# On the distribution of the components of multicurves of given type

Viveka Erlandsson  School of Mathematics, University of Bristol, and
Department of Mathematics and Statistics, UiT The Arctic University of Norway
<EMAIL_ADDRESS>and  Juan Souto  IRMAR, Université de Rennes
<EMAIL_ADDRESS>
###### Abstract.

We study the distribution of the individual components of a random multicurve
under the action of the mapping class group.

##  1\. Introduction

Throughout this paper let $\Sigma$ be a compact, connected, orientable surface
of genus $g$ and $r$ boundary components satisfying $2g+r\geqslant 3$. We
denote by $\|\cdot\|$ the $\ell^{1}$-norm on Euclidean space and by
$\Delta_{k}=\\{\vec{x}\in\mathbb{R}^{k}_{\geqslant 0},\ \|\vec{x}\|=1\\}$ the
standard simplex—note that the dimension of $\Delta_{k}$ is $k-1$. Finally,
independently of the concrete ambient space, we denote by $\delta_{p}$ the
Dirac probability measure centered on $p$.

### Distribution of vectors of lengths and ratios

Under a $k$-multicurve on $\Sigma$ we consider an ordered tuple
$\vec{\gamma}=(\gamma_{1},\dots,\gamma_{k})$ of (primitive, essential) curves,
weighted if you so wish, or even multicurves, that is, a formal linear
combination of curves with positive coefficients. If $\phi$ is any reasonable
notion of length, then we will use polar coordinates to represent the vector
encoding the lengths $\phi(\gamma_{1}),\dots,\phi(\gamma_{k})$ of the
individual components:

$\mathbb{L}_{\phi}(\vec{\gamma})=\left(\frac{1}{\sum_{i}\phi(\gamma_{i})}(\phi(\gamma_{1}),\dots,\phi(\gamma_{k})),\sum_{i}\phi(\gamma_{i})\right)\in\Delta_{k}\times\mathbb{R}_{\geqslant
0}.$

The mapping class group $\operatorname{Map}(\Sigma)$ acts on the set of
$k$-multicurves and the goal of this paper is to look at the distribution of
the vectors $\mathbb{L}_{\phi}(\vec{\gamma})$ where $\vec{\gamma}$ ranges over
the mapping class group orbit
$\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$ of some fixed
$k$-multicurve. More specifically we will study the asymptotic behavior of the
measures

$\underline{m}(\phi,\vec{\gamma}^{o},L)=\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\mathbb{L}_{\phi}(\frac{1}{L}\vec{\gamma})}$

on $\Delta_{k}\times\mathbb{R}_{\geqslant 0}$, where

${\bf
M}_{\phi,\vec{\gamma}^{o}}(L)=\left\\{(\gamma_{i})_{i}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\
\middle|\ \sum_{i}\phi(\gamma_{i})\leqslant L\right\\}.$

This problem was first studied by Mirzakhani in [22] in the case that
$\phi=\ell$ is the length function of a hyperbolic structure on $\Sigma$ and
that $\vec{\gamma}^{o}$ is a pants decomposition. Later, building on
Mirzakhani’s work, Arana-Herrera [2] and Liu [14] studied the case that
$\vec{\gamma}^{o}$ is a general simple $k$-multicurve, that is a
$k$-multicurve whose components are simple and disjoint of each other. For
example, they proved that for simple $\vec{\gamma}^{o}$ the measures
$\underline{m}(\ell,\vec{\gamma}^{o},L)$ have a limit when $L\to\infty$, and
that in fact the limit does not depend on the chosen hyperbolic metric. Our
first result is that this is true for all $k$-multicurves, simple or not, and
all $\phi$ in a very large class of length functions:

######  Theorem 1.1.

For every $k$-multicurve $\vec{\gamma}^{o}=(\gamma_{1},\dots,\gamma_{k})$ in
$\Sigma$ there is a probability measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ on
$\Delta_{k}$ such that for any continuous, positive, homogenous function
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ on the space of currents
on $\Sigma$ we have

$\lim_{L\to\infty}\underline{m}(\phi,\vec{\gamma}^{o},L)=\mathfrak{p}_{\vec{\gamma}^{o}}\otimes\left((6g-6+2r)\cdot
t^{6g-7+2r}\cdot d{\bf t}\right)$

where $d{\bf t}$ stands for the standard Lebesgue measure on
$\mathbb{R}_{\geqslant 0}$. In particular, the limiting distribution is
independent of $\phi$.

Since
$\mathfrak{p}_{\vec{\gamma}^{o}}(U)=\big{(}\mathfrak{p}_{\vec{\gamma}^{o}}\otimes(6g-6+2r)\cdot
t^{6g-7+2r}d{\bf t}\big{)}(U\times[0,1])$ we get the following from Theorem
1.1:

######  Corollary 1.2.

With notation as in Theorem 1.1 we have

$\lim_{L\to\infty}\frac{|\\{\vec{\gamma}\in{\bf M}_{\phi,\gamma^{o}}(L)\text{
with }\frac{1}{\sum_{i}\phi(\gamma_{i})}(\phi(\gamma_{i}))_{i}\in U\\}|}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}=\mathfrak{p}_{\vec{\gamma}^{o}}(U)$

for every $U\subset\Delta_{k}$ open with
$\mathfrak{p}_{\vec{\gamma}^{o}}(\partial\overline{U})=0$.∎

In some cases the measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ can be explicitly
calculated and this is actually what Mirzakhani, Arana-Herrera, and Liu do for
simple $k$-multicurves. For example, if $\vec{P}^{o}$ is a pants decomposition
we have

(1.1)
$\mathfrak{p}_{\vec{P}^{o}}(U)=\frac{(6g-7+2r)!}{\sqrt{3g-3+r}}\cdot\int_{U}\prod
x_{i}\ d\operatorname{Leb}_{\Delta_{3g-3+r}}$

where $\operatorname{Leb}_{\Delta_{3g-3+r}}$ is the Lebesgue measure on the
simplex (see Example 6 in Section 8 below). From here we get statements like
the following: for any continuous, positive and homogenous function
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ and any $\varepsilon$ we
have that

$\lim_{L\to\infty}\frac{|\\{\vec{P}\in{\bf M}_{\phi,\vec{P}^{o}}(L)|\text{
there is }i\text{ with
}\phi(\gamma_{i})\leq\varepsilon\cdot\frac{L}{3g-3+r}\\}|}{|{\bf
M}_{\phi,\vec{P}^{o}}(L)|}\leq\varepsilon$

for any pants decomposition $\vec{P}^{o}$. It might be worth noticing that for
$\varepsilon$ very small, this bound is pretty course since we can also bound
the limit from above by $(6g-6+2r)\cdot\varepsilon^{2}$ which in fact is
asymptotically correct as $\varepsilon\to 0$.

Calculating the measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ for a general
$\vec{\gamma}^{o}$, simple or not, is in principle feasible (see Proposition
8.1), but it becomes very fast very complicated. Deep down, the reason why
$\mathfrak{p}_{\vec{\gamma}^{o}}$ is in principle computable is because it is
the push-forward under a polyhedral map $\mathbb{S}^{6g-7+2r}\to\Delta_{k}$ of
a measure in the Lebesgue class with PL-density. On the other hand, there is
no bound independent of $\vec{\gamma}^{o}$ for the complexity of the
polyhedral map we just mentioned (see Example 8 in Section 8).

Recall now that there is a plethora of continuous, homogenous and positive
functions on the space of currents [7, 9, 16] such as for example the length
function associated to any negatively curved Riemannian metric, the extremal
length associated to any conformal structure, the word length on
$\pi_{1}(\Sigma)$ associated to certain specific sets of generators, the
stable length associated to any discrete and cocompact action of
$\pi_{1}(\Sigma)$ on a geodesic metric space, etc, etc… Still, the conclusion
of Theorem 1.1 is that the asymptotic distribution of the vectors
$\mathbb{L}_{\phi}(\vec{\gamma})$ for $\vec{\gamma}$ in a mapping class group
orbit does not change if we replace $\phi$ by another function $\psi$. On the
other hand it is very sensitive to the choice of $\vec{\gamma}^{o}$. Now, the
roles of these two factors flip when we consider the distribution

(1.2) $r(\psi/\phi,\vec{\gamma}^{o},L)=\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}$

of the vectors

$\operatorname{ratio}_{\psi/\phi}((\gamma_{i})_{i})=\left(\frac{\psi(\gamma_{i})}{\phi(\gamma_{i})}\right)_{i}\in\mathbb{R}_{\geqslant
0}^{k}$

whose entries are the ratios of the $\psi$\- and $\phi$-lengths of the
individual components of the $k$-multicurve in question.

######  Theorem 1.3.

For any two continuous, homogenous, positive functions
$\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ there is a
probability measure $\mathfrak{r}_{\psi/\phi}$ on $\mathbb{R}_{\geqslant 0}$
with

$\operatorname{diag}_{*}(\mathfrak{r}_{\psi/\phi})=\lim_{L\to\infty}\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}r(\psi/\phi,\vec{\gamma}^{o},L)$

for any $k$-multicurve $\vec{\gamma}^{o}$. Here
$\operatorname{diag}:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}^{k}$
is the diagonal map $\operatorname{diag}(t)=(t,\dots,t)$.

######  Remark.

In (1.2) we take a sum over the subset ${\bf
M}_{\phi,\vec{\gamma}^{o}}(L)\subset\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$
because we want to be working with probability measures, or at least with
Radon measures. If we had summed over the whole orbit then we would have
obtained measures which are not locally finite.

While the measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ from Theorem 1.1 is rather
well-behaved, the measure $\mathfrak{r}_{\psi/\phi}$ from Theorem 1.3 can
present all sorts of pathologies. For example, while its support is always a
(possibly degenerate) interval in $\mathbb{R}_{>0}$, it can well have
infinitely many atoms (see Example 4 in Section 8).

### Distribution of multicurves

While Mirzakhani, Arana-Herrera, and Liu rely on the dynamics of the
earthquake flow, we use a rather different approach. Thinking of multicurves
as currents, we think of $k$-multicurves as $k$-multicurrents, that is, as
elements in the product
$\mathcal{C}(\Sigma)^{k}=\mathcal{C}(\Sigma)\times\dots\times\mathcal{C}(\Sigma)$,
and we study the distribution of the orbit
$\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$ in
$\mathcal{C}(\Sigma)^{k}$. More precisely, we will study the measures

$m(\vec{\gamma}^{o},L)=\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\vec{\gamma}}$

as $L\to\infty$. Our main theorem, Theorem 1.4 below, asserts that these
measures converge to some measure as $L\to\infty$, but before being able to
state it we need some notation. First, consider the map

(1.3)
$\mathbb{I}:\mathcal{C}(\Sigma)\times\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant
0}^{k},\ \ \
\mathbb{I}:(\sigma,(\mu_{i})_{i})\mapsto(\iota(\sigma,\mu_{i}))_{i}$

where $\iota(\cdot,\cdot)$ is the geometric intersection form. Now, given a
$k$-multicurve $\vec{\gamma}^{o}$, denote by $\mathcal{D}_{\vec{\gamma}^{o}}$
some fundamental domain for the action
$\operatorname{Stab}(\vec{\gamma}^{o})\curvearrowright\mathcal{M}\mathcal{L}(\Sigma)$
of the stabilizer of $\vec{\gamma}^{o}$ on the space of measured laminations
(see Proposition 2.1 for details) and consider the measure
$\mathfrak{q}_{\vec{\gamma}^{o}}$ on $\Delta_{k}$ given by

(1.4)
$\mathfrak{q}_{\vec{\gamma}^{o}}(U)=\frac{1}{\mathfrak{b}_{g,r}}\cdot\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U))$

where $\mathfrak{m}_{\operatorname{Thu}}$ is the Thurston measure on
$\mathcal{M}\mathcal{L}(\Sigma)$, where $\mathfrak{b}_{g,r}$ is a constant
that depends only on the topological type of the surface (see Section 2.2 for
details), and where $\operatorname{cone}(U)=\\{tu\text{ with }t\in[0,1],u\in
U\\}$. Consider also the map

$\mathbb{D}:\Delta\times\mathcal{M}\mathcal{L}(\Sigma)\to\mathcal{C}(\Sigma)^{k},\
\ \mathbb{D}:((a_{i}),\lambda)\mapsto(a_{1}\lambda,\dots,a_{k}\lambda),$

of which we think as a thickened diagonal embedding. With this notation in
place we have:

######  Theorem 1.4.

For any $k$-multicurve $\vec{\gamma}^{o}=(\gamma_{i})_{i}$ in $\Sigma$, we
have

(1.5)
$\mathbb{D}_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})=\lim_{L\to\infty}m(\vec{\gamma}^{o},L)$

in the weak-*-topology on Radon measures on $\mathcal{C}(\Sigma)^{k}$.

