# Dual spaces of geodesic currents

Luca de Rosa  Department of Mathematics
ETH Zurich
Rämistrasse 101, 8092 Zurich
Switzerland<EMAIL_ADDRESS>and  Dídac Martínez-Granado  Department
of Mathematics
University of Luxembourg
Av. de la Fonte 6, Esch-sur-Alzette, L-4364, Luxembourg
<EMAIL_ADDRESS>
###### Abstract.

Every geodesic current on a hyperbolic surface has an associated dual space.
If the current is a lamination, this dual embeds isometrically into a real
tree. We show that, in general, the dual space is a Gromov hyperbolic metric
tree-graded space, and express its Gromov hyperbolicity constant in terms of
the geodesic current. In the case of geodesic currents with no atoms and full
support, such as those coming from certain higher rank representations, we
show the duals are homeomorphic to the surface. We also analyze the
completeness of the dual and the properties of the action of the fundamental
group of the surface on the dual. Furthermore, we compare two natural
topologies in the space of duals.

###### Contents

  1. 1 Introduction
  2. 2 Background
  3. 3 Dual space of a geodesic current
  4. 4 Continuity of the projection
  5. 5 Examples
  6. 6 Hyperbolicity
  7. 7 Decomposition theorem
  8. 8 Actions
  9. 9 Completeness
  10. 10 Topology
  11. A Equivalence of notions of tree graded spaces

##  1\. Introduction

An $\mathbb{R}$-tree is a geodesic metric space where any two points are
connected by a unique arc isometric to a closed interval in $\mathbb{R}$. A
measured lamination $\lambda$ on a surface $X$ defines a (dual)
$\pi_{1}(X)$-action on an $\mathbb{R}$-tree as follows. Lift the lamination to
the universal cover $\widetilde{X}$ and define the pseudo-distance between two
points by considering the measure of the set of geodesics intersecting the
geodesic segment connecting them. This turns out to define a $0$-hyperbolic
space $X_{\lambda}$, that embeds isometrically into an $\mathbb{R}$-tree,
called the tree dual to the measured lamination. There are several equivalent
formulations of this construction: see [MS91, Wol98, Kap09]. We explore the
equivalence of these with our construction in Subsection 5.1. It follows from
the construction of the dual $X_{\lambda}$ of a measured lamination that the
translation length of an element $g\in\pi_{1}(X)$ is equal to the intersection
number of the measured lamination with $[g]$, the homotopy class represented
by $g$.

First introduced by Bonahon in his seminal paper [Bon86], geodesic currents
can be understood as an extension of measured laminations where the geodesics
in the support of the measure are allowed to intersect each other.

The above construction of dual space of a lamination can be extended to any
geodesic current $\mu$ on a compact hyperbolic surface $X$ (possibly with non-
empty geodesic boundary).

######  Definition (Dual space of a geodesic current).

A geodesic current $\mu$ induces a pseudo-distance on $\widetilde{X}$ given by

$d_{\mu}(\overline{x},\overline{y})=\frac{1}{2}\left\\{\mu(G[\overline{x},\overline{y}))+\mu(G(\overline{x},\overline{y}])\right\\}.$

where $G[\overline{x},\overline{y})$ denotes the set of hyperbolic geodesics
of $\widetilde{X}$ transverse to the geodesic segment
$[\overline{x},\overline{y})$. The dual space of $\mu$, denoted by $X_{\mu}$,
is defined as the metric quotient of on $\widetilde{X}$ under this this
pseudo-distance. The set of all dual spaces of geodesic currents will be
denoted by $\mathcal{D}(X)$.

This space was introduced by Burger-Iozzi-Parreau-Pozzetti in [BIPP21a]. We
note that $X_{\mu}$ depends on the choice of hyperbolic structure $X$ (see
Subsection 3.3). $X_{\mu}$ might not, a priori, be $0$-hyperbolic, but we show
that, in fact, it is always $\delta$-hyperbolic for some $\delta\geq 0$ that
can be described in terms of $\mu$. If $B$ denotes a _box_ of geodesics in
$\widetilde{X}$ given by a product of two intervals on
$\partial\widetilde{X}$, $B^{\perp}$ denotes the _opposite box_ given by the
complementary intervals, and $\mathcal{B}$ the family of all boxes $B$, we
prove the following result.

######  Theorem A (Hyperbolicity).

Let $\mu$ be a geodesic current on $X$, and let

$\delta_{\mu}=\sup_{B\in\mathcal{B}}\min{\\{\mu(B),\mu(B^{\perp})\\}}.$

Then the dual space $X_{\mu}$ is a $\delta_{\mu}$-hyperbolic space, and
$\delta_{\mu}$ is the optimal hyperbolicity constant.

It follows from this that $X_{\mu}$ is $0$-hyperbolic if and only if $\mu$ is
a measured lamination (see Corollary 6.13). Theorem A is stated as Theorem
6.12.

For the following, compare Figure 1.1.

Figure 1.1.  The figure shows a sketch of a geodesic current $\mu$ with two
components $\mu_{1}$ (type 1, blue, left) and $\mu_{2}$ (type 2, green,
right), separated by a simple multi-curve of one single component (type 3,
red, middle).

Even though, by the above, $X_{\mu}$ is not in general isometric to an
$\mathbb{R}$-tree, it has a structure that resembles that of an
$\mathbb{R}$-tree, as follows. In [BIPP21a], Burger-Iozzi-Parreau-Pozzetti
showed that there exists a simple multi-curve $m$ associated to $\mu$, called
_special multi-curve_ , given by disjoint simple closed geodesics
$(s_{j})_{j=1}^{k}$, that decompose $X$ into sub-surfaces with geodesic
boundary $X_{1},\cdots,X_{n}$ such that the geodesic current $\mu$ on $X$ can
be written as a sum

$\mu=\sum_{i=1}^{n}\mu_{i}+\sum_{j=1}^{k}a_{j}s_{j},$

where $a_{j}\in\mathbb{R}$ and each $\mu_{i}$ is supported on $X_{i}$, and
where the weights $a_{j}$ could be $0$. Moreover, for each $i$ such that
$\mu_{i}\neq 0$, precisely one of the following holds:

  * •

$\mathrm{sys}_{X_{i}}(\mu_{i})>0$ (_type 1_);

  * •

the current $\mu_{i}$ is a non-discrete measured lamination (_type 2_).

The _systole_ of a current $\mu$ on a surface with possibly non empty boundary
$X$ is given by

$\mathrm{sys}_{X}(\mu)=\inf_{c}i(\mu,c)$

where the infimum is taken over all closed geodesics contained in the interior
of $X$. A non-discrete measured lamination is one without closed components
(see Subsection 2.2 for details, and Section 7 for more on decomposition).

Parallel to this result, we obtain a decomposition theorem for the dual space
$X_{\mu}$ of $\mu$ as a (metric) tree graded space. A _tree graded space $X$_,
in the sense of Drutu-Sapir [DS05], is a geodesic metric space together with a
family of distinguished subsets $\mathcal{P}$ called pieces. Intuitively, $X$
is assembled from $\mathcal{P}$ by attaching them along an $\mathbb{R}$-tree
$T$ that acts as a “central spine”, in such a way that any two pieces
intersect each other at most at one point along $T$. This notion was suggested
to us by A.Parreau and B. Pozzetti. In fact, in [BIPP21c, Section 6] Burger,
Iozzi, Parreau and Pozzetti relate $\mathbb{R}^{2}$-tree-graded spaces to the
dual of a current given as a sum of two transverse measured laminations.

Our dual spaces are not endowed with a geodesic structure in general, so we
introduce the more general notion of _metric_ tree-graded space, and prove the
following.

######  Theorem B (Dual structure theorem).

The dual space $X_{\mu}$ is a metric tree-graded space whose underlying tree
is the dual tree of the special multi-curve $m$, and the pieces are the dual
spaces $(X_{i})_{\mu_{i}}$ of the currents $\mu_{i}$ on the sufaces $X_{i}$.

See Theorem 7.12 and Section 7 for a precise statement and definitions.

Compare Figure 1.2 for a part of a sketch of a geometric realization of the
dual space $X_{\mu}$ and a hint of its tree graded structure, where $\mu$ is
the current illustrated in Figure 1.1. The Figure shows three pieces of
$X_{\mu}$, two peripheral ones corresponding to the current $\mu_{2}$, and
one, central, corresponding to $\mu_{1}$, as well as two edges of the tree
$T$.

Figure 1.2.  The figure shows a sketch of the dual space $X_{\mu}$
corresponding to the current in Figure 1.1. The sketch is superimposed on the
support of $\mu$ (the lifts of the geodesics on $X$ to the universal cover
$\widetilde{X}$), that has been faded out so that a (geometric realization of
the) dual stands out.

Dual spaces come equipped with a natural action of $\pi_{1}(X)$, induced from
the action of such group in the universal cover $\widetilde{X}$. We relate the
properties of the action to the properties of the geodesic current.

######  Theorem C (Action).

Given any geodesic current $\mu$ on a surface $X$, the fundamental group
$\pi_{1}(X)$ acts by isometries on the dual space $X_{\mu}$, and it does so:

  1. (1)

Cobounded.

  2. (2)

Properly if and only if $\mu$ has no components of type 2 in its
decomposition.

  3. (3)

Freely if and only if $\mu$ has only one component in its decomposition.

Theorem C is stated as a series of smaller results in Section 8. We also study
the metric completeness of the dual spaces.

######  Theorem D (Completeness).

Let $\mu$ be a geodesic current on $X$ with no atoms. Then the dual space
$X_{\mu}$ is metrically complete if and only if $\mu$ has no components of
type 2 in its decomposition.

Theorem D appears stated as Theorem 9.5, and its proof spans Section 9. That
theorem also analyzes the atomic case, where the dual is complete if and only
if it is a multi-curve.

The dual space $X_{\mu}$ comes equipped with the natural projection map
$\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$. We study its continuity properties.

######  Theorem E (Continuity of projection).

Given a geodesic current $\mu$ on $X$, the projection
$\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$ satisfies:

  1. (1)

The projection $\pi_{\mu}$ is continuous if and only if $\mu$ has no atoms;

  2. (2)

If $\mu$ has no atoms and is filling, then $\pi_{\mu}$ is closed;

  3. (3)

If $\mu$ has no atoms and has full support, then $\pi_{\mu}$ is a
homeomorphism.

Theorem E appears as a series of Propositions in Section 4. Theorem E, in
conjunction with Theorem C, shows that $X_{\mu}/\pi_{1}(X)$ is homeomorphic to
$X$ for geodesic currents coming from certain higher rank representations of
$\pi_{1}(X)$ (precisely, for those coming from _positively ratioed
representations_ in the sense of [MZ19]). In the case a real convex projective
structure $Z=\Omega/\pi_{1}(Z)$ on $S$, we use this result to induce an
isometry between the dual space with underlying metric $Z$ and the surface $S$
equipped with the Hilbert metric coming from $Z$ (see Proposition 5.13).

We also study and relate two natural topologies in the space of geodesic
currents $\operatorname{Curr}(X)$ and in the space of duals $\mathcal{D}(X)$.
The space $\mathcal{D}(X)$ can be equipped with the equivariant Gromov-
Hausdorff topology, first introduced and studied by Paulin [Pau88]. This is a
variation of the Gromov-Hausdorff topology that bakes in the action of a
group. On the other hand, the space of currents $\mathrm{Curr}(X)$ is
naturally endowed with the weak∗-topology. We prove

######  Theorem F (Topologies).

The map $\Psi\colon\operatorname{Curr}(X)\to\mathcal{D}(X)$, defined by
$\mu\mapsto X_{\mu}$, is a continuous injection.

Theorem F appears stated as Theorem 10.3. Our theorem partially extends a
result of Paulin [Pau89] from the setting of $\mathbb{R}$-trees to more
general $\delta$-hyperbolic spaces within the class $\mathcal{D}(X)$.

###  1.1. Outline

In Section 2 we introduce geodesic currents and give some examples whose duals
we will study later on in Section 5. We also describe the weak∗-topology on
the space of currents and give a convenient family of neighborhoods that will
play a role in Section 10.

In Section 3 we introduce the dual of a geodesic current and relate it to the
notion of measured wall spaces [CD17].

In Section 4 we show that the natural projection map from the universal cover
of the surface to the dual space is continuous when the current has no atoms,
and a homeomorphism when the current is non-atomic and has full support. We
also show that if $\mu$ has atoms, the projection map is neither lower nor
upper-semicontinuous.

In Section 5 we explore other natural examples of such duals other than the
$\mathbb{R}$-trees coming from measured laminations (which are discussed in
detail in Section 5.1). For example, we study the dual space of two
intersecting measured laminations, and relate it to the concept of core of
trees previously introduced by Guirardel in [Gui05].

We also show how for geodesic currents coming from certain Anosov
representations, known as positively ratioed, such as strictly convex
projective structures, the duals are homeomorphic to the surface $X$. In fact,
in the case of a Hitchin representation $\rho$ in
$\operatorname{\mathit{PSL}}(3,\mathbb{R})$, the associated strictly convex
projective structure $\Omega_{\rho}$ is isometric to the dual
$X_{\mu_{\Omega}}$, when $X$ is the Hilbert metric on $S$.

