$ without moving the points in
$\SS$. Let $\mathfrak{z}_{t}:\Sigma\to X_{q_{2}}$ be the map
$(f_{2}\circ\psi_{t})|_{\Sigma}$. Notice in particular that this map coincides
with $\mathfrak{z}_{2}$ on $\SS$. Define

$\gamma:[0,1]\to\mathcal{H}_{\mathrm{m}},\
t\mapsto(X_{q_{2}},\omega_{q_{2}},\mathfrak{z}_{t},f_{2}\circ\psi_{t}).$

The path $\gamma$ connects $(q_{2},f_{2})$ to $(q_{1},f_{1})$ while staying in
a single fiber of $\tilde{p}_{\SS}$. ∎

If we let ${\mathrm{Res}}_{\SS}:H^{1}(S,\Sigma;\mathbb{C})\to
H^{1}(S,\SS;\mathbb{C})$ be the canonical restriction map induced by the
inclusion $\SS\subset\Sigma$, we have

(17)
$\displaystyle\mathrm{dev}\circ\tilde{p}_{\SS}={\mathrm{Res}}_{\SS}\circ\mathrm{dev},$

where we denote the developing maps of $\mathcal{H}_{\mathrm{m}}$ and
$\mathcal{H}_{\mathrm{m}}(\kappa_{\mathfrak{S}})$ by the same symbol
$\mathrm{dev}$. Denote by $\mathfrak{R}_{\SS}$ the kernel of
${\mathrm{Res}}_{\SS}$ and let

$Z_{\SS}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathfrak{R}_{\SS}\cap
Z.$

If $v\in H^{1}(S,\Sigma;\mathbb{C})$, we will denote by $v|_{\SS}$ the image
of $v$ under ${\mathrm{Res}}_{\SS}$. Let
$\\{\xi^{*}:\xi\in\Sigma\\}\subset\mathfrak{R}$ be as in §2.3. Any element of
$\mathfrak{R}_{\SS}$ (resp. $Z_{\SS}$) is a linear combination with complex
(resp. real) coefficients of the $\xi^{*}$ where $\xi$ ranges over
$\Sigma\smallsetminus\SS$.

If $\xi\in\SS,$ we can think of $\xi^{*}$ simultaneously as an element of
$H^{1}(S,\Sigma;\mathbb{C})$ and $H^{1}(S,\SS;\mathbb{C})$, and this
identification defines a natural embedding

(18) $\iota:\ker\left(H^{1}(S,\SS;\mathbb{R}^{2})\to
H^{1}(S;\mathbb{R}^{2})\right)\to\mathfrak{R};$

that is, between the Rel subspaces corresponding to $\mathcal{H}_{\SS}$ and
$\mathcal{H}$.

######  Proposition 4.2.

Let $\mathcal{N}\subset\mathcal{H}_{\SS}$ be an invariant subvariety. Then
$\mathcal{M}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}p_{\SS}^{-1}(\mathcal{N})$
is an invariant subvariety. Furthermore, if $\mu$ is a $U$-invariant ergodic
Radon probability measure on $\mathcal{M}$ that assigns zero mass to the set
of surfaces with horizontal saddle connections, is invariant under
$\mathrm{Rel}_{v}$ for any $v\in Z_{\SS}$, and such that
${p_{\SS}}_{*}\mu=m_{\mathcal{N}}$, then $\mu=m_{\mathcal{M}}$.

###### Proof.

We start by proving that $\mathcal{M}$ is an invariant variety. Let
$Y\subset\mathcal{H}_{\mathrm{m}}(\kappa_{\SS})$ be an irreducible component
of $\pi_{\SS}^{-1}(\mathcal{N})$ and let $W$ be the linear subspace of
$H^{1}(S,\SS;\mathbb{C})$ on which it is modeled. Let $V\subset
H^{1}(S,\Sigma;\mathbb{C})$ be the pre-image of $W$ under the map
${\mathrm{Res}}_{\SS}$ and let $X$ be the pre-image of $Y$ under
$\tilde{p}_{\SS}$. We claim that $X$ is an equilinear manifold. Indeed, it
follows from Proposition 4.1 that $X$ is connected and it follows from (17)
that $X$ is open in $\mathrm{dev}^{-1}(V)$. Since $X$ is closed, we conclude
that $X$ coincides with a connected component of $\mathrm{dev}^{-1}(V)$. Since
$\mathrm{Mod}(S,\Sigma)$ acts trivially on $\mathfrak{R}$, the induced action
on $V$ of the stabilizer in $\mathrm{Mod}(S,\Sigma)$ of $X$ acts by
determinant 1 endomorphisms. This proves that $X$ is an equilinear manifold.
We deduce from (16) that

$\pi^{-1}(\mathcal{M})=\bigcup\tilde{p}_{\SS}^{-1}(Y),$

where $Y$ ranges over the set of irreducible components of
$\pi_{\SS}^{-1}(\mathcal{N})$. Furthermore, this union is locally finite as
the collection of irreducible components of $\pi_{\SS}^{-1}(\mathcal{N})$ is
itself locally finite. This proves that $\mathcal{M}$ is an invariant variety,
and it is easily verified that any irreducible component of
$\pi^{-1}(\mathcal{M})$ is of the form $\tilde{p}_{\SS}^{-1}(Y)$, where $Y$ is
an irreducible component of $\pi_{\SS}^{-1}(\mathcal{N})$.

The second assertion follows from Proposition 2.6. To see this, note that the
embedding $\iota$ defined in (18) induces an embedding of the spaces
$\mathfrak{R}_{\mathcal{N}}$ and $Z_{\mathcal{N}}$ in the spaces
$\mathfrak{R}_{\mathcal{M}}$ and $Z_{\mathcal{M}}$ respectively, satisfying

(19) $p_{\SS}\circ\mathrm{Rel}_{\iota(x)}=\mathrm{Rel}_{x}\circ p_{\SS}\text{
for all }x\in Z_{\mathcal{N}},$

and

(20) $Z_{\mathcal{M}}=\iota(Z_{\mathcal{N}})\oplus Z_{\SS}.$

We have assumed that $\mu$ is invariant under $Z_{\SS}$, and that
$p_{\SS*}\mu=m_{\mathcal{N}}$, which is invariant under $Z_{\mathcal{N}}.$
Considering the disintegration of $\mu$ into its conditional measures with
respect to the map $p_{\SS}$, which is unique up to a null set, and using (19)
and (20), we see that $\mu$ is invariant under $\mathrm{Rel}_{x}$ for any
$x\in Z_{\mathcal{M}}.$ ∎

######  Theorem 4.3.

Let $\mathcal{N}\subset\mathcal{H}_{\SS}$ be an invariant subvariety and let
$\mu$ be an ergodic $U$-invariant measure on $\mathcal{H}$ such that
${p_{\SS}}_{*}\mu=m_{\mathcal{N}}$. Then there is an invariant subvariety
$\mathcal{M}\subset\mathcal{H}$ and $x\in Z$ such that
$\mu={\mathrm{Rel}_{x}}_{*}m_{\mathcal{M}}$.

The proof will require the drift argument presented in §3.1. We will use some
of the notations and computations used in the proof of Proposition 3.1.

###### Proof.

Let $\mu$ be an ergodic $U$-invariant measure, let $N_{\mu}$ be the group as
in (8) and let $N_{\mu}^{\circ}$ be its connected component of identity. We
will first prove Theorem 4.3 under the additional assumption that

(21) $\displaystyle N^{\circ}_{\mu}\cap Z_{\SS}=\\{0\\}.$

We claim that for any $\varepsilon>0$, we can find a compact set
$L\subset\mathcal{H}$ with $\mu(L)>1-\varepsilon$ that intersects every fiber
of $p_{\SS}$ in at most a finite set. Indeed, by inner regularity of $\mu$,
let $K$ be a compact set in $\Omega_{\mu}$ of measure bigger than
$1-\varepsilon$. Combining Egorov’s theorem and Birkhoff’s ergodic theorem, we
can find a compact set $L\subset\Omega_{\mu}$ and $T_{0}>0$ so that for any
$T>T_{0}$ and any $q\in L,$

(22) $\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q\in K\\}|>1-2\varepsilon.$

Suppose there is $q\in\mathcal{N}$ such that
$L_{q}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}L\cap
p_{\SS}^{-1}(\\{q\\})$ is infinite. By compactness, there is sequence of
points $q_{n}\in L_{q}$ converging to a point $q_{\infty}\in L_{q}$ with
$q_{n}\neq q_{\infty}$. It follows from (17) that for $n$ large enough, there
is $w_{n}\in\mathfrak{R}_{\SS}$ such that $q_{n}=q_{\infty}\pluscirc w_{n}$
and $w_{n}\to 0$. It follows from Proposition 2.5 and (21) that for large
values of $n$ the vector $w_{n}$ is not in $Z_{\SS}$. If $\varepsilon$ is
small enough, we can argue as in the proof of Proposition 3.1 and show that
for any $n$ large enough, there is $s_{n}>0$ such that both $u_{s_{n}}q_{n}$
and $u_{s_{n}}q_{\infty}$ belong to $K$ and

$\frac{\eta\varepsilon}{4}\leq\|x_{n}+s_{n}y_{n}\|\leq\eta\varepsilon,$

where $x_{n}$ and $y_{n}$ are such that $w_{n}=x_{n}+\mathbf{i}y_{n}$,
$\|\cdot\|$ is some norm on $\mathfrak{R}_{\mathcal{N}}$, and $\eta$ is small
enough so that $\mathrm{Rel}_{w}(q)$ is well-defined for any
$w\in\mathfrak{R}$ with $\|w\|<\eta\varepsilon$. By (5),

$u_{s_{n}}q_{n}=u_{s_{n}}q_{\infty}\pluscirc(x_{n}+s_{n}y_{n}+\mathbf{i}y_{n}).$

Using the compactness of $K$, by taking the limit of a convergent subsequence,
as in the proof of Proposition 3.1, we obtain
$q_{\infty}^{\prime}\in\Omega_{\mu}$ and $x\in Z_{\SS}$ with
$0<\|x\|\leq\varepsilon$ such that $q_{\infty}^{\prime}\pluscirc
w\in\Omega_{\mu}$. By Proposition 2.5, this implies that $\mu$ is invariant
under $\mathrm{Rel}_{x}$. Since $\varepsilon$ can be taken arbitrarily small,
this contradicts (21), showing that $L_{q}$ is finite for $q\in\mathcal{N}.$

Now, let $\nu{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}{p_{\SS}}_{*}\mu$
and let $q\mapsto\nu_{q}$ be the fiber-wise measures obtained by
disintegration of $\mu$ along the fibers of $p_{\SS}$. The fact that $\mu$ is
$U$-invariant implies that there is a $\nu$-conull set $E\subset\mathcal{N}$
such that for any $s\in\mathbb{R}$ and any $q\in E$ we have

(23) ${u_{s}}_{\ast}\nu_{q}=\nu_{u_{s}q}.$

Let $\nu_{q}=\nu_{q}^{(at)}+\nu_{q}^{(na)}$ be the decomposition of $\nu_{q}$
into its atomic and non-atomic parts. That is, $\nu_{q}^{(at)}$ is the
restriction of $\nu_{q}$ to its collection of atoms. By (23) and the
ergodicity of $\nu$ we find that for $\nu$-a.e. $q$, either
$\nu_{q}=\nu_{q}^{(at)}$ or $\nu_{q}=\nu_{q}^{(na)}.$ Suppose by contradiction
that the latter holds, and consider once again the compact set $L$. We have
shown that $\nu(L)>0$ and $L_{q}$ is finite for any $q\in\mathcal{N}$. Then we
have $\nu_{q}(L)=\nu_{q}^{(na)}(L)=0$ for $\nu$-a.e. $q,$ and thus
$\mu(L)=\int_{\mathcal{N}}\nu_{q}(L)\ d\nu(q)=0,$ a contradiction.

Thus there is a conull set $E\subset\mathcal{N}$ such that $\nu_{q}$ is an
atomic measure whenever $q\in E$. By ergodicity and Equation (23) it also
follows that the number of atoms and their mass can be assumed to be constant
on $E$. We denote by $N$ this number of atoms, so each $q\in E$ has mass
$1/N$, and we denote by $\left\\{q^{(1)},\ldots,q^{(N)}\right\\}$ the atoms of
$\nu_{q}$.

We now claim that for any compact $L$ of $\mathcal{M}$ with
$\mu(L)>1-\frac{1}{N}$, there are $q\in L$, a sequence of times $s_{n}>0$
converging to $0$ and a sequence of $w_{n}\in\mathfrak{R}_{\SS}$ converging to
$0$ such that

(24) $q_{n}=v_{s_{n}}q\pluscirc w_{n}\in L,$

where $v_{s}=\exp\left(\begin{smallmatrix}0&&0\\\
s&&0\end{smallmatrix}\right)$. Indeed, let $L$ be such a compact set. For any
$q\in E$ we either have $\nu_{q}(L)=1$ or $\nu_{q}(L)\leq 1-\frac{1}{N}$.
Since $\mu(L)>1-\frac{1}{N}$, the set $\\{q\in E:\nu_{q}(L)=1\\}$ must have
positive measure and by Luzin’s theorem, it contains a set $E_{0}$ of positive
measure so that the restriction of the measurable function
$q\mapsto\left\\{q^{(1)},\dots,q^{(N)}\right\\}$ to $E_{0}$ is continuous. As
the measure $\nu$ is $G$-invariant, the set

$E_{1}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left\\{q\in
E_{0}:v_{-\frac{1}{n}}q\in E_{0}\ \text{for infinitely many}\ n\right\\}$

has positive measure. Let $q\in E_{1}$, so that $q\in E_{0}$ and there is a
decreasing sequence $s_{n}$ converging to $0$ such that
$q_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}v_{s_{n}}q\in E_{0}$. By
definition of $E_{1}$, for any $j\in\\{1,\dots,N\\}$ the points $q_{n}^{(j)}$
are in $L$ and so is $q^{(1)}$. Since $p_{\SS}$ is $G$-equivariant, we know
that $v_{s_{n}}q^{(1)}$ is in the same fiber of $p_{\SS}$ as
$\left\\{q_{n}^{(1)},\dots,q_{n}^{(N)}\right\\}$. By continuity and up to
extracting a further subsequence, there is a $j_{0}\in\\{1,\dots,N\\}$ and
$w_{n}\in Z_{\SS}$ such that $q_{n}^{(j_{0})}=v_{s_{n}}q^{(1)}\pluscirc w_{n}$
and $w_{n}\to 0$, which proves the claim.