To prove Theorem 1.4 we will rely on the measure convergence theorem we proved
in [10] (see Section 2.2 below), and in fact we will also coarsely follow the
strategy of its proof. First we show that the family of measures
$(m(\vec{\gamma}^{o},L))_{L}$ is precompact and that every sublimit
$\mathfrak{n}$ as $L\to\infty$ is supported by the image of $\mathbb{D}$. We
then use the measure convergence theorem from [10] to prove that any such
sublimit is of the form
$\mathfrak{n}=\mathbb{D}_{*}(\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}})$
for some measure $\mathfrak{q}$ on $\Delta_{k}$. Then we prove a specific
version of a weaker form of Theorem 1.1, one for the function
$\phi(\vec{\gamma})=\mathbb{I}(\sigma^{o},\vec{\gamma})$ for some filling
multicurve satisfying a certain technical condition. We can use this version
of Theorem 1.1 to identify the measure $\mathfrak{q}$, proving that it is
identical to the measure $\mathfrak{q}_{\vec{\gamma}^{o}}$ given in (1.4).
Since the measure $\mathfrak{q}$, and hence the sublimit $\mathfrak{n}$, does
not depend on the specific sequence used to extract the sublimit, we get that
the actual limit exits. This concludes the sketch of the proof of Theorem 1.4.

Once we have proved Theorem 1.4, the earlier mentioned results follow easily.
For example, the measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ from Theorem 1.1 is
the probability measure associated to $\mathfrak{q}_{\vec{\gamma}^{o}}$:

$\mathfrak{p}_{\vec{\gamma}^{o}}=\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|}\cdot\mathfrak{q}_{\vec{\gamma}^{o}},$

where, in the context of measures, $\|\cdot\|$ stands for the total measure.
The measure $\mathfrak{r}_{\psi/\phi}$ is, up to scaling, nothing other than
the push-forward under $\lambda\mapsto\frac{\psi(\lambda)}{\phi(\lambda)}$ of
the restriction of $\mathfrak{m}_{\operatorname{Thu}}$ to the set of measured
laminations with $\phi(\lambda)\leqslant 1$:

$\mathfrak{r}_{\psi/\phi}=\frac{1}{\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L},\
\phi(\lambda)\leqslant
1\\})}\cdot\left(\frac{\psi(\cdot)}{\phi(\cdot)}\right)_{*}\big{(}\mathfrak{m}_{\operatorname{Thu}}|_{\\{\lambda\in\mathcal{M}\mathcal{L},\
\phi(\lambda)\leqslant 1\\}}\big{)}.$

This paper is organized as follows: After some preliminaries in Section 2 we
prove in Section 3 the weaker form of a version Theorem 1.1 mentioned earlier.
In Section 4 we prove the subconvergence of the measures
$m(\vec{\gamma}^{o},L)$ as $L\to\infty$ and that every sublimit is of the form
$\mathbb{D}_{*}(\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}})$ for
some measure $\mathfrak{q}$ on $\Delta_{k}$. Combining the results of Section
3 and Section 4 we prove Theorem 1.4 in Section 5. Theorem 1.1 and Corollary
1.2 are proved in Section 6 and Theorem 1.3 in Section 7. In the final Section
8 we discuss the measures $\mathfrak{p}_{\vec{\gamma}^{o}}$ and
$\mathfrak{r}_{\psi/\psi}$ in some concrete cases.

### Acknowledgements

We thank Mingkun Liu for helpful discussions: without his help we would still
be computing integrals.

##  2\. Some background

As in the introduction, let $\Sigma$ be a compact, connected and orientable
surface of genus $g$ with $r$ boundary components satisfying $2g+r\geqslant
3$. We also fix some arbitrary hyperbolic metric on $\Sigma$ with respect to
which the boundary is totally geodesic. As is customary, we identify curves
with their free homotopy classes and with closed geodesics in $\Sigma$. We
will always assume that curves are primitive and that they are essential in
the sense that they cannot be homotoped into $\partial\Sigma$, nor into a
point. We stress that we do not assume that curves are simple.

### Warning

Although the results from the introduction are valid in all generality, we
will assume from now on that $\Sigma$ is non-exceptional, meaning that $(g,r)$
is not one of the following: $(0,3),(1,1),(1,2)$ and $(2,0)$. In other words,
we are ruling out those surfaces where the mapping class group is either
finite or has non-trivial center. The center does not cause any fundamental
difficulties, but it involves further complicating an already pretty exuberant
notation. In any case, the reader who wants to consider one of the exceptional
surfaces will have no difficulty modifying the arguments. At the end of the
day we rely heavily on [10], where we explicitly dealt with the center. No
center from now on!

###  2.1. Currents

Currents on $\Sigma$ are nothing other than $\pi_{1}(\Sigma)$-invariant Radon
measures on the set of geodesics on the universal cover of $\Sigma$. We endow
the space $\mathcal{C}(\Sigma)$ of currents with the weak-*-topology. The set
$\mathcal{C}_{K}(\Sigma)$ of currents supported by the set of geodesics
contained in the preimage of some compact
$K\subset\Sigma\setminus\partial\Sigma$ is closed. On the other hand, the set
$\mathcal{C}_{0}(\Sigma)=\cup_{K}\mathcal{C}_{K}(\Sigma)$ of compactly
supported currents is not. Currents were introduced by Bonahon in [4, 5, 6],
but instead of referring to the original papers or to other sources (we still
recommend the very nice paper [1]) we will mostly follow the terminology and
notation from [10].

When working with currents, it often does not really matter what currents
actually are: what matters is that they exist, that some well known objects
are currents, and that the space of currents has some very nice properties. We
summarize what we will need here:

(1) The space $\mathcal{C}(\Sigma)$ of currents is a locally compact
metrizable topological space. It is moreover a linear cone in a topological
vector space, meaning that one can add currents and that one can multiply
currents by positive numbers, and that these two operations are associative,
distributive, and continuous.

(2) Curves and measured laminations are currents. In fact, the embedding
$\mathcal{M}\mathcal{L}(\Sigma)\hookrightarrow\mathcal{C}(\Sigma)$ is a
homeomorphism onto its image, a subset of $\mathcal{C}_{K}(\Sigma)$ for some
suitable compact set $K\subset\Sigma\setminus\partial\Sigma$. On
$\mathcal{M}\mathcal{L}(\Sigma)$ there is a natural
$\operatorname{Map}(\Sigma)$-invariant measure, the Thurston measure
$\mathfrak{m}_{\operatorname{Thu}}$, which scales with power $6g-6+2r$, that
is $\mathfrak{m}_{\operatorname{Thu}}(t\cdot
U)=t^{6g-6+2r}\cdot\mathfrak{m}_{\operatorname{Thu}}(U)$. When we push forward
$\mathfrak{m}_{\operatorname{Thu}}$ by the embedding from
$\mathcal{M}\mathcal{L}(\Sigma)$ to $\mathcal{C}(\Sigma)$ we get a measure to
which we still refer to by the same name and which we denote by the same
symbol $\mathfrak{m}_{\operatorname{Thu}}$. Masur proved that the set of
uniquely ergodic measured laminations which intersect every curve has full
Thurston measure [17] and that $\mathfrak{m}_{\operatorname{Thu}}$ is ergodic
with respect to $\operatorname{Map}(\Sigma)$ action [18]. Note that there are
different reasonable normalizations for the Thurston measure—here we follow
the one coming from constructing it as a scaling limit (see [10, Theorem
4.16]).

(3) There is a continuous bilinear map

$\iota:\mathcal{C}(\Sigma)\times\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant
0},$

the geometric intersection form, extending the geometric intersection number
of both curves and measured laminations. Indeed, measured laminations are
characterized as being exactly those currents $\lambda$ with vanishing self-
intersection number $\iota(\lambda,\lambda)=0$.

(4) A compactly supported current $\mu\in\mathcal{C}_{0}(\Sigma)$ is filling
if we have $\iota(\mu,\lambda)>0$ for every non-zero compactly supported
current $\lambda\in\mathcal{C}_{0}(\Sigma)$. We denote the set of all filling
currents by $\mathcal{C}_{\operatorname{fill}}(\Sigma)$. A multicurve is
filling if the associated current is. In topological terms, a multicurve is
filling if and only if the geodesic representative of its support cuts the
surface into a collection of disks and annuli, each one of the latter
containing a boundary component.

A key property of filling currents is the following: Given
$K\subset\Sigma\setminus\partial\Sigma$ compact and
$\mu\in\mathcal{C}(\Sigma)$ filling, then the set
$\\{\lambda\in\mathcal{C}_{K}(\Sigma)\text{ with }\iota(\mu,\lambda)\leqslant
L\\}$ is compact for all $L$.

(5) The mapping class group acts continuously on $\mathcal{C}(\Sigma)$ by
linear automorphisms preserving the intersection form. For every curve
$\gamma$ there is a compact set $K\subset\Sigma\setminus\partial\Sigma$ with
$\operatorname{Map}(\Sigma)\cdot\gamma\subset\mathcal{C}_{K}(\Sigma)$.

Besides the basic properties of currents and the space of currents that we
just listed, the following theorem of Mondello and the first author [8] plays
an important role here:

######  Theorem (Erlandsson-Mondello).

The action of $\operatorname{Map}(\Sigma)$ on the open set
$\mathcal{C}_{\operatorname{fill}}(\Sigma)$ of filling currents is discrete in
the sense that for every $A\subset\mathcal{C}_{\operatorname{fill}}(\Sigma)$
compact there are only finitely many $\phi\in\operatorname{Map}(\Sigma)$ with
$\phi(A)\cap A\neq\varnothing$.

In [10] we explained how to construct fundamental domains for the action of
certain subgroups of the mapping class group on certain subspaces of the space
of filling currents. Below we will be working in exactly the same setting as
there but, since nothing changes, it might be a good idea to state it in
generality.

So, let us suppose that $\Gamma\subset\operatorname{Map}(\Sigma)$ is a
subgroup and $\mathcal{U}\subset\mathcal{C}_{\operatorname{fill}}(\Sigma)$ is
a $\Gamma$-invariant subset of the space of filling currents. Our aim is to
construct a fundamental domain for the action
$\Gamma\curvearrowright\mathcal{U}$. To do so let us fix a filling multicurve
$\sigma^{o}$ satisfying

(2.6)
$\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\iota(\sigma^{o},\lambda)=\iota(\sigma^{o},\phi(\lambda))\\})=0$

for every $\phi\in\operatorname{Map}(\Sigma)$—we proved in [10, Lemma 8.3]
that such multicurves exist. Once we have fixed $\sigma^{o}$ we can consider
the set

(2.7) $\mathcal{D}_{\Gamma}=\\{\lambda\in\mathcal{U}\text{ with
}\iota(\lambda,\sigma^{o})<\iota(\lambda,\phi(\sigma^{o}))\text{ for all
}\phi\in\Gamma\setminus\operatorname{Id}\\}.$

Repeating word-by-word the argument of [10, Proposition 8.4] we get that
$\mathcal{D}_{\Gamma}$ is a fundamental domain in the following sense:

######  Proposition 2.1.

The set $\mathcal{D}_{\Gamma}$ is open in $\mathcal{U}$ and satisfies
$\mathcal{D}_{\Gamma}\cap\phi(\mathcal{D}_{\Gamma})=\varnothing$ for all
$\phi\in\Gamma\setminus\\{\operatorname{Id}\\}$. On the other hand, the
translates of its closure $\overline{\mathcal{D}}_{\Gamma}$ cover
$\mathcal{U}$, that is
$\mathcal{U}=\cup_{\phi\in\Gamma}\phi(\overline{\mathcal{D}}_{\Gamma})$.
Finally, we also have that
$\overline{\mathcal{D}}_{\Gamma}\setminus\mathcal{D}_{\Gamma}$ has vanishing
Thurston measure:
$\mathfrak{m}_{\operatorname{Thu}}(\overline{\mathcal{D}}_{\Gamma}\setminus\mathcal{D}_{\Gamma})=0$.∎

###  2.2. Curve counting

Given a multicurve $\gamma^{o}$ in $\Sigma$, Mirzakhani studied in [21, 22]
the asymptotic growth of the cardinality of the set

${\bf
M}_{\Sigma,\gamma^{o}}(L)=\\{\gamma\in\operatorname{Map}(\Sigma)\cdot\gamma^{o}\text{
with }\ell_{\Sigma}(\gamma)\leqslant L\\}$

where $\ell_{\Sigma}$ stands for the hyperbolic length. She proved in [21] for
simple multicurves and in [22] for general ones that the limit

(2.8) $\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}|{\bf M}_{\Sigma,\gamma^{o}}(L)|$

exists and is positive.

######  Remark.

In fact Mirzakhani gave the value for the limit above but we prefer to not do
it explicitly now: we are going to express it shortly in a somewhat different
way than she did and we do not want to risk causing any confusion.