In Section 6, we prove that duals are $\delta$-hyperbolic and moreover relate
the optimal $\delta$-hyperbolicity constant to the geodesic current. Using
this relation, we give inequalities between the $\delta$-hyperbolicity
constants of $X_{\mu}$ and the duals of the subcurrents $\mu_{i}$ in its
structural decomposition (in the sense of [BIPP21a]).

In Section 7, we prove that, using the decomposition theorem for geodesic
currents proven in [BIPP21a], one can obtain a corresponding decomposition for
the dual space as a tree graded space (in the sense of Drutu-Sapir [DS05]).
Strictly speaking, we need to develop a notion of metric tree graded space,
because of the lack, in general, of a geodesic structure on $X_{\mu}$.

In Section 8, we prove that the action of the fundamental group on the dual
space is cobounded, it is proper if and only if the current has no filling
measured lamination components (type 2), and it is free if and only if the
current has only one component in its decomposition.

In Section 9 we prove that, for a non-atomic $\mu$, the dual space $X_{\mu}$
is complete if and only if $\mu$ has no components of type 2 in its
decomposition. We also analyze the case where the current has atoms, and show
it is complete if and only if $\mu$ is a multi-curve.

In Section 10 we relate two natural topologies in the space of duals. One the
one hand, the axis topology, given in terms of translation lengths, is
directly related to the weak∗-topology on currents. On the other hand, one can
also consider the equivariant Gromov-Hausdorff topology, previously introduced
by Paulin in [Pau88]. We show that the weak∗-topology is finer than Gromov-
Hausdorff topology. Explicitly, we prove that the map sending a geodesic
current to its dual is a continuous injection when the space of currents is
equipped with the weak∗-topology and the space of duals is equipped with the
equivariant Gromov-Hausdorff topology. This partially generalizes work of
Paulin in [Pau89] for $\mathbb{R}$-trees within $\mathcal{D}(X)$. We also
discuss connections with recent work of Oregón-Reyes [OR22] and Jenya Sapir
[Sap22].

Cantrell and Oregón-Reyes [CR23] fit the notion of dual spaces of geodesic
currents in the general framework of boundary metric structures. These are
left-invariant hyperbolic pseudo-metrics on a non-elementary hyperbolic group
satisfying the _bounded backtracking property_. Such property has also been
studied independently by Kapovich and the second author in upcoming work
[KMG24], where they use it to construct an extension to geodesic currents for
the stable length of such actions, as well as other natural notions of length,
and relate it to the concept of small action of groups on $\mathbb{R}$-trees.

###  1.2. Acknowledgments

We thank Marc Burger, Indira Chatterji, Michael Kapovich, Giuseppe Martone,
Eduardo Oregón-Reyes, Anne Parreau, Beatrice Pozzetti, Jenya Sapir and Dylan
Thurston for useful conversations. The first author would like to thank
Raphael Appenzeller, Benjamin Brück, Matthew Cordes, Xenia Flamm and Francesco
Fournier Facio for insightful conversations. The second author acknowledges
support from U.S. National Science Foundation grants DMS 1107452, 1107263,
1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR
Network), and from the Luxembourg National research Fund AFR/Bilateral-
ReSurface 22/17145118, and thanks Marc Burger for his invitation to ETH
Zürich, during which a portion of this work was done.

##  2\. Background

Table 1 outlines the main notation in this paper, unless we explicitly state
otherwise.

Notation | Meaning
---|---
$S$ | compact topological surface
$X$ | compact $\delta$-hyperbolic geodesic surface
$\Gamma,\pi_{1}(X)$ | deck transformation group of $X$
$\widetilde{X}$ | universal cover of $X$
$X_{\mu}$ | dual of $\mu$
$\overline{x},\overline{K}$ | points/sets in $\widetilde{X}$ (as opposed to in $X_{\mu}$)
$x,K$ | points/sets in $X_{\mu}$
$\widetilde{X_{\lambda}},T_{\lambda}$ | geometric realization of $X_{\lambda}$
$\mathcal{G}(\widetilde{X})$ | bi-infinite geodesics on $\widetilde{X}$
$\gamma$ | geodesic in $\mathcal{G}(\widetilde{X})$
$\lambda$ | measured lamination
$\mu$ | geodesic current
$s$ | simple closed curve
$c$ | closed curve
$\mathit{Curves}(X)$ | (weighted) multi-curves
$\mathcal{ML}(X)$ | measured laminations on $X$
$\operatorname{Curr}(X)$ | geodesic currents on $X$
$\mathbb{P}\operatorname{Curr}(X)$ | projective geodesic currents on $X$
$\mathcal{D}(X)$ | space of duals
$G$ | general group (mostly hyperbolic)
Table 1. Notation for the objects related to surfaces, curves, geodesic
currents and duals.

In this section we introduce the basic concepts we will explore in this paper:
the definition and basic properties of geodesic current, some examples of
geodesic currents that will feature in this paper, and the weak∗-topology for
geodesic currents. This will motivate the central object of study, in Section
3, the _dual of a geodesic current_.

###  2.1. Geodesic Currents and the Intersection Form

Let $S$ be a closed connected orientable topological surface of negative Euler
characteristic and genus $g$. Let $p\colon\widetilde{S}\to S$ denote the
universal cover of $S$. For any $\delta$-hyperbolic geodesic metric on $S$, we
will denote by $X$ the surface $S$ equipped with said metric. $\widetilde{X}$
will denote $\widetilde{S}$ with the induced geodesic metric on
$\widetilde{S}$. Let $\partial\widetilde{X}$ denote the Gromov boundary of
$\widetilde{X}$. This compactifies $\widetilde{X}$ to a closed disk, and the
action of $\pi_{1}(X)$ on $\widetilde{X}$ by isometries extends to an action
by homeomorphisms on $\widetilde{X}\cup\partial\widetilde{X}$ [BH11, Chapter
III.H.3, 3.7 Proposition]. Let $\mathcal{G}(\widetilde{X})$ denote the space
of bi-infinite unoriented geodesics in $\widetilde{X}$, given by the quotient
of the space of unit speed parameterized geodesics with the compact-open
topology, where we furthermore forget the parameterization. Any
$\gamma\in\mathcal{G}(\widetilde{X})$ has unique endpoints at infinity
$(\gamma_{+},\gamma_{-})$, and the map
$\partial\colon\mathcal{G}(\widetilde{X})\to(\partial\widetilde{X}\times\partial\widetilde{X}-\Delta)/\mathbb{Z}_{2}$
sending $\gamma$ to its endpoints is a continuous, closed,
$\pi_{1}(X)$-equivariant surjective map [BL18, Proposition 2.2]. It need not
be injective: for example, in $CAT(0)$ metrics, it can have non-degenerate
intervals as fibers. In this paper, we will restrict our attention to geodesic
metrics on $\widetilde{X}$ that are proper, $\delta$-hyperbolic and for which
the map $\partial$ is a homeomorphism. In particular, given two endpoints at
infinity there is a unique element of $\mathcal{G}(\widetilde{X})$ with those
endpoints. We will furthermore assume that given any two points in
$\widetilde{X}$, there is a unique (unoriented, unparameterized) geodesic
segment between them. Any negatively curved Riemannian metric satisfies these
assumptions, including hyperbolic metrics. However, there are other noteworthy
examples such as Hilbert metrics on $\widetilde{X}$ coming from endowing $S$
with the Hilbert metric obtained by quotienting a strictly convex projective
domain $\Omega$ by a discrete and cocompact representation of $\pi_{1}(S)$
into $PGL(3,\mathbb{R})$ [dlH93, Page 99]. See more details in Section 5.
Unless stated otherwise, _the metric $X$ will be a hyperbolic structure on
$S$_, so that we can identify $X$ with the quotient $\widetilde{X}/\Gamma$ for
some $\Gamma=\pi_{1}(X)\leq{\operatorname{\mathit{PSL}}(2,\mathbb{R})}$
Fuchsian subgroup.

There are several equivalent definitions of geodesic currents. For an
excellent account we refer the reader to [ES22, Chapter 3]. We will mainly use
the following one.

######  Definition 2.1 (Geodesic current).

A _geodesic current_ on $X$ is a positive $\Gamma$-invariant locally finite
Radon measure on the set $\mathcal{G}(\widetilde{X})$ of unoriented
unparameterized bi-infinite geodesics in $\widetilde{X}$, which we identify by
their endpoints in the boundary at infinity. Let

$(\partial\widetilde{X})^{(2)}=\\{(x,y)\in(\partial\widetilde{X})^{2}\colon
x\neq y\\}.$

We define

$\mathcal{G}(\widetilde{X})\coloneqq(\partial\widetilde{X})^{(2)}/\sim$

where $(x,y)\sim(y,x)$. We denote with $\mathrm{Curr}(X)$ the set of all
geodesic currents on $X$.

We say that an interval in $\partial\widetilde{X}$ (or a geodesic segment
$\tau$ in $\widetilde{X}$) is _degenerate_ if it is a singleton. We say it is
_non-degenerate_ otherwise.

We will consider two types of subsets of geodesics: boxes and transversals.

######  Definition 2.2 (Box of geodesics).

Let $I_{a,b}$ denote a _generalized ordered interval_ in
$\partial\widetilde{X}$, i.e., any of the following non-empty and possibly
degenerate intervals: $(a,b),[a,b),(a,b],(a,b)$, where $a,b$ are ordered
counter-clockwise in $\partial\widetilde{X}$.

We define a _box of geodesics_ as any subset of $\mathcal{G}(\widetilde{X})$
of the type $B=I_{a,b}\times I_{c,d}$. Let $\mathcal{B}$ denote the family of
all boxes of geodesics.

######  Definition 2.3 (Transversal of geodesics).

Given a geodesic segment $\tau$ in $\widetilde{X}$ (which could be
degenerate), $G(\tau)$ denotes the subset of geodesics in
$\mathcal{G}(\widetilde{X})$ intersecting $\tau$ transversely, i.e., all
$\gamma\in\mathcal{G}(\widetilde{X})$ so that $\gamma\cap\tau=\\{p\\}$, where
$p\in\tau$. Given any subset $A$ of $\widetilde{X}$ which contains at least
one non-degenerate geodesic segment, $G(A)$ denotes the set of geodesics
$\gamma\in\mathcal{G}(\widetilde{X})$ intersecting transversely at least one
non-degenerate geodesic segment $\tau\subseteq A$. We will refer to these sets
$G(\tau)$ and $G(A)$ as _transversals_.

Note that as subsets of geodesics, a transversal $G(\tau)$, for $\tau$ a
geodesic segment, is not contained in one single box of geodesics, but it can
be contained in a union of two boxes. Also, if $\tau$ is a non-degenerate
geodesic segment, then $G(\tau)$ contains a non-degenerate box of geodesics.

Geodesic currents provide a unifying generalization of notions such as simple
closed curves, hyperbolic structures on $X$, and geodesic laminations, among
many others. For an introduction on the subject we refer to the seminal paper
[Bon88] by Bonahon.

Denote by
$\mathfrak{I}\subset\mathcal{G}(\widetilde{X})\times\mathcal{G}(\widetilde{X})$
the open set consisting of pairs of transversely intersecting geodesics in
$\widetilde{X}$. We endow $\mathfrak{I}$ with the subspace topology from
$\mathcal{G}(\widetilde{X})\times\mathcal{G}(\widetilde{X})$. Notice that the
diagonal $\Gamma$-action on
$\mathcal{G}(\widetilde{X})\times\mathcal{G}(\widetilde{X})$ descends to a
free, properly discontinuous and cocompact action on $\mathfrak{I}$ [ES22,
Page 44], and hence the projection
$\pi\colon\mathfrak{I}\to\mathfrak{I}/\Gamma$ is a topological covering.

Given two geodesic currents $\mu,\nu\in\operatorname{Curr}(X)$, they induce a
$\Gamma$-invariant product measure $\mu\times\nu$ on
$\mathcal{G}(\widetilde{X})\times\mathcal{G}(\widetilde{X})$, and hence on
$\mathfrak{I}$. This measure descends to a measure on $\mathfrak{I}/\Gamma$
via the covering $\pi\colon\mathfrak{I}\to\mathfrak{I}/\Gamma$. We will still
indicate such measure with $\mu\times\nu$, for notation simplicity.

######  Definition 2.4 (Intersection form).

The _intersection form_ evaluated on the two currents
$\mu,\nu\in\operatorname{Curr}(X)$ is the total volume of
$\mathfrak{I}/\Gamma$ in the measure $\mu\times\nu$

$i(\mu,\nu):=\left(\mu\times\nu\right)\left(\mathfrak{I}/\Gamma\right).$

###  2.2. Boundaries and filling currents

In most of the paper we will assume $X$ is a closed hyperbolic surface, even
though most of our results only use the cocompactness of $\pi_{1}(X)$ on
$\widetilde{X}$, and so are also true for surfaces with geodesic boundary. The
only time we will explicitly refer to surfaces with boundary, though, will be
when working with subsurfaces. We notice that certain notions such as
Liouville currents only make sense for closed surfaces.