We now employ the drift argument. Namely we take $K,K_{0}$ as in the proof of
Proposition 3.1. We can assume that the compact set $L$ of the previous
paragraph satisfies $L\subset K_{0}$ and that the sequence $q_{n}$ in $L$ is
as in (24). Then the drift argument shows that the measure $\mu$ is invariant
under a one-parameter group $\rho$ satisfying (12). Because $w\in Z_{\SS}$ and
by (21) we have $a\neq 0$. Using $B$-invariance and [EM18] as in the proof of
Theorem 3.3, we see that there is $x\in Z$ such that
$\mu={\mathrm{Rel}_{x}}_{\ast}m_{\mathcal{M}}$ for some invariant subvariety
$\mathcal{M}\subset\mathcal{H}$. This concludes the proof under the assumption
(21).

When assumption (21) does not hold, we proceed as follows. Let
$\SS^{\prime}\subset\Sigma$ be a subset of minimal cardinality containing
$\SS$, and for which

(25) $N^{\circ}_{\mu}\cap Z_{\SS^{\prime}}=\\{0\\}.$

Condition (25) is clearly satisfied for $\SS^{\prime}=\Sigma,$ so the
collection of subsets for which (25) holds is not empty. We define the map

$p_{\SS^{\prime},\SS}:\mathcal{H}_{\SS^{\prime}}\to\mathcal{H}_{\SS},\
q\mapsto(X_{q},\omega_{q},{\mathfrak{z}}|_{\SS}).$

Notice that this map is exactly the map defined in (14) in the case where the
source is the stratum $\mathcal{H}(\kappa_{\SS^{\prime}})$. By construction,
we have $p_{\SS}=p_{\SS^{\prime},\SS}\circ p_{\SS^{\prime}}$. Let
$\mathcal{N}^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}p^{0-1}_{\SS^{\prime},\SS}(\mathcal{N}).$
It follows from Proposition 4.2 applied to $\mathcal{H}_{\SS^{\prime}}$ and
$p_{\SS^{\prime},\SS}$ (in place of $\mathcal{H}$ and $p_{\SS}$) that
$\mathcal{N}^{\prime}$ is an invariant subvariety. To conclude the proof of
Theorem 4.3, it is enough to prove that
${p_{\SS^{\prime}}}_{*}\mu=m_{\mathcal{N}^{\prime}}$. Indeed, in that case we
can repeat the argument presented in the first part of the proof, with
$\SS^{\prime}$ in place of $\SS$.

We now set
$\nu{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}{p_{\SS^{\prime}}}_{\ast}\mu$
and prove $\nu=m_{\mathcal{N}^{\prime}}$. We shall use Proposition 4.2 applied
once again to $p_{\SS^{\prime},\SS}$ in place of $p_{\SS}$. Since
$p_{\SS^{\prime},\SS}\circ p_{\SS^{\prime}}=p_{\SS}$ and by our assumption
that ${p_{\SS}}_{*}\mu=m_{\mathcal{N}}$, we have
${p_{\SS^{\prime},\SS}}_{*}\nu=m_{\mathcal{N}}$. By the second assertion of
the Proposition, it is enough to show that the measure $\nu$ is invariant
under the flow $\mathrm{Rel}_{v}$ for any $v\in Z_{\SS^{\prime},\SS}$, where

$Z_{\SS^{\prime},\SS}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\ker\Big{(}H^{1}(S,\SS^{\prime};\mathbb{C})\to
H^{1}(S,\SS;\mathbb{C})\Big{)}.$

Let $\\{\xi^{*}:\xi\in\Sigma\\}$ be as in §2.3. Then the space
$Z_{\SS^{\prime},\SS}$ is spanned by the restriction to
$H^{1}(S,\SS^{\prime};\mathbb{C})$ of the $\xi^{*}$ (which formally is the
vector ${\mathrm{Res}}_{\SS^{\prime}}(\xi^{*})$) where $\xi$ ranges over the
points in $\SS^{\prime}\smallsetminus\SS$. Let
$\xi_{0}\in\SS^{\prime}\smallsetminus\SS$. By the minimality of
$\SS^{\prime}$, we have that

$\dim\big{(}N_{\mu}^{\circ}\cap
Z_{\SS^{\prime}\smallsetminus\\{\xi_{0}\\}}\big{)}>0.$

Let $x\in N_{\mu}^{\circ}\cap Z_{\SS^{\prime}\smallsetminus\\{\xi_{0}\\}}$ be
a non-zero element. We can write

$x=\lambda_{\xi_{0}}\xi_{0}^{*}+\sum_{\xi\in\Sigma\smallsetminus\SS^{\prime}}\lambda_{\xi}\xi^{*}$

and we claim that $\lambda_{\xi_{0}}\neq 0$. Indeed, otherwise we would have
that $x\in Z_{\SS^{\prime}}$ which, in combination with (25), would imply
$x=0$. By rescaling we can assume $\lambda_{\xi_{0}}=1$. Now, notice that the
restriction to $H^{1}(S,\SS^{\prime};\mathbb{C})$ of
$\sum_{\xi\in\Sigma\smallsetminus\SS^{\prime}}\lambda_{\xi}\xi^{*}$ is equal
to $0$ and thus we have
${\mathrm{Res}}_{\SS^{\prime}}(x)={\mathrm{Res}}_{\SS^{\prime}}(\xi_{0}^{*})$.
It follows from (17) and the fact that $x\in N_{\mu}$ that

${\mathrm{Rel}_{t\cdot{\mathrm{Res}}_{\SS}(\xi_{0}^{*})}}_{*}\nu={p_{\SS^{\prime}}}_{*}({\mathrm{Rel}_{t\cdot
x}}_{*}\mu)={p_{\SS^{\prime}}}_{*}\mu=\nu,$

hence $\nu$ is indeed invariant by $\mathrm{Rel}_{v}$ for any $v\in
Z_{\SS^{\prime},\SS}$. ∎

######  Remark 4.4.

The special case of Theorem 4.3 in which $\mathcal{N}$ is a closed
$\mathrm{GL}_{2}^{+}(\mathbb{R})$-orbit is the main result of [EMM06]. In the
same paper, the authors deduce that the orbit closure of a translation surface
that is a translation branched cover of a Veech surface satisfies the weak
classification of $U$-invariant measures. We recall that a translation cover
of $q_{0}$ is a pair $(q,\rho)$ where $\rho:X_{q}\to X_{q_{0}}$ is a
holomorphic map such that $\rho^{*}\omega_{q_{0}}=\omega_{q}$ and that $q_{0}$
is a Veech surface if its orbit is closed. Analogously, it should be possible
to deduce from Theorem 4.3 that the orbit-closure of a translation branched
cover of a surface in a rank one invariant subvariety of Rel dimension one,
satisfies weak classification of $U$-invariant measures. This would require
pinning down the relationship, as dynamical systems for the $U$-action,
between the spaces $p_{\SS}^{-1}(\mathcal{N})$ of Proposition 4.2 (adding
marked points to surfaces in an invariant subvariety $\mathcal{N}$), and the
space of branched translation covers of a fixed topological type of surfaces
in $\mathcal{N}$. See [SW08, Section 3] for a related discussion of the moduli
space of a branched covers.

For $\mathcal{M}=p_{\SS}^{-1}(\mathcal{N})$ as in Proposition 4.2, the
invariant subvarieties $\mathcal{M}^{\prime}\subset\mathcal{M}$ such that
$p_{\SS}(\mathcal{M}^{\prime})=\mathcal{N}$ correspond to finite
$\mathrm{GL}_{2}^{+}(\mathbb{R})$-equivariant configurations of marked points
over $\mathcal{N}$. Such configurations have been classified, see [EFW18,
AW21].

As a corollary we obtain that adding marked points to invariant subvarieties
that satisfy themselves the weak classification of $U$-invariant measures,
results in more such invariant subvarieties. In particular, we have:

######  Corollary 4.5.

Let $\mathcal{M}\subset\mathcal{H}$ be an invariant subvariety such
$p_{\Sigma_{0}}(\mathcal{M})$ has rank one and Rel-dimension one, where
$\Sigma_{0}$ is as in (13). Then $\mathcal{M}$ satisfies the weak
classification of $U$-invariant measures.

##  5\. Nondivergence of the horocycle flow and frequency of passage near
singular sets

The main results of this section are Proposition 5.7 and Proposition 5.8,
which uniformly bound the amount of time a horocycle trajectory can spend
either near the support of one of the measures described in §2.5, or near the
set of surfaces with a horizontal saddle connection of bounded length. A
special case of Proposition 5.7 was obtained in [BSW16, §10]. We also
establish a similar result for large circles in Proposition 5.12 that is used
for the proof of Theorem 7.4.

###  5.1. Singular sets and long horocycle trajectories

By a well-known compactness criterion, the sets

$\\{q\in\mathcal{H}:\text{any saddle connection on }q\text{ has length
}\geq\varepsilon\\}$

are an exhaustion of $\mathcal{H}$ by compact sets. We will be interested in
giving upper bounds on the amount of time a horocycle trajectory can spend
outside large compact sets. For measurable $A\subset\mathbb{R}$, we denote the
Lebesgue measure of $A$ by $|A|.$ Let $\mathcal{M}$ be an invariant subvariety
and let $\mathcal{M}_{r}$ denote the surfaces in $\mathcal{M}$ with no
horizontal saddle connections of length less than $r$.

######  Theorem 5.1 (Quantitative nondivergence of the horocycle flow,
[MW02]).

There are constants $C>0$, $\rho_{0}$ and $\alpha>0$ such that for every
$q\in\mathcal{M}$, every $\rho\in(0,\rho_{0}]$ and every $T>0$, if for every
saddle connection $\delta$ on $q$ we have
$\max_{s\in[0,T]}\|\mathrm{hol}_{u_{s}q}(\delta)\|\geq\rho$, then for any
$0<\varepsilon<\rho$,

(26) $\displaystyle|\\{s\in[0,T]:u_{s}q\ \text{has a saddle connection of
length}<\varepsilon\\}|<C\left(\frac{\varepsilon}{\rho}\right)^{\alpha}T.$

In particular,

  * (I)

For any $\varepsilon>0$ and any compact $L\subset\mathcal{M}$, there is a
compact $K$ such that for any $T>0$ and any $q\in L$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q\notin K\\}|<\varepsilon.$

  * (II)

For any $\varepsilon>0$ and $r>0$, there is a compact $K\subset\mathcal{M}$
such that for any $q\in\mathcal{M}_{r}$ there is a $T_{0}>0$ such that for all
$T>T_{0}$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q\notin K\\}|<\varepsilon.$

It will be useful to introduce a scalar product $\langle\cdot|\cdot\rangle$ on
$Z_{\mathcal{M}}$ and we denote by $\|\cdot\|$ the induced norm on
$Z_{\mathcal{M}}$. For $R>0$ and an invariant subvariety
$\mathcal{N}\varsubsetneq\mathcal{M}$, we denote

(27)
$B^{\perp}_{\mathcal{N}}(R){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left\\{x\in
Z_{\mathcal{N}}^{\perp}:\|x\|\leq R\right\\}.$

From [MW14, Thm. 11.2] or [BSW16, Thm. 6.1] we have:

######  Lemma 5.2.

There is a constant $\sigma>0$, depending on the scalar product on
$Z_{\mathcal{M}}$, such that for any $R>0$, if $x\in Z_{\mathcal{M}}$ with
$\|x\|<R$ and $q\in\mathcal{M}_{\sigma R}$ then $\mathrm{Rel}_{x}(q)$ is well-
defined.

######  Proposition 5.3.

Let $\mathcal{N}\subset\mathcal{M}$ be an invariant subvariety and let
$Z^{\prime}$ be any linear subspace of $Z_{\mathcal{M}}$ transverse to
$Z_{\mathcal{N}}$. Then for any $R>0$, there is $r>0$ such that the map

(28) $\displaystyle\mathcal{N}_{r}\times\\{x\in Z^{\prime}:\|x\|\leq
R\\}\to\mathcal{M},\ \ \ \ \ (q,x)\mapsto\mathrm{Rel}_{x}(q)$

is well-defined and injective.

###### Proof.

Let $K$ be a compact subset of $\mathcal{M}$ as in item (II) of Theorem 5.1
for $r=1$ and $\varepsilon=\frac{1}{2}$. Since $K$ is compact, there is
$\ell\in(0,1)$ such that any surface $q\in K$ does not have horizontal saddle
connections shorter than $\ell$. For any $x\in Z^{\prime}$, the vector field
$\vec{x}$ is transverse to $\mathcal{N}$. From this, and from Lemma 5.2, we
find that there is $\varepsilon^{\prime}<\frac{\ell}{\sigma}$ such that the
map

$K\times\\{x\in Z^{\prime}:\|x\|\leq\varepsilon^{\prime}\\}\to\mathcal{M},\ \
\ \ \ (q,x)\mapsto\mathrm{Rel}_{x}(q)$

is well-defined and injective.