In [10] we obtained the existence of (2.8) from a measure convergence theorem
(as did Mirzakhani already in [21] in the case that $\gamma^{o}$ is simple).
More precisely, denoting as always by $\delta_{x}$ the Dirac probability
measure centered on $x$, we considered the measures

(2.9)
$m^{L}_{\gamma^{o}}=\frac{1}{L^{6g-6+2r}}\sum_{\gamma\in\operatorname{Map}(\Sigma)\cdot\gamma^{o}}\delta_{\frac{1}{L}\gamma}$

on the space of currents $\mathcal{C}(\Sigma)$ and proved that they converge
when $L\to\infty$:

######  Theorem.

[10, Theorem 8.1] Given any multicurve $\gamma^{o}\in\mathcal{C}(\Sigma)$ let
$\mathcal{D}_{\operatorname{Stab}(\gamma^{o})}$ be a fundamental domain for
the action of $\operatorname{Stab}(\gamma^{o})$ on the set

$\mathcal{C}_{\gamma^{o}}=\\{\lambda\in\mathcal{C}_{0}(\Sigma)\text{ with
}\lambda+\gamma^{o}\text{ filling}\\}$

and set

$\mathfrak{c}(\gamma^{o})=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{D}_{\operatorname{Stab}(\gamma^{o})},\
\iota(\gamma^{o},\lambda)\leqslant 1\\}).$

Then we have

(2.10)
$\lim_{L\to\infty}m^{L}_{\gamma^{o}}=\frac{\mathfrak{c}(\gamma^{o})}{\mathfrak{b}_{g,r}}\cdot\mathfrak{m}_{\operatorname{Thu}}.$

where $\mathfrak{b}_{g,r}$ depends just on the genus and number of boundary
components of $\Sigma$.

######  Remark.

The constant $\mathfrak{b}_{g,r}$ in the theorem can actually be expressed in
the following two ways. In [10] we defined it as

$\mathfrak{b}_{g,r}=\sum_{\gamma\in\operatorname{Map}(\Sigma)\backslash\mathcal{M}\mathcal{L}_{\mathbb{Z}}(\Sigma)}\mathfrak{c}(\gamma)$

where $\mathcal{M}\mathcal{L}_{\mathbb{Z}}(\Sigma)$ is the set of all simple
integrally weighted multicurves, but closer to what Mirzakhani did in [21], it
can also be expressed as

$\mathfrak{b}_{g,r}=2^{2g-3+r}\cdot\int_{\mathcal{M}_{g,r}}\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\ell_{X}(\lambda)\leqslant 1\\})\ dX$

where the integral is taken with respect to the Weil-Petersson metric on
$\mathcal{M}_{g,r}$, the moduli space of hyperbolic surfaces of genus $g$ and
$r$ cusps. See [10, Proposition 8.8] for the relationship between our and
Mirzakhani’s constants.

From the measure convergence theorem above we obtained that for any positive,
homogenous and continuous function
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ we have convergence when
in (2.8) we replace the hyperbolic length function $\ell_{\Sigma}$ by the
function $\phi$. This applies in particular to functions like
$\lambda\mapsto\iota(\sigma^{o},\lambda)$ for some filling current
$\sigma^{o}$. Since this is the setting we are going to be working in we state
this precisely. For a filling current $\sigma^{o}$ consider the set

(2.11) ${\bf
M}_{\sigma^{o},\gamma^{o}}(L)=\\{\gamma\in\operatorname{Map}(\Sigma)\cdot\gamma^{o}\text{
with }\iota(\sigma^{o},\gamma)\leqslant L\\}.$

Then we have

(2.12) $\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}|{\bf
M}_{\sigma^{o},\gamma^{o}}(L)|=\frac{\mathfrak{c}(\gamma^{o})}{\mathfrak{b}_{g,r}}\cdot
B(\sigma^{o})$

where
$B(\sigma^{o})=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\iota(\sigma^{o},\lambda)\leqslant 1\\})$.

###  2.3. $k$-multicurrents

For lack of a better name, we refer to the elements
$\vec{\mu}=(\mu_{1},\dots,\mu_{k})$ in $\mathcal{C}(\Sigma)^{k}$ as
$k$-multicurrents. Note for example that $k$-multicurves are
$k$-multicurrents. As the reader has already seen, we indicate such
$k$-multicurves and $k$-multicurrents either by writing out the tuple, maybe
in the short form $(\mu_{i})_{i}$, or by decorating them with a little arrow
on top. The mapping class group acts diagonally on the space of
$k$-multicurrents:

$\phi((\mu_{i})_{i})=(\phi(\mu_{i}))_{i}$

The mapping class group equivariant map

$\pi:\mathcal{C}(\Sigma)^{k}\to\mathcal{C}(\Sigma),\ \
\pi((\mu_{i})_{i})=\sum_{i}\mu_{i}$

will play a key role in this paper, but before going any further let us agree
on some terminology: we will refer to the entries $\mu_{1},\dots,\mu_{k}$ of a
$k$-multicurrent $\vec{\mu}=(\mu_{1},\dots,\mu_{k})\in\mathcal{C}^{k}(\Sigma)$
as its components and we will say that the $k$-multicurrent $\vec{\mu}$
represents the current $\pi(\vec{\mu})$. Note that a $k$-multicurrent which
represents a multicurve is nothing other than a $k$-multicurve.

Although it is very important that we work with general currents, we will be
especially interested on those $k$-multicurrents representing measured
laminations. Let us prove two simple lemmas that will look self-evident to
experts:

######  Lemma 2.2.

A $k$-multicurrent in
$\vec{\mu}=(\mu_{1},\dots,\mu_{k})\in\mathcal{C}^{k}(\Sigma)$ represents a
measured lamination if and only if all the components $\mu_{i}$ are also
measured laminations satisfying $\iota(\mu_{i},\mu_{j})=0$ for all $i$ and
$j$.

###### Proof.

By assumption we have that

$0=\iota\left(\pi(\vec{\mu}),\pi(\vec{\mu})\right)=\iota\left(\sum_{i}\mu_{i},\sum_{j}\mu_{j}\right)=\sum_{i,j}\iota(\mu_{i},\mu_{j})$

and since the intersection form only takes non-negative values it follows that
$\iota(\mu_{i},\mu_{j})=0$ for all $i,j$. Now the claim follows from the fact
that measured laminations are characterized as being the only currents with
vanishing self-intersection number. ∎

Recall now that a measured lamination is uniquely ergodic if its support only
carries a single transverse measure up to scaling.

######  Lemma 2.3.

If a $k$-multicurrent $\vec{\mu}\in\mathcal{C}^{k}(\Sigma)$ represents an
uniquely ergodic measured lamination $\lambda$ then there is a point
$(a_{i})_{i}\in\Delta_{k}$ in the standard $k$-simplex with
$\vec{\mu}=(a_{1}\lambda,\dots,a_{k}\lambda)$.

###### Proof.

Each component $\mu_{i}$ of $\vec{\mu}$ represents a measured lamination with
support contained in that of $\lambda$. It follows thus from the assumption
that $\lambda$ is uniquely ergodic that there is some $a_{i}\geqslant 0$ with
$\mu_{i}=a_{i}\lambda$. The assumption that the vector
$(a_{1}\lambda,\dots,a_{k}\lambda)$ represents $\lambda$ implies that $\sum
a_{i}=1$, meaning that $(a_{1},\dots,a_{k})\in\Delta_{k}$. ∎

##  3\. A first convergence result

Still with the same notation let $\Sigma$ be a compact orientable connected
surface with genus $g$ and $r$ boundary components, and recall that we are
just dealing with the case that $\Sigma$ is non-exceptional. In this section
we prove a specific case of a close relative of Theorem 1.1, but first let us
recall the map (1.3) from the introduction:

$\mathbb{I}:\mathcal{C}(\Sigma)\times\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant
0}^{k},\ \mathbb{I}(\sigma,(\mu_{i})_{i})=(\iota(\sigma,\mu_{i})_{i}).$

This is what we prove:

######  Proposition 3.1.

Let $\vec{\gamma}^{o}$ be a $k$-multicurve in $\Sigma$, let $\sigma^{o}$ be a
filling multicurve in $\Sigma$ satisfying (2.6), and let
$\mathcal{D}=\mathcal{D}_{\operatorname{Stab}(\vec{\gamma}^{o})}$ be as in
(2.7). We have

$\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}})\right)=\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\mathbb{I}(\sigma^{o},\vec{\gamma})}$

in the weak-*-topology on measures on $\mathbb{R}_{\geqslant 0}^{k}$. Here
$B(\sigma^{o})=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\iota(\sigma^{o},\lambda)\leqslant 1\\})$.

###### Proof.

Denote the measures inside the limit by

$\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)=\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}(\mathbb{I}(\sigma^{o},\vec{\gamma}))}.$

Note that these measures are supported on $\mathbb{R}_{\geq 0}^{k}$ as opposed
to $\Delta_{k}\times\mathbb{R}_{\geq 0}$ which is the case for the measures
$\underline{m}(\phi,\vec{\gamma}^{0},L)$ in Theorem 1.1, and this is the
reason why we use a different decoration on $m$.

Since we have
$\mathbb{I}(\sigma^{o},\phi(\vec{\gamma}))=\mathbb{I}(\phi^{-1}(\sigma^{o}),\vec{\gamma})$
we can, for any choice of a set $\Theta\subset\operatorname{Map}(\Sigma)$ of
representatives of classes
$\operatorname{Map}(\Sigma)/\operatorname{Stab}(\vec{\gamma}^{o})$, rewrite
them as

$\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)=\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\phi\in\Theta}\delta_{\frac{1}{L}\phi^{-1}(\sigma^{o})}\right).$

We now get from Proposition 2.1 that $\Theta$ can be chosen to satisfy

$\\{\phi\in\operatorname{Map}(\Sigma)|\phi^{-1}(\sigma^{o})\in\mathcal{D}\\}\subset\Theta\subset\\{\phi\in\operatorname{Map}(\Sigma)|\phi^{-1}(\sigma^{o})\in\overline{\mathcal{D}}\\}.$

Indeed, the first inclusion follows from the fact that every
$\operatorname{Stab}(\vec{\gamma}^{o})$-orbit meets $\mathcal{D}$ at most
once, and the second because $\overline{\mathcal{D}}$ meets every orbit at
least once. Anyways, from here we get the following comparisons for the
measure $\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)$:

(3.13)
$\begin{split}\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)\geqslant\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\sigma\in(\operatorname{Map}(\Sigma)\cdot\sigma^{o})\cap\mathcal{D}}\delta_{\frac{1}{L}\sigma}\right)\\\
\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)\leqslant\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\sigma\in(\operatorname{Map}(\Sigma)\cdot\sigma^{o})\cap\overline{\mathcal{D}}}\delta_{\frac{1}{L}\sigma}\right)\end{split}$

From (2.10) applied to $\sigma^{o}$, we get that

$\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\sigma\in\operatorname{Map}(\Sigma)\cdot\sigma^{o}}\delta_{\frac{1}{L}\sigma}=\frac{\mathfrak{c}(\sigma^{o})}{\mathfrak{b}_{g,r}}\mathfrak{m}_{\operatorname{Thu}}.$

Taking into account that
$\mathfrak{m}_{\operatorname{Thu}}(\overline{\mathcal{D}}\setminus\mathcal{D})=0$
we deduce that

$\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\sigma\in(\operatorname{Map}(\Sigma)\cdot\sigma^{o})\cap\mathcal{D}}\delta_{\frac{1}{L}\sigma}=\frac{\mathfrak{c}(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}})$

and that

$\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\sigma\in(\operatorname{Map}(\Sigma)\cdot\sigma^{o})\cap\overline{\mathcal{D}}}\delta_{\frac{1}{L}\sigma}=\frac{\mathfrak{c}(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\overline{\mathcal{D}}}).$

Using once again that $\overline{\mathcal{D}}\setminus\mathcal{D}$ has
vanishing Thurston measure, we get that these two limits actually agree. From
the continuity of $\mathbb{I}(\cdot,\vec{\gamma}^{0})$ and from (3.13) we get
thus that

(3.14)
$\lim_{L\to\infty}\overline{m}(\sigma^{o},\vec{\gamma}^{o},L)=\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{\mathfrak{c}(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}})\right).$

The claim follows once we note that $B(\sigma^{o})=\mathfrak{c}(\sigma^{o})$
because (2.6) implies that $\sigma^{o}$ has trivial stabilizer. ∎

##  4\. Subconvergence and disintegration of sublimits

As all along, let $\Sigma$ be a non-exceptional, compact, connected,
orientable surface with genus $g$ and $r$ boundary components. Given a
$k$-multicurve $\vec{\gamma}^{o}\in\mathcal{C}(\Sigma)^{k}$ we consider as in
the introduction the measures

$m(\vec{\gamma}^{o},L)=\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\vec{\gamma}}$

on $\mathcal{C}(\Sigma)^{k}$. We are interested in their behavior when
$L\to\infty$. Theorem 1.4 asserts indeed that the measures
$m(\vec{\gamma}^{o},L)$ converge when $L\to\infty$ to a measure of some
specific form. Our goal in this section is to prove that this is the case if
we allow ourselves to pass to subsequences:

######  Proposition 4.1.

Let $\vec{\gamma}^{o}=(\gamma_{i})_{i}\in\mathcal{C}(\Sigma)^{k}$ be a
$k$-multicurve and consider any sequence $L_{s}\to\infty$. There is a
subsequence $(L_{s_{j}})_{j}$ such that the limit

$\mathfrak{n}=\lim_{j\to\infty}m(\vec{\gamma}^{o},L_{s_{j}})$

exists in the weak-*-topology. Moreover, the limiting measure $\mathfrak{n}$
is the push-forward of $\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}}$
under the map

(4.15)
$\mathbb{D}:\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\to\mathcal{C}(\Sigma)^{k},\
\ ((a_{i}),\lambda)\mapsto(a_{1}\lambda,\dots,a_{k}\lambda)$

for some measure $\mathfrak{q}$ on the standard $k$-simplex
$\Delta_{k}\subset\mathbb{R}_{\geqslant 0}^{k}$.