Given a compact hyperbolic surface $X$ with geodesic boundary, the space of
_internal geodesic currents_ is the subspace
$\operatorname{Curr}_{0}(X)\subseteq\operatorname{Curr}(X)$ consisting of
currents not supported on lifts of boundary geodesics. If $\mu$ is only
supported on lifts of boundary parallel geodesics, we say it is a _boundary
geodesic current_. By the definition of intersection number of geodesic
currents (see also [ES22, Exercise 3.11]), $i(\mu,\nu)=0$ for every non-
trivial $\nu\in\operatorname{Curr}(X)$ if and only if $\mu$ is a boundary
geodesic current. In fact, any geodesic current can be written uniquely as a
sum of a boundary current and an internal current. Moreover, the subspace of
boundary currents is closed, and the subspace of internal currents is dense in
$\operatorname{Curr}(X)$ (see the proof of all these claims in [EM18, Lemma
2.12]). We define a _filling geodesic current_ as any internal current $\mu$
so that $i(\mu,\nu)>0$ for every non-trivial internal current $\nu$.

Let $p\colon\widetilde{X}\to X$ denote the universal covering projection, $Y$
a subsurface of $X$ with totally geodesic boundary, and suppose that $\mu$ is
a geodesic current on $X$ so that $p(\operatorname{supp}(\mu))\subset
Y-\partial Y$. We will say $\mu$ is _filling in a subsurface_ $Y$ of $X$ if
for every non-trivial current $\nu\in\operatorname{Curr}(X)$ so that
$p(\operatorname{supp}(\mu))\subset Y-\partial Y$, we have $i(\mu,\nu)>0$. An
example of a filling geodesic current on $X$ is a _filling multi-curve_ ,
i.e., a multi-curve $c$, so that $X-c$ is a disjoint union of topological
disks and once-punctured disks. In fact, a multi-curve $c$ is a filling multi-
curve if and only if its associated geodesic current is filling in the sense
of geodesic currents [ES22, Exercise 3.13]. Another example, if $X$ is closed,
is the Liouville current (see Examples 2.11).

###  2.3. Examples of geodesic currents

In this section we recall some examples of geodesic currents that will appear
in the forthcoming sections.

####  2.3.1. Weighted multi-curves

Let $K\subset S$ be a smooth non-necessarily-connected 1-manifold without
boundary, together with a map $c$ from $K$ to $S$. We say that $c$ is
_trivial_ if the image of $c$ is contained in a disk (so it is _null-
homotopic_) or it is _boundary parallel_ , i.e. homotopic to a component of
$\partial S$. A _multi-curve_ is an equivalence class of maps $c$ as above
where $c$ and $c^{\prime}$ are considered equivalent if they are related by
_homotopy_ within the space of all maps from $K$ to $S$ (not necessarily
immersions), _reparametrization_ of the 1-manifold and dropping trivial
components. If the domain $K$ of $c$ is connected, we will call $\gamma$ a
_curve_.

We write $\mathit{Curves}(S)$ for the space of all curves on $S$. There is a
bijection between curves $c$ and conjugacy classes of $g$ and $g^{-1}$, for
$g,g^{-1}\in\pi_{1}(X)$. A _weighted multi-curve_
$c=\lambda_{1}c_{1}+\cdots+\lambda_{n}c_{n}$, is a multi-curve $\cup_{i}c_{i}$
where each component (curve) $c_{i}$ is equipped with a non-negative real
number $\lambda_{i}$. The geodesic current corresponding to $c$ is the sum of
weighted Dirac measures on $\mathcal{G}(\widetilde{X})$ supported on
$\cup_{i}\Gamma\tilde{\gamma}_{i}\subseteq\mathcal{G}(\widetilde{X})$, where
$\tilde{\gamma_{i}}$ is a lift of the geodesic in the class of $c_{i}$.

In fact, a geodesic current $\mu$ has atoms as a measure if and only if
$\mu=\nu+\delta$, where $\delta$ is a non-trivial weighted multi-curve. First,
we consider the case of “$0$-dimensional atoms”, i.e. atoms whose topological
dimension in the space $\mathcal{G}(\widetilde{X})$ is $0$.

######  Lemma 2.5 ([Mar16, Proposition 8.2.7]).

Let $\mu\in\operatorname{Curr}(X)$. If $\mu(\\{\gamma\\})>0$ for
$\gamma\in\mathcal{G}(\widetilde{X})$, then $\gamma$ is a lift of a closed
geodesic.

In fact, $1$-dimensional atoms also come from weighted multi-curves. For
$z,x,y\in\partial\widetilde{X}$ cyclically ordered we define a _pencil_
$P(z,[x,y])\subseteq\mathcal{G}(\widetilde{X})$ to be the set of all geodesics
with one endpoint $z$ and the other endpoint in $[x,y]$. When we just write
$P(z)$ we mean the pencil consisting of all geodesics with $z$ as one of their
endpoint. We prove the following characterization.

######  Lemma 2.6.

A geodesic current $\mu$ has an atom if and only if there exists $z\in\partial
X$ so that the pencil $P(z)$ of geodesics at $z$, satisfies $\mu(P(z))>0$.
Moreover, $\mu(P(z))<\infty$.

###### Proof.

If $\mu$ has an atom $\gamma$, then $\mu(P(\gamma^{+}))>0$. On the other hand,
suppose that $\mu$ has no atoms, but $\mu(P(z))>0$ for some $z\in\partial X$.
Then, by [Mar16, Proposition 8.2.8], there exists a closed geodesic $c$ so
that $z=\gamma_{+}$. Let $P(z,[x,y])$ be the set of geodesics with one
endpoint at $z$ and the other within $[x,y]\subset\partial X$, where we assume
that $\gamma_{-}\in[x,y]$. Note that, by the north-south dynamics of $\gamma$,
$P(z)=\cup_{n\geq 0}\gamma^{n}P(z,[x,y])=\gamma^{n}P(z,[x,y])$. Then, taking
measures, by $\pi_{1}(X)$-invariance, we have
$\mu(P(z))=\mu(\gamma^{n}P(z,[x,y]))=\mu(P(z,[x,y]))$. Thus, by assumption it
follows $\mu(P(z,[x,y]))>0$. On the other hand,
$\\{(\gamma_{+},\gamma_{-})\\}=\cap_{n\geq 0}\gamma^{-n}P(z,[x,y])$. Thus, by
continuity of measures from below ([Hal50, Theorem D],
$\mu(\gamma)=\lim_{n}\mu(\gamma^{-n}P(z,[x,y]))$. By assumption,
$\mu(\gamma)=0$, and by $\pi_{1}(X)$-invariance,
$\lim_{n}\mu(\gamma^{-n}P(z,[x,y]))=\mu(P(z,[x,y]))$. Thus, it follows
$\mu(P(z,[x,y]))=0$, a contradiction. Note that since
$\mu(P(z))=\mu(\gamma^{n}P(z,[x,y]))=\mu(P(z,[x,y]))$, and $P(z,[x,y])$ is a
compact set of geodesics, $\mu(P(z))$ is finite. ∎

A consequence of the above lemma is the following, which will not be used in
the sequel but worth recording.

######  Lemma 2.7.

Let $\mu$ be a geodesic current, and $\gamma\in\operatorname{supp}\mu$. The
geodesic $\gamma$ is not an atom of $\mu$ if and only if for all
$\varepsilon>0$, there exists $\delta>0$ so that
$\mu(N_{\delta}(\gamma))<\varepsilon$, where $N_{\delta}$ is the
$\delta$-neighborhood of $\gamma$ in $\mathcal{G}(\widetilde{X})$.

###### Proof.

If $\gamma$ is an atom, then the condition is obviously violated.

If $\gamma$ is not an atom, denote with $\gamma_{-}$ and $\gamma_{+}$ its
endpoints, and consider the pencil
$P=P\left(\gamma_{-},(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon)\right)$.
We know from Lemma 2.6 that $\mu(P)>0$ if and only if it contains the axis of
some non-zero element of $\Gamma$ which projects to a closed geodesic in $X$.
Suppose that this is the case, i.e. there exists a geodesic line $l\in P$ such
that $p(l)$ is a closed geodesic in $X$. Then $l$ is an atom for $\mu$,
contradicting the fact that $\gamma$ is in the support of $\mu$, and is
asymptotic to $l$. It follows that
$\mu(P(\gamma_{-},(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon))=0$.

Now we define a sequence of boxes as follows: start with

$B_{1}=N_{\varepsilon}(\gamma)=(\gamma_{-}-\varepsilon,\gamma_{-}+\varepsilon)\times(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon)$

and define

$B_{n}=(\gamma_{-}-\varepsilon/n,\gamma_{-}+\varepsilon/n)\times(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon)$

by pinching one of the intervals of the corresponding box of geodesics so that
$\cap_{n=1}^{\infty}B_{n}=P(\gamma_{-},(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon)$.
By continuity of measures from below, we have
$\lim_{n}\mu(\cap_{i=1}^{n}B_{i})\to\mu(P(\gamma_{-},(\gamma_{+}-\varepsilon,\gamma_{+}+\varepsilon))=0$.
For any $\varepsilon>0$, if we take $\delta$ so that
$N_{\delta}(\gamma)\subset B_{n}$, and $\mu(B_{n})<\varepsilon$, then we have
$\mu(N_{\delta}(\gamma))<\varepsilon$, as wanted. ∎

Finally, any geodesic current can be approximated by a sequence of weighted
multi-curves.

######  Proposition 2.8 ([Bon86, Proposition 4.4]).

The subset of geodesic currents coming from weighted multi-curves is dense
with respect to the weak∗-topology of currents.

We discuss the weak∗-topology in Subsection 2.4.

####  2.3.2. Measured laminations

We start with the definition of measured geodesic lamination:

######  Definition 2.9.

A _geodesic lamination_ $\Lambda$ is a set of disjoint simple complete
geodesics in $X$, whose union is a closed subset of $X$. A _transverse
measure_ for $\Lambda\subset X$ is a family $\lambda$ of locally finite Borel
measures ${\lambda}_{\alpha}$ on each arc $\alpha\subset X$ transverse to
$\lambda$, such that

  1. (1)

For every $\alpha$ transverse arc, the support of $\lambda_{\alpha}$ is
$\alpha\cap\Lambda$;

  2. (2)

If $\alpha^{\prime}\subset\alpha$ is a sub-arc of $\alpha$, then the measure
$\lambda_{\alpha^{\prime}}$ is the restriction of $\lambda_{\alpha}$;

  3. (3)

For every $\alpha$ transverse arc, the measure $\lambda_{\alpha}$ is invariant
through isotopies of transverse arcs.

A _measured geodesic lamination_ is a geodesic lamination together with a
transverse measure.

It is a well-known fact (see [Bon88, Proposition 17]) that measured
laminations can be embedded into the space of geodesic currents (see [AL17,
Lemma 4.4]).

In fact, the image of the above embedding is characterized as those geodesic
currents $\eta$ so that $i(\alpha,\alpha)=0$.

######  Proposition 2.10 ([Bon88, Proposition 14]).

The image of the embedding $(\Lambda,\mu)\mapsto\eta_{\mu}$ consists of
geodesic currents $\eta$ so that $i(\alpha,\alpha)=0$.

When $X$ has boundary, there are multiple types of measured laminations one
can consider (see [PH92, 1.8], [Kap09, Chapter 11] for several treatments). In
this paper, we will only consider measured laminations whose associated
geodesic currents are internal currents, so they are in
$\operatorname{Curr}_{0}(X)$. When working in a subsurface $Y\subset X$, we
will only consider internal measured laminations within that subsurface, in
the sense that, for $p\colon\widetilde{X}\to X$ the universal covering
projection, we have $p(\operatorname{supp}(\mu))\subset Y-\partial Y$. We say
that a measured lamination is _discrete_ if it is a simple multi-curve, i.e.,
all the leaves of the support of the lamination $\Lambda$ are simple closed
geodesics. We say it is a _non-discrete measured lamination_ otherwise. Non-
discrete measured laminations are equivalent to type 2 subcurrents in the
structural decomposition theorem for geodesic currents (see Section 7). For
any such measured lamination $\lambda$, there is a minimal (with respect to
inclusion) subsurface $Y$ of $X$ that contains
$p(\operatorname{supp}(\lambda))$, and so that for every internal closed curve
$c$ in $Y$, we have $i(\lambda,c)>0$. Some authors choose to call these
measured laminations “filling”, but that would clash with our choice of
“filling” for geodesic currents. Observe that a measured lamination is never
filling in the sense of geodesic currents, since $i(\lambda,\lambda)=0$.