Given $R>0$, let
$r{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{R}{\varepsilon^{\prime}}$
and $t{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\log r$, so that
$e^{t}\varepsilon^{\prime}=R\text{ and }r=e^{t}.$ Suppose
$q,q^{\prime}\in\mathcal{N}_{r}$ and $x,x^{\prime}\in\\{z\in
Z^{\prime}:\|z\|\leq R\\}$ are such that
$\mathrm{Rel}_{x}(q)=\mathrm{Rel}_{x^{\prime}}(q^{\prime})$ (it follows from
Lemma 5.2 and the choice of $\varepsilon^{\prime}$ that $\mathrm{Rel}_{x}(q)$
and $\mathrm{Rel}_{x^{\prime}}(q^{\prime})$ are well-defined). By (3) and (4),
one has $g_{-t}q,g_{-t}q^{\prime}\in\mathcal{N}_{1}$, and thus, by definition
of $K$ there is $s>0$ such that $u_{s}g_{-t}q$ and $u_{s}g_{-t}q^{\prime}$
both belong to $K$. By (5),

$\displaystyle\mathrm{Rel}_{e^{-t}x}(u_{s}g_{-t}q)=u_{s}g_{-t}\mathrm{Rel}_{x}(q)=u_{s}g_{-t}\mathrm{Rel}_{x^{\prime}}(q^{\prime})=\mathrm{Rel}_{e^{-t}x^{\prime}}(u_{s}g_{-t}q^{\prime}).$

By definition $e^{-t}x$ and $e^{-t}x^{\prime}$ belong to $\\{z\in
Z^{\prime}:\|z\|\leq\varepsilon^{\prime}\\}$, and thus $x=x^{\prime}$ and
$q=q^{\prime}$. ∎

######  Remark 5.4.

Let $\mathcal{N}\subset\mathcal{M}$ be an invariant subvariety. Proposition
5.3 implies that for any $R>0$, there is $r>0$ such that the map

(29) $\displaystyle\mathcal{N}_{r}\times
B^{\perp}_{\mathcal{N}}(R)\to\mathcal{M},\ \ \ \ \
(q,x)\mapsto\mathrm{Rel}_{x}(q)$

is well-defined and injective.

The following useful statement was derived from Theorem 5.1 in [BSW16, Prop.
10.5].

######  Proposition 5.5.

Let $L$ be a compact subset of $\mathcal{M}_{\infty}$, $\varepsilon>0$ and
$r>0$. There is an open neighborhood $\mathcal{W}$ of $L$ and a compact
$\widehat{\mathcal{W}}\subset\mathcal{M}_{r}$ containing $\mathcal{W}$ such
that for any $q\in\mathcal{H}$ and any interval $I\subset\mathbb{R}$ for which
$u_{s}q\in\mathcal{W}$ for some $s\in I$, we have

$\displaystyle\left|\left\\{s\in
I:u_{s}q\notin\widehat{\mathcal{W}}\right\\}\right|<\varepsilon|I|.$

By combining Proposition 5.3 and Proposition 5.5, we get the following:

######  Proposition 5.6.

Let $\mathcal{N}\subset\mathcal{M}$ be an invariant subvariety. For any $R>0$,
$\varepsilon>0$ and any compact $K\subset\mathcal{N}_{\infty}$, there is
$\delta>0$, a neighborhood $\mathcal{W}$ of $K$ and a compact set
$\widehat{\mathcal{W}}\subset\mathcal{N}$ containing $\mathcal{W}$, such that:

  * •

the map

$\displaystyle\widehat{\mathcal{W}}\times B^{\perp}_{\mathcal{N}}(R)\times
B^{\perp}_{\mathcal{N}}(\delta)\to\mathcal{M},\ \ \ \ \
(q,x,y)\mapsto\mathrm{Rel}_{x+\mathbf{i}\cdot y}(q)$

is well-defined, continuous and injective;

  * •

for any interval $I\subset\mathbb{R}$ and any $q\in\mathcal{M}$ the following
is satisfied: if $s\in I$ is such that $u_{s}q\in\mathcal{W}$, then

$\displaystyle\left|\left\\{s\in
I:u_{s}q\notin\widehat{\mathcal{W}}\right\\}\right|<\varepsilon|I|.$

###### Proof.

By Proposition 5.3, there exists $r>0$ such that the map
$\Phi:\mathcal{N}_{r}\times B^{\perp}_{\mathcal{N}}(R)\to\mathcal{M}$ defined
by (29) is well-defined and injective. Let $\mathcal{W}$ and
$\widehat{\mathcal{W}}$ be the sets provided by Proposition 5.5, corresponding
to this $r$ and to $L=K$, where $K$ is as in the statement of the Proposition.
Since $\widehat{\mathcal{W}}\subset\mathcal{N}_{r}$, it follows that the
restriction of $\Phi$ to $\widehat{\mathcal{W}}\times
B^{\perp}_{\mathcal{N}}(R)$ is still injective. Since $\widehat{\mathcal{W}}$
is compact, and since for any $y\in Z_{\mathcal{N}}^{\perp}$, $\mathbf{i}\cdot
y$ is transverse to $\mathcal{N}$, there is $\delta_{0}>0$ such that for any
$\delta\leq\delta_{0}$, the map

$\displaystyle\widehat{\mathcal{W}}\times B^{\perp}_{\mathcal{N}}(R)\times
B^{\perp}_{\mathcal{N}}(\delta)\to\mathcal{M},\ \ \ \ \
(q,x,y)\mapsto\mathrm{Rel}_{\mathbf{i}\cdot y}(\mathrm{Rel}_{x}(q))$

is well-defined, continuous and injective. Finally, it follows from [BSW16,
Prop. 4.5] that for any $(q,x,y)\in\widehat{\mathcal{W}}\times
B^{\perp}_{\mathcal{N}}(R)\times B^{\perp}_{\mathcal{N}}(\delta)$ we have
$\mathrm{Rel}_{\mathbf{i}\cdot
y}(\mathrm{Rel}_{x}(q))=\mathrm{Rel}_{x+\mathbf{i}\cdot y}(q)$, which
concludes the proof. ∎

For any $r>0$ and any $K\subset\mathcal{N}$, denote

$Z(K,R){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left\\{\mathrm{Rel}_{x}(q):q\in
K,\,x\in B^{\perp}_{\mathcal{N}}(R)\cap Z^{(q)}\right\\}.$

######  Proposition 5.7 (Frequency of passage near Rel pushes of invariant
subvarieties).

Let $\mathcal{N}\varsubsetneq\mathcal{M}$ be an invariant subvariety, let
$r>0$, let $K$ be a compact subset of $\mathcal{N}_{\infty}$ and let
$\varepsilon>0$. There is a neighborhood $\mathcal{U}$ of $Z(K,r)$ and $R>r$
such that for any compact subset $L$ of $\mathcal{M}_{R}\smallsetminus
Z(\mathcal{N},R)$, there is $T_{0}>0$ such that for any $T>T_{0}$ and any
$q\in L$, we have

(30) $\displaystyle\frac{1}{T}\
|\\{s\in[0,T]:u_{s}q\in\mathcal{U}\\}|<\varepsilon.$

###### Proof.

Let $R^{\prime}>r$ be large enough so that

(31) $\displaystyle R^{\prime}>\frac{(8+\varepsilon)(r+1)}{\varepsilon}.$

By Proposition 5.6, there is an open neighborhood $\mathcal{W}$ of $K$, a
compact $\widehat{\mathcal{W}}\subset\mathcal{N}$ that contains $\mathcal{W}$,
and $\delta>0$, such that the map

$\displaystyle\bar{\Phi}:\widehat{\mathcal{W}}\times
B^{\perp}_{\mathcal{N}}(R^{\prime})\times
B^{\perp}_{\mathcal{N}}(\delta)\to\mathcal{M},\ \
\bar{\Phi}(q,x,y){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Rel}_{x+\mathbf{i}\cdot
y}(q)$

is well-defined, continuous and injective, and such that for any interval
$I\subset\mathbb{R}$ and $s\in I$ and $q\in\mathcal{N}$ with
$u_{s}q\in\mathcal{W}$, we have

(32) $\left|\left\\{s\in
I:u_{s}q\notin\widehat{\mathcal{W}}\right\\}\right|<\frac{\varepsilon}{4}|I|.$

Denote the interior of $B^{\perp}_{\mathcal{N}}(R^{\prime})$ by
$\mathrm{int}(B^{\perp}_{\mathcal{N}}(R^{\prime}))$ (that is, use strict
inequality in (27)), and let

$\displaystyle\mathcal{V}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathcal{W}\times\mathrm{int}(B^{\perp}_{\mathcal{N}}(r+1))\times\mathrm{int}(B^{\perp}_{\mathcal{N}}(\delta))\
\ \ \text{ and }\ \ \
\mathcal{U}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\bar{\Phi}(\mathcal{V}).$

Since $\mathcal{M}$ has rank one, and since $\bar{\Phi}$ is a homeomorphism
onto its image, $\mathcal{U}$ is open in $\mathcal{M}$. It contains $Z(K,r)$
by construction. Let $q\in\mathcal{M}\smallsetminus Z(\mathcal{N},R^{\prime})$
and let

$I_{\mathcal{U},q}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{t>0:u_{t}q\in\mathcal{U}\\}.$

By injectivity of the map $\bar{\Phi}$, for any $t\in I_{\mathcal{U},q}$ there
is a unique triple $(q_{t},x_{t},y_{t})\in\mathcal{V}$ such that
$u_{t}q=\bar{\Phi}(q_{t},x_{t},y_{t})$. Since $q\notin
Z(\mathcal{N},R^{\prime})$ we have $y_{t}\neq 0$. For any $t\in
I_{\mathcal{U},q}$, let

(33) $\displaystyle\mathcal{J}_{t}$
$\displaystyle=t+\\{s\in\mathbb{R}:\|x_{t}+sy_{t}\|\leq R^{\prime}\\}$ (34)
$\displaystyle\mathcal{I}_{t}$
$\displaystyle=t+\\{s\in\mathbb{R}:\|x_{t}+sy_{t}\|<r+1\\}.$

Since $\mathcal{I}_{t}$ and $\mathcal{J}_{t}$ are sub-level sets of real
quadratic polynomials, they are intervals.

We claim that if $t\in I_{\mathcal{U},q}$ and $I$ is an interval that contains
one of the endpoints of $\mathcal{J}_{t}$, we have

(35) $|\mathcal{I}_{t}\cap I|<\frac{\varepsilon}{4}|\mathcal{J}_{t}\cap I|.$

Indeed, define
$p(s){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\|x_{t}+sy_{t}\|^{2}=\|x_{t}\|^{2}+2s\langle
x_{t},y_{t}\rangle+s^{2}\|y_{t}\|^{2}$ and for any $C\geq 0$, define

(36)
$\displaystyle\Delta_{C}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}4\langle
x_{t},y_{t}\rangle^{2}-4\|y_{t}\|^{2}(\|x_{t}\|^{2}-C^{2})).$

This is the discriminant of the real quadratic polynomial whose zeroes are
points at which $\|x_{t}+sy_{t}\|=C.$ Since $\|x_{t}+sy_{t}\|\geq 0$ for all
$s$, we have $\Delta_{0}\leq 0$ and since $R^{\prime}>r+1$, by the assumption
that $I$ contains one of the endpoints of $\mathcal{J}_{t}$ we have
$\Delta_{r+1}>0$. From the formula for the roots of a quadratic polynomial,
for any $C\geq r+1$ we have

$p(s)\leq C^{2}\ \Longleftrightarrow\ s\in-\frac{\langle
x_{t},y_{t}\rangle}{\|y_{t}\|^{2}}+\bigg{[}\frac{-\sqrt{\Delta_{C}}}{2\|y_{t}\|^{2}},\frac{\sqrt{\Delta_{C}}}{2\|y_{t}\|^{2}}\bigg{]}.$

By construction, we have that $s+t\in\mathcal{J}_{t}$ if and only if $p(s)\leq
R^{\prime 2}$ and $s+t\in\mathcal{I}_{t}$ if and only if $p(s)<(r+1)^{2}$.
Remembering that $\Delta_{r+1}=\Delta_{0}+4\|y_{t}\|^{2}(r+1)^{2}$ and that
$\Delta_{0}\leq 0$, we have

(37) $\frac{|\mathcal{I}_{t}\cap I|}{|\mathcal{J}_{t}\cap
I|}\leq\frac{\sqrt{\Delta_{r+1}}}{\|y_{t}\|^{2}}\,\cdot\,\frac{2\|y_{t}\|^{2}}{\sqrt{\Delta_{R^{\prime}}}-\sqrt{\Delta_{r+1}}}.$

Indeed, (37) is obvious in case $\mathcal{I}_{t}\cap I=\varnothing$; in case
it is not, $I$ contains a subinterval bounded by an endpoint of
$\mathcal{I}_{t}$ and an endpoint of $\mathcal{J}_{t}$, so an interval in
which $P(s)\in\left((r+1)^{2},R^{\prime 2}\right]$. This gives a lower bound
for the denominator $|\mathcal{J}_{t}\cap I|$, and as an upper bound for the
numerator we use $|\mathcal{I}_{t}|.$ We obtain

$\displaystyle\frac{|\mathcal{I}_{t}\cap I|}{|\mathcal{J}_{t}\cap I|}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq: we
compute}}}{{\leq}}2\frac{\sqrt{\Delta_{r+1}}}{\sqrt{\Delta_{R^{\prime}}}-\sqrt{\Delta_{r+1}}}$
$\displaystyle=2\frac{\sqrt{\Delta_{r+1}}({\sqrt{\Delta_{R^{\prime}}}+\sqrt{\Delta_{r+1}}})}{4\|y_{t}\|^{2}(R^{\prime
2}-(r+1)^{2})}=\frac{\Delta_{r+1}+\sqrt{\Delta_{r+1}\Delta_{R^{\prime}}}}{2\|y_{t}\|^{2}(R^{\prime
2}-(r+1)^{2})}$
$\displaystyle\leq\frac{\Delta_{r+1}+\sqrt{\Delta_{0}^{2}+16\|y_{t}\|^{4}R^{\prime
2}(r+1)^{2}}}{2\|y_{t}\|^{2}(R^{\prime 2}-(r+1)^{2})}$
$\displaystyle\leq\frac{\Delta_{r+1}-\Delta_{0}+4\|y_{t}\|^{2}R^{\prime}(r+1)}{2\|y_{t}\|^{2}(R^{\prime
2}-(r+1)^{2})}$ $\displaystyle\leq 2\frac{(r+1)^{2}+R^{\prime}(r+1)}{R^{\prime
2}-r^{2}}=\frac{2(r+1)}{R^{\prime}-r}\stackrel{{\scriptstyle\eqref{eq: def of
R'}}}{{<}}\frac{\varepsilon}{4}.$

This proves (35).