Starting with the proof, fix a compact set
$K\subset\Sigma\setminus\partial\Sigma$ such that $\mathcal{C}_{K}(\Sigma)$
contains the mapping class group orbit
$\operatorname{Map}(\Sigma)\cdot\gamma_{i}$ of every component of
$\vec{\gamma}$, and note that $m(\vec{\gamma}^{o},L)$ is supported by the
subset
$\mathcal{C}_{K}(\Sigma)^{k}=\mathcal{C}_{K}(\Sigma)\times\dots\times\mathcal{C}_{K}(\Sigma)$
of $\mathcal{C}(\Sigma)^{k}$. With this notation in place we prove next that
the measures $m(\vec{\gamma}^{o},L)$ form a relatively compact family.

######  Lemma 4.2.

Every sequence $L_{s}\to\infty$ has a subsequence $(L_{s_{j}})_{j}$ such that
the limit

$\mathfrak{n}=\lim_{j}m(\vec{\gamma}^{o},L_{s_{j}})$

exists in the weak-*-topology on the space of Radon measures on
$\mathcal{C}_{K}(\Sigma)^{k}$. Moreover, the measure $\mathfrak{n}$ is mapping
class group invariant.

###### Proof.

Note first that since all the measures $m(\gamma^{o},L)$ are mapping class
group invariant, the same will be true for any putative limit $\mathfrak{n}$.

Now, since $\mathcal{C}_{K}(\Sigma)$ is locally compact, so is the product
$\mathcal{C}_{K}(\Sigma)^{k}$. It follows that to prove that our sequence of
measures $(m(\vec{\gamma}^{o},L_{s}))_{s}$ has a convergent subsequence, it
suffices to show that the sequence of real numbers
$(m(\vec{\gamma}^{o},L_{s})(W))_{s}$ is bounded for every compact set $W$.
Note now that the restriction to $\mathcal{C}_{K}(\Sigma)$ of the length
function $\ell:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ associated to
some hyperbolic metric on $\Sigma$ is proper, and hence so is also the
function

$\hat{\ell}:\mathcal{C}_{K}(\Sigma)^{k}\to\mathbb{R}_{\geqslant 0},\ \
\hat{\ell}(\vec{\mu})=\sum_{i}\ell(\mu_{i})=\ell(\pi(\vec{\mu})),$

and fix some $R$ with $\hat{\ell}(W)\subset[0,R]$. The restriction of
$\pi:\mathcal{C}(\Sigma)^{k}\to\mathcal{C}(\Sigma)$ to
$\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$ is $c_{0}$-to-1 for some
finite $c_{0}$ and it follows that

$\displaystyle m(\vec{\gamma}^{o},L)(W)$ $\displaystyle\leqslant
m(\vec{\gamma}^{o},L)\big{(}\\{(\vec{\mu})\in\mathcal{C}_{K}(\Sigma)^{k}\text{
with }\hat{\ell}(\vec{\mu})\leqslant R\\}\big{)}$
$\displaystyle=\frac{1}{L^{6g-6+2r}}\left|\left\\{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\text{
with }\hat{\ell}\left(\frac{1}{L}\vec{\gamma}\right)\leqslant
R\right\\}\right|$
$\displaystyle=\frac{c_{0}}{L^{6g-6+2r}}\cdot\left|\left\\{\pi(\vec{\gamma})\in\operatorname{Map}(\Sigma)\cdot\pi(\vec{\gamma}^{o})\text{
with }\ell\left(\frac{1}{L}\pi(\vec{\gamma})\right)\leqslant
R\right\\}\right|$
$\displaystyle=\frac{c_{0}}{L^{6g-6+2r}}\cdot\left|\left\\{\pi(\vec{\gamma})\in\operatorname{Map}(\Sigma)\cdot\pi(\vec{\gamma}^{o})\text{
with }\ell(\pi(\vec{\gamma}))\leqslant R\cdot L\right\\}\right|.$

Now we get from [10, Theorem 5.19] that there is some $A>0$, depending on the
multicurve $\pi(\vec{\gamma}^{o})$ and on $R$, with

(4.16)
$|\\{\pi(\vec{\gamma})\in\operatorname{Map}(\Sigma)\cdot\pi(\vec{\gamma}^{o})\text{
with }\ell(\pi(\vec{\gamma}))\leqslant R\cdot L\\}|\leqslant A\cdot(R\cdot
L)^{6g-6+2r}$

for all large $L$. Having proved that $m(\vec{\gamma}^{o},L)(W)\leqslant
c_{0}\cdot A\cdot R^{6g-6+2r}$ for all sufficiently large $L$, we are done. ∎

From now on let $\mathfrak{n}$ be as in Lemma 4.2. Our next goal is to show
that $\mathfrak{n}$ is supported by the subset of $k$-multicurrents
representing a measured lamination.

######  Lemma 4.3.

The limiting measure $\mathfrak{n}$ is supported by
$\pi^{-1}(\mathcal{M}\mathcal{L}(\Sigma))\subset\mathcal{C}(\Sigma)^{k}$.

###### Proof.

With the same notation as in the proof of Lemma 4.2 we let
$\hat{\ell}(\vec{\mu})=\sum\ell(\mu_{i})$, let $c_{0}$ and $A$ satisfy (4.16),
and stress that there is some $K$ such that all the measures
$m(\vec{\gamma},L)$, and hence also the sublimit $\mathfrak{n}$, are supported
by $\mathcal{C}_{K}(\Sigma)^{k}$. Now, noting that for all
$\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$ we have

$\iota(\pi(\vec{\gamma}),\pi(\vec{\gamma}))=\iota(\pi(\vec{\gamma}^{o}),\pi(\vec{\gamma}^{o}))$

we get for all fixed but otherwise arbitrary $R>0$ that

$\displaystyle\int_{\hat{\ell}^{-1}[0,R)}$
$\displaystyle\iota(\pi(\vec{\mu}),\pi(\vec{\mu}))\
d\mathfrak{n}(\mu)\leqslant$
$\displaystyle\leqslant\limsup_{L\to\infty}\int_{\hat{\ell}^{-1}[0,R]}\iota(\pi(\vec{\mu}),\pi(\vec{\mu}))\
dm(\vec{\gamma}^{o},L)(\mu)$
$\displaystyle=\limsup_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\tiny{\begin{array}[]{l}\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\\\
\hat{\ell}(\gamma)\leqslant
RL\end{array}}}\iota\left(\pi\left(\frac{1}{L}\vec{\gamma}\right),\pi\left(\frac{1}{L}\vec{\gamma}\right)\right)$
$\displaystyle=\limsup_{L\to\infty}\left(\frac{|\\{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\text{
with }\hat{\ell}(\vec{\gamma})\leqslant
RL\\}|}{L^{6g-6+2r}}\cdot\frac{\iota(\pi(\vec{\gamma}^{o}),\pi(\vec{\gamma}^{o}))}{L^{2}}\right)$
$\displaystyle\leqslant\limsup_{L\to\infty}\left(c_{0}\cdot A\cdot
R^{6g-6+2r}\cdot\frac{\iota(\pi(\vec{\gamma}^{o}),\pi(\vec{\gamma}^{o}))}{L^{2}}\right)=0.$

Having proved that

$\int_{\hat{\ell}^{-1}[0,R)}\iota(\pi(\vec{\mu}),\pi(\vec{\mu}))\
d\mathfrak{n}(\mu)=0$

for all $R$, we deduce that the sublimit $\mathfrak{n}$ is supported by the
set of those $\vec{\mu}\in\mathcal{C}^{k}(\Sigma)$ with
$\iota(\pi(\vec{\mu}),\pi(\vec{\mu}))=0$. Our claim follows from Lemma 2.2. ∎

Our next goal is to refine Lemma 4.3, showing that $\mathfrak{n}$ is indeed
supported by the image of the map (4.15).

######  Lemma 4.4.

The limiting measure $\mathfrak{n}$ is supported by
$\mathbb{D}(\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma))$ where
$\mathbb{D}$ is the map (4.15).

###### Proof.

As in the proof of Lemma 4.2 let $c_{0}$ be such that the restriction of
$\pi:\mathcal{C}(\Sigma)^{k}\to\mathcal{C}(\Sigma)$ to the orbit
$\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}$ is $c_{0}$-to-1, set
$\eta^{o}=\pi(\vec{\gamma}^{o})$ and note that

$\pi_{*}(m(\vec{\gamma}^{o},L))=\frac{c_{0}}{L^{6g-6+2r}}\sum_{\eta\in\operatorname{Map}(\Sigma)\cdot\eta^{o}}\delta_{\frac{1}{L}\eta}.$

With the same notation as in (2.9) this reads as

$\pi_{*}(m(\vec{\gamma}^{o},L))=c_{0}\cdot m^{L}_{\eta^{o}}$

Now, we get from the continuity of $\pi$ and (2.10) that

(4.17)
$\pi_{*}(\mathfrak{n})=c_{0}\cdot\frac{c(\eta^{o})}{\mathfrak{b}_{g,r}}\cdot\mathfrak{m}_{\operatorname{Thu}}.$

As already mentioned, it is a result of Masur [17] that the set $\mathcal{E}$
of uniquely ergodic measured laminations has full Thurston measure. We get
thus from (4.17) that $\pi^{-1}(\mathcal{E})$ has full $\mathfrak{n}$-measure.
Since Lemma 2.3 implies that $\pi^{-1}(\mathcal{E})$ is contained in the image
of (4.15), the claim follows. ∎

We are now ready to finish the proof of Proposition 4.1:

###### Proof of Proposition 4.1.

Let us recap what we know so far. We get from Lemma 4.2 that the sequence
$(m(\vec{\gamma}^{o},L_{s}))_{s}$ is precompact and that the accumulation
points are mapping class group invariant. Denoting by

$\mathfrak{n}=\lim_{i}m(\vec{\gamma}^{o},L_{s_{i}})$

the limit of some convergent subsequence $(m(\vec{\gamma}^{o},L_{s_{i}}))_{i}$
we get from Lemma 4.4 that $\mathfrak{n}$ is supported by the image of the map

$\mathbb{D}:\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\to\mathcal{C}(\Sigma)^{k},\
\ ((a_{i}),\lambda)\mapsto(a_{1}\lambda,\dots,a_{k}\lambda).$

Since this map is a homeomorphism onto its image, we can think of the limiting
measure $\mathfrak{n}$ as being the push-forward under $\mathbb{D}$ of a
measure $\hat{\mathfrak{n}}$ on
$\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)$. What is left to do is to
prove that $\hat{\mathfrak{n}}$ is the product of a measure on $\Delta_{k}$
and the Thurston measure.

Before doing that note however that the map $\mathbb{D}$ is mapping class
group equivariant, where the action of $\operatorname{Map}(\Sigma)$ on
$\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)$ is trivial on the first
factor and the standard one on the second. Invariance of the measure
$\mathfrak{n}$ and equivariance of the embedding $\mathbb{D}$ imply that also
$\hat{\mathfrak{n}}$ is mapping class group invariant. Note also that the
composition of $\mathbb{D}$ with the projection map $\pi$ has a very simple
form:

$(\pi\circ\mathbb{D})((a_{i}),\lambda)=\lambda.$

Coming now to the meat of the proof, suppose that $U\subset\Delta_{k}$ is a
Borel set and consider the restriction $\hat{\mathfrak{n}}_{U}$ of
$\hat{\mathfrak{n}}$ to $U\times\mathcal{M}\mathcal{L}$. The measure
$\hat{\mathfrak{n}}_{U}$ is mapping class group invariant because both
$U\times\mathcal{M}\mathcal{L}$ and $\hat{\mathfrak{n}}$ are. This implies
that also $(\pi\circ\mathbb{D})_{*}(\hat{\mathfrak{n}}_{U})$ is mapping class
group invariant. The push-forward
$(\pi\circ\mathbb{D})_{*}(\hat{\mathfrak{n}}_{U})$ is evidently absolutely
continuous with respect to
$(\pi\circ\mathbb{D})_{*}(\hat{\mathfrak{n}})=\pi_{*}(\mathfrak{n})$ and hence
with respect to $\mathfrak{m}_{\operatorname{Thu}}$ because of (4.17). It
follows thus from the ergodicity of the Thurston measure [18] that
$(\pi\circ\mathbb{D})_{*}(\hat{\mathfrak{n}}_{U})$ is a constant multiple of
$\mathfrak{m}_{\operatorname{Thu}}$. Let
$\mathfrak{q}(U)\in\mathbb{R}_{\geqslant 0}$ be the unique number with

(4.18)
$(\pi\circ\mathbb{D})_{*}(\mathfrak{n}_{U})=\mathfrak{q}(U)\cdot\mathfrak{m}_{\operatorname{Thu}}$

It is evident that $\mathfrak{q}(\varnothing)=0$ and that the map
$\mathfrak{q}$ from the Borel $\sigma$-algebra of $\Delta_{k}$ to
$\mathbb{R}_{\geqslant 0}$ is $\sigma$-additive. In other words,
$\mathfrak{q}$ is a measure. It remains to check that $\hat{\mathfrak{n}}$
actually agrees with $\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}}$.
To see that this is the case note that for any
$V\subset\mathcal{M}\mathcal{L}$ we have
$(U\times\mathcal{M}\mathcal{L})\cap(\pi\circ\mathbb{D})^{-1}(V)=U\times V$.
This means that we have

$\displaystyle\hat{\mathfrak{n}}(U\times V)$
$\displaystyle\stackrel{{\scriptstyle\text{definition}}}{{=}}\hat{\mathfrak{n}}_{U}(U\times
V)=(\pi\circ\mathbb{D})_{*}(\mathfrak{n}_{U})(V)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq define
measure}}}{{=}}\mathfrak{q}(U)\cdot\mathfrak{m}_{\operatorname{Thu}}(V)=(\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}})(U\times
V)$

as we needed to prove. ∎

##  5\. Proof of Theorem 1.4

Oh surprise, surprise, in this section we prove Theorem 1.4. In a nutshell,
what we need to prove is that the measure $\mathfrak{q}$ in Proposition 4.1
does not depend on the individual sequence. First we recall some notation from
the introduction: $\mathfrak{q}_{\vec{\gamma}^{o}}$ is the measure on
$\Delta_{k}$ given by

$\mathfrak{q}_{\vec{\gamma}^{o}}(U)=\frac{1}{\mathfrak{b}_{g,r}}\cdot\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U))$

where

$\mathbb{I}:\mathcal{C}(\Sigma)\times\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant
0}^{k},\ \ \
\mathbb{I}:(\sigma,(\mu_{i})_{i})\mapsto(\iota(\sigma,\mu_{i})_{i}),$

where
$\mathcal{D}_{\vec{\gamma}}=\mathcal{D}_{\operatorname{Stab}(\vec{\gamma})}$
is as in Proposition 2.1, and where $\operatorname{cone}(U)=\\{tu\text{ with
}t\in[0,1],u\in U\\}$ is the cone with basis $U$.