####  2.3.3. Liouville current

######  Example 2.11.

Given a box $B=[a,b]\times[c,d]\subset\mathcal{G}(\widetilde{X})$, the
Liouville current can be explicitly defined as follows. Consider the
hyperbolic cross ratio on the upper half space $\mathbb{H}^{2}$, defined by
taking, for any box of geodesics $B=[a,b]\times[c,d]$ in
$\mathcal{G}(\mathbb{H}^{2})$, the expression

$L(B)=\left|\log\frac{|a-c||b-d|}{|a-d||b-c|}\right|.$

Let $[(Y,\varphi)]\in\operatorname{Teich}(X)$, where $Y$ is some hyperbolic
structure on $X$, a priori distinct from $X$. Since $Y$ is a hyperbolic
structure, we have a $\pi_{1}(Y)$-invariant isometry
$I\colon\mathbb{H}^{2}\to\widetilde{Y}$. We consider the following measure on
$\mathcal{G}(\widetilde{Y})$, given by $\mathcal{L}_{Y}\coloneqq I_{*}(L)$.
From the definition of $\mathcal{L}_{Y}$ and the intersection number of
geodesic currents, one can check the following property

$i(\mathcal{L}_{Y},c)=\ell_{Y}(g)$

where $\ell_{Y}$ denotes the hyperbolic length (see [Bon88, Proposition 14]).
In fact, $\mathcal{L}_{Y}$ is characterized by this property, by [Ota90,
Théorème 2]. We will call this the _intersection property_.

A stronger (a priori) property of $\mathcal{L}_{Y}$, which also follows from
its definition, and fully characterizes $\mathcal{L}_{Y}$ (see [Mar16,
Proposition 8.1.12]), is the following.

######  Definition 2.12.

Let $X$ be a compact surface of genus $g\geq 2$ endowed with a length geodesic
metric $Y$. A geodesic current $\mu\in\operatorname{Curr}(X)$ satisfies the
_Crofton property_ if

$\mu(G[x,y])=\ell_{\widetilde{Y}}([x,y])$

for every $x,y\in\widetilde{Y}$, where $[x,y]$ denotes a geodesic from $x$ to
$y$, and $\ell_{\widetilde{Y}}$ is the lift of $\ell_{Y}$ to the universal
cover.

Such property is named after Crofton since it is a special case of the Crofton
formula for integral geometry [San04, 19].

Now, since $\varphi\colon X\to Y$ is a quasi-conformal marking from the base
hyperbolic structure $X$ to another hyperbolic structure $Y$, we can define a
geodesic current $\mathcal{L}_{Y}^{X}$ in $\operatorname{Curr}(X)$ as follows.
Since the marking $\varphi$ induces a $\pi_{1}(X)$-equivariant homeomorphism
$\varphi\colon\widetilde{X}\to\widetilde{Y}$, we put

$\mathcal{L}_{Y}^{X}\coloneqq\varphi^{-1}_{*}\mathcal{L}_{Y}=\varphi^{-1}_{*}\circ
I_{*}(L).$

The geodesic current $\mathcal{L}_{Y}^{X}$ has full support and has no atoms,
and it is defined as the _Liouville current_ associated to $[(Y,\varphi)]$
(see [Bon88, Page 145]).

Otal, in [Ota90, Page 155], extended the construction of Liouville current
$\mathcal{L}_{Y}$ to any negatively curved Riemannian metric $Z$ on $X$ (not
necessarily of constant curvature $-1$). Otal’s current, $\mathcal{L}_{Z}$,
also satisfies the Crofton property

$\mathcal{L}_{Z}(G[x,y])=\ell_{Z}([x,y])$

for any $Z$-geodesic segment $[x,y]$ in $\widetilde{Z}$, where here $G[x,y]$
denotes the set of $Z$-geodesics intersecting $[x,y]$ transversely.

####  2.3.4. Geodesic currents coming from Anosov representations

Let $G$ be a real, connected, non-compact, semisimple, linear Lie group. Let
$K$ denote a maximal compact subgroup of $G$, so that $V=G/K$ is the
Riemannian symmetric space of $G$. Let $[P]$ be the conjugacy class of a
parabolic subgroup $P\subset G$. Then there is a notion of _$[P]$ -Anosov
representation_ $\rho\colon\pi_{1}(X)\to G$; see, for example, Kassel’s notes
[Kas18, Section 4]. When $\operatorname{rank}_{\mathbb{R}}(G)=1$ there is
essentially one class $[P]$, so we can simply refer to them as Anosov
representations, and they can be defined as those injective representations
$\rho\colon\pi_{1}(X)\to G$ where $\Gamma\coloneqq\rho(\pi_{1}(X))$ preserves
and acts cocompactly on some nonempty convex subset of $X$. Examples of these
are Fuchsian and quasi-Fuchsian representations. In general rank, the
conjugacy classes of parabolic subgroups of $G$ correspond to subsets $\theta$
of the set of restricted simple roots $\Delta$ of $G$. For a given
$[P]$-Anosov representation and each $\alpha\in\theta$, one can define a curve
functional on oriented curves

$\ell_{\alpha}^{\rho}\colon\mathcal{C}(X)\to\mathbb{R}_{\geq 0}$

by considering the $\log$ of the diagonal matrix of eigenvalues of $\rho(g)$
and composing it with $\alpha+i(\alpha)$, where $\alpha\in\Delta$ is a root,
and $\iota(\alpha)$ denotes the root obtained by acting by the negative of the
largest element in the Weyl group. See [MZ19, Section 2] for details. Martone
and Zhang show that for a certain subset of Anosov representations called
_positively ratioed_ [MZ19, Definition 2.21], there exists a geodesic current
$\mu_{\rho}$ so that

$i(\mu_{\rho},[g])=\ell^{\rho}_{\alpha}(g)$

for all $g\in\pi_{1}(X)$. The construction goes through interpreting geodesic
currents as generalized positive cross-ratios, an observation that was already
used by Hamenstaedt [Ham97, Lemma 1.10], [Ham99, Section 2]. This class
includes two types of representations of interest: Hitchin representations and
maximal representations. In this paper we will only consider _Hitchin
representations_ , i.e., a representation
$\rho\colon\pi_{1}(X)\to\operatorname{\mathit{SL}}(n,\mathbb{R})$ which may be
continuously deformed to a composition of the irreducible representation of
$\operatorname{\mathit{PSL}}(2,\mathbb{R})$ into
$\operatorname{\mathit{PSL}}(n,\mathbb{R})$ with a discrete faithful
representation of $\pi_{1}(X)$ into
$\operatorname{\mathit{PSL}}(2,\mathbb{R})$.

Continuity of the cross-ratio is crucial in Martone-Zhang’s construction of
positive cross-ratios. From the geodesic current viewpoint, it translates into
the fact that their associated geodesic currents have no atoms. In fact, the
following can be extracted from [MZ19, Page 17]).

######  Lemma 2.13.

For $\rho\colon\pi_{1}(X)\to G$ a positively ratioed Anosov representation,
the associated geodesic current $\mu_{\rho}$ is non-atomic and has full
support.

Recently, Burger-Iozzi-Parreau-Pozzeti [BIPP21b, Proposition 4.3] have lifted
the continuity assumption in the generalized cross-ratio, thus extending the
construction of such currents beyond positively ratioed representations: see
[BP21a] and [BP21b]. Their associated currents can, in general, have atoms.

In Subsection 5.3, we discuss in more detail the case of Hitchin
representations for $\operatorname{\mathit{SL}}(3,\mathbb{R})$, their
connection to convex projective structures and their associated dual spaces.

###  2.4. Weak∗-topology of currents

As a space of Radon measures on $\mathcal{G}(\widetilde{X})$, it is natural to
endow the space of geodesic currents $\operatorname{Curr}(X)$ with the _weak
∗-topology on geodesic currents_, defined by the family of semi-norms

$|\alpha|_{\xi}=\int_{\mathcal{G}(\widetilde{X})}\xi\alpha$

for $\alpha\in\operatorname{Curr}(X)$, as $\xi$ ranges over all continuous
function $\xi\colon\mathcal{G}(\widetilde{X})\to\mathbb{R}$ with compact
support. The space $\operatorname{Curr}(X)$ is second countable and completely
metrizable (see [ES22, Proposition A.9]. Thus, the topology can be specified
via sequential convergence.

The intersection number
$i\colon\operatorname{Curr}(X)\times\operatorname{Curr}(X)\to\mathbb{R}$ is
continuous with respect to this topology (see [Bon86, Proposition 4.5]).

In fact, the weak∗ topology coincides with the topology of intersection
numbers, by [DLR10, Theorem 11], which essentially follows by work of Otal in
[Ota90, Théorème 2].

######  Theorem 2.14.

A sequence of geodesic currents $(\mu_{i})$ converges $\mu_{i}\to\mu$ in the
weak∗-topology if and only if $i(\mu_{i},c)\to i(\mu,c)$.

###  2.5. Systole of a geodesic current

Given a geodesic current $\mu$ on $X$, we define the _systole of $\mu$_ as

$\operatorname{sys}(\mu)\coloneqq\inf\\{i(\mu,c):c\in\mathit{Curves}(X)\\}$

We point out that, as a function on geodesic currents with the weak∗-topology,
$\operatorname{sys}$ is a continuous function (see [BIPP21a, Corollary
1.5(1)]).

Given a subsurface $Y$ of $X$ with totally geodesic boundary and $\mu$ a
geodesic current with $p(\operatorname{supp}(\mu))\subset Y$, we define the
_systole of_ $\mu$ _relative to_ $Y$, as follows,

$\mathrm{sys}_{Y}(\mu):=\inf\\{i(\mu,c):c\in\mathit{Curves}(Y-\partial Y)\\}.$

##  3\. Dual space of a geodesic current

In this section we define and prove the basic properties of the dual space of
a current.

We start by recalling some facts about pseudo-metric spaces. A _pseudo-metric_
on a set $Y$ is a map $d\colon Y\times Y\to\mathbb{R}$ satisfying the symmetry
and triangle inequalities. Points $x\neq y$ with $d(x,y)=0$ are allowed. A
pseudo-metric space $Y$ with pseudo-metric $d$ has a canonical metric space
quotient $Y/\sim$. It is given by the equivalence classes for the equivalence
relation identifying $x$ and $y$ in $Y$ if and only if $d(x,y)=0$, and endowed
with the induced metric coming from $d$. We call this the _metric quotient_ of
$Y$. In most of this paper, we will let $Y=\widetilde{X}$ be the universal
cover of the surface $X$ equipped with the pullback metric on $X$, and the
pseudo-distance will be defined from a geodesic current as defined below. We
will decorate the points in $\widetilde{X}$ with an overline, as in
$\overline{x}$.

###  3.1. The dual space of a geodesic current

A geodesic current $\mu\in\operatorname{Curr}(X)$ induces a pseudo-distance on
$\widetilde{X}$ given by

$d_{\mu}(\overline{x},\overline{y})=\frac{1}{2}\left\\{\mu(G[\overline{x},\overline{y}))+\mu(G(\overline{x},\overline{y}])\right\\}.$

Note that the pseudo-distance $d_{\mu}$ is _straight_ (see [BIPP21a,
Proposition 4.1], in the sense that it is additive on hyperbolic geodesic
lines. Precisely, let $\overline{x},\overline{y},\overline{z}\in\gamma$ be
three points lying in the order $\overline{x}<\overline{y}<\overline{z}$ on a
hyperbolic geodesic $\gamma\subseteq\widetilde{X}$, then we have

$d_{\mu}(\overline{x},\overline{z})=d_{\mu}(\overline{x},\overline{y})+d_{\mu}(\overline{y},\overline{z}).$

A non-straight version of this pseudo-distance was first considered by
Glorieux in [Glo17].

######  Definition 3.1 (Dual space of a geodesic current).

For $\overline{x},\overline{y}\in\widetilde{X}$, consider the equivalence
under the pseudo-metric, $\overline{x}\sim\overline{y}$ if and only if
$d_{\mu}(\overline{x},\overline{y})=0$. The metric quotient
$X_{\mu}\coloneqq\widetilde{X}/\sim$ will be called the _dual space of the
geodesic current_ $\mu$.

Given $X$, we denote by $\mathcal{D}(X)$ the set of all duals of geodesic
currents on $X$.

When $\mu$ is a measured lamination, then it is known that $X_{\mu}$ is a
$0$-hyperbolic space and, hence, it can be isometrically embedded in a unique
$\mathbb{R}$-tree $\widehat{X_{\mu}}$ (see also 5.1). It follows that
$X_{\mu}$ can be endowed with a geodesic structure via such embedding
$X_{\mu}\hookrightarrow\widehat{X_{\mu}}$.

######  Remark 3.2.

In the remaining of the paper, when $\mu$ is a measured lamination, we will
often denote $\widehat{X_{\mu}}$ simply with $\mathcal{T}(\mu)$, to emphasise
that it is an $\mathbb{R}$-tree.

###  3.2. Geodesic structure

In this subsection we explore how to define a geodesic structure on $X_{\mu}$
when $\mu$ is not necessarily a measured lamination, i.e. when the support of
$\mu$ is allowed to have intersections. In particular, we will construct an
isometric embedding $X_{\mu}\hookrightarrow\widehat{X_{\mu}}$ when $\mu$ is a
multi-curve with self intersections, and when $\mu$ has no atoms at all. The
mixed case, i.e., when the current has both atomic and non-atomic parts, or
the case when the current is purely atomic with a non-discrete set of atoms,
will not be discussed in this project and will be fleshed out in a sequel to
this project with Anne Parreau.