Now, we claim that if $s\in\mathcal{J}_{t}\smallsetminus\mathcal{I}_{t}$ is
such that $u_{s}q\in\mathcal{U}$ then
$u_{s-t}q_{t}\notin\widehat{\mathcal{W}}$. Indeed, we have
$u_{s}q=\mathrm{Rel}_{x_{t}+(s-t)y_{t}+\mathbf{i}\cdot y_{t}}(u_{s-t}q)$ by
(5) and thus, if $u_{s-t}q\in\widehat{\mathcal{W}}$, then
$u_{s}q=\bar{\Phi}(u_{s-t}q,x_{t}+(s-t)y_{t},y_{t})$. But the fact that
$s\notin\mathcal{I}_{t}$ yields a contradiction to the injectivity of
$\bar{\Phi}$.

For $I$ containing an endpoint of $\mathcal{J}_{t}$, we deduce

$\displaystyle|\mathcal{J}_{t}\cap I_{\mathcal{U},q}\cap I|=\ \ $
$\displaystyle|\mathcal{I}_{t}\cap I_{\mathcal{U},q}\cap
I|+|\big{(}\mathcal{J}_{t}\smallsetminus\mathcal{I}_{t}\big{)}\cap
I_{\mathcal{U},q}\cap I|$ $\displaystyle\leq\ \ $
$\displaystyle|\mathcal{I}_{t}\cap I|+\left|\left\\{s\in\mathcal{J}_{t}\cap
I:u_{s}q_{t}\notin\widehat{\mathcal{W}}\right\\}\right|$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq: our claim},\eqref{eq: the
interior}}}{{<}}$ $\displaystyle\frac{\varepsilon}{4}|\mathcal{J}_{t}\cap
I|+\frac{\varepsilon}{4}|\mathcal{J}_{t}\cap
I|=\frac{\varepsilon}{2}|\mathcal{J}_{t}\cap I|.$

Now, let $\sigma$ be as in Lemma 5.2, let
$c{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\max(1,\sigma)$ and let
$R{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}cR^{\prime}$. We will show
that $\mathcal{U}$ and $R$ satisfy the required conclusions. Indeed, let $L$
be a compact subset of $\mathcal{M}_{R}\smallsetminus Z(\mathcal{N},R)$. For
any $q\in L$, we will denote by $\mathcal{J}_{t}=\mathcal{J}_{q,t}$ the
interval as in (33) to stress the dependence on $q$. We claim that there is
$\delta_{0}$ such that for any $q\in L$ and for any $t\in I_{\mathcal{U},q}$,
if $0\in\mathcal{J}_{q,t}$ then $\|y_{t}\|\geq\delta_{0}$. Indeed, suppose by
contradiction that there is a sequence $q_{n}\in L$ together with $t_{n}\in
I_{\mathcal{U},q_{n}}$ such that $0\in\mathcal{J}_{q,t_{n}}$. Let
$(q_{n}^{\prime},x_{n},y_{n})\in\mathcal{V}$ such that
$u_{t_{n}}q_{n}=\bar{\Phi}(q^{\prime}_{n},x_{n},y_{n})$ and assume that
$\|y_{n}\|\to 0$. Since $0\in\mathcal{J}_{q,t_{n}}$, we have

$q_{n}=u_{-t_{n}}\bar{\Phi}(q^{\prime}_{n},x_{n},y_{n})=u_{-t_{n}}q^{\prime}_{n}\pluscirc(x_{n}-t_{n}y_{n}+\mathbf{i}y_{n})$

with $\|x_{n}-t_{n}y_{n}\|\leq R^{\prime}$. Passing to a subsequence, we can
assume that $q_{n}$ converges to some $q_{\infty}\in L$ and that
$x_{n}-t_{n}y_{n}$ converges to some $x_{\infty}$ satisfying
$\|x_{\infty}\|\leq R^{\prime}$. In particular,
$q_{\infty}\pluscirc(-x_{\infty},0)$ is well-defined by Lemma 5.2 and it
follows from [BSW16, Prop 4.6] that $q_{n}\pluscirc-(x_{n}-t_{n}y_{n},y_{n})$
is a sequence in $\mathcal{N}$ that converges to
$q_{\infty}\pluscirc(-x_{\infty},0)$. Since $\mathcal{N}$ is closed, this
shows that $q_{\infty}$ belongs to $Z(\mathcal{N},R)$, which is a
contradiction and our claim is proved.

Let $T>0$, let $q\in L$ and let $t\in I_{\mathcal{U},q}$ satisfy
$[0,T]\cap\mathcal{J}_{q,t}\neq\emptyset$. If $[0,T]$ does not contain one of
the endpoints of $\mathcal{J}_{q,t}$, then it is contained in
$\mathcal{J}_{q,t}$. In particular, $0\in\mathcal{J}_{q,t}$ and thus
$\|y_{t}\|\geq\delta_{0}$. It follows also that

$T<\inf(\mathcal{J}_{q,t})+|\mathcal{J}_{q,t}|<|\mathcal{J}_{q,t}|<\frac{\sqrt{\Delta_{R^{\prime}}}}{\|y_{t}\|^{2}}\leq
2\|y_{t}\|R^{\prime}\leq\frac{R}{\delta_{0}}.$

Let
$T_{0}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{R}{\delta_{0}}$.
We proved that for any $q\in L$, any $T>T_{0}$ and any $t\in
I_{\mathcal{U},q}$ such that $[0,T]\cap\mathcal{J}_{q,t}\neq\emptyset$, we
have that $[0,T]$ contains one of the endpoints of $\mathcal{J}_{q,t}$. The
set $[0,T]\cap I_{\mathcal{U},q}$ is covered by the sets
$(\mathcal{J}_{q,t})_{t\in I_{\mathcal{U},q}}$. From this cover, we can
extract a subcover $(\mathcal{J}_{j})$ such that any $q\in[0,T]$ is covered at
most twice, and hence

$\displaystyle|[0,T]\cap I_{\mathcal{U},q}|$
$\displaystyle\leq\sum_{j}|[0,T]\cap I_{\mathcal{U}}\cap\mathcal{J}_{j}|$
$\displaystyle<\frac{\varepsilon}{2}\sum_{j}|[0,T]\cap\mathcal{J}_{j}|\leq\varepsilon
T.$

∎

######  Proposition 5.8 (Frequency of passage near the horizontal saddle
connection locus).

Let $\varepsilon>0$ and let $r>0$. There is an open set $\mathcal{U}$ that
contains $\mathcal{M}\smallsetminus\mathcal{M}_{r}$ and $R>r$ such that for
any compact subset $L$ of $\mathcal{M}_{R}$, there is $T_{0}>0$ such for any
$T>T_{0}$ and any $q\in L$, we have

(38) $\displaystyle\frac{1}{T}\
|\\{s\in[0,T]:u_{s}q\in\mathcal{U}\\}|<\varepsilon.$

###### Proof.

Let $C$, $\rho_{0}$ and $\alpha$ as in Theorem 5.1, let
$\varepsilon^{\prime}<\rho_{0}$ be small enough so that
$C(\frac{\varepsilon^{\prime}}{\rho_{0}})^{\alpha}<\varepsilon$, let
$K\subset\mathcal{M}$ be the compact set of surfaces whose shortest saddle
connection has length $\geq\varepsilon^{\prime}$, let
$t>\log(\frac{r}{\varepsilon^{\prime}})$, let $R=\rho_{0}e^{t}$ and finally
let $\mathcal{U}=g_{t}(K^{c})$. Notice that
$\mathcal{M}\smallsetminus\mathcal{M}_{r}\subset\mathcal{U}$.

Now let $L$ be any compact subset of $\mathcal{M}_{R}$. By compactness, there
is $\eta>0$ such that for any $q\in L$ and any saddle connection $\delta$ on
$q$ with $\mathrm{hol}_{q}(\delta)=(x,y)$, we either have $|x|>R$ or
$|y|>\eta$. Let $T^{\prime}_{0}$ be large enough so that
$T^{\prime}_{0}e^{t}\eta\geq 2\rho_{0}$. This condition ensures for any $q\in
L$, the length of any saddle connection on $g_{-t}q$ will eventually become
larger than $\rho_{0}$ after applying the horocycle flow up to time $T_{0}$.
It follows from Theorem 5.1 that for any $T>T^{\prime}_{0}$ and any $q\in L$

$\displaystyle\frac{1}{T}\ |\\{s\in[0,T]:u_{s}g_{-t}q\in
K^{c}\\}|<\varepsilon.$

Finally, let $T_{0}=e^{2t}T_{0}^{\prime}$, let $q\in L$ and let $T>T_{0}$.
Then we have

$\displaystyle\frac{1}{T}\int_{0}^{T}\mathbf{1}_{\mathcal{U}}(u_{s}q)\,ds=\frac{1}{T}\int_{0}^{T}\mathbf{1}_{K^{c}}(g_{-t}u_{s}q)\,ds$
$\displaystyle=$
$\displaystyle\frac{1}{Te^{-2t}}\left|\left\\{s\in\left[0,Te^{-2t}\right]:u_{s}g_{-t}q\notin
K\right\\}\right|<\varepsilon.$

∎

###  5.2. Singular sets and large circles

We extend the Euclidean norm on $Z_{M}$ to a Euclidean norm on
$\mathfrak{R}_{\mathcal{M}}$, i.e., for any $v=x+\mathbf{i}\cdot
y\in\mathfrak{R}_{\mathcal{M}}$, we let $\|v\|=\sqrt{\|x\|^{2}+\|y\|^{2}}$.
Notice that $r_{\theta}$ acts on $\mathfrak{R}_{\mathcal{M}}$ by isometry.

######  Proposition 5.9.

Let $\mathcal{N}\subset\mathcal{M}$ be an invariant subvariety and let
$q\notin\mathcal{N}$. There exists $\tilde{\delta}>0$ such that for all $t>0$,
if there exists $\theta\in[0,2\pi]$ such that
$g_{t}r_{\theta}q=q^{\prime}\pluscirc v$ with $q^{\prime}\in\mathcal{N}$ and
$v\in\mathfrak{R}^{\perp}_{\mathcal{N}}$, then
$\|g_{-t}v\|\geq\tilde{\delta}$.

###### Proof.

Since $q\notin\mathcal{N}$ and $\mathcal{N}$ is closed, it follows by
continuity of Rel flows that there is $\tilde{\delta}>0$ such that for any
$w\in\mathfrak{R}^{\perp}_{\mathcal{N}}$ with $\|w\|<\tilde{\delta}$ we have
that $\mathrm{Rel}_{w}(q)$ is well-defined and does not belong to
$\mathcal{N}$. Let $t>0$ and assume that there are $\theta\in[0,2\pi]$ and
$q^{\prime}\in\mathcal{N}$ such that $g_{t}r_{\theta}q=q^{\prime}\pluscirc v$.
Let $w{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}-r_{-\theta}g_{-t}v.$ If
$\|g_{-t}v\|<\tilde{\delta}$ then we also have $\|w\|<\tilde{\delta}$, and
thus $q\pluscirc w$ is well-defined. But by (5), $q\pluscirc
w=r_{-\theta}g_{-t}q^{\prime}\in\mathcal{N}$ and we get a contradiction with
the definition of $\tilde{\delta}$. This implies that
$\|g_{-t}v\|\geq\tilde{\delta}$. ∎

Let $n$ be a positive integer. We consider the diagonal action of
$\mathrm{GL}_{2}^{+}(\mathbb{R})$ on $\mathbb{C}^{n}$ where
$\mathrm{GL}_{2}^{+}(\mathbb{R})$ acts on each copy of
$\mathbb{C}\simeq\mathbb{R}^{2}$ by linear transformations. For any
$v\in\mathbb{C}^{n}$, we let

(39) $\displaystyle E_{t,v}=\\{g_{t}r_{\theta}v:\theta\in[0,2\pi]\\}.$

We will need the following:

######  Lemma 5.10.

For any $0\leq\eta_{1}\leq\eta_{2}\leq 2$ and $0<\varepsilon<1$ we have

(40)
$\arccos(\varepsilon^{2}\eta_{1}-1)-\arccos(\varepsilon^{2}\eta_{2}-1)\leq\varepsilon(\mathrm{arccos}(\eta_{1}-1)-\mathrm{arccos}(\eta_{2}-1)).$

###### Proof.

Use $2\varepsilon x-\varepsilon^{2}x^{2}>\varepsilon(2x-x^{2})$ in conjunction
with the following computation:

$\displaystyle\arccos(\varepsilon^{2}\eta_{1}-1)-\arccos(\varepsilon^{2}\eta_{2}-1)=\int_{\varepsilon^{2}\eta_{1}-1}^{\varepsilon^{2}\eta_{2}-1}\frac{1}{\sqrt{1-x^{2}}}\
dx$ $\displaystyle=$
$\displaystyle\int_{\varepsilon^{2}\eta_{1}}^{\varepsilon^{2}\eta_{2}}\frac{1}{\sqrt{2x-x^{2}}}\
dx=\varepsilon^{2}\int_{\eta_{1}}^{\eta_{2}}\frac{1}{\sqrt{2\varepsilon^{2}x-\varepsilon^{4}x^{2}}}\
dx$ $\displaystyle\leq$
$\displaystyle\varepsilon\int_{\eta_{1}}^{\eta_{2}}\frac{1}{\sqrt{2x-x^{2}}}\
dx=\varepsilon(\arccos(\eta_{1}-1)-\arccos(\eta_{2}-1)).$