######  Theorem 1.4.

For any $k$-multicurve $\vec{\gamma}^{o}=(\gamma_{i})_{i}$ in $\Sigma$, we
have

(5.19)
$\mathbb{D}_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})=\lim_{L\to\infty}m(\vec{\gamma}^{o},L)$

in the weak-*-topology on Radon measures on $\mathcal{C}(\Sigma)^{k}$.

###### Proof.

Let us explain the basic strategy of the proof. First, we get from Proposition
4.1 that there are subsequences $(L_{n})$ such that the limit

(5.20)
$\mathfrak{n}=\lim_{n\to\infty}\frac{1}{L_{n}^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L_{n}}\vec{\gamma}}$

exists and is of the form

(5.21)
$\mathfrak{n}=\mathbb{D}_{*}(\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}})$

for some measure $\mathfrak{q}$ on $\Delta$. Since our family of measures is
pre-compact by Lemma 4.2, all we need to do to prove the existence of the
limit (5.19) is to show that the measure $\mathfrak{n}$ from (5.20) does not
depend on the specific sequence $(L_{n})$. Because of the specific form of
$\mathfrak{n}$ it suffices to prove that the measure $\mathfrak{q}$ does not
depend on the sequence. To do that we will compute $\mathfrak{q}$.

Once the strategy of the proof is clear, let us get into the matter. We start
by fixing a filling multicurve $\sigma^{o}$ in $\Sigma$ satisfying (2.6) and
note that

$\mathbb{I}(\sigma^{o},\cdot)_{*}\left(\frac{1}{L_{n}^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L_{n}}\vec{\gamma}}\right)=\frac{1}{L_{n}^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L_{n}}\mathbb{I}(\sigma^{o},\vec{\gamma})}.$

From Proposition 3.1 we get that

$\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})\right)=\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\mathbb{I}(\sigma^{o},\vec{\gamma})}.$

Now, from (5.20) and the continuity of $\mathbb{I}(\sigma^{o},\cdot)$ we get
that

(5.22)
$\mathbb{I}(\sigma^{o},\cdot)_{*}\mathfrak{n}=\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})\right).$

In light of (5.21) we can write this as

(5.23)
$\big{(}\mathbb{I}(\sigma^{o},\cdot)\circ\mathbb{D}\big{)}_{*}(\mathfrak{q}\otimes\mathfrak{m}_{\operatorname{Thu}})=\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\left(\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot\mathfrak{m}_{\operatorname{Thu}}\right).$

To write this in a nicer way we will work in polar coordinates:

$\operatorname{polar}:\mathbb{R}^{k}_{\geqslant
0}\to\Delta_{k}\times\mathbb{R}^{0}_{\geqslant 0},\ \
\operatorname{polar}(\vec{x})=\left(\frac{1}{\|\vec{x}\|}\vec{x},\|\vec{x}\|\right).$

The reason to do so is that we have

$\displaystyle\operatorname{polar}\circ\mathbb{I}(\sigma^{o},\cdot)\circ\mathbb{D}$
$\displaystyle:\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\to\Delta_{k}\times\mathbb{R}_{\geqslant
0}$
$\displaystyle\operatorname{polar}\circ\mathbb{I}(\sigma^{o},\cdot)\circ\mathbb{D}$
$\displaystyle:((a_{i}),\lambda)\mapsto((a_{i}),\iota(\sigma^{o},\lambda))$

and we then get from (5.23) that

$\mathfrak{q}\otimes\big{(}\iota(\sigma^{o},\cdot)_{*}\mathfrak{m}_{\operatorname{Thu}}\big{)}=\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot\big{(}\operatorname{polar}\circ\mathbb{I}(\cdot,\vec{\gamma}^{o})\big{)}_{*}\left(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}}\right)$

Evaluating both sides of this equality at the set $U\times[0,1]$ we have

$\mathfrak{q}(U)\cdot\big{(}\iota(\sigma^{o},\cdot)_{*}\mathfrak{m}_{\operatorname{Thu}}\big{)}[0,1]=\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot\big{(}\operatorname{polar}\circ\mathbb{I}(\cdot,\vec{\gamma}^{o})\big{)}_{*}\left(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}}\right)(U\times[0,1])$

All that is left is cleaning up a bit. Indeed, note that

$\big{(}\iota(\sigma^{o},\cdot)_{*}\mathfrak{m}_{\operatorname{Thu}}\big{)}[0,1]=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)|\iota(\sigma^{o},\lambda)\leqslant
1\\})=B(\sigma^{o})$

and that

$\big{(}\operatorname{polar}\circ\mathbb{I}(\cdot,\vec{\gamma}^{o})\big{)}_{*}\left(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}}\right)(U\times[0,1])=\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U))$

where
$\operatorname{cone}(U)=\operatorname{polar}^{-1}(U\times[0,1])=\\{tu\text{
with }t\in[0,1]\text{ and }u\in U\\}$ is the cone with base $U$. Combining the
last three equalities we get

$\mathfrak{q}(U)\cdot
B(\sigma^{o})=\frac{B(\sigma^{o})}{\mathfrak{b}_{g,r}}\cdot\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U)),$

from where we get the identity

(5.24)
$\mathfrak{q}(U)=\frac{1}{\mathfrak{b}_{g,r}}\cdot\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U)).$

We have proved that $\mathfrak{q}$ is the measure given in (1.4), that is
$\mathfrak{q}=\mathfrak{q}_{\vec{\gamma}^{o}}$. In particular $\mathfrak{q}$,
and hence the sublimit $\mathfrak{n}$, does not depend on the specific
sequence $(L_{n})$. Summing up, we have proved that

$\mathbb{D}_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})=\lim_{n\to\infty}\frac{1}{L_{n}^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L_{n}}\vec{\gamma}}$

for any sequence $L_{n}\to\infty$ such that the limit on the right exists.
Since our family of measures is pre-compact by Lemma 4.2, it follows that the
actual limit exists. This concludes the proof of Theorem 1.4. ∎

We note that the measure $\mathfrak{q}_{\vec{\gamma}^{o}}$ is 0 when
restricted to the boundary of $\Delta_{k}$. Indeed, note that the preimage
under $\mathbb{I}(\cdot,\vec{\gamma}^{o})$ of $\partial\Delta_{k}$ is the
union of the sets $\\{\lambda\in\mathcal{C}(\Sigma)\text{ with
}\iota(\lambda,\gamma_{i})=0\\}$ for $i=1,\ldots k$ and as we pointed out
earlier, each of these sets has 0 Thurston measure. We record this for later
use:

######  Lemma 5.1.

For every $k$-multicurve $\vec{\gamma}^{o}$, we have
$\mathfrak{q}_{\vec{\gamma}^{o}}(\partial\Delta_{k})=0.$ ∎

##  6\. Assymptotic distributions of length vectors

In this section we derive Theorem 1.1 from Theorem 1.4. First we start with a
more general but weaker version of Theorem 1.1:

######  Proposition 6.1.

With notation as in Theorem 1.4 let
$F:\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant 0}^{m}$ be a continuous,
positive and homogenous map and let

(6.25)
$\mathfrak{c}(F)\stackrel{{\scriptstyle\text{def}}}{{=}}\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}(\\{\vec{x}\in\mathbb{R}^{k}_{\geqslant
0}\text{ with }\|\vec{x}\|\leqslant 1\\}).$

Denoting by

${\bf
M}_{F,\gamma^{o}}(L)=\\{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\text{
with }\|F(\vec{\gamma})\|\leqslant L\\}$

the set of all $k$-multicurrents $\vec{\gamma}$ in the orbit of
$\vec{\gamma}^{o}$ whose image under $F$ has at most norm $1$, we have

$\frac{1}{\mathfrak{c}(F)}\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}=\lim_{L\to\infty}\frac{1}{|{\bf
M}_{F,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}F(\vec{\gamma})}$

with respect to the weak-*-topology on measures on $\mathbb{R}_{\geqslant
0}^{m}$.

Here we say that the map $F$ is positive if it maps non-zero $k$-multicurrents
to non-zero vectors, and we say that it is homogenous if
$F(t\cdot\vec{\mu})=t\cdot F(\vec{\mu})$ for all $t\geqslant 0$ and all
$\vec{\mu}\in\mathcal{C}(\Sigma)^{k}$. Positivity implies that $|{\bf
M}_{F,\vec{\gamma}^{o}}(L)|<\infty$ for all $L$.

###### Proof.

From the homogeneity of $F$ we get that
$F(\frac{1}{L}\vec{\gamma})=\frac{1}{L}F(\vec{\gamma})$ and hence that

(6.26)
$F_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\vec{\gamma}}\right)=\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}F(\vec{\gamma}))}.$

Evaluating the right side we get that

$\left(F_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\vec{\gamma}}\right)\right)(\\{\vec{x}\in\mathbb{R}^{k}_{\geqslant
0}\text{ with }\|\vec{x}\|\leqslant 1\\})=\frac{|{\bf
M}_{F,\vec{\gamma}^{o}}(L)|}{L^{6g-6+2r}}$

Now, Theorem 1.4 and the continuity of $F$ imply that

(6.27)
$(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})=\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}F(\vec{\gamma}))}.$

Moreover, homogeneity of $F$ implies that
$(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})$
has the same scaling behavior as the Thurston measure:

$\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}(L\cdot
U)=L^{6g-6+2r}\cdot\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}(U)$

for all $U\subset\mathbb{R}_{\geqslant 0}^{k}$. This implies that the
$(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})$-measure
of the set $\\{\|F(\cdot)\|=1\\}$ vanishes and hence, by the convergence of
the measures and the Portmanteau theorem, that

$\lim_{L\to\infty}\frac{|{\bf
M}_{F,\vec{\gamma}^{o}}(L)|}{L^{6g-6+2r}}=\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}(\\{\vec{x}\in\mathbb{R}^{k}_{\geqslant
0}\text{ with }\|\vec{x}\|\leqslant 1\\})\stackrel{{\scriptstyle\eqref{eq I am
getting lost}}}{{=}}\mathfrak{c}(F).$

It follows thus that, up to paying the price of multiplying by the constant
$\mathfrak{c}(F)$, we can replace in the limit in (6.27) the scaling factor
$L^{6g-6+2r}$ by $|{\bf M}_{F,\vec{\gamma}^{o}}(L)|$. When we do that we get

$\frac{1}{\mathfrak{c}(F)}\big{(}(F\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}=\lim_{L\to\infty}\frac{1}{|{\bf
M}_{F,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}\vec{\gamma}}$

This concludes the proof of Proposition 6.1. ∎

Shortly, when we prove Theorem 1.1, we will consider maps
$\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant 0}^{k}$ of a very specific
form. Note however that Proposition 6.1 allows for many different kinds of
maps. For example, if we fix two negatively curved Riemannian metrics $X,Y$ on
$\Sigma$, a conformal structure $Z$ on $\Sigma$, and a generating set $S$ for
the fundamental group of $\pi_{1}(\Sigma)$ then we could apply the proposition
to the following map:

$F:\mathcal{C}(\Sigma)^{2}\to\mathbb{R}^{4},\ \
F(\mu_{1},\mu_{2})=\left(\begin{array}[]{c}-\ell_{X}(\mu_{1})+\operatorname{Lip}(Y,X)\cdot\ell_{Y}(\mu_{1})\\\
\ell_{X}(\mu_{1})+\ell_{X}(\mu_{2})-\ell_{X}(\mu_{1}+\mu_{2})\\\
\operatorname{Ext}_{Z}(\mu_{1}+\mu_{2})\\\
\|\mu_{1}\|_{S}+\|\mu_{2}\|_{S}-\|\mu_{1}+\mu_{2}\|_{S}\end{array}\right)$

where $\ell_{X}(\cdot)$ and $\ell_{Y}(\cdot)$ are the length function
associated to the Riemannian metrics $X$ and $Y$, where
$\operatorname{Lip}(Y,X)$ is the smallest possible Lipschitz constant of maps
$(\Sigma,Y)\to(\Sigma,X)$ homotopic to the identity, where
$\operatorname{Ext}_{Z}(\cdot)$ is the extremal length associated to the
conformal structure $Z$, and where $\|\cdot\|_{S}$ is the stable norm on
$\pi_{1}(\Sigma)$ associated to the word metric given by $S$. We are not sure
why anybody would want to consider such maps, but we just wanted to point out
that it would be possible.