####  3.2.1. Multi-curve case

Let now $\mu$ be a weighted multi-curve. The support of $\mu$ is union of
finitely many discrete orbits of lifts of closed geodesics. In this case, the
dual space $X_{\mu}$ is in general not an $\mathbb{R}$-tree, but it can still
be isometrically embedded in a graph, hence in a geodesic space, as follows.

Let $x,y\in X_{\mu}$ be two point in the dual space of $\mu$. They correspond
to two subsets $R_{x}=\pi_{\mu}^{-1}(x),R_{y}=\pi_{\mu}^{-1}(y)$ of
$\widetilde{X}$. Note that $R_{x}$ and $R_{y}$ can be a connected component of
$\widetilde{X}\setminus\mathrm{supp}(\mu)$, a geodesic, a geodesic segment, or
just a point.

######  Definition 3.3.

We say that $x$ and $y$ are _adjacent_ in $X_{\mu}$ if there exist
$\overline{x}\in\pi_{\mu}^{-1}(x)$, $\overline{y}\in\pi_{\mu}^{-1}(y)$ and a
geodesic segment $[\overline{x},\overline{y}]$ in $\widetilde{X}$ that doesn’t
intersect transversely any geodesic of $\mathrm{supp}(\mu)$ in its interior
$(\overline{x},\overline{y})$, and intersects lifts of closed geodesics only
in one of its endpoints $\overline{x}$ or $\overline{y}$.

In Figure 3.1 we can see a ‘zoomed-in’ of a configuration of geodesics in the
support of $\mu$ in $\widetilde{X}$. According to definition 3.3 the point
$\pi_{\mu}(x)$ is adjacent to $\pi_{\mu}(y)$, but not adjacent to
$\pi_{\mu}(z)$. The point $\pi_{\mu}(v)$ is adjacent to $\pi_{\mu}(w)$ but not
adjacent to $\pi_{\mu}(x),\pi_{\mu}(y)$ or $\pi_{\mu}(z)$. Note that the
additional condition that $[\overline{x},\overline{y}]$ intersects in only one
of its endpoint lifts of closed geodesics is motivated by the fact that we
want the region corresponding to $\pi_{\mu}(x)$ to be adjacent to the points
$\pi_{\mu}(y)$ and $\pi_{\mu}(u)$ corresponding to boundary geodesics, but we
don’t want $\pi_{\mu}(y)$ and $\pi_{\mu}(u)$ to be adjacent.

Figure 3.1.

If $v_{1}$ and $v_{2}$ in $X_{\mu}$ are adjacent, we add an edge $e_{1,2}$
joining them of length $d_{\mu}(v_{1},v_{2})$. This defines a graph
$\widehat{X_{\mu}}$ that has as vertices the points of $X_{\mu}$, and edges
realising the adjacency between points. This embeds $X_{\mu}$ isometrically
into a connected graph $\widehat{X_{\mu}}$, and hence a geodesic space. We
stress that this construction is by no means canonical, since the notion of
adjacency could be defined in many geometrically meaningful different ways. In
the next Example 3.4 we will see the geometric realisation $\widehat{X_{\mu}}$
of a current $\mu=\alpha+\beta$ supported on a pair of intersecting simple
closed curves.

######  Example 3.4.

Consider the 1-punctured torus $X=X_{1,1}$ and the current $\mu=\alpha+\beta$
whose support is given by the two orthogonally intersecting simple closed
geodesics $\alpha$ and $\beta$. Up to pre-composing with an isometry of
$\widetilde{X}$, the lift in $\widetilde{X}$ of $\alpha$ and $\beta$ in yellow
and green respectively, is as in Fig. 3.2. In that figure we also see the
points of $X_{\mu}$, i.e the equivalence classes with respect to the pseudo-
distance $d_{\mu}$.

Notice that each point in $X_{\mu}$ corresponds to either a geodesic segment
or to an unbounded complementary region. Finally, we can embed $X_{\mu}$ in
its geometric realisation graph $\widehat{X_{\mu}}$, which is sketched in
purple in Figure 3.2.

Figure 3.2. A sketch of $\widehat{X_{\mu}}$ when $\mu$ is a union of two
intersecting simple closed curves $\alpha$ and $\beta$ in a once-punctured
torus. The orbits $\Gamma\tilde{\alpha}$ and $\Gamma\tilde{\beta}$ are denoted
in green and yellow, respectively. The support of $\mu$ is precisely the union
of the two orbits. The blue dots represent the points of $X_{\mu}$.

####  3.2.2. Non-atomic case

Assuming $\mu$ has no atoms, we will show $X_{\mu}$ is a geodesic space.

A metric space $(X,d)$ is called _Menger convex_ if, for every $x,y\in X$,
there exists $z\in X$ so that $d(x,z)=d(y,z)=\frac{1}{2}d(x,y)$. The following
Lemma can be found in [Pap14, Theorem 2.6.2].

######  Lemma 3.5.

Let $X$ be a proper metric space. $X$ is geodesic if and only if it is Menger
convex.

######  Proposition 3.6.

If $\mu$ is a geodesic current without atoms, then $X_{\mu}$ is a geodesic
metric space.

###### Proof.

Since $\mu$ has no atoms, by Proposition 4.13, $\pi_{\mu}$ is continuous.
Furthermore, by Proposition 8.13, $X_{\mu}$ is proper. Thus, it suffices to
check Menger convexity, by Lemma 3.5. Given any two points $x,y\in X_{\mu}$,
let $\overline{x}\in\pi_{\mu}^{-1}(x)$ and $\overline{y}\in\pi_{\mu}^{-1}(y)$,
and let $I$ be the geodesic segment connecting $\overline{x}$ and
$\overline{y}$. $\pi_{\mu}(I)$ is connected, and thus there exists
$\overline{z}\in I$ so that $z=\pi_{\mu}(\overline{z})$ satisfies the
condition of the statement. ∎

It would be interesting to see which conditions to impose on $\mu$ in order to
obtain sharper convexity properties. We say that a $\delta$-hyperbolic with
midpoints is _convex_ if, for every triple of points $p,q,r$, if $m_{1}$
denotes the midpoint between $p$ and $q$ and $m_{2}$ denotes the midpoint
between $p$ and $r$, then we have $d(m_{1},m_{2})\leq\frac{1}{2}d(q,r)$. What
are the conditions on $\mu$ ensuring that $X_{\mu}$ is convex? Convexity
properties of this type (and stronger) are useful to guarantee sequential pre-
compactness in the setting of the equivariant Gromov-Hausdorff topology as
well as separation properties for this topology (see Section 10 and [Pau88,
Chapter 4]).

###  3.3. Dependence on the metric structure

To define the space of geodesic currents $\operatorname{Curr}(X)$, we fixed a
$\delta$-hyperbolic geodesic structure $X$ on $S$. Given a homeomorphism
between two different structures $f\colon X\to X^{\prime}$, we get a
homeomorphism
$\widetilde{f}\colon\partial\widetilde{X}\to\partial\widetilde{X^{\prime}}$ by
[CB88, Lemma 3.7] extending $f$ to the boundary. This induces a homeomorphism
between the corresponding spaces of geodesics and, by pushforward of measures,
induces a homeomorphism
$\widetilde{f}_{*}\colon\operatorname{Curr}(X)\to\operatorname{Curr}(X^{\prime})$.
In particular, this induces an action of the mapping class group on
$\operatorname{Curr}(X)$. For any two structures $X$, we have a commutative
diagram of bijections

$\operatorname{Curr}(X)$$\mathcal{D}(X)$$\operatorname{Curr}(X^{\prime})$$\mathcal{D}(X^{\prime})$.

The left vertical arrow is a homeomorphism from the above paragraph.

From Theorem 10.3, it will follow that, with respect to the marked equivariant
Gromov-Hausdorff topology on $\mathcal{D}(X)$ (see Section 10), the two
horizontal maps are homeomorphisms. Thus, we have a commutative diagram of
homeomorphisms. Observe that the isometry class of $X_{\mu}$ does depend on
the choice of the underlying hyperbolic structure $X$. However, for a fixed
(unweighted) multi-curve $\mu$ the dual spaces $X_{\mu}$, for different
choices of hyperbolic structure $X$, embed as 2-dimensional faces of a
$\mathit{CAT}(0)$-cube complex $\mathcal{S}(\mu)$ independent on the choice of
hyperbolic structure: the _Sageev complex_ (see [AG17, 3] for a description in
this setting, and [CN05, Sag95, Sag14] in bigger generality). It would be
interesting to see if there is an analogous construction of such complex for
an arbitrary geodesic current.

###  3.4. Measured wall spaces

We point out that the dual space of a geodesic current is an example of a
measured wall space, in the sense of [CMV04].

Given a set $X$, a _wall_ of $X$ is a partition $X=h\cup h^{c}$ where $h$ is
any subset of $X$ and $h^{c}$ denotes its complement. A collection
$\mathcal{H}$ is called a collection of half-spaces if for every
$h\in\mathcal{H}$ the complementary subset $h^{c}$ is also in $\mathcal{H}$.
Let $\mathcal{W}_{\mathcal{H}}$ denote the collection of pairs $w=(h,h^{c})$
with $h\in\mathcal{H}$. We say that $h$ and $h^{c}$ are the two half-spaces
bounding the wall $w$. We say that a wall $w=(h,h^{c})$ separates two disjoint
subsets $A,B$ in $X$ if $A\subset h$ and $B\subset h^{c}$ or vice-versa and
denote by $\mathcal{W}(A|B)$ the set of walls separating $A$ and $B$. In
particular, $W(A|\emptyset)$ is the set of walls $w=(h,h^{c})$ such that
$A\subset h$ or $A\subset h^{c}$, hence
$\mathcal{W}(\emptyset|\emptyset)=\mathcal{W}$. We use $\mathcal{W}(x|y)$ to
denote $\mathcal{W}(\\{x\\}|\\{y\\})$.

######  Definition 3.7 (Space with measured walls).

A space with measured walls is a 4-tuple $(X,\mathcal{W},\mathcal{A},\mu)$
where $\mathcal{W}$ is a collection of walls, $\mathcal{A}$ is a
$\sigma$-algebra of subsets in $\mathcal{W}$ and $\mu$ is a measure on
$\mathcal{A}$ so that for every two points $x,y\in X$, the set of separating
walls $\mathcal{W}(x|y)$ is in $\mathcal{A}$ and has finite measure. We let
$d_{\mu}(x,y)=\mu(\mathcal{W}(x|y))$, and we call it the wall-pseudo metric.

It is easy to see that $(\widetilde{X},d_{\mu})$ is a measured wall space on
$\widetilde{X}$, where $\mathcal{W}$ are oriented geodesics of
$\widetilde{X}$, $\mathcal{A}$ is the Borel $\sigma$-algebra of oriented
geodesics, and $W\subset\mathcal{W}$, the measure on walls $\mu^{\prime}$ is
given by $\mu^{\prime}\coloneqq\frac{1}{2}\mu(W)$. This normalization factor
is introduced to account for the fact that we get two copies of each geodesic
in the support of the current in the space of walls.

##  4\. Continuity of the projection

In this section we study the semicontinuity and continuity properties of the
natural metric quotient projection map $\pi_{\mu}\colon\widetilde{X}\to
X_{\mu}$.

###  4.1. Measure theory results

We recall some measure theory results that will be used in this section. Given
a sequence of subsets $(A_{n})$, $A_{n}\subset X$, we have the following
definitions:

(4.1) $\liminf_{n}A_{n}\coloneqq\cup_{n\geq 1}\cap_{j\geq n}A_{j}$

and

(4.2) $\limsup_{n}A_{n}\coloneqq\cap_{n\geq 1}\cup_{j\geq n}A_{j}.$

The Morgan laws immediately give the following identity

(4.3) $\liminf_{n}A_{n}=\left(\limsup_{n}A_{n}^{c}\right)^{c}.$

The following result is standard and can be found, for example, in [Hal50,
Theorem D] and [Hal50, Theorem E].

######  Lemma 4.4.

If $(A_{n})$ is a sequence of measurable sets in $X$, so that
$A_{n+1}\subseteq A_{n}$, then for any measure $\mu$ on $X$, we have

$\lim_{n}\mu(A_{n})=\mu(\cap_{n}A_{n}).$

This property is called _continuity of measures from below_. If $(A_{n})$ is a
sequence of measurable sets in $X$, so that $A_{n}\subseteq A_{n+1}$ so that,
for some $N>0$, $\mu(A_{n})<\infty$ for $n\geq N$, then for any measure $\mu$
on $X$, we have

$\lim_{n}\mu(A_{n})=\mu(\cup_{n}A_{n}).$

This property is called _continuity of measures from above_.