∎

For any $r>0$, let $B(r)$ be the ball of radius $r$ in $\mathbb{R}^{n}$. Let
$T,\delta>0$ and define

$\displaystyle
B(T,\delta){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{x+\mathbf{i}\cdot
y\in\mathbb{C}^{n}:(x,y)\in B(T)\times B(\delta)\\}.$

######  Proposition 5.11.

Let $v\in\mathbb{C}^{n}$, let $T,\delta>0$ and let $0<\varepsilon<1$. If
$E_{t,v}$ is not contained in $B(T,\delta)$ then

$\displaystyle|\\{\theta\in[0,2\pi]:g_{t}r_{\theta}v\in B(\varepsilon
T,\varepsilon\delta)\\}|<\varepsilon|\\{\theta\in[0,2\pi]:g_{t}r_{\theta}v\in
B(T,\delta)\\}|.$

###### Proof.

We first observe that, up to replacing $T$ and $\delta$ by $e^{-t}T$ and
$e^{t}\delta$, we can assume that $t=0$ and up to replacing $v$ by
$r_{\frac{\pi}{2}}v$ we can assume that $T\leq\delta$. Write
$v=(x_{1}+\mathbf{i}y_{1},\dots,x_{n}+\mathbf{i}y_{n})$ and
$r_{\theta}v=(x_{\theta},y_{\theta})$. Let $R=\|v\|$ and let
$\theta_{0}\in[0,2\pi]$ be such that

$\displaystyle\cos(2\theta_{0})\left(\sum_{i=1}^{n}x_{i}y_{i}\right)-\sin(2\theta_{0})\left(\sum_{i=1}^{n}\frac{x_{i}^{2}-y_{i}^{2}}{2}\right)=0$
$\displaystyle\cos(2\theta_{0})\left(\sum_{i=1}^{n}\frac{x_{i}^{2}-y_{i}^{2}}{2}\right)+\sin(2\theta_{0})\left(\sum_{i=1}^{n}x_{i}y_{i}\right)\geq
0.$

Let $r\geq 0$ such that

$r^{2}=2\left(\cos(2\theta_{0})\left(\sum_{i=1}^{n}\frac{x_{i}^{2}-y_{i}^{2}}{2}\right)+\sin(2\theta_{0})\left(\sum_{i=1}^{n}x_{i}y_{i}\right)\right).$

Notice that $r\leq R$ and

$\displaystyle\|x_{\theta+\theta_{0}}\|^{2}$
$\displaystyle=\frac{r^{2}}{2}\cos(2\theta)+\frac{R^{2}}{2}$
$\displaystyle\|y_{\theta+\theta_{0}}\|^{2}$
$\displaystyle=-\frac{r^{2}}{2}\cos(2\theta)+\frac{R^{2}}{2}.$

The degenerate case where $r=0$ is trivial so we can assume $r>0$ and up to
replacing $v$ by $r_{\theta_{0}}v$ we can assume that $\theta_{0}=0$. It
follows that for any $T^{\prime},\delta^{\prime}>0$, we have that
$r_{\theta}v\in B(T^{\prime},\delta^{\prime})$ if and only if

$\displaystyle-\frac{2\delta^{\prime
2}-R^{2}}{r^{2}}\leq\cos(2\theta)\leq\frac{2T^{\prime 2}-R^{2}}{r^{2}}.$

Let
$\eta_{1}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}-\frac{2\delta^{2}-(R^{2}+r^{2})}{r^{2}}$
and
$\eta_{2}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{2T^{2}-(R^{2}-r^{2})}{r^{2}}$.
The assumption that $E_{0,v}$ is not contained in $B(T,\delta)$ implies that
$\eta_{2}\leq 2$. Notice that
$\left[-\frac{2(\varepsilon\delta)^{2}-R^{2}}{r^{2}},\frac{2(\varepsilon
T)^{2}-R^{2}}{r^{2}}\right]\subset[\varepsilon^{2}\eta_{1}-1,\varepsilon^{2}\eta_{2}-1]$.
Using Lemma 5.10,

$\displaystyle|\\{\theta\in[0,2\pi]:g_{t}r_{\theta}v\in B(\varepsilon
T,\varepsilon\delta)\\}|$
$\displaystyle<\arccos(\max(0,\varepsilon^{2}\eta_{1})-1)-\arccos(\varepsilon^{2}\eta_{2}-1)$
$\displaystyle<\varepsilon(\arccos(\max(0,\eta_{1})-1)-\arccos(\eta_{2}-1))$
$\displaystyle<\varepsilon|\\{\theta\in[0,2\pi]\ |\ g_{t}r_{\theta}v\in
B(T,\delta)\\}|.$

∎

For any $r,\delta>0$ and any invariant subvariety
$\mathcal{N}\subset\mathcal{M}$, we denote

$\displaystyle
B_{\mathcal{N}}^{\perp}(r,\delta){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{x+\mathbf{i}\cdot
y\in\mathfrak{R}_{\mathcal{M}}:(x,y)\in B_{\mathcal{N}}^{\perp}(r)\times
B_{\mathcal{N}}^{\perp}(\delta)\\}$

######  Proposition 5.12 (Proportion of large circles near singular sets).

Let $\mathcal{N}\varsubsetneq\mathcal{M}$ be an invariant subvariety, let
$r>0$, let $K$ be a compact subset of $\mathcal{N}_{\infty}$ and let
$\varepsilon>0$. There is a neighborhood $\mathcal{U}$ of $Z(K,r)$ such that
for any $q\in\mathcal{M}\smallsetminus\mathcal{N}$, there is $t_{0}>0$ such
that for any $t>t_{0},$

$\displaystyle\frac{1}{2\pi}|\\{\theta\in[0,2\pi]:g_{t}r_{\theta}q\in\mathcal{U}\\}|<\varepsilon.$

###### Proof.

Let $R>0$ be large enough so that

$\displaystyle R>\frac{8}{\varepsilon}r,$

and let $\tilde{\delta}$ be as in Proposition 5.9. A straightforward extension
of [BSW16, Prop 12.5] to arbitrary rank one invariant subvarieties gives the
following analogue of Proposition 5.6: There exist $t_{0}>0$,
$\delta<\frac{\tilde{\delta}}{\sqrt{2}}$, a neighborhood $\mathcal{W}$ of $K$
and a compact set $\widehat{\mathcal{W}}\subset\mathcal{N}$ such that

  * •

the map

$\displaystyle\bar{\Phi}:\widehat{\mathcal{W}}\times
B^{\perp}_{\mathcal{N}}(R)\times
B^{\perp}_{\mathcal{N}}(\delta)\to\mathcal{M},\ \
\bar{\Phi}(q,x,y){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Rel}_{x+\mathbf{i}\cdot
y}(q)$

is well-defined, continuous and injective;

  * •

for any $t>t_{0}$ and any interval $I\subset
J{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\left[\frac{\pi}{2}-\frac{\varepsilon\pi}{4},\frac{\pi}{2}+\frac{\varepsilon\pi}{4}\right]\cup\left[\frac{3\pi}{2}-\frac{\varepsilon\pi}{4},\frac{3\pi}{2}+\frac{\varepsilon\pi}{4}\right]$,
if $q^{\prime}\in\mathcal{N}$ and $\theta\in I$ satisfy
$g_{t}r_{\theta}q^{\prime}\in\mathcal{W}$, then

(41) $\displaystyle|\\{\alpha\in
I:g_{t}r_{\alpha}g_{-t}q^{\prime}\notin\widehat{\mathcal{W}}\\}|<\frac{\varepsilon}{8}|I|.$

Let

$\displaystyle\mathcal{V}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathcal{W}\times\mathrm{int}(B^{\perp}_{\mathcal{N}}(\frac{\varepsilon}{8}R)\times
B^{\perp}_{\mathcal{N}}(\frac{\varepsilon}{8}\delta))\ \ \ \text{ and }\ \ \
\mathcal{U}=\bar{\Phi}(\mathcal{V}).$

Since $\mathcal{M}$ has rank one, and since $\bar{\Phi}$ is a homeomorphism
onto its image, $\mathcal{U}$ is open in $\mathcal{M}$, and contains $Z(K,r)$.
Let $t>t_{0}$ and let

$\displaystyle
I_{\mathcal{U},q}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\\{\theta\in[0,2\pi]:g_{t}r_{\theta}q\in\mathcal{U}\\}.$

We need to verify that $|I_{\mathcal{U},q}|<2\pi\varepsilon$. Since
$|J|<\varepsilon\pi$, it is enough to show that $|I_{\mathcal{U},q}\cap
J^{c}|<\pi\varepsilon$. For any $\theta\in I_{\mathcal{U},q}$, let
$(q_{\theta},x_{\theta},y_{\theta})\in\mathcal{V}$ such that
$g_{t}r_{\theta}q=\bar{\Phi}(q_{\theta},x_{\theta},y_{\theta})$ and define

$\displaystyle\mathcal{I}_{\theta}$
$\displaystyle{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\theta+\\{\alpha\in[0,2\pi]:g_{t}r_{\alpha}g_{-t}(x_{\theta}+\mathrm{i}y_{\theta})\in
B_{\mathcal{N}}^{\perp}(\frac{\varepsilon}{8}R,\frac{\varepsilon}{8}\delta)\\},$
$\displaystyle\mathcal{J}_{\theta}$
$\displaystyle{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\theta+\\{\alpha\in[0,2\pi]:g_{t}r_{\alpha}g_{-t}(x_{\theta}+\mathrm{i}y_{\theta})\in
B_{\mathcal{N}}^{\perp}(R,\delta)\\}.$

By definition, we know that
$\|g_{-t}(x_{\theta}+\mathrm{i}y_{\theta})\|>\tilde{\delta}$. In particular,
there is $\alpha\in[0,2\pi]$ such that
$r_{\alpha}g_{-t}(x_{\theta}+\mathrm{i}y_{\theta})=x+\mathrm{i}y$ with
$\|y\|^{2}>\frac{\tilde{\delta}^{2}}{2}>\delta^{2}$ and thus
$\alpha\notin\mathcal{J}_{\theta}$. Identifying $\mathbb{C}^{n}$ and
$\mathfrak{R}^{\perp}_{\mathcal{N}}$ with a
$\mathrm{GL}^{+}(2,\mathbb{R})$-equivariant isometry, it follows from
Proposition 5.11 that

(42)
$\displaystyle|\mathcal{I}_{\theta}|<\frac{\varepsilon}{8}|\mathcal{J}_{\theta}|.$

We claim that if
$\alpha\in\mathcal{J}_{\theta}\smallsetminus\mathcal{I}_{\theta}$ then either
$g_{t}r_{\alpha-\theta}g_{-t}q_{\theta}\notin\widehat{\mathcal{W}}$ or
$g_{t}r_{\alpha}q\notin\mathcal{U}$. Indeed, let
$\alpha\in\mathcal{J}_{\theta}$ and suppose
$g_{t}r_{\alpha-\theta}g_{-t}q_{\theta}\in\widehat{\mathcal{W}}$. It follows
from (5) that

$\displaystyle g_{t}r_{\alpha}q$
$\displaystyle=(g_{t}r_{\alpha-\theta}g_{-t})g_{t}r_{\theta}q$
$\displaystyle=(g_{t}r_{\alpha-\theta}g_{-t})(q_{\theta}\pluscirc(x_{\theta}+\mathbf{i}y_{\theta}))$
$\displaystyle=g_{t}r_{\alpha-\theta}g_{-t}q_{\theta}\pluscirc
g_{t}r_{\alpha-\theta}g_{-t}(x_{\theta}+\mathbf{i}y_{\theta}),$

and thus by injectivity of $\bar{\Phi}$, it follows that
$g_{t}r_{\alpha}q\notin\mathcal{U}$, which proves our claim. Using (41) and
(42), we obtain:

$\displaystyle|I_{\mathcal{U},q}\cap\mathcal{J}_{\theta}\cap J^{c}|$
$\displaystyle=|I_{\mathcal{U},q}\cap\mathcal{I}_{\theta}\cap
J^{c}|+|I_{\mathcal{U},q}\cap\mathcal{J}_{\theta}\smallsetminus\mathcal{I}_{\theta}\cap
J^{c}|$
$\displaystyle\leq|\mathcal{I}_{\theta}|+\left|\left\\{\alpha\in\mathcal{J}_{\theta}\cap
J^{c}:g_{t}r_{\alpha-\theta}g_{-t}q_{\theta}\notin\widehat{\mathcal{W}}\right\\}\right|$
$\displaystyle<\frac{\varepsilon}{8}|\mathcal{J}_{\theta}|+\frac{\varepsilon}{8}|\mathcal{J}_{\theta}\cap
J^{c}|\leq\frac{\varepsilon}{4}|\mathcal{J}_{\theta}|.$

We have shown that the set $I_{\mathcal{U},q}\cap J^{c}$ is covered by the
$(\mathcal{J}_{\theta})_{\theta\in I_{\mathcal{U},q}}$, and from this cover we
can extract a sub-cover $(\mathcal{J}_{\theta_{l}})_{l}$ such that any point
is covered at most twice. This yields

$|I_{\mathcal{U},q}\cap
J^{c}|\leq\sum_{l}|I_{\mathcal{U},q}\cap\mathcal{J}_{l}\cap
J^{c}|<\frac{\varepsilon}{2}\sum_{l}|\mathcal{J}_{l}|\leq\varepsilon\pi.$

∎

##  6\. From measure classification to equidistribution of orbits

For any rank-one invariant subvariety $\mathcal{M}$, we denote by
$\mathcal{S}_{\mathcal{M}}$ the set of surfaces $q\in\mathcal{M}$ without
horizontal saddle connections for which there is a proper invariant subvariety
$\mathcal{N}\varsubsetneq\mathcal{M}$ and $x\in Z_{\mathcal{N}}^{\perp}$ such
that $\mathrm{Rel}_{x}(q)\in\mathcal{N}$. By Proposition 1.3 in [EM18], there
are at most countably many invariant subvarieties of $\mathcal{M}$. We assume
there are countably many, and denote them by
$\left(\mathcal{N}^{(i)}\right)_{i\in\mathbb{N}}$ (the reader will have no
difficulty in adapting our arguments to the case in which there are finitely
many). With this notation,

$\displaystyle\mathcal{S}_{\mathcal{M}}=\underset{(i,j)\in\mathbb{N}\times\mathbb{N}}{\bigcup}Z\left(\mathcal{N}^{(i)},j\right);$

in particular, the set $\mathcal{S}_{\mathcal{M}}$ is measurable.