After this comment, let us prove Theorem 1.1, but first we recall the notation
used in the introduction: Given a continuous, homogenous and positive function
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ and a $k$-multicurve
$\vec{\gamma}$ we are setting

$\mathbb{L}_{\phi}(\vec{\gamma})=\left(\frac{1}{\sum_{i}\phi(\gamma_{i})}(\phi(\gamma_{1}),\dots,\phi(\gamma_{k})),\sum_{i}\phi(\gamma_{i})\right)\in\Delta_{k}\times\mathbb{R}_{\geqslant
0}.$

The theorem is about the convergence of the measures

$\underline{m}(\phi,\vec{\gamma}^{o},L)=\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\mathbb{L}_{\phi}(\frac{1}{L}\vec{\gamma})}$

on $\Delta_{k}\times\mathbb{R}_{\geqslant 0}$, where

${\bf
M}_{\phi,\vec{\gamma}^{o}}(L)=\left\\{(\gamma_{i})_{i}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}\
\middle|\ \sum_{i}\phi(\gamma_{i})\leqslant L\right\\}.$

This is what we have to prove:

######  Theorem 1.1.

For every $k$-multicurve $\vec{\gamma}^{o}=(\gamma_{1},\dots,\gamma_{k})$ in
$\Sigma$ there is a probability measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ on
$\Delta_{k}$ such that for any continuous, positive, homogenous function
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ on the space of currents
on $\Sigma$ we have

$\lim_{L\to\infty}\underline{m}(\phi,\vec{\gamma}^{o},L)=\mathfrak{p}_{\vec{\gamma}^{o}}\otimes\left((6g-6+2r)\cdot
t^{6g-7+2r}\cdot d{\bf t}\right)$

where $d{\bf t}$ stands for the standard Lebesgue measure on
$\mathbb{R}_{\geqslant 0}$. In particular, the limiting distribution is
independent of $\phi$.

###### Proof.

Applying Proposition 6.1 to the map

$F_{\phi}:\mathcal{C}(\Sigma)^{k}\to\mathbb{R}_{\geqslant 0}^{k},\ \
F_{\phi}((\mu_{i})_{i})=(\phi(\mu_{i})_{i})$

we get that

(6.28)
$\frac{1}{\mathfrak{c}(F_{\phi})}\big{(}(F_{\phi}\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}=\lim_{L\to\infty}\frac{1}{|{\bf
M}_{F_{\phi},\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}F_{\phi}(\vec{\gamma})}$

Let us calculate $\mathfrak{c}(F_{\phi})$:

######  Lemma 6.2.

We have $\mathfrak{c}(F_{\phi})=\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot
B(\phi)$ where, as before,
$B(\phi)=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant 1\\})$.

###### Proof.

Noting that

$(F_{\phi}\circ\mathbb{D})((a_{i})_{i},\lambda)=(a_{i}\cdot\phi(\lambda))_{i}=\phi(\lambda)\cdot(a_{i})_{i}$

and hence that

$\|(F_{\phi}\circ\mathbb{D})((a_{i})_{i},\lambda)\|=\phi(\lambda)\cdot\|(a_{i})_{i}\|=\phi(\lambda)$

we have

$\displaystyle\mathfrak{c}(F_{\phi})$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq I am getting
lost}}}{{=}}\big{(}(F_{\phi}\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}(\\{\vec{x}\in\mathbb{R}^{k}_{\geqslant
0}\text{ with }\|\vec{x}\|\leqslant 1\\})$
$\displaystyle=(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})(\\{((a_{i}),\lambda)\in\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\|(F_{\phi}\circ\mathbb{D})((a_{i}),\lambda)\|\leqslant 1\\})$
$\displaystyle=(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})(\\{((a_{i}),\lambda)\in\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant 1\\})$
$\displaystyle=(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})(\Delta_{k}\times\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant 1\\})$
$\displaystyle=\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot B(\phi)$

as we wanted to prove. ∎

Considering again our version of polar coordinates

$\operatorname{polar}:\mathbb{R}_{\geqslant
0}^{k}\to\Delta_{k}\times\mathbb{R}_{k\geqslant 0},\ \
\operatorname{polar}(\vec{x})=\left(\frac{1}{\|\vec{x}\|}\cdot\vec{x},\|\vec{x}\|\right),$

note that we have

$\operatorname{polar}(F_{\phi}((\mu_{i})_{i}))=\operatorname{polar}(\phi(\mu_{1}),\dots,\phi(\mu_{k}))=\mathbb{L}_{\phi}((\mu_{i})_{i})$

and hence that

$\underline{m}(\phi,\vec{\gamma}^{o},L)=\operatorname{polar}_{*}\left(\frac{1}{|{\bf
M}_{F_{\phi},\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in\operatorname{Map}(\Sigma)\cdot\vec{\gamma}^{o}}\delta_{\frac{1}{L}F_{\phi}(\vec{\gamma})}\right).$

From (6.28), Lemma 6.2, and the continuity of $\operatorname{polar}$ we get
thus that

(6.29) $\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot
B(\phi)}\big{(}(\operatorname{polar}\circ
F_{\phi}\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})\big{)}=\lim_{L\to\infty}\underline{m}(\phi,\vec{\gamma}^{o},L).$

What we have to do is to calculate the measure on the left side in (6.29) and
what helps us is that the map

(6.30) $\operatorname{polar}\circ
F_{\phi}\circ\mathbb{D}:\Delta_{k}\times\mathcal{M}\mathcal{L}(\Sigma)\to\Delta_{k}\times\mathbb{R}_{\geqslant
0}$

is very simple:

$\displaystyle(\operatorname{polar}\circ
F_{\phi}\circ\mathbb{D})((a_{i})_{i},\lambda)$
$\displaystyle=(\operatorname{polar}\circ F_{\phi})((a_{i}\cdot\lambda)_{i})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}\operatorname{polar}((a_{i}\cdot\phi(\lambda))_{i})\stackrel{{\scriptstyle(**)}}{{=}}((a_{i}),\phi(\lambda))$

where in (*) we used homogeneity of $\phi$ and in (**) the fact that
$\sum_{i}a_{i}=1$ because $(a_{i})_{i}\in\Delta_{k}$. In other words we have
that $\operatorname{polar}\circ
F_{\phi}\circ\mathbb{D}=\operatorname{Id}\times\phi$ and hence that

$(\operatorname{polar}\circ
F_{\phi}\circ\mathbb{D})_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})=\operatorname{Id}_{*}(\mathfrak{q}_{\vec{\gamma}^{o}})\otimes\phi_{*}(\mathfrak{m}_{\operatorname{Thu}})=\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\phi_{*}(\mathfrak{m}_{\operatorname{Thu}}).$

We record what we have proved so far:

(6.31) $\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot
B(\phi)}\cdot\left(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\phi_{*}(\mathfrak{m}_{\operatorname{Thu}})\right)=\lim_{L\to\infty}\underline{m}(\phi,\vec{\gamma}^{o},L).$

The next observation is that it is easy to give a formula for
$\phi_{*}(\mathfrak{m}_{\operatorname{Thu}})$.

######  Lemma 6.3.

We have
$\phi_{*}(\mathfrak{m}_{\operatorname{Thu}})=B(\phi)\cdot(6g-6+2r)\cdot
t^{6g-7+2r}\cdot d{\bf t}$ where $d{\bf t}$ stands for the standard Lebesgue
measure on $\mathbb{R}_{\geqslant 0}$.

###### Proof.

Note that it suffices to show that the measures on the left and the right
agree when evaluated on all intervals $[0,T]$. On the right side we get the
value

$B(\phi)\cdot(6g-6+2r)\int_{0}^{T}t^{6g-7+2r}d{\bf t}=B(\phi)\cdot
T^{6g-6+2r}$

and on the left we get

$\displaystyle\phi_{*}\mathfrak{m}_{\operatorname{Thu}}[0,T]$
$\displaystyle=\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant T\\})$
$\displaystyle\stackrel{{\scriptstyle(*)}}{{=}}\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant 1\\})\cdot T^{6g-6+2r}$
$\displaystyle=B(\phi)\cdot T^{6g-6+2r}$

where in (*) we used once again that $\phi$ is homogenous as well as the
scaling behavior of $\mathfrak{m}_{\operatorname{Thu}}$. Having found that
both measures assign the same value to the interval $[0,T]$ for all $T$, we
get that they both agree. ∎

Continuing with the proof of Theorem 1.1 note that combining (6.31) and Lemma
6.3 we get that

$\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|}\cdot\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\big{(}(6g-6+2r)\cdot
t^{6g-7+2r}d{\bf
t}\big{)}=\lim_{L\to\infty}\underline{m}(\phi,\vec{\gamma}^{o},L).$

Theorem 1.1 follows when we set
$\mathfrak{p}_{\vec{\gamma}^{o}}=\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|}\cdot\mathfrak{q}_{\vec{\gamma}^{o}}$.
∎

##  7\. Asymptotic distribution of vectors of ratios

We now prove Theorem 1.3. Once again we recall the notation from the
introduction. Given two continuous, homogenous, positive functions
$\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ and a
$k$-multicurve $\vec{\gamma}$ we denote by

$\operatorname{ratio}_{\psi/\phi}((\gamma_{i})_{i})=\left(\frac{\psi(\gamma_{i})}{\phi(\gamma_{i})}\right)_{i}\in\mathbb{R}_{\geqslant
0}^{k}$

the vector whose entries are the ratios of the $\psi$ and $\phi$ values of
each component of $\vec{\gamma}$. What we care about is the limit of the
measures

$r(\phi/\psi,\vec{\gamma}^{o},L)=\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}$

as $L\to\infty$.

######  Theorem 1.3.

For any two continuous, homogenous, positive functions
$\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ there is a
probability measure $\mathfrak{r}_{\psi/\phi}$ on $\mathbb{R}_{\geqslant 0}$
with

$\operatorname{diag}_{*}(\mathfrak{r}_{\psi/\phi})=\lim_{L\to\infty}\frac{1}{|{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)|}\underline{r}(\phi/\psi,\vec{\gamma}^{o},L)$

for any $k$-multicurve $\vec{\gamma}^{o}$. Here
$\operatorname{diag}:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}^{k}$
is the diagonal map $\operatorname{diag}(t)=(t,\dots,t)$.

###### Proof.

First note that, as in the proof of Theorem 1.1 we get, for every $\phi$, that

$|{\bf M}_{\phi,\vec{\gamma}^{o}}(L)|\sim\mathfrak{c}(\phi)\cdot
L^{6g-6+2r}=\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot B(\phi)\cdot
L^{6g-6+2r},$

meaning that for every $\psi$ we have

$r(\phi/\psi,\vec{\gamma}^{o},L)\sim\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot
B(\phi)}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})},$

where we have used $\sim$ to indicate that the expressions are asymptotic as
$L\to\infty$. It follows that it suffices to consider the measures on the
right side.