######  Lemma 4.5.

Let $\mu$ be a (non-necessarily finite) measure on a measurable space $X$, and
let $(A_{n})$ be a sequence of measurable sets. Then

$\liminf_{n}\mu(A_{n})\geq\mu(\liminf_{n}A_{n}).$

Moreover, if for some $n$, $\mu\left(\cup_{j\geq n}A_{j}\right)$ is finite, we
have

$\limsup_{n}\mu(A_{n})\leq\mu(\limsup_{n}A_{n}).$

###### Proof.

For the first inequality, let $B_{n}\coloneqq\cap_{j\geq n}A_{j}$. Note that
$B_{n}\subset B_{n+1}$, and thus by continuity of measures from below 4.4, we
get

$\mu(\lim_{n}B_{n})=\lim_{n}\mu(B_{n}).$

Note that

$\lim_{n}B_{n}=\cup_{n\geq 1}B_{n}=\liminf_{n}A_{n}.$

Since $\cap_{j\geq n}A_{n}\subset A_{k}$ for $k\geq n$, we have

$\mu\left(\cap_{j\geq n}A_{n}\right)\subset\inf\\{\mu(A_{k})\colon k\geq
n\\}.$

Thus, letting $n\to\infty$, we get

$\mu\left(\liminf A_{n}\right)\leq\liminf_{n}\mu(A_{n}).$

The other inequality follows from continuity of measures from above 4.4, where
the hypothesis of $\mu\left(\cup_{j\geq n}A_{j}\right)$ being finite for some
$n$ is crucial. ∎

The following result will play a crucial role in understanding the behaviour
of $d_{\mu}$ in the presence of atoms.

######  Proposition 4.6.

Let $X=\Gamma\backslash\widetilde{X}$ be a hyperbolic structure. Let $G[x]$
denote the set of geodesics through $x$. If $x$ does not lie on the axis of an
element $g\in\Gamma-\\{e\\}$, then

$\mu(G[x])=0.$

###### Proof.

Let $\mu$ be a geodesic current, which can be though of as flip invariant
geodesic flow invariant measure on $T^{1}X$ (see, for example,[ES22, Chapter
3]). In $T^{1}X$, the set $G[x]$ corresponds to the fiber at $x\in X$ of the
canonical projection $\pi\colon T^{1}X\to X$. Let $K\coloneqq\pi^{-1}(x)$. Let
$f$ be the 1-time map $\phi_{1}\colon T^{1}X\to T^{1}X$, where $\phi_{t}$
denotes the geodesic flow. By Poincaré recurrence theorem, we have

$\mu(K)=\mu(K^{r})$

where $K^{r}$ is the set of $f$-recurrent points $v\in K$, i.e. the set of
points $v\in K$ so that there exists positive integer $n_{v}>0$ so that
$f^{n_{v}}(v)\in K$. For each $v$, consider the subsegment of a geodesic flow
trajectory $r_{v}$ given by $[v,\phi_{n^{v}}(v)]$. Note that $\pi(r^{v})$ is a
geodesic segment from $x$ to $x$ on $X$ of integral length $n^{v}$. The map
$\psi\colon v\mapsto\pi(r_{v})$ is clearly injective, since $v$ is the tangent
vector of the initial point of $\pi(r_{v})$. Let $\mathcal{R}$ be the set of
all such geodesic segments $\pi(r_{v})$ as $v$ ranges over $K^{r}$. We claim
that $\mathcal{R}$ is countable. First, we note that given a path from $p$ to
$p$, there is a unique geodesic path in the homotopy class relative to $p$,
and it has the minimal length among such paths. Consider the map $\varphi$
sending $r_{v}$ to its path-length $n^{v}$. We claim this map has finite
fibers: in any Riemannian manifold there are finitely many relative homotopy
classes of loops based at $p$ of length at most $L$. Since each $r_{v}$ is
geodesic, and the geodesic in the base-point homotopy class is unique, the set
of paths in $\mathcal{R}$ of length equal to $n^{v}$ is finite. Let
$\mathcal{N}$ be the subset of natural numbers appearing as $n_{v}$ for some
$v\in K^{r}$. Note that $\mathcal{R}=\cup_{n\in\mathcal{N}}\varphi^{-1}(n)$,
and thence $\mathcal{R}$ is a countable union of finite sets, so it is
countable. Since $\psi$ is an injection, $K^{r}$ is countable. Thus, using
countable additivity of $\mu$, $\mu(K^{r})$ can be written as a sum of
measures of singletons, which are all 0 except if $v$ is contained in a lift
of a closed geodesic. ∎

######  Proposition 4.7.

If $\mu$ is a geodesic current without atoms, then the function

$f(\cdot,\cdot)\coloneqq
d_{\mu}(\pi_{\mu}(\cdot),\pi_{\mu}(\cdot))\colon\widetilde{X}\times\widetilde{X}\to\mathbb{R}$

is a continuous function.

###### Proof.

Let $\left(\overline{x_{n}},\overline{y_{n}}\right)$ be a sequence in
$\widetilde{X}\times\widetilde{X}$ converging to
$(\overline{x},\overline{y})$. We first show that

(4.8)
$\liminf_{n}d_{\mu}(\pi_{\mu}(\overline{x_{n}}),\pi_{\mu}(\overline{y_{n}}))\geq
d_{\mu}(\pi_{\mu}(\overline{x}),\pi_{\mu}(\overline{y}))$

Note that, since $\mu$ has no atoms, we have, by Proposition 4.6,

$d_{\mu}(\pi_{\mu}(\overline{x_{n}}),\pi_{\mu}(\overline{y_{n}}))=\mu(G(\overline{x_{n}},\overline{y_{n}})).$

for all $n$.

######  Claim 4.9.

If $\gamma\in G(\overline{x},\overline{y})$, then $\gamma\in\cup_{N\geq
1}\cap_{n\geq N}G(\overline{x_{n}},\overline{y_{n}})$.

###### Proof.

Since $\gamma\in G(\overline{x},\overline{y})$, $\gamma$ intersects
$(\overline{x},\overline{y})$ transversely at some
$\overline{z}\in(\overline{x},\overline{y})$. Since
$(\overline{x_{n}},\overline{y_{n}})$ is converging in Hausdorff distance to
$(\overline{x},\overline{y})$, we have that for $n_{0}$ large enough, $\gamma$
must intersect transversely all $(\overline{x_{n}},\overline{y_{n}})$ for all
$n\geq n_{0}$. ∎

From Claim 4.9 and Lemma 4.5, it follows that

$\mu(G(\overline{x},\overline{y}))\leq\mu\left(\cup_{N\geq 1}\cap_{n\geq
N}G(\overline{x_{n}},\overline{y_{n}})\right)\leq\liminf_{n}\mu(G(\overline{x_{n}},\overline{y_{n}}))$

from which Equation 4.8 follows. We will now prove

(4.10)
$\limsup_{n}d_{\mu}(\pi_{\mu}(\overline{x_{n}}),\pi_{\mu}(\overline{y_{n}}))\leq
d_{\mu}(\pi_{\mu}(\overline{x}),\pi_{\mu}(\overline{y})).$

######  Claim 4.11.

If $\gamma$ is a geodesic disjoint from $(\overline{x},\overline{y})$, then
$\gamma\notin\limsup_{n}G(\overline{x_{n}},\overline{y_{n}})$.

###### Proof.

Let $A_{n}=G(\overline{x_{n}},\overline{y_{n}})$. We are assuming $\gamma$ is
a geodesic disjoint from $(x,y)$, and we want to show
$\gamma\notin\limsup_{n}A_{n}$, which, by Equation 4.3 is equivalent to
showing $\gamma\in\liminf_{n}A_{n}^{c}$, i.e., to showing that there exists
$N>0$ so that for all $n\geq N$, $\gamma\notin A_{n}$. This follows, since
$\gamma$ and $(\overline{x},\overline{y})$ are a definite distance apart, and
$(\overline{x_{n}},\overline{y_{n}})$ is converging to
$(\overline{x},\overline{y})$ in the Hausdorff distance of $\widetilde{X}$.
Thus, for $n_{0}$ large enough, $(\overline{x_{n}},\overline{y_{n}})$ is also
disjoint from $\gamma$, for all $n\geq n_{0}$. ∎

A geodesic which is not in $G(\overline{x},\overline{y})$ is either disjoint
from $(\overline{x},\overline{y})$, or it is the geodesic $\gamma_{x,y}$
determined by the points $x$ and $y$. Claim 4.11 can be phrased as follows:

$\limsup_{n}A_{n}\subset G(\overline{x},\overline{y})-\\{\gamma_{x,y}\\}.$

Since $\mu$ has no atoms, we have

$\limsup_{n}\mu(A_{n})\leq\mu(\limsup_{n}A_{n})\leq\mu(G(\overline{x},\overline{y})-\\{\gamma_{x,y}\\})=\mu(G(\overline{x},\overline{y})),$

where we have used the second equation in Lemma 4.5 in the first inequality,
and the last equation plus the monotonicity of $\mu$ in the second inequality.
It remains to explain why the application of LemmaLemma 4.5 is justified.
Since $\overline{x_{n}}\to\overline{x}$ and $\overline{y_{n}}\to\overline{y}$,
there exists $n_{0}$ so that for $n\geq n_{0}$,
$(\overline{x_{n}},\overline{y_{n}})\subset B$ for some compact ball $B$. Then

$\cup_{n\geq n_{0}}A_{n}\subset G(B).$

Since $B$ is compact, $\mu(G(B))$ is finite, and thus the application of the
Lemma is justified. This shows Equation 4.10 and finishes the proof. ∎

######  Proposition 4.12.

Let $\mu\in\operatorname{Curr}(X)$. If $\mu$ has atoms, then $f(\cdot,\cdot)$
is neither lower semicontinuous nor upper semicontinuous.

###### Proof.

Suppose $\mu$ is a geodesic current with atoms. By Lemma 2.5 and Lemma 2.6,
the only 0-dimensional and 1-dimensional atoms of a geodesic current are
concentrated on lifts of closed geodesics and pencils containing lifts of
closed geodesics, respectively. Let then $\gamma$ be a lift of a closed
geodesic in the support of $\mu$. Let $[\overline{x},\overline{y})$ be a
subsegment of $\gamma$, short enough so that no other lift of a closed
geodesic in the support of $\mu$ intersects it transversely. Let
$\overline{y_{n}}$ be a sequence of points outside of $\gamma$ converging to
$\overline{y}$. Then, if we denote
$\mu(G[\overline{x},\overline{y_{n}}))=\varepsilon_{1}$, we have
$\varepsilon_{1}>0$ for all $n$, while
$\mu(G[\overline{x},\overline{y}))=\varepsilon_{2}$, where
$\varepsilon_{1}>\varepsilon_{2}\geq 0$. This shows that

$\limsup_{n}f(\overline{x_{n}},\overline{y_{n}})>f(x,y)$

so $f$ is not upper semicontinuous. Observe that this construction can also be
adapted for an open interval $(\overline{x},\overline{y})$, by considering
open segments $(\overline{x_{n}},\overline{y_{n}})$ given by sequences of
points $(\overline{x_{n}})$ and $(\overline{y_{n}})$, where
$\overline{x_{n}}\to\overline{x}$ and $\overline{y_{n}}\to\overline{y}$ from
distinct sides of $\gamma$. This ensures that
$(\overline{x_{n}},\overline{y_{n}})$ crosses $\gamma$ transversely for all
$n$, while $\gamma$ does not intersect $(\overline{x},\overline{y})$
transversely. Let’s show how lower semicontinuity fails. Let $\gamma$ be again
a lift of a closed geodesic in the support of $\mu$. Suppose that $\gamma$
intersects a segment $[\overline{x},\overline{y})$ transversely at
$\overline{x}$, and let $\overline{x_{n}}$ be a sequence converging to
$\overline{x}$ so that
$[\overline{x_{n}},\overline{y})\subset[\overline{x_{n+1}},\overline{y})$ for
all $n$. Then, $\mu(G[\overline{x_{n}},\overline{y}))\leq\varepsilon_{1}$ for
all $n$ and for some $\varepsilon_{1}\geq 0$, while, if
$\mu(G[\overline{x},\overline{y}))=\varepsilon_{2}$, we have that
$0\leq\varepsilon_{1}<\varepsilon_{2}$. This shows that

$\liminf_{n}f(\overline{x_{n}},\overline{y_{n}})<f(x,y),$

so $f$ is not lower semicontinuous. ∎

Combining both Propositions 4.7 and 4.12, we get the following.

######  Proposition 4.13.

A geodesic current $\mu\in\operatorname{Curr}(X)$ has no atoms if and only if
the map $\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$ is continuous.