The following result is independent of the weak classification of
$U$-invariant measure and holds in arbitrary rank-one invariant subvarieties.
It contrasts with a surprising phenomenon in $\mathcal{H}(1,1)$ discovered in
[CSW20, Theorem 1.2].

######  Theorem 6.1.

Let $\mathcal{M}$ be a rank-one invariant subvariety, let
$q\in\mathcal{M}_{\infty}\smallsetminus\mathcal{S}_{\mathcal{M}}$ and

$\mu_{T}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{1}{T}\int_{0}^{T}u_{s\ast}\delta_{q}\,ds.$

Then any weak-$\ast$ limit of $(\mu_{T})_{T>0}$ is a $U$-invariant Radon
probability measure that assigns zero measure to
$\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and to
$\mathcal{S}_{\mathcal{M}}$.

###### Proof.

Let $\nu$ be a weak-$\ast$ limit of a convergent subsequence
$(\mu_{T_{n}})_{n>0}$.

Step 1. We show that $\nu(\mathcal{M})=1$. It is enough to show that for any
$\varepsilon>0$, there is a compact $K\subset\mathcal{M}$ such that
$\nu(K)\geq 1-\varepsilon$. We take $K$ as in item (II) in Theorem 5.1 for
$r=1$. By assumption $q$ does not have any horizontal saddle connections and
in particular, no horizontal saddle connection shorter than $r$. Thus there is
$T_{0}>0$ such that for any $T>T_{0}$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q\notin K\\}|<\varepsilon.$

It follows that for $n$ for which $T_{n}>T_{0}$, we have $\mu_{T_{n}}(K)\geq
1-\varepsilon$. Since $K$ is compact, by a well-known property of weak-*
convergence, we obtain $\nu(K)\geq 1-\varepsilon$.

Step 2. We now show that
$\nu(\mathcal{M}\smallsetminus\mathcal{M}_{\infty})=0$. For any $i>0$, let
$K_{i}\subset\mathcal{M}_{\infty}$ be the set of surfaces whose saddle
connections have length bounded below by $\frac{1}{i}$, and which have
horizontal saddle connections, all of whose lengths are bounded above by $i$.
By construction the sets $(K_{i})_{i>0}$ exhaust
$\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and thus it is enough to show
that for any $i>0$ and any $\varepsilon>0$, $\nu(K_{i})<\varepsilon.$ By a
well-known property of weak-* convergence, it suffices to show that for each
$\varepsilon>0$ and each $i$, there is an open set $\mathcal{U}$ that contains
$K_{i}$, such that $\mu_{T}(\mathcal{U})<\varepsilon$ for any $T$ large
enough. Once again, let $K$ be as in item (II) in Theorem 5.1 for $r=1$. Since
the lengths of all saddle connection are bounded below on the compact set $K$,
there is $t>0$ such that $g_{-t}(K_{i})\subset K^{c}$. We denote by
$\mathcal{U}$ the set $g_{t}(K^{c})$ and let $q_{t}=g_{-t}q$. Since $q_{t}$
does not have horizontal saddle connection either, there is a $T_{0}>0$ such
that for any $T>T_{0}$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{t}\notin K\\}|<\varepsilon.$

Now, let $T_{1}$ be large enough so that $e^{-2t}T_{1}>T_{0}$ and let
$T>T_{1}$. Then, using the commutation relation
$g_{-t}u_{s}q=u_{s^{\prime}}g_{t}q$, where
$s^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}e^{-2t}s$, we
obtain

$\displaystyle\mu_{T}(\mathcal{U})$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\mathbf{1}_{\mathcal{U}}(u_{s}q)\,ds$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\mathbf{1}_{K^{c}}(g_{-t}u_{s}q)\,ds$
$\displaystyle=\frac{1}{Te^{-2t}}\left|\left\\{s^{\prime}\in\left[0,Te^{-2t}\right]:u_{s^{\prime}}q_{t}\notin
K\right\\}\right|<\varepsilon.$

Step 3. We now show that for any invariant subvariety
$\mathcal{N}\varsubsetneq\mathcal{M}$ and $r>0$ we have
$\nu(Z(\mathcal{N},r))=0$. By Step 2, it is enough to show that for any
$\varepsilon>0$ and any compact $K\subset\mathcal{N}_{\infty}$, there is an
open set $\mathcal{U}$ that contains $Z(K,r)$ such that
$\mu_{T}(\mathcal{U})<\varepsilon$ for any $T$ large enough. This follows from
Proposition 5.7, applied to $L=\\{q\\}$. ∎

######  Theorem 6.2 (Genericity).

Suppose $\mathcal{M}$ satisfies the weak classification of $U$-invariant
measures. The orbit of any surface $q\in\mathcal{M}$ without horizontal saddle
connections is generic for some measure as in §2.5. That is, for any
$q\in\mathcal{M}_{\infty}$, there are $x\in Z_{\mathcal{M}}$, an invariant
subvariety $\mathcal{N}\subset\mathcal{M}$ and
$q^{\prime}\in\mathcal{N}_{\infty}$ such that $q=\mathrm{Rel}_{x}(q^{\prime})$
and

$\frac{1}{T}\int_{0}^{T}u_{s\ast}\delta_{q}\,ds\underset{T\to\infty}{\longrightarrow}\mathrm{Rel}_{x\ast}m_{\mathcal{N}}\
\ \ \ \text{ (w.r.t. the weak-* topology).}$

######  Remark 6.3.

It is instructive to compare Theorem 6.2 with [CSW20, Thm. 1.4], which gives
examples of surfaces in the stratum $\mathcal{H}(1,1)$ which are not generic
for any measure. Note that $\mathcal{H}(1,1)$ is not of rank one. It should
also be noted that the genericity result in [BSW16, Thm. 11.1] is stronger
than Theorem 6.2, as it does not assume that $q$ has no horizontal saddle
connections. This is possible since it relies on a stronger measure
classification result.

###### Proof.

Let $\mathcal{N}$ be the invariant subvariety of smallest dimension amongst
the collection of all invariant subvarieties $\mathcal{O}$ such that there is
$\xi\in Z_{\mathcal{M}}$ satisfying $\mathrm{Rel}_{\xi}(q)\in\mathcal{O}$.
Such a pair $(\mathcal{N},\xi)$ is unique by Proposition 5.3. For any $T>0$,
we define

$\displaystyle\nu_{T}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{1}{T}\int_{0}^{T}u_{s\ast}\delta_{\mathrm{Rel}_{\xi}(q)}\,ds.$

The measure $\nu_{T}$ is supported on $\mathcal{N}$. By definition of
$\mathcal{N}$, we have that
$\mathrm{Rel}_{\xi}(q)\notin\mathcal{S}_{\mathcal{N}}$ and since $q$ does not
have horizontal saddle connections we have that
$\mathrm{Rel}_{\xi}(q)\in\mathcal{N}_{\infty}$. It follows from Theorem 6.1
(applied to $\mathcal{N}$ and $\mathrm{Rel}_{\xi}(q)$) that any limit $\nu$ of
a convergent subsequence of $\nu_{T}$ is a Radon probability measure that
assigns zero measure to the set of surfaces with horizontal saddle
connections, as well as to the singular set $\mathcal{S}_{\mathcal{N}}$. We
decompose the measure $\nu$ into $U$-ergodic components; that is,

$\displaystyle\nu=\int_{\mathcal{M}}\nu_{p}\ d\nu(p)$

where there is a $\nu$-conull set $E$ comprised of surfaces $p\in\mathcal{M}$
for which $\nu_{p}$ is an ergodic $U$-invariant probability Radon measure that
assigns zero measure to $\mathcal{N}\smallsetminus\mathcal{N}_{\infty}$ and to
$\mathcal{S}_{\mathcal{N}}$. Let $p\in E$. Since $\mathcal{M}$ satisfies the
weak classification of $U$-invariant measures, $\nu_{p}$ is a measure obtained
by pushing a measure $m_{\mathcal{N}^{\prime}}$ on an invariant subvariety
$\mathcal{N}^{\prime}$ by an element of real Rel. Since $\nu_{p}$ is supported
in $\mathcal{N}$ and $\nu_{p}(\mathcal{S}_{\mathcal{N}})=0$, it must be that
$\nu_{p}=m_{\mathcal{N}}$. Therefore also $\nu=m_{\mathcal{N}}$, and since
$\nu$ was an arbitrary subsequential limit of $\nu_{T}$, we have that
$\nu_{T}\longrightarrow_{T\to\infty}m_{\mathcal{N}}$ (with respect to the
weak-* topology).

This amounts to saying that
$q^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Rel}_{\xi}(q)$
is generic for $m_{\mathcal{N}}$. From Proposition 2.5 we obtain that
$q=\mathrm{Rel}_{-\xi}(q^{\prime})$ is generic for
$\mathrm{Rel}_{-\xi\ast}m_{\mathcal{N}}$, which is exactly what we wanted with
$x=-\xi$. ∎

The next result shows that the convergence in Theorem 6.2 holds uniformly on
compact sets, outside finitely many obvious exceptions. The statement is
modeled on [DM93, Thm. 3] and will be useful in [CWY]. It is new even in the
case of eigenform loci in $\mathcal{H}(1,1)$ studied in [BSW16].

######  Theorem 6.4 (Uniform equidistribution).

Suppose $\mathcal{M}$ satisfies the weak classification of $U$-invariant
measures, let $K\subset\mathcal{M}$ be compact, let $\varepsilon>0$ and let
$f\in C_{c}(\mathcal{M})$. There are finitely many invariant subvarieties
$\mathcal{N}^{(1)},\ldots,\mathcal{N}^{(k)}$ contained in $\mathcal{M}$ and
$r>0$ such that for any compact $L\subset
K\cap\mathcal{M}_{r}\smallsetminus\bigcup_{i=1}^{k}Z\left(\mathcal{N}^{(i)},r\right)$,
there is $T_{0}>0$ such that for all $T>T_{0}$ and all $q\in L$, we have

$\displaystyle\left|\frac{1}{T}\int_{0}^{T}f(u_{s}q)\,ds-\int_{\mathcal{M}}f\
dm_{\mathcal{M}}\right|<\varepsilon.$

###### Proof.

Let $\left(\mathcal{N}^{(i)}\right)_{i\in\mathbb{N}^{\ast}}$ be the collection
of all invariant subvarieties properly contained in $\mathcal{M}$ (if the
collection is finite we index it so that $\mathcal{N}^{(i)}$ is eventually
constant). Suppose by way of contradiction that there are a compact set
$K\subset\mathcal{M}$, a function $f\in C_{c}(\mathcal{M})$ and
$\varepsilon_{0}>0$ such that the following holds:

Contradicting assumption:  For all $n\geq 1$ and $r>0$, there is a compact
$L\subset
K\cap\mathcal{M}_{r}\smallsetminus\bigcup_{i=1}^{n}Z\left(\mathcal{N}^{(i)},r\right)$
such that for all $S>0$, there is $T>S$ and $q\in L$ such that:

(43) $\displaystyle\left|\frac{1}{T}\int_{0}^{T}f(u_{s}q)\
ds-\int_{\mathcal{M}}f\ dm_{\mathcal{M}}\right|\geq\varepsilon_{0}.$

For any $i\in\mathbb{N}$, let $(K_{i,j})_{j\in\mathbb{N}}$ be an exhaustion by
compact subsets of $\mathcal{N}^{(i)}_{\infty}$ (the set of surfaces without
horizontal saddle connections in $\mathcal{N}^{(i)}$), let

$\displaystyle
K_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\underset{i\leq
n}{\bigcup}Z(K_{i,n},n),$

and let $L_{n}$ be the compact subset comprised of surfaces that have a
horizontal saddle connection of length at most $n$ and whose shortest saddle
connection is longer than $\frac{1}{n}$. We claim that there is a sequence of
surfaces $(q_{n})_{n\in\mathbb{N}}$ in $K$, a sequence of open sets
$(\mathcal{U}_{n})_{n\in\mathbb{N}}$, and a sequence of times
$(T_{n})_{n\in\mathbb{N}}$ with $T_{n}>n$ such that:

  1. (i)

for any $n$, $K_{n}\cup L_{n}\subset\mathcal{U}_{n}$;

  2. (ii)

for any $n$,

(44)
$\displaystyle\left|\frac{1}{T_{n}}\int_{0}^{T_{n}}f(u_{s}q_{n})\,ds-\int_{\mathcal{M}}f\
dm_{\mathcal{M}}\right|\geq\varepsilon_{0};$

  3. (iii)

for any $n>0$, for all $0<i\leq n$,

(45)
$\displaystyle\frac{1}{T_{n}}|\\{s\in[0,T_{n}]:u_{s}q_{n}\in\mathcal{U}_{i}\\}|\leq\frac{1}{i}.$

Indeed, for any $n\geq 1$ and any $0<i\leq n$, let $\mathcal{U}_{i,n}$ and
$R_{i,n}>n$ be as in Proposition 5.7 for $K=K_{i,n}$, $r=n$ and
$\varepsilon=\frac{1}{2n^{2}}$. Let also $\mathcal{W}_{n}$ and
$R_{n}^{\prime}$ be be the $\mathcal{U}$ and $R$ given by Proposition 5.8 for
$\varepsilon=\frac{1}{2n}$ and $r=n$. Now, define
$R_{n}=\max(R_{1,n},\ldots,R_{n,n},R_{n}^{\prime})$ and
$\mathcal{U}_{n}=\bigcup_{i\leq n}\mathcal{U}_{i,n}\cup\mathcal{W}_{n}$. By
construction, $\mathcal{U}_{n}$ contains $K_{n}\cup L_{n}$ and satisfies the
following property: for any compact subset $L$ of
$\mathcal{M}_{R_{n}}\smallsetminus\bigcup_{i\leq
n}Z\left(\mathcal{N}^{(i)},R_{n}\right)$, there is $S_{n}>0$ such that

(46) $\displaystyle\forall q\in L,\ \forall T>S_{n},\
\frac{1}{T}|\\{t\in[0,T]:u_{s}q\in\mathcal{U}_{n}\\}|\leq\frac{1}{n}.$

Now, let $n\geq 1$, let $r_{n}=\max(R_{1},\dots,R_{n})$ and let $L$ be as in
the contradicting assumption for $n$ and $r_{n}$. For any $i\leq n$, the set
$L$ is a compact subset of $\mathcal{M}_{R_{i}}\smallsetminus\bigcup_{j\leq
n}Z\left(\mathcal{N}^{(j)},R_{i}\right)$ and let $S_{i}$ be as in (46). We
define $S=\max(S_{1},\dots,S_{n},n)$. By the contradicting assumption, there
is $q_{n}\in L$ together with $T_{n}>S$ such that (44) is satisfied. By
definition of $T_{n}$, the surface $q_{n}$ satisfies (45). Since $r_{n}>n$,
this concludes the construction of $(q_{n})_{n\in\mathbb{N}}$.