Well, note that the map which sends the $k$-multicurve $\vec{\gamma}$ to
$\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})$ extends continuously to the
space of $k$-multicurrents with non-zero entries:

$\operatorname{ratio}_{\psi/\phi}:(\mathcal{C}(\Sigma)\setminus\\{0\\})^{k}\to\mathbb{R}_{\geqslant
0}^{k},\ \
\operatorname{ratio}_{\psi/\phi}((\mu_{i})_{i})=\left(\frac{\psi(\mu_{i})}{\phi(\mu_{i})}\right)_{i}$

and that this map has the following properties:

(7.32) $\displaystyle\operatorname{ratio}_{\phi/\psi}(L\cdot\vec{\mu})$
$\displaystyle=\operatorname{ratio}_{\phi/\psi}(\vec{\mu}),\text{ and}$ (7.33)
$\displaystyle(\operatorname{ratio}_{\psi/\phi}\circ\mathbb{D})((a_{i}),\lambda)$
$\displaystyle=\operatorname{diag}\left(\frac{\psi(\lambda)}{\phi(\lambda)}\right).$

From (7.32) we get that

$\displaystyle\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}$
$\displaystyle\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}=$
$\displaystyle=\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\frac{1}{L}\vec{\gamma})}$
$\displaystyle=(\operatorname{ratio}_{\psi/\phi})_{*}\left(\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\frac{1}{L}\vec{\gamma}}\right)$
$\displaystyle=(\operatorname{ratio}_{\psi/\phi})_{*}\left(m(\vec{\gamma},L)|_{\\{\vec{\mu}\in\mathcal{C}(\Sigma)^{k}\text{
with }\|\phi(\mu)\|\leqslant 1\\}}\right)$

where $m(\vec{\gamma},L)$ is the measure considered in Theorem 1.4. The
convergence of the measures $m(\vec{\gamma},L)$, the continuity of
$\operatorname{ratio}_{\psi/\phi}$ on its domain
$(\mathcal{C}(\Sigma)\setminus\\{0\\})^{k}$, and the fact that
$\mathfrak{q}_{\vec{\gamma}^{o}}(\partial\Delta_{k})=0$ (Lemma 5.1) imply thus
that

$\displaystyle\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}$
$\displaystyle\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}=$
$\displaystyle=(\operatorname{ratio}_{\psi/\phi})_{*}\left((\mathbb{D}_{*}(\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}}))|_{\\{\mu\in\mathcal{C}(\Sigma)^{k}|\phi(\mu)\leqslant
1\\}}\right)$
$\displaystyle=(\operatorname{ratio}_{\psi/\phi}\circ\mathbb{D})_{*}\left((\mathfrak{q}_{\vec{\gamma}^{o}}\otimes\mathfrak{m}_{\operatorname{Thu}})|_{\Delta_{k}\times\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)|\phi(\lambda)\leqslant
1\\}}\right).$

Using (7.33), we can rewrite this as

$\lim_{L\to\infty}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}=\\\
=\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot\left(\operatorname{diag}\circ\left(\frac{\psi(\cdot)}{\phi(\cdot)}\right)\right)_{*}\left(\mathfrak{m}_{\operatorname{Thu}}|_{\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)|\phi(\lambda)\leqslant
1\\}}\right).$

Scaling by the missing factors we get thus that

$\displaystyle\lim_{L\to\infty}r(\phi/\psi,\vec{\gamma}^{o},L)$
$\displaystyle=\lim_{L\to\infty}\frac{1}{\|\mathfrak{q}_{\vec{\gamma}^{o}}\|\cdot
B(\phi)}\frac{1}{L^{6g-6+2r}}\sum_{\vec{\gamma}\in{\bf
M}_{\phi,\vec{\gamma}^{o}}(L)}\delta_{\operatorname{ratio}_{\psi/\phi}(\vec{\gamma})}$
$\displaystyle=\operatorname{diag}_{*}\left(\left(\frac{\psi(\cdot)}{\phi(\cdot)}\right)_{*}\left(\frac{1}{B(\phi)}\mathfrak{m}_{\operatorname{Thu}}|_{\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)|\phi(\lambda)\leqslant
1\\}}\right)\right)$

and the claim follows when we set

(7.34)
$\mathfrak{r}_{\psi/\phi}=\left(\frac{\psi(\cdot)}{\phi(\cdot)}\right)_{*}\left(\frac{1}{B(\phi)}\mathfrak{m}_{\operatorname{Thu}}|_{\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)|\phi(\lambda)\leqslant
1\\}}\right).$

We have proved Theorem 1.3. ∎

##  8\. Examples

In this final section we discuss the measures
$\mathfrak{p}_{\vec{\gamma}^{o}}$ and $\mathfrak{r}_{\psi/\phi}$ in some
concrete examples. In a nutshell we get that the measure
$\mathfrak{p}_{\vec{\gamma}^{o}}$ is much better behaved than the measure
$\mathfrak{r}_{\psi/\phi}$. In our examples this is made apparent by the fact
that while we calculate $\mathfrak{p}_{\vec{\gamma}^{o}}$ in some concrete
cases, all results about $\mathfrak{r}_{\psi/\phi}$ describe different
pathologies.

### Some pathological examples of the measure $\mathfrak{r}_{\psi/\phi}$

From (7.34) we get an explicit description of the measure
$\mathfrak{r}_{\psi/\phi}$ for two given continuous, homogenous and positive
functions $\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$. We want
however a different formula. First note that $\phi$ determines a probability
measure on the space $\mathcal{P}\mathcal{M}\mathcal{L}(\Sigma)$ of projective
measured laminations via the formula

$\mathfrak{n}_{\phi}(U)=\frac{1}{B(\phi)}\mathfrak{m}_{\operatorname{Thu}}(\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma)\text{
with }\phi(\lambda)\leqslant 1\text{ and }[\lambda]\in U\\})$

where $[\lambda]$ stands for the projective class of $\lambda$. Noting that
the function $\frac{\psi(\cdot)}{\phi(\cdot)}$ descends to a well-defined map

$\frac{\psi(\cdot)}{\phi(\cdot)}:\mathcal{P}\mathcal{M}\mathcal{L}(\Sigma)\to\mathbb{R}_{\geqslant
0}$

it is easy to see that we can rewrite $\mathfrak{r}_{\psi/\phi}$ as

(8.35)
$\mathfrak{r}_{\psi/\phi}=\left(\frac{\psi(\cdot)}{\phi(\cdot)}\right)_{*}\mathfrak{n}_{\phi}.$

Since $\mathcal{P}\mathcal{M}\mathcal{L}(\Sigma)$ is compact and connected, we
get that the support of $\mathfrak{r}_{\psi/\phi}$ is a possibly degenerate
interval $[a,b]\subset\mathbb{R}_{>0}$. Our first observation, a not very
surprising one, is that $\mathfrak{r}_{\psi/\phi}$ can well be absolutely
continuous with respect to Lebesgue measure:

######  Example 1.

There are two filling weighted multicurves $\alpha,\beta$ such that
$\mathfrak{r}_{\iota(\beta,\cdot)/\iota(\alpha,\cdot)}$ is absolutely
continuous with respect to the Lebesgue measure. To construct our two
multicurves we start by choosing a filling multicurve $\sigma$ and a
collection $\gamma_{1},\dots,\gamma_{n}$ of curves with the property that
whenever we have $\lambda,\lambda^{\prime}\in\mathcal{M}\mathcal{L}(\Sigma)$
with

$\iota(\gamma_{i},\lambda)=\iota(\gamma_{i},\lambda^{\prime})\text{ for all
}i=1,\dots,n$

then $\lambda=\lambda^{\prime}$. This means in particular that the map

$\mathbb{I}=\mathbb{I}(\cdot,(\gamma_{i})):\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma),\
\iota(\sigma,\lambda)=1\\}\to\mathbb{R}^{n},\ \
\mathbb{I}(\lambda)=(\iota(\gamma_{i},\lambda))_{i}$

is an embedding. In fact, if we endow $\mathcal{M}\mathcal{L}(\Sigma)$ with
its standard PL-structure, then $\mathbb{S}$ is a PL-homeomorphism onto a PL-
sphere in Euclidean space. Moreover, if we identify
$\mathcal{P}\mathcal{M}\mathcal{L}(\Sigma)\simeq\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma),\
\iota(\sigma,\lambda)=1\\}$ by sending the class $[\lambda]$ to its unique
representative $\lambda$ with $\iota(\sigma,\lambda)=1$, then
$\mathbb{I}_{*}(\mathfrak{n}_{\iota(\sigma,\cdot)})$ is a measure in the
Lebesgue class associated to the PL-structure. It follows that for generic
choices of $(a_{1},\dots,a_{n})\in\mathbb{R}^{n}_{\geq 0}$ the push forward of
$\mathbb{I}_{*}(\mathfrak{n}_{\iota(\sigma,\cdot)})$ under the linear map
$\rho_{(a_{1},\dots,a_{n})}:\mathbb{R}^{n}\to\mathbb{R}$ given by
$\rho_{(a_{1},\dots,a_{n})}(x_{1},\dots,x_{n})\mapsto\sum a_{i}x_{i}$ is a
measure which is absolutely continuous with respect to the Lebesgue measure.
Now, the point of all this is that

$\frac{\iota(\sigma+\sum
a_{i}\gamma_{i},\cdot)}{\iota(\sigma,\cdot)}:\mathcal{P}\mathcal{M}\mathcal{L}(\Sigma)\simeq\\{\lambda\in\mathcal{M}\mathcal{L}(\Sigma),\
\iota(\sigma,\lambda)=1\\}\to\mathbb{R}$

satisfies

$\frac{\iota(\sigma+\sum
a_{i}\gamma_{i},\cdot)}{\iota(\sigma,\cdot)}=1+(\rho_{(a_{1},\dots,a_{n})}\circ\mathbb{I}).$

We can thus take $\alpha=\sigma$ and $\beta=\sigma+\sum a_{i}\gamma_{i}$ for
generic choices of $a_{i}>0$.

In some sense Example 1 is just an example of what one would expect
generically to happen. It is however clear that $\mathfrak{r}_{\psi/\phi}$ is
not always absolutely continuous with respect to the Lebesgue measure. A silly
example of this would be to take $\phi=\psi$ because in this case our measure
is a Dirac measure $\delta_{1}$ centered at $1\in\mathbb{R}_{\geqslant 0}$.
There are however much more interesting examples:

######  Example 2.

There are two distinct filling multicurves $\alpha,\beta$ such that
$\mathfrak{r}_{\iota(\alpha,\cdot)/\iota(\beta,\cdot)}=\delta_{1}$ is a Dirac
measure. Indeed, Horowitz [11] proved that there are plenty of pairs of curves
$\alpha,\beta$ which have the same length in any hyperbolic metric and it is
easy to get them filling: just get two in a pair of pants and then take an
immersion of the pair of pants in the surface $\Sigma$ in such a way that each
curve in the pair of pants is sent to a filling curve. Anyways, in [13]
Leininger proved that any one of Horowitz’s pairs of curves $\alpha,\beta$
also satisfies that $\iota(\alpha,\lambda)=\iota(\beta,\lambda)$ for all
$\lambda\in\mathcal{M}\mathcal{L}(\Sigma)$. It follows that the map
$\operatorname{ratio}_{\iota(\alpha,\cdot)/\iota(\beta,\cdot)}$ is constant
$1$. The claim follows thus from (8.35).

To construct more pathological examples recall that, when we choose a
hyperbolic metric on $\Sigma$, the space of currents can be identified with
the space $\mathcal{M}_{flip-flow}(\Sigma)$ of measures on the unit tangent
bundle $T^{1}\Sigma$ which are invariant under the geodesic flip and the
geodesic flow (see for example [10, Theorem 3.4]). It follows that if
$f:T^{1}\Sigma\to\mathbb{R}_{+}$ is any continuous function then

$\phi_{f}:\mathcal{C}(\Sigma)\simeq\mathcal{M}_{flip-
flow}(\Sigma)\to\mathbb{R}_{+},\ \phi_{f}(\mu)=\int_{T^{1}\Sigma}f\ d\mu$

is a continuous, homogenous and positive function on the space of currents.
Now, the key observation is that, because of the Birman-Series Theorem [3],
the set $\mathcal{X}$ of those vectors in $T^{1}\Sigma$ which are tangent to
some simple geodesic is a closed set of Hausdorff dimension $1$. It follows
that there are plenty of continuous functions $f:T^{1}\Sigma\to\mathbb{R}_{+}$
with support disjoint of $\mathcal{X}$ and for any such we have
$\phi_{f}(\lambda)+\phi(\lambda)=\phi(\lambda)$ for all
$\lambda\in\mathcal{M}\mathcal{L}(\Sigma)$ and any
$\phi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$. We thus have the
following:

######  Example 3.

There are plenty of pairs of distinct continuous, homogenous, positive
functions $\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{+}$ with
$\psi(\lambda)=\phi(\lambda)$ for all
$\lambda\in\mathcal{M}\mathcal{L}(\Sigma)$. For any such pair we have
$\mathfrak{r}_{\phi,\psi}=\delta_{1}$.

Now one can use the same idea to construct all sorts of examples. For
instance, let $\lambda_{1},\lambda_{2}\in\mathcal{M}\mathcal{L}(\Sigma)$ be
two maximal measured laminations whose supports are mutually transversal
geodesic laminations and let $U_{1},U_{2}\subset T^{1}\Sigma$ be open
neighborhoods of the supports of $\lambda_{1}$ and $\lambda_{2}$ respectively.
Then let $K_{i}\subset U_{i}$ be a compact neighborhood of the support of
$\lambda_{i}$ and let $f:T^{1}\Sigma\to\mathbb{R}_{+}$ be a continuous
function with $f|_{K_{1}}$ constant $2$ and $f|_{K_{2}}$ constant $3$. The map
$\operatorname{ratio}_{\phi_{f}/\phi_{\bf 1}}$ takes then the value $2$ in a
neighborhood of $\lambda_{1}$ and $3$ in a neighborhood of $\lambda_{2}$. It
follows that $\mathfrak{r}_{\phi_{f}/\phi_{\bf 1}}$ has at least two atoms,
one at $2$ and one at $3$.

Evidently one can use this construction to get any number of atoms. In fact,
even to get infinitely many atoms. And clearly one has an enormous flexibility
when constructing $f$.

######  Example 4.

For any $k=1,2,\dots,\infty$ there are plenty of pairs of distinct continuous,
homogenous and positive functions
$\phi,\psi:\mathcal{C}(\Sigma)\to\mathbb{R}_{\geqslant 0}$ such that
$\mathfrak{r}_{\psi/\phi}$ has $k$ atoms.