###### Proof.

If $\mu$ has no atoms, then the implication follows from Propositions 4.7 and
4.12. Namely, for $\overline{x}\in\widetilde{X}$, consider the continuous
function $g_{\overline{x}}(\cdot)\coloneqq f(\overline{x},\cdot)$. Since this
function is continuous at $\overline{x}$, for every $\varepsilon>0$, we can
find $\delta_{\overline{x},\varepsilon}$ so that if
$d_{\widetilde{X}}(\overline{x},\overline{y})<\delta$, we have
$g_{\overline{x}}(\overline{y})<\varepsilon$, i.e.,
$d_{\mu}(\pi_{\mu}(\overline{x}),\pi_{\mu}(\overline{y}))<\varepsilon$, which
proves continuity of $\pi_{\mu}$ at $\overline{x}$. If $\mu$ has atoms, then
let $\bar{x}\in\gamma$, for $\gamma$ an atom of $\mu$. Then, there exist an
$\varepsilon>0$ and a sequence of points $\overline{x_{n}}\notin\gamma$, so
that $d_{\mu}(\pi_{\mu}(\bar{x_{n}}),\pi_{\mu}(\bar{x}))>\varepsilon$ as
$\overline{x_{n}}\to\overline{x}$, so $\pi_{\mu}$ is not continuous. ∎

Proposition 4.13 might come as a surprise for those familiar with the
classical work on $\mathbb{R}$-trees dual to measured foliations/laminations
in [Wol98], where the natural projection map is always continuous. We
emphasize that when $\mu$ is a measured lamination, one can eliminate the
discontinuities of $\pi_{\mu}$ created by the atoms by a process of blow-up,
as described in [Kap09, Defintion 11.27].

We can combine the above results with the results in Section 8, to obtain the
following.

######  Proposition 4.14.

Given a geodesic current $\mu$ on $X$, the projection
$\pi_{\mu}\colon\widetilde{X}\to X_{\mu}$ satisfies:

  1. (1)

If $\mu$ has no atoms and no subcurrents of type 2 in its decomposition then
$\pi_{\mu}$ is a $\pi_{1}(X)$-equivariant closed map.

  2. (2)

If $\mu$ has no atoms and has full support, then $\pi_{\mu}$ is a
$\pi_{1}(X)$-equivariant homeomorphism.

###### Proof.

Since $\mu$ has no atoms, by Proposition 4.7, $\pi_{\mu}$ is continuous and by
Proposition 8.13, $X_{\mu}$ is a proper metric space. Therefore, by Lemma 9.4,
it is locally compact. By Lemma 8.9 the action of $\pi_{1}(X)$ on $X_{\mu}$ is
cocompact and by Lemma 8.19, the action is proper. Hence, the quotient
$X_{\mu}/\pi_{1}(X)$ is Hausdorff (and compact). By $\pi_{1}(X)$ equivariance
of $\pi_{\mu}$, the descended map $\overline{\pi_{\mu}}\colon X\to
X_{\mu}/\pi_{1}(X)$ is a well-defined continuous surjective map from the
compact surface $X$ to the Hausdorff space $X_{\mu}/\pi_{1}(X)$, so it is a
closed map. If moreover $\mu$ has full support, then $\pi_{\mu}$ is injective,
and hence it is a homeomorphism. By equivariance, this implies the same
results for $\pi_{\mu}$.∎

##  5\. Examples

In this section we study examples of dual spaces associated to the geodesic
currents discussed in Subsection 2.3

###  5.1. The dual tree of a measured geodesic lamination

The first example of dual space of a geodesic current is the so-called _dual
tree $\mathcal{T}(\lambda)$ of a lamination $\lambda$_. Such space has been
well studied (for example in [MS91]) before the notion of dual space of a
geodesic current was introduced. We recall here the original construction by
Morgan–Shalen and show that the dual tree $\mathcal{T}(\lambda)$ of a
lamination coincides with the dual space $X_{\lambda}$ of the current
$\lambda$.

Let $(\lambda,\mu)$ be a measured lamination (recall Definition 2.9) on a
hyperbolic surface $X$ with support $\lambda=\mathrm{supp}(\mu)$ and
transverse measure $\mu$. Denote with $(\widetilde{\lambda},\tilde{\mu})$ the
lifted measured lamination on $\widetilde{X}$, and define $\mathfrak{C}$ the
set of connected components of
$\widetilde{X}\setminus\operatorname{supp}\lambda$, each of which is called a
_complementary region_ of $\lambda$.

Let $c_{0},c_{1}\in\mathfrak{C}$. We define a metric on $\mathfrak{C}$ as
follows:

$d_{(\lambda,\mu)}(c_{0},c_{1})=\inf\mu(\gamma)$

where the $\inf$ is taken over all quasi-transverse arcs $\gamma$ such that
$\gamma(0)=x_{0}\in c_{o}$ and $\gamma(1)=x_{1}\in c_{1}$. A _quasi-transverse
arc_ is an arc intersecting transversely each leaf of the lamination at most
once.

Morgan and Shalen proved the following

######  Theorem 5.1 ([MS91] Lemma 5).

Denote with $\mathfrak{C}$ the set of complementary regions of
$\tilde{\lambda}$. There exists an $\mathbb{R}$-tree $\mathcal{T}(\lambda)$
called the _dual tree of $\lambda$_ and an isometric embedding
$\psi:\mathfrak{C}\hookrightarrow\mathcal{T}(\lambda)$ such that:

  1. (1)

$\psi(\mathfrak{C})$ spans $\mathcal{T}(\lambda)$.

  2. (2)

Any point $x\in\mathcal{T}(\lambda)\setminus\psi(\mathfrak{C})$ is an edge
point, i.e. separates $\mathcal{T}(\lambda)$ in two connected components.

  3. (3)

The action $\pi_{1}(X)$ on $\mathfrak{C}$ extends uniquely to an action by
isometries of $\pi_{1}(X)$ on $\mathcal{T}(\lambda)$.

Moreover, if $T$ and $T^{\prime}$ are two $\mathbb{R}$-trees satisfying the
above properties, then there exists an equivariant isometry $T\to T^{\prime}$
with respect to the $\pi_{1}(X)$-action.

On the other hand in Section 6 we show that the dual space $X_{\lambda}$ of
the current $\lambda$ is $0$-hyperbolic as well, and hence embeds
isometrically into an $\mathbb{R}$-tree $\widehat{X_{\lambda}}$ [Eva08,
Theorem 3.38].

######  Lemma 5.2.

The $\mathbb{R}$-trees $\widehat{X_{\lambda}}$ and $T(\lambda)$ are isometric.

###### Proof.

The action of $\pi_{1}(X)$ on both $\mathbb{R}$-trees satisfy the hypotheses
of [CM87, Theorem 3.7] and they the same length functions, hence, there exists
an equivariant isometry between them. ∎

By Lemma 5.2, we will often refer with a slight abuse of notation to
$X_{\lambda}$ as the dual tree $\mathcal{T}(\lambda)$ of $\lambda$.

It is worth noticing that there are other equivalent definitions of the dual
tree of a lamination, for example the one given by Kapovich in [Kap09, Section
11.12] or by Wolf in [Wol98], which uses the notion of measured foliations.
For a comparison between these definitions we refer the reader to [DR23].

###  5.2. Guirardel core: a filling sum of two measured laminations on the
surface

Let $X$ be a compact hyperbolic surface (possibly with boundary), and consider
$\alpha,\beta$ two measured laminations so that $\alpha+\beta$ is filling as a
geodesic current. For concreteness, the reader might want to assume that
$\alpha$ and $\beta$ are simple closed curves, so that the multi-curve
$\mu=\alpha\cup\beta$ is filling (this can always be achieved, see [FM12,
1.3.2] for the argument in genus 2).

Given a finitely generated group $G$ and two $\mathbb{R}$-trees $T_{1},T_{2}$
equipped with isometric actions, the _Guirardel’s core_ $\mathcal{C}$ is the
smallest subset of $T_{1}\times T_{2}$ which is

  1. (1)

$\pi_{1}(X)$-invariant

  2. (2)

closed and connected

  3. (3)

For every $x_{0}\in T_{1}$ and every $x_{1}\in T_{2}$, both
$\mathcal{C}\cup(\\{x_{0}\\}\times T_{2})$ and
$\mathcal{C}\cup(T_{1}\times\\{x_{1}\\})$ are convex.

We commend the reader to the paper [Gui05, Section 2.2, Example 3] for more
details. When $G$ is the fundamental group of a closed surface, and
$T_{1},T_{2}$ are the $\mathbb{R}$-trees $\widehat{X_{\alpha}}$ and
$\widehat{X_{\beta}}$, we claim that $X_{\mu}$ embeds isometrically into the
Guirardel’s core $\mathcal{C}$ of the product of trees $\widehat{X_{\alpha}}$
and $\widehat{X_{\beta}}$. In this case, the core can be described as follows.
Let $\pi_{\alpha}\colon\widetilde{X}\to\widehat{X_{\alpha}}$ and
$\pi_{\beta}\colon\widetilde{X}\to\widehat{X_{\beta}}$ be the composition of
the natural projection maps with the isometric embedding of the dual into
their $\mathbb{R}$-tree, and define
$f\colon\widetilde{X}\to\widehat{X_{\alpha}}\times\widehat{X_{\beta}}$ by
$(\pi_{\alpha}(\overline{x}),\pi_{\beta}(\overline{x}))$. Guirardel’s core
corresponds to $\mathcal{C}=f(\widetilde{X})$. We now define a map
$\overline{f}$ from $X_{\mu}$ to $\mathcal{C}$, by picking, for every $x\in
X_{\mu}$, a point $\overline{x}\in\pi_{\mu}^{-1}(x)$, and defining
$\overline{f}(x)\coloneqq f(\bar{x})$.

######  Lemma 5.3.

The map $\overline{f}:X_{\mu}\to\mathcal{C}$ is a homeomorphism.

###### Proof.

We firstly show that $\overline{f}$ is well-defined. Indeed, suppose that
$x\sim y$, i.e., $d_{\mu}(x,y)=0$. This happens in the following cases:

  1. (1)

$x,y$ are the same point;

  2. (2)

$x,y$ are in the same complementary region of $\mu$;

  3. (3)

$x,y$ are in the same lift of $\alpha$ and not separated by any lift of
$\beta$, or on the same lift of $\beta$ and not separated by any lift of
$\alpha$.

In any of these cases, we see that $\overline{f}(x)=\overline{f}(y)$. In fact,
one can check that those conditions also characterize the set of pairs $(x,y)$
so that $\overline{f}(x)=\overline{f}(y)$, which proves injectivity.
Surjectivity is clear from the definition. Continuity of $\overline{f}$ and
its inverse with respect to the topology of $X_{\mu}$ follows from the fact
that, as pseudo-metrics, we have

$d_{\mu}(\bar{x},\bar{y})=d_{\alpha}(\bar{x},\bar{y})+d_{\beta}(\bar{x},\bar{y}).$

∎

######  Remark 5.4.

In his work, Guirardel goes on to prove that $i(\alpha,\beta)$ is equal to the
volume of $\mathcal{C}/\pi_{1}(X)$, where volume is defined by taking the
supremum, over finite trees (convex hull of finitely many points)
$K_{1},K_{2}$ of $T_{1},T_{2}$ of the product of Lebesgue measures
$\mu_{K_{1}}\times\mu_{K_{2}}$ (a finite tree is simplicial, and the Lebesgue
measure of a simplicial tree is induced from the Lebesgue measure on the
edges, since each edge is isometric to an interval of $\mathbb{R}$). On the
other hand,
$i(\alpha,\beta)=\frac{1}{2}i(\alpha+\beta,\alpha+\beta)=\frac{1}{2}i(\mu,\mu)$.
It would be interesting to see if one can recover the self-intersection number
of $\mu$ as some sort of volume of $X_{\mu}/\pi_{1}(X)$. Compare this with
Example 5.4, where we show that when $\mu$ is the Liouville current of $X$,
$X/\pi_{1}(X)$ is isometric to the hyperbolic surface $X$, and, on the other
hand, $i(\mu,\mu)=\pi^{2}|\chi(X)|$ (by [Bon88, Proposition 15], so the volume
on the dual is not quite the hyperbolic area of $X$, but rather a multiple of
it. We refer to [BIPP21c, Definition 3.4] as well as [Ouy19, Section 6.5] for
a construction of a space on which our space embeds isometrically (provided
one makes a choice of $\ell_{1}$ metric on the product).

###  5.3. Duals for positively ratioed Anosov representations

In this example we assume $X$ is a closed hyperbolic surface and we go back to
the class of geodesic currents introduced in Subsection 2.3.4 namely,
positively ratioed Anosov representations. First, we have the following
immediate observation.

######  Lemma 5.5.

Let $\rho\colon\pi_{1}(X)\to G$ be a positively ratioed representation and
$\mu_{\rho}$ its associated geodesic current as in 2.3.4. Then
$X_{\mu_{\rho}}/\pi_{1}(X)$ is homeomorphic to $X$.

###### Proof.

By Lemma 2.13, geodesic currents associated to positively ratioed Anosov
representations have no atoms and full support. Thus, by Proposition 4.14, the
result follows. ∎

######  Remark 5.6.

Note that there are points in the boundary of the Hitchin component
corresponding to geodesic currents that might have atoms or might not have
full support (see [BIPP21a] and [BIPP21b]). In fact, as we mentioned in
Subsection 2.3.4, in the paper [BIPP21b] the notion of positively ratioed
representations is generalized to representations whose cross-ratios need not
be continuous (and, thus, whose associated geodesic currents might have
atoms). In the sequel to this project with Anne Parreau, we will endow the
dual spaces coming from these representations with a geodesic structure.