Now define a sequence of measures by

$\displaystyle\mu_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{1}{T_{n}}\int_{0}^{T_{n}}\delta_{u_{s}q_{n}}\,ds.$

Our goal is to show that this sequence of measures weak-* converges to
$m_{\mathcal{M}}$, which will contradict (44). By passing to a subsequence, we
can suppose that $(\mu_{n})_{n>0}$ converges to some measure $\nu$, and we
need to show $\nu=m_{\mathcal{M}}$. We follow the strategy used in the proof
Theorem 6.1.

Step 1. We show that $\nu(\mathcal{M})=1$. It is enough to show that for any
$\varepsilon>0$, there is a compact $K_{0}\subset\mathcal{M}$ such that
$\mu_{n}(K_{0})\geq 1-\varepsilon$ for any $n$ large enough. We can choose
$K_{0}$ and $T_{0}$ as in item (I) in Theorem 5.1 for $r=1$ and $L=K$. For any
$n>T_{0}$ we have $T_{n}>T_{0}$ and thus

$\displaystyle\mu_{n}(K_{0})=\frac{1}{T_{n}}|\\{t\in[0,T_{n}]:u_{t}q_{n}\in
K_{0}\\}|\geq 1-\varepsilon.$

Step 2. We now want to show that
$\nu(\mathcal{M}\smallsetminus\mathcal{M}_{\infty})=0$. By construction, the
$(L_{i})_{i>0}$ exhaust $\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and
thus it is enough to show that for any $\varepsilon>0$ and $i>0$, there is an
open set $\mathcal{U}$ that contains $L_{i}$ and such that
$\mu_{n}(\mathcal{U})<\varepsilon$ for any $n$ large enough. Let

$N>\max\left(i,\frac{1}{\varepsilon}\right)$

and set $\mathcal{U}=\mathcal{U}_{N}$. By (i) and the definition of $N$,
$\mathcal{U}_{N}$ contains $L_{i}$ and for any $n>N$ we have from (45) that

$\displaystyle\mu_{n}(\mathcal{U})=\frac{1}{T_{n}}|\\{t\in[0,T_{n}]:u_{s}q_{n}\in\mathcal{U}_{N}\\}|\leq\frac{1}{N}<\varepsilon.$

Step 3. We now prove that if $\mathcal{N}\varsubsetneq\mathcal{M}$ is an
invariant subvariety and $r>0$, then $\nu(Z(\mathcal{N},r))=0$. Let $i>0$ such
that $\mathcal{N}=\mathcal{N}^{(i)}$. We recall that we had an exhaustion by
nested compact sets $(K_{i,j})_{j>0}$ of $\mathcal{N}^{(i)}_{\infty}$. It is
enough to show that for any $\varepsilon>0$ and $j>0$, there is a open set
$\mathcal{U}$ that contains $Z(K_{i,j},r)$ and such that
$\mu_{n}(\mathcal{U})<\varepsilon$ for any $n$ large enough. Let

$N>\max\left(i,j,r,\frac{1}{\varepsilon}\right)$

and set $\mathcal{U}=\mathcal{U}_{N}$. By (i) and the definition of $N$,
$\mathcal{U}_{N}$ contains $Z(K_{i,j},r)$ and for any $n>N$ we have from (45)
that

$\displaystyle\mu_{n}(\mathcal{U})=\frac{1}{T_{n}}|\\{t\in[0,T_{n}]:u_{s}q_{n}\in\mathcal{U}_{N}\\}|\leq\frac{1}{N+1}<\varepsilon.$

Step 4. Since for any $n>0$, we have $T_{n}>n$, the measure $\nu$ is a
$U$-invariant Radon measure. It follows from Step 1 that it is a probability
measure, from Step 2 that it assigns measure zero to the set of surfaces that
have a horizontal saddle connection and from Step 3 that it assigns measure
zero to the set $\mathcal{S}_{\mathcal{M}}$. Let

$\displaystyle\nu=\int_{\mathcal{M}}\nu_{p}\ d\nu(p)$

be the decomposition of $\nu$ into $U$-invariant ergodic measures. There is a
$\nu$-conull set $E$ of surfaces $p\in\mathcal{M}$ for which $\nu_{p}$ is an
ergodic $U$-invariant probability Radon measure that assigns measure zero to
the set of surfaces with horizontal saddle connections and such that
$\nu_{p}(\mathcal{S}_{\mathcal{M}})=0$. By the weak classification of
$U$-invariant measure, for every $p\in E$ we have $\nu_{p}=m_{\mathcal{M}}$,
and hence $\nu=m_{\mathcal{M}}$. This is a contradiction to (44), concluding
the proof. ∎

##  7\. Equidistribution results for sequences of measures

In this section we state and prove several equidistribution results regarding
limits of sequences of measures. The first result extends [BSW16, Thm. 1.5].

######  Theorem 7.1.

Suppose $\mathcal{M}$ satisfies the weak classification of $U$-invariant
measures, let $\mathcal{L}\subset\mathcal{M}$ be an invariant subvariety and
let $x\in Z_{\mathcal{M}}$. There are an invariant subvariety
$\mathcal{N}\subset\mathcal{M}$ that contains $\mathcal{L}$ and $x\in
Z_{\mathcal{N}}$ such that, in the weak-* topology,
${\mathrm{Rel}_{tx}}_{\ast}m_{\mathcal{L}}\to m_{\mathcal{N}}$ as
$t\to\infty$.

For the proof of Theorem 7.1 we will need the following:

######  Lemma 7.2.

Suppose $\mathcal{N}_{1},\mathcal{N}_{2}\varsubsetneq\mathcal{M}$ are two
invariant subvarieties. If there is $r>0$ such that
$m_{\mathcal{N}_{1}}(Z(\mathcal{N}_{2},r))>0$ then
$\mathcal{N}_{1}\subset\mathcal{N}_{2}$.

###### Proof.

Since $m_{\mathcal{N}_{1}}$ is $U$-invariant and ergodic, we have that
$m_{\mathcal{N}_{1}}(Z(\mathcal{N}_{2},r))=1$. For any $t$, since
$m_{\mathcal{N}_{1}}$ is $g_{t}$-invariant and
$g_{-t}(Z(\mathcal{N}_{2},r))=Z(\mathcal{N}_{2},e^{-t}r)$, we have
$m_{\mathcal{N}_{1}}(Z(\mathcal{N}_{2},e^{-t}r))=1.$ It follows that
$(\mathcal{N}_{2})_{\infty}=\bigcap_{n>0}Z(\mathcal{N}_{2},re^{-n})$ also has
full measure for $m_{\mathcal{N}_{1}}$. Since $m_{\mathcal{N}_{1}}$ assigns
zero measure to surfaces with horizontal saddle connections, we deduce that
$m_{\mathcal{N}_{1}}(\mathcal{N}_{2})=1$. This implies that $\mathcal{N}_{2}$
contains $\mathcal{N}_{1}=\mathrm{supp}\,m_{\mathcal{N}_{1}}$. ∎

###### Proof of Theorem 7.1.

Let $\mathcal{C}(\mathcal{L},x)$ be the collection of invariant subvarieties
$\mathcal{N}$ of $\mathcal{M}$ that contain $\mathcal{L}$ and such that $x\in
Z_{\mathcal{N}}$, and let $\mathcal{N}\in\mathcal{C}(\mathcal{L},x)$ be of
smallest dimension. We will show that
$\mu_{t}={\mathrm{Rel}_{tx}}_{\ast}m_{\mathcal{L}}$ converges to
$m_{\mathcal{N}}$ as $t\to\infty$.

It is enough to show that if
$\mu_{n}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mu_{t_{n}}$ is a
convergent subsequence then its limit is $m_{\mathcal{N}}$. Denote by $\nu$
the limit of $\mu_{n}$. By construction,
$\mathrm{supp}\,\nu\subset\mathcal{N}$.

Step 1. We show that $\nu$ is a probability measure. It is enough to show that
for any $\varepsilon>0$ there is compact $K\subset\mathcal{M}$ such that
$\mu_{n}(K)\geq 1-\varepsilon$ for $n$ large enough. We choose $K$ as in
Theorem 5.1, item (II) for $r=1$. The measure $m_{\mathcal{L}}$ assigns full
measure to surfaces without horizontal saddle connections, and since this set
is invariant under $\mathrm{Rel}_{tx}$ we have
$\mu_{n}(\mathcal{M}_{\infty})=1$. Therefore there is
$q_{n}\in\mathcal{M}_{\infty}$ that is Birkhoff generic for the indicator
function $\mathbf{1}_{K}$ with respect to $\mu_{n}$. By the choice of $K$, for
any $T$ large enough,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}\in K\\}|>1-\varepsilon.$

By Birkhoff genericity, the left-hand side tends to $\mu_{n}(K)$ as
$T\to\infty$ and thus $\mu_{n}(K)\geq 1-\varepsilon$.

Step 2. We now show that
$\nu(\mathcal{M}\smallsetminus\mathcal{M}_{\infty})=0$. For any $i>0$, let
$K_{i}$ be the closure of the set of surfaces whose saddle connection lengths
are bounded below by $\frac{1}{i}$, and which have at least one horizontal
saddle connection, with the lengths of all horizontal saddle connections
bounded above by $i$. By construction the $(K_{i})_{i>0}$ exhaust
$\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and thus it is enough to show
that for any $i>0$ and any $\varepsilon>0$, there is an open set $\mathcal{U}$
that contains $K_{i}$ and such that $\mu_{n}(\mathcal{U})\leq\varepsilon$ for
any $n$ large enough. Once again, let $K$ be as in item (II) in Theorem 5.1
for $r$ as above. Since the length of saddle connections is bounded below on
compact sets and $K$ is compact, there is a $t>0$ such that
$g_{-t}(K_{i})\subset K^{c}$, so that

$K_{i}\subset\mathcal{U}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}g_{t}(K^{c}).$

Let $n$ be large enough so that $t_{n}>t$. We have
$\mu_{n}(\mathcal{M}\smallsetminus\mathcal{M}_{re^{t_{n}}})=0$ and thus there
is a surface $q_{n}$ without horizontal saddle connections shorter than
$re^{t_{n}}$ that is Birkhoff generic for $\mathbf{1}_{K^{c}}$ with respect to
$\mu_{n}$. The surfaces
$q_{n}^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}g_{-t}q_{n}$ do
not have horizontal saddle connections shorter that $r$ and thus there is
$T_{0}>0$ such that for any $T>T_{0}$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}^{\prime}\notin
K\\}|<\varepsilon.$

Now, let $T_{1}$ be large enough so that $e^{-2t}T_{1}>T_{0}$. For any
$T>T_{1}$ we have

$\displaystyle\frac{1}{T}\int_{0}^{T}\mathbf{1}_{\mathcal{U}}(u_{s}q_{n})\,ds=\frac{1}{T}\int_{0}^{T}\mathbf{1}_{K^{c}}(g_{-t}u_{s}q_{n})\
ds$ $\displaystyle=$
$\displaystyle\frac{1}{Te^{-2t}}\left|\left\\{s\in\left[0,Te^{-2t}\right]:u_{s}q_{n}^{\prime}\notin
K\right\\}\right|<\varepsilon.$

By Birkhoff genericity the quantity on the left-hand side converges to
$\mu_{n}(\mathcal{U})$ as $T\to\infty$ and thus
$\mu_{n}(\mathcal{U})\leq\varepsilon$.

Step 3. Let $\mathcal{N}^{\prime}\varsubsetneq\mathcal{N}$ be an invariant
subvariety and let $r>0$. We want to show that
$\nu(Z(\mathcal{N}^{\prime},r)=0$. It is enough to show that for any compact
subset $K\subset\mathcal{N}^{\prime}_{\infty}$ and $\varepsilon>0$ there is an
open set $\mathcal{U}$ that contains $Z(K,r)$ such that
$\mu_{n}(\mathcal{U})\leq\varepsilon$ for $n$ large enough. Let $\mathcal{U}$
and $R>r$ be as in Proposition 5.7.