It is evident that in general the measure $\mathfrak{r}_{\psi/\phi}$ can be
pretty badly behaved and that it seems that one can get any pathology one can
think of. It might be however interesting to see how
$\mathfrak{r}_{\psi/\phi}$ behaves if we restrict $\phi,\psi$ to belonging to
some natural class, such as length functions of hyperbolic metrics.

### Some calculations of $\mathfrak{p}_{\vec{\gamma}^{o}}$

Let us recall now that the measure $\mathfrak{p}_{\vec{\gamma}^{o}}$ is the
probability measure on $\Delta_{k}$ proportional to the measure (1.4) given,
up to a constant, by

(8.36)
$U\mapsto\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{\gamma}^{o}}})(\operatorname{cone}(U))$

where
$\mathcal{D}_{\vec{\gamma}^{o}}=\mathcal{D}_{\operatorname{Stab}(\vec{\gamma}^{o})}$
is a fundamental domain for the action of
$\operatorname{Stab}(\vec{\gamma}^{o})$ on $\mathcal{M}\mathcal{L}(\Sigma)$,
and where as always $\operatorname{cone}(U)$ is the cone with basis $U$. It is
thus clear that to calculate $\mathfrak{p}_{\vec{\gamma}^{o}}$ one needs to
understand

  1. (1)

how to build a fundamental domain $\mathcal{D}_{\vec{\gamma}^{o}}$ for the
action
$\operatorname{Stab}(\vec{\gamma}^{o})\curvearrowright\mathcal{M}\mathcal{L}(\Sigma)$,
and

  2. (2)

the restriction of the map $\mathbb{I}(\cdot,\vec{\gamma}^{o})$ to
$\mathcal{D}_{\vec{\gamma}^{o}}$.

Well, there are good news and bad news. The good news is that this is in
principle feasible, and the bad news is that as soon as $\vec{\gamma}$ gets
complicated then anybody agreeing to do it must be either slightly unaware of
the amount of work involved, or under the influence. Let us still consider
some specific cases:

######  Example 5.

Let $\Sigma$ be a one-holed torus, let $\alpha^{o},\beta^{o}$ be simple curves
which intersect once, and set $\vec{\gamma}^{o}=(\alpha^{o},\beta^{o})$. The
map

$\mathbb{I}(\cdot,\vec{\gamma}^{o}):\mathcal{M}\mathcal{L}(\Sigma)\to\mathbb{R}_{\geqslant
0}^{2}$

is 2-to-1 on $\mathbb{R}_{>0}^{2}$ and 1-to-1 elsewhere. From here, and the
definition of the Thurston measure, we get that

$\mathbb{I}(\cdot,\vec{\gamma}^{o})_{*}\mathfrak{m}_{\operatorname{Thu}}=2\cdot\operatorname{Leb}_{\mathbb{R}^{2}}$

where $\operatorname{Leb}_{\mathbb{R}^{2}}$ is the standard Lebesgue measure.
We thus get from (8.36) that

$\mathfrak{p}_{\vec{\gamma}^{o}}(U)=\frac{1}{\sqrt{2}}\operatorname{Leb}_{\Delta_{2}}(U)$

for every $U\subset\Delta_{2}$ measurable.

In the previous example and in the sequel, we denote by
$\operatorname{Leb}_{\Delta_{k}}$ the standard Lebesgue measure on
$\Delta_{k}$, that is the one obtained by integrating the Riemannian volume
form.

######  Example 6.

Suppose that $\vec{P}^{o}$ is a labeled pants decomposition. The stabilizer of
$\vec{P}^{o}$ is nothing other than the group
$\mathbb{T}\subset\operatorname{Map}(\Sigma)$ consisting of multi-twists along
the components of $\vec{P}^{o}$ and it is easy to find a fundamental domain
$\mathcal{D}_{\vec{P}^{o}}$ for the action of $\mathbb{T}$ on
$\mathcal{M}\mathcal{L}(\Sigma)$: it is intuitively the set of measured
laminations whose leaves twist at most once around the components of
$\vec{P}^{o}$, and this can be made formal using Dehn-Thurston coordinates. In
those coordinates, the map $\mathbb{I}(\cdot,\vec{P}^{o})$ is just the
projection onto half of the coordinates. We get then from there that

$\mathbb{I}(\cdot,\vec{P}^{o})_{*}(\mathfrak{m}_{\operatorname{Thu}}|_{\mathcal{D}_{\vec{P}^{o}}})(V)=\frac{1}{2^{2g-3+r}}\int_{V}\prod
x_{i}\ dx_{1}\ldots dx_{3g-3+r}$

for any $V\subset\mathbb{R}_{\geqslant 0}^{3g-3+r}$ (see for example [10,
Exercise 12.1], although this computation is in one way or another already in
Mirzakhani’s thesis [20], and we would not be surprised if there were earlier
sources) meaning that

$\displaystyle\mathfrak{q}_{\vec{P}^{o}}(U)$
$\displaystyle=\frac{1}{\mathfrak{b}_{g,r}}\cdot\frac{1}{2^{2g-3+r}}\int_{\operatorname{cone}(U)}\prod
x_{i}\ dx_{1}\ldots dx_{3g-3+r}$
$\displaystyle=\frac{1}{\mathfrak{b}_{g,r}}\cdot\frac{1}{2^{2g-3+r}}\cdot\frac{1}{2(3g-3+r)^{3/2}}\int_{U}\prod
x_{i}\ d\operatorname{Leb}_{\Delta_{3g-3+r}}$

for any $U\subset\Delta_{3g-3+r}$.

Now, taking into account that
$\mathfrak{p}_{\vec{P}^{o}}=\frac{\mathfrak{q}_{\vec{P}^{o}}}{\|\mathfrak{q}_{\vec{P}^{o}}\|}$
and that

$\int_{\Delta_{3g-3+r}}\prod
x_{i}\,d\operatorname{Leb}_{\Delta_{3g-3+r}}=\frac{\sqrt{3g-3+r}}{(6g-7+2r)!}$

we get that

$\mathfrak{p}_{\vec{P}^{o}}(U)=\frac{(6g-7+2r)!}{\sqrt{3g-3+r}}\cdot\int_{U}\prod
x_{i}\ d\operatorname{Leb}_{\Delta_{3g-3+r}}$

for all $U\subset\Delta_{3g-3+r}$.

This concludes the discussion for the pair of pants.

The real reason why one can calculate $\mathfrak{q}_{\vec{P}^{o}}$ for a
labeled pants decomposition is that it is not hard to give a specific
fundamental domain for the action of $\operatorname{Stab}(\vec{P}^{o})$ on
$\mathcal{M}\mathcal{L}(\Sigma)$. Explicit fundamental domains can also be
given for general simple multicurves $\vec{\gamma}^{o}$, and we could thus use
them to calculate $\mathfrak{q}_{\vec{\gamma}^{o}}$ in such cases. But one can
also do something else. Indeed, relying on the earlier mentioned results of
Arana-Herrera [2] and Liu [14] we get from Corollary 1.2 the following:

######  Example 7.

If $\vec{\gamma}^{o}$ is a simple $k$-multicurve then we have for every
$U\subset\Delta_{k}$

$\mathfrak{p}_{\vec{\gamma}^{o}}(U)=\int_{U}P_{\vec{\gamma}^{o}}d\operatorname{Leb}_{\Delta_{k}}$

for some specific homogenous polynomial $P_{\vec{\gamma}^{o}}$ of degree
$6g-6+2r-k$.

######  Remark.

From Arana-Herrera [2] and Liu [14] one gets an expression for the
coefficients of the polynomial $P_{\vec{\gamma}^{o}}$ in terms of intersection
numbers of Chern classes of line bundles on the Deligne-Mumford
compactification of moduli space. If on the other hand one uses the
fundamental domain implicit in the proof of [10, Theorem 11.2] then the
coefficients of $P_{\vec{\gamma}^{o}}$ are expressed in terms of Thurston
volumes of the set of measured laminations carried by some specific train
tracks. Using work of Norbury [23] and Kontsevich [12], one can reconcile the
two points of view along the lines of what is done in [15].

In all these examples the measure is in the Lebesgue class and has full
support. This is evidently not true in general, basically because there is no
reason for the map
$\mathbb{I}(\cdot,\vec{\gamma}^{o}):\mathcal{M}\mathcal{L}(\Sigma)\to\mathbb{R}_{\geqslant
0}^{k}$ to be surjective. Indeed, this map is PL with respect to the standard
PL-structure of $\mathcal{M}\mathcal{L}(\Sigma)$ and this implies that it
cannot possibly be surjective if
$k>6g-6+2r=\dim(\mathcal{M}\mathcal{L}(\Sigma))$. Anyways, since the action
$\operatorname{Stab}(\vec{\gamma}^{o})\curvearrowright\mathcal{M}\mathcal{L}(\Sigma)$
also admits a PL-fundamental domain, we get the following:

######  Proposition 8.1.

For every $k$-multicurve $\vec{\gamma}^{o}=(\gamma_{1},\dots,\gamma_{k})$
there are finitely many linear maps
$L_{1},\dots,L_{n}:\mathbb{R}^{6g-6+2r}\to\mathbb{R}^{k}$ mapping
$\Delta_{6g-6+2r}$ into $\Delta_{k}$ such that $\mathfrak{p}_{\vec{\gamma}}$
is a linear combination with positive coefficients of the measures
$(L_{1})_{*}\operatorname{Leb}_{\Delta_{6g-6+2r}},\dots,(L_{n})_{*}\operatorname{Leb}_{\Delta_{6g-6+2r}}$.

###### Sketch of proof.

We will only prove Proposition 8.1 in the case that each component
$\gamma_{1},\dots,\gamma_{k}$ is a non-simple curve and that $\sum\gamma_{i}$
has trivial stabilizer. This fully suffices to get the gist of the proof and
the reader used to working with train-tracks will have no difficulty filling
in the details of the remaining cases. Well, fixing a hyperbolic metric on
$\Sigma$ we get from our additional assumptions on $\vec{\gamma}^{o}$ that
there is some $\varepsilon$ so that all the intersections between a leaf of a
measured lamination and one of the components $\gamma_{i}$ happen with at
angle greater than $4\varepsilon$. Take now finitely many maximal train-tracks
which carry all of $\mathcal{M}\mathcal{L}(\Sigma)$ and such that the set of
measured laminations carried by two of them has negligible measure (take for
example the standard train tracks from [24]). Now, splitting these train
tracks finitely many times we can replace this initial collection by a new
collection $\tau_{1},\dots,\tau_{n}$ of train tracks which besides having
these same properties, also have geodesic curvature $<\varepsilon$ and meet
the components of $\vec{\gamma}^{o}$ with angle greater than $4\varepsilon$.
This implies that if we identify the set $\mathcal{M}\mathcal{L}(\tau_{i})$ of
measured laminations carried by $\tau_{i}$ with the set $W(\tau_{i})$ of
positive solutions of the switch equations, then the restriction of
$\mathbb{I}(\cdot,\vec{\gamma}^{o})$ to $\mathcal{M}\mathcal{L}(\tau_{i})$ is
linear. The claim follows because
$\mathcal{M}\mathcal{L}(\tau_{i})=W(\tau_{i})$ is simply the intersection of a
linear space with the positive quadrant and because the Thurston measure on
$\mathcal{M}\mathcal{L}(\tau_{i})$ is proportional to the standard Lebesgue
measure on the polyhedral cone $W(\tau_{i})$. ∎

Now, the reader probably suspects that if we consider sequences
$(\vec{\gamma}_{n})_{n}$ of $k$-multicurves which become more and more
complicated, then the measures $\mathfrak{p}_{\vec{\gamma}_{n}}$ can become
much wilder. This is true:

######  Example 8.

Suppose that $\Sigma$ is a one-holed torus, let $\alpha^{o},\beta^{o}$ be two
simple curves which intersect once, fix a complete hyperbolic metric $X$ on
the interior of $\Sigma$, let $(\eta_{n}^{o})_{n}$ be a sequence of weighted
geodesics whose associated geodesic flow invariant measures converge to the
Liouville measure of $X$, and set

$\vec{\gamma}_{n}^{o}=(\alpha^{o},\beta^{o},\eta_{n}^{o}).$

The key observation is that the maps $\mathbb{I}(\cdot,\vec{\gamma}_{n}^{o})$
converge uniformly on compacta to the map

$\mathbb{I}_{\infty}:\mathcal{M}\mathcal{L}(\Sigma)\to\mathbb{R}^{3},\ \
\lambda\mapsto(\iota(\alpha^{o},\lambda),\iota(\beta^{o},\lambda),\ell_{X}(\lambda))$

and hence that measures
$\mathbb{I}(\cdot,\vec{\gamma}_{n}^{o})_{*}\mathfrak{m}_{\operatorname{Thu}}$
converge to $(\mathbb{I}_{\infty})_{*}\mathfrak{m}_{\operatorname{Thu}}$. From
the work of McShane–Rivin [19] we get that the support of
$(\mathbb{I}_{\infty})_{*}\mathfrak{m}_{\operatorname{Thu}}$ is the cone over
the union of two continuous maps $[0,1]\to\Delta_{3}$, each one of which fails
to be smooth in a dense set of points. We deduce that there is no $N$ such
that all measures $\mathfrak{p}_{\vec{\gamma}_{n}^{o}}$ are supported by $N$
linear images of the simplex $\Delta_{2}$.

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