Suppose first that $\mathcal{L}$ is not contained in $\mathcal{N}^{\prime}$.
If $\mu_{n}(Z(\mathcal{N}^{\prime},R))>0$ then
$m_{\mathcal{L}}(Z(\mathcal{N}^{\prime},R+t_{n}\|x\|))>0$ and thus by Lemma
7.2, we get that $\mathcal{L}\subset\mathcal{N}^{\prime}$, which is a
contradiction. It follows that $\mu_{n}(Z(\mathcal{N}^{\prime},R))=0$ and thus
by Birkhoff’s Theorem, one can find a point $q_{n}\in\mathcal{M}\smallsetminus
Z(\mathcal{N}^{\prime},R)$ such that

$\displaystyle\mu_{n}(\mathcal{U})=\underset{T\to\infty}{\lim}\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}\in\mathcal{U}\\}|.$

Applying Proposition 5.7 to $L=\\{q_{n}\\}$ shows that for all $T$ large
enough,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}\in\mathcal{U}\\}|<\varepsilon.$

This shows that $\mu_{n}(\mathcal{U})\leq\varepsilon$.

Suppose now that $\mathcal{L}\subset\mathcal{N}^{\prime}$. By definition of
$\mathcal{N}$, we know that $x\notin Z_{\mathcal{N}^{\prime}}$. It thus
follows from Proposition 5.3 that the map

$\displaystyle\Phi:\mathcal{N}_{\infty}^{\prime}\times\mathbb{R}\to\mathcal{N},\
\ \ \ \
\Phi(q,x){\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\mathrm{Rel}_{tx}(q)$

is injective. This means that for any $t_{n}$ large enough such that
$t_{n}\|x\|>R$, the surface $q_{n}=\mathrm{Rel}_{t_{n}x}(q_{n}^{\prime})$ does
not belong to $Z(\mathcal{N},R)$ and thus applying once more Proposition 5.7
to $L=\\{q_{n}\\}$ shows that $\mu_{n}(\mathcal{U})\leq\varepsilon$ for $n$
large enough.

To deduce now that $\nu=m_{\mathcal{N}}$, we use the ergodic decomposition
argument which we used in Step 4 of the Proof of Theorem 6.4. ∎

######  Theorem 7.3.

Suppose $\mathcal{M}$ satisfies the weak classification of $U$-invariant
measures and let $\mu$ be a $U$-invariant ergodic Radon probability measure on
$\mathcal{M}$. Then there is an invariant subvariety
$\mathcal{N}\subset\mathcal{M}$ such that ${g_{t}}_{\ast}\mu$ converges to
$m_{\mathcal{N}}$ as $t\to\infty$. If we furthermore assume that
$\mu(\mathcal{M}_{\infty})=1$, then there is an invariant subvariety
$\mathcal{L}\subset\mathcal{N}$ such that ${g_{t}}_{\ast}\mu$ converges to
$m_{\mathcal{L}}$ as $t\to-\infty$.

Remark. If $\mu$ is an ergodic $U$-invariant measure with
$\mu(\mathcal{M}_{\infty})<1$ then by ergodicity $\mu$-a.e $q\in\mathcal{M}$
has a horizontal saddle connection and ${g_{t}}_{\ast}\mu$ converges to the
zero measure as $t\to-\infty$.

###### Proof of Theorem 7.3.

We first prove the first assertion. We assume first that for any invariant
subvariety $\mathcal{N}\varsubsetneq\mathcal{M}$, we have
$\mu(\mathcal{N})=0$. In that case, we are going to show that
${g_{t}}_{\ast}\mu$ converges to $m_{\mathcal{M}}$ as $t\to\infty$. Let
$\mu_{n}={g_{t_{n}}}_{\ast}\mu$ be a convergent subsequence when $t\to\infty$
and let $\nu$ be its limit. If generic surfaces for $\mu$ have some horizontal
saddle connections, let $r$ be the length of the shortest of them. Otherwise,
let $r=1$.

Step 1. We show that $\nu(\mathcal{M})=1$. Let $\varepsilon>0$. It is enough
to show that there is a compact $K$ such that $\mu_{n}(K)\geq 1-\varepsilon$.
Let $K$ be a compact as in item (II) of Theorem 5.1 with $r,\varepsilon$ as
above. The measure $\mu_{n}$ assigns measure zero to surfaces with a saddle
connection shorter than $r$ and thus there is a surface $q_{n}$ that is
Birkhoff generic for the indicator function $\mathbf{1}_{K}$ with respect to
$\mu_{n}$. By definition of $K$, for any $T$ large enough,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}\in K\\}|\geq 1-\varepsilon.$

By Birkhoff genericity, the quantity on the left-hand side converges to
$\mu_{n}(K)$ as $T\to\infty$ and thus $\mu_{n}(K)\geq 1-\varepsilon$.

Step 2. We now show that
$\nu(\mathcal{M}\smallsetminus\mathcal{M}_{\infty})=0$. For any $i>0$, let
$K_{i}$ be the closure of the set of surfaces whose saddle connection lengths
are bounded below by $\frac{1}{i}$, and which have at least one horizontal
saddle connection, with the lengths of all horizontal saddle connections
bounded above by $i$. By construction the $(K_{i})_{i>0}$ exhaust
$\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and thus it is enough to show
that for any $i>0$ and any $\varepsilon>0$, there is an open set $\mathcal{U}$
that contains $K_{i}$ and such that $\mu_{n}(\mathcal{U})\leq\varepsilon$ for
any $n$ large enough. Once again, let $K$ be as in item (II) in Theorem 5.1
for $r$ as above. Since the length of saddle connections is bounded below on
compact sets and $K$ is compact, there is a $\tau>0$ such that
$g_{\tau}(K_{i})\subset K^{c}$, so that

$K_{i}\subset\mathcal{U}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}g_{\tau}(K^{c}).$

Let $n$ be large enough so that $t_{n}>\tau$. We have
$\mu_{n}(\mathcal{M}_{re^{t_{n}}})=0$ and thus there is a surface $q_{n}$
without horizontal saddle connections shorter than $re^{t_{n}}$ that is
Birkhoff generic for $\mathbf{1}_{K^{c}}$ with respect to $\mu_{n}$. The
surfaces
$q_{n}^{\prime}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}g_{-\tau}q_{n}$
do not have horizontal saddle connections shorter that $r$ and thus there is
$T_{0}>0$ such that for any $T>T_{0}$,

$\displaystyle\frac{1}{T}|\\{s\in[0,T]:u_{s}q_{n}^{\prime}\notin
K\\}|<\varepsilon.$

Now let $T_{1}$ be large enough so that $e^{-2t}T_{1}>T_{0}$. For any
$T>T_{1}$ we have

$\displaystyle\frac{1}{T}\int_{0}^{T}\mathbf{1}_{\mathcal{U}}(u_{s}q_{n})\,ds=\frac{1}{T}\int_{0}^{T}\mathbf{1}_{K^{c}}(g_{-\tau}u_{s}q_{n})\
ds$ $\displaystyle=$
$\displaystyle\frac{1}{Te^{-2\tau}}\left|\left\\{s\in\left[0,Te^{-2\tau}\right]:u_{s}q_{n}^{\prime}\notin
K\right\\}\right|<\varepsilon.$

By Birkhoff genericity the quantity on the left-hand side converges to
$\mu_{n}(\mathcal{U})$ as $T\to\infty$ and thus
$\mu_{n}(\mathcal{U})\leq\varepsilon$.

Step 3. We now show that for any invariant subvariety
$\mathcal{N}\varsubsetneq\mathcal{M}$ and $r>0$ we have
$\nu(Z(\mathcal{N},r))=0$. It is enough to show that for any compact subset
$K$ of $\mathcal{N}_{\infty}$ and $\varepsilon>0$, there is a open set
$\mathcal{U}$ that contains $Z(K,r)$ and such that
$\mu_{n}(\mathcal{U})<\varepsilon$ for any $n$ large enough. We claim that
there is $r_{0}>0$ such that $\mu(Z(\mathcal{N},r_{0}))=0$. Indeed, suppose
that for any $n>1$, we have $\mu(Z(\mathcal{N},\frac{1}{n}))>0$. Since $\mu$
is ergodic and the sets $Z(\mathcal{N},r)$ are $U$-invariant, we actually have
$\mu\left(Z(\mathcal{N},\frac{1}{n})\right)=1$ and thus
$\mathcal{N}_{\infty}=\bigcap_{n>1}Z(\mathcal{N},\frac{1}{n})$ also has full
measure, contradicting the assumption on $\mu$ made at the beginning of the
proof. This proves the claim.

According to Proposition 5.7, there is $R>r$ and a neighborhood $\mathcal{U}$
such that for any compact subset $L$ of $\mathcal{M}\smallsetminus
Z(\mathcal{N},R)$, there is $T_{0}>0$ such that for any $T>T_{0}$,

$\displaystyle\frac{1}{T}\
|\\{s\in[0,T]:u_{s}q\in\mathcal{U}\\}|<\varepsilon.$

Let $n$ be large enough so that $e^{-t_{n}}R<r_{0}$ and let $q_{n}$ be a
generic point for the measure $\mu_{n}$. If $q_{n}\in Z(\mathcal{N},R)$ then
the measure $\mu$ is supported on $Z(\mathcal{N},e^{-t_{n}}R)$ and thus
$\mu(Z(\mathcal{N},r_{0}))=1$, a contradiction. Thus we can apply Proposition
5.7 to $L=\\{q_{n}\\}$ to find $T_{0}$ such that for any $T>T_{0}$, we have

$\displaystyle\frac{1}{T}\
|\\{s\in[0,T]:u_{s}q_{n}\in\mathcal{U}\\}|<\varepsilon.$

By the Birkhoff Ergodic Theorem, the left-hand side converges to
$\mu_{n}(\mathcal{U})$ when $T$ tends to $+\infty$ and thus
$\mu_{n}(\mathcal{U})<\varepsilon$ for any $n$ large enough. We prove as in
the final step of the proof of Theorem 6.4 that $\nu=m_{\mathcal{M}}$.

General case. Suppose now $\mu$ is any $U$-invariant ergodic Radon probability
measure and consider the collection $\mathcal{C}(\mu)$ of all the invariant
subvarieties that have positive measure for $\mu$. Then in fact, by
ergodicity, $\mu(\mathcal{N})=1$ for any $\mathcal{N}\in\mathcal{C}(\mu)$. Let
$\mathcal{N}_{0}\in\mathcal{C}(\mu)$ of smallest dimension. We claim that
there is no $\mathcal{N}_{1}\in\mathcal{C}(\mu)$ which is properly contained
in $\mathcal{N}_{0}$. Indeed, if $\mathcal{N}_{1}\in\mathcal{C}(\mu)$ with
$\mathcal{N}_{1}\subset\mathcal{N}_{0}$ then
$\dim\mathcal{N}_{1}=\dim\mathcal{N}_{0}$ by choice of $\mathcal{N}_{0}$, so
that $\mathcal{N}_{1}$ is open in $\mathcal{N}_{0}$. This implies that
$m_{\mathcal{N}_{0}}(\mathcal{N}_{1})>0$, which implies by ergodicity and
$G$-invariance that $m_{\mathcal{N}_{0}}=m_{\mathcal{N}_{1}}$ and hence
$\mathcal{N}_{0}=\mathcal{N}_{1}$. Thus we can now apply the first part of the
proof to $\mathcal{N}_{0}$ in place of $\mathcal{M}$, which proves that
${g_{t}}_{\ast}\mu$ converges to $m_{\mathcal{N}_{0}}$.

We now prove the second assertion of the statement. We assume that
$\mu(\mathcal{M}_{\infty})=1$. Since $\mathcal{M}$ satisfies the weak
classification of $U$-invariant measures, it follows that there is an
invariant subvariety $\mathcal{L}\subset\mathcal{M}$ and $x\in
Z_{\mathcal{M}}$ such that $\mu={\mathrm{Rel}_{x}}_{\ast}m_{\mathcal{L}}$.
Notice that for any $t\in\mathbb{R}$, we have
${g_{t}}_{\ast}\mu={\mathrm{Rel}_{e^{t}x}}_{\ast}m_{\mathcal{L}}$. It follows
from Theorem 7.1 and the first part of the proof that
$\mathcal{L}\subset\mathcal{N}$. Finally, it is clear that ${g_{t}}_{\ast}\mu$
converges to $m_{\mathcal{L}}$ when $t\to-\infty$. ∎

######  Theorem 7.4.

Suppose $\mathcal{M}$ satisfies the weak classification of horocycle invariant
measures. Let $q\in\mathcal{M}$,
$\mathcal{N}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\overline{Gq}\subset\mathcal{M}$,

$\displaystyle\mu_{q}{\,\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\,}\frac{1}{2\pi}\int_{0}^{2\pi}{r_{\theta}}_{\ast}\delta_{q}\
d\theta,$

and let $\mu_{t,q}$ denote the ‘pushed circle average’ ${g_{t}}_{\ast}\mu$.
Then $\mu_{t,q}\underset{t\to\infty}{\longrightarrow}m_{\mathcal{N}}.$

###### Proof.

Up to replacing $\mathcal{M}$ by $\overline{Gq},$ we may assume that $q$ is
not contained in any invariant subvariety $\mathcal{N}\subsetneq\mathcal{M}$.
Let $\nu$ be a weak-$\ast$ limit of a convergent subsequence
$(\mu_{t_{n},q})_{n>0}$. We want to show that $\nu=m_{\mathcal{M}}$.

Step 1. We want to show that $\nu(\mathcal{M})=1$. It is enough to show that
for any $\varepsilon>0$, there is a compact $K\subset\mathcal{M}$ such that
$\nu(K)>1-\varepsilon$. This follows from [EM01, Theorem 5.2].

Step 2. We now want to show that
$\nu(\mathcal{M}\smallsetminus\mathcal{M}_{\infty})=0$. By construction, the
$(L_{i})_{i>0}$ exhaust $\mathcal{M}\smallsetminus\mathcal{M}_{\infty}$ and
thus it is enough to show that for any $\varepsilon>0$ and $i>0$, there is a
open set $\mathcal{U}$ that contains $L_{i}$ and such that