In what follows we specialize to the case
$G=\operatorname{\mathit{SL}}(3,\mathbb{R})$, and consider only Hitchin
representations. By work of Choi-Goldman [CG93], to every
$\operatorname{\mathit{SL}}(3,\mathbb{R})$ Hitchin representation corresponds,
in a one-to-one fashion, a strictly convex real projective structure
$\Omega_{\rho}$, that we define now.

A _strictly convex real projective surface_ is a quotient $Z=\Omega/\Gamma$
where $\Omega\subseteq\mathbb{RP}^{2}$ is a strictly convex domain of the real
projective plane, and $\Gamma<SL(3,\mathbb{R})$ is a discrete group of
projective transformations acting properly on $\Omega$. Thus, $Z$ is the
topological surface $S$ equipped with a Hilbert metric. Let $\widetilde{Z}$ be
the induced metric in the universal cover of $S$. With this choice, the
compactification $\widetilde{Z}\cup\partial Z$ can be identified with
$\Omega\cup\partial\Omega$ and thus a geodesic current on $Z$ is a
$\pi_{1}(Z)$-invariant locally finite Borel measure on the set
$(\partial\Omega\times\partial\Omega-\Delta)/\mathbb{Z}_{2}$.

We define a proper complete path metric $d_{\Omega}$ which is invariant under
$\mathrm{Aut}(\Omega)$, called the _Hilbert metric_ associated to a convex
$\mathbb{RP}^{2}$-structure $Z$. Let $p,q\in\Omega$, the projective geodesic
through $p$ and $q$ in $\Omega$ defines a pair of points $a$ and $b$ on
$\partial\Omega$. We define the following complete metric (see, for example
[Bea99, Theorem 2.1]) on $\Omega$

(5.7)
$d_{\Omega}(p,q)\coloneqq\left\\{\begin{matrix}\log{\mathrm{cr}(a,p,q,b)}&\mbox{
if }p\neq q\\\ 0&p=q\end{matrix}\right.$

where $a,p,q,b$ appear in this order along the projective line and
$\mathrm{cr}(a,p,q,b)$ denotes the projective cross-ratio. This metric
descends to the strictly convex projective structure $Z$. Furthermore, the
projective cross-ratio also induces a geodesic current $\mu_{\Omega}$, by
setting

$\mu_{\Omega}(B)=\log\mathrm{cr}(a,b,c,d)$

where $B=I_{a,b}\times I_{c,d}\subseteq\mathcal{G}(\widetilde{Z})$ is a box of
geodesics with vertices $a,b,c,d\in\partial Z$, see for example [BIPP21b,
Subsection 4.1] for details. The current $\mu_{\Omega}$ satisfies the relation
$\ell_{\rho}^{H}(c)=i(\mu_{\Omega},c)$ for every closed curve
$c\in\mathit{Curves}(Z)$, where $\ell_{\rho}^{H}$ is the Hilbert length
function associated to $\rho$, which in this setting is given by

$\ell_{\rho}^{H}\colon\Gamma\to\mathbb{R}:g\mapsto\inf_{z\in\widetilde{Z}}d_{\Omega}(z,gz)$

or, equivalently, by

$\log{\frac{\lambda_{1}(\rho(g))}{\lambda_{3}(\rho(g))}}$

where $\lambda_{i}(\rho(g))$ is the $i$-th eigenvalue of
$g\in\operatorname{\mathit{SL}}(3,\mathbb{R})$.

We show now that $\mu_{\Omega}$ satisfies the _Crofton property_ in the sense
introduced in Subsection 2.11. First, we recall some classical results in
projective geometry of dimension 2.

Let $\Omega$ be an arbitrary convex domain of the projective plane
$\mathbb{R}P^{2}$. Busemann [Bus55] introduced an additive non-negative
measure $\sigma$ on the set of projective lines on $\Omega$ satisfying

  1. (1)

For any $C\subset\Omega$ so that $\overline{C}\cap\partial\Omega=\emptyset$,
$\sigma(G(C))<\infty$

  2. (2)

$\sigma(G[x])=0$ for any $x\in\Omega$

  3. (3)

$\sigma(G(S))>0$, where $S\subset\Omega$ contains a non-degenerate projective
line segment.

Using this measure $\sigma$, he defined a _$\sigma$ -metric_ on $\Omega$ given
by

$d_{\sigma}(x,y)=\sigma(G[x,y]).$

It turns out that, in dimension two, any continuous metric on $\Omega$ for
which the projective lines are geodesics can be realized as such a
$\sigma$-metric [Pog73]. The Hilbert metric $d_{\Omega}$ defined above
satisfies these assumptions, and thus, there exists a measure $\sigma$ on the
set of projective lines on $\Omega$ satisfying

$d_{\Omega}(x,y)=\sigma(G[x,y]).$

######  Proposition 5.8.

For every pair $\overline{x},\overline{y}\in\Omega$,

(5.9)
$\ell_{\Omega}([\overline{x},\overline{y}])=\mu_{\Omega}(G[\overline{x},\overline{y}]).$

###### Proof.

By the discussion above, we know that

$d_{\Omega}([\overline{x},\overline{y}])=\sigma(G[\overline{x},\overline{y}]).$

Observe that, since $\pi_{1}(Z)$ acts by projective isometries,
$g(G[\overline{x},\overline{y}])=G[g(\overline{x}),g(\overline{y})]$. Thus, by
the above equation

$\sigma(g(G[\overline{x},\overline{y}]))=\sigma(G[g(\overline{x}),g(\overline{y})])=\ell_{\Omega}([g(\overline{x}),g(\overline{y})])=\ell_{\Omega}([\overline{x},\overline{y}]).$

This shows that

$\sigma(g(G[\overline{x},\overline{y}]))=\mu(G[\overline{x},\overline{y}]),$

for every $g\in\pi_{1}(Z)$, and every $\overline{x},\overline{y}\in Z$. Since
the sets $G[\overline{x},\overline{y}]$ generate the Borel sigma algebra of
geodesics (see [MZ19, Lemma A.2], where the proof is given assuming $X$ is
hyperbolic, but the same proof works verbatim in the setting of $X$ Hilbert
strictly convex), this shows $\sigma$ is $\pi_{1}(Z)$-invariant. It follows
that $\sigma$ is a geodesic current on $\mathcal{G}(\widetilde{Z})$. Let $g$ a
loxodromic element with axis a projective line $L_{g}$. Let $\overline{x}\in
L_{g}$. Then

$\ell_{\Omega}(g)=d_{\Omega}([\overline{x},g\overline{x}])=\sigma(G[\overline{x},g\overline{x}])=i(\sigma,g).$

On the other hand, we also have

$\ell_{\Omega}(g)=d_{\Omega}([\overline{x},g\overline{x}])=\mu_{\Omega}(G[\overline{x},g\overline{x}])=i(\mu_{\Omega},g).$

Thus, by [Ota90, Théorème 2], $\mu_{\Omega}=\sigma$. ∎

Let us now fix a Hitchin representation
$\rho\colon\Gamma=\pi_{1}(X)\to\operatorname{\mathit{SL}}(3,\mathbb{R})$.
Martone-Zhang show in [MZ19] that $\rho$ induces a geodesic current
$\mu_{\rho}$ on $\mathcal{G}(X)$ such that

$i(\mu_{\rho},c)=\ell_{\Omega_{\rho}}(c)$

for every $c\in\mathit{Curves}(X)$. Moreover, the Hitchin representation
$\rho$ induces a _boundary map_

$\psi\colon\partial\widetilde{X}\to\partial\Omega.$

This boundary map is the extension to the boundary at infinity of the
developing map of the strictly convex real projective structure.

######  Proposition 5.10.

Given $X$ a closed hyperbolic surface and a Hitchin representation
$\rho\colon\pi_{1}(X)\to\operatorname{\mathit{SL}}(3,\mathbb{R})$, let
$\mathit{dev}\colon\widetilde{X}\to\Omega$ be the developing map of the
associated convex projective structure. Let $\psi$ denote the extension to the
boundary at infinity of this homeomorphism

$\psi\colon\partial\widetilde{X}\to\partial\Omega.$

If $[\gamma]\in\pi_{1}(X)$, then $\psi(\gamma^{\pm})=(\rho([\gamma]))^{\pm}$.

###### Proof.

The space $\Omega$ is $\delta$-hyperbolic if and only if it is a strictly
convex divisible domain [Ben04, Theorem 4.5]. Thus, the extension of
$\mathit{dev}$ to the boundaries at infinity is well-defined. Let us recall
that points in the boundary at infinity are equivalence classes of sequences
of points, where the equivalence classes identify two sequences if they are at
bounded distance. The map $\psi$ hence associates to a point
$x=[(x_{n})]\in\partial\widetilde{X}$ the point
$[(\mathrm{dev}(x_{n}))]\in\partial\Omega$. Let us fix a basepoint
$o\in\widetilde{X}$ and let $[\gamma]\in\pi_{1}(X)$, then we can write
$\gamma^{+}\in\partial\widetilde{X}$ as $\gamma^{+}=[{(\gamma^{n}\cdot o)}]$.
We have that

$\psi(\gamma^{+})=\psi[(\mathrm{dev}(\gamma^{n}\cdot
o))]=[(\rho(\gamma^{n})\cdot\mathrm{dev}(o))]=\rho([\gamma])^{+}$

where we have used in the second equality the $\Gamma$-equivariance of the
developing map. ∎

In fact, we have the following.

######  Lemma 5.11.

The current $\mu_{\Omega}$ is the push-forward of $\mu_{\rho}$ via $\psi$, the
extension to the boundary of the developing map.

###### Proof.

Let $\gamma_{-}$ and $\gamma_{+}$ be the repelling and attractive fixed points
in $\partial\widetilde{X}$ of the deck transformation corresponding to
$[\gamma]\ni c$. By Proposition 5.10, we have that
$\gamma^{\prime}_{+}=\psi(\gamma_{+})$ corresponds to the attracting point of
$\rho(\gamma)$, and $\gamma^{\prime}_{-}=\psi(\gamma_{-})$ corresponds to the
repelling point of $\rho(\gamma)$. Then we have, on the one hand, if
$\gamma_{-}^{\prime}\gamma_{+}^{\prime}$ denotes the projective line
determined by the points at infinity $\gamma_{-}^{\prime}$ and
$\gamma_{+}^{\prime}$, and $x\in\gamma_{-}^{\prime}\gamma_{+}^{\prime}$, we
have

$\displaystyle\ell_{\Omega}(c)$ $\displaystyle=\mu_{\Omega}(G[x,\gamma x])$
$\displaystyle=i(\mu_{\Omega},\gamma)=\mu_{C}([\gamma^{\prime}_{-},\gamma^{\prime}_{+}]\times[z,\gamma
z]).$

By $\pi_{1}(X)$-equivariance of $\psi$, we get

$\displaystyle i(\psi_{*}\mu_{\rho},\gamma)$
$\displaystyle=\psi_{*}\mu_{\rho}([\gamma^{\prime}_{-},\gamma^{\prime}_{+}]\times[z,\gamma
z])=$
$\displaystyle=\mu_{\rho}([\psi^{-1}\gamma^{\prime}_{-},\psi^{-1}\gamma^{\prime}_{+}]\times[\psi^{-1}(z),\gamma\psi^{-1}(z)])$
$\displaystyle=\mu_{\rho}([\gamma_{-},\gamma_{+}]\times[\psi^{-1}(z),\gamma\psi^{-1}(z)])=\ell_{\Omega_{\rho}}(\gamma).$

Since $i(\mu_{\Omega_{\rho}},\gamma)=\ell_{\Omega_{\rho}}(\gamma)$, by [Ota90,
Théorème 2] we have $\mu_{\Omega}=\psi_{*}\mu_{\rho}$. ∎

At this point, we observe that the results obtained in this paper for
$X_{\mu}$ equipped with an underlying hyperbolic structure $X$ would also hold
if the underlying metric was a Hilbert metric coming from a strictly convex
projective structure on the surface. In view of this, one could have defined,
from the get go, dual spaces $X_{\mu}$ in the setting of Hilbert metrics
coming from strictly convex real projective structures $X$ on the surface $S$.
We choose not to write things in such generality in this paper, since the only
place we allow $X$ to be a Hilbert metric (as opposed to a hyperbolic metric)
is in this example.

######  Proposition 5.12.

The dual space $(X_{\mu_{\Omega}},d_{\mu_{\Omega}})$ obtained by letting $X$
be the Hilbert metric induced by a convex projective structure $\Omega_{\rho}$
satisfies Theorems A, B, C, E and F.

###### Proof.

We list the properties used about the metric $X$ used throughout the paper,
justify why they are also satisfied when $X$ is a Hilbert metric coming from a
strictly convex projective structure, and point precisely to where they have
been used.

  1. (1)

Geodesics are unique and they are the same as length minimizers. This is true
for $X$ coming from a strictly convex projective structure, as shown in (see
[Bea99]). This is assumed in the choice of metric in Section 2.1, in
Propositions 4.6 and 4.7, as well as Section 6, Section 8 and Section 9.

  2. (2)

There exists a continuous geodesic flow for $X$, and geodesic currents can be
seen also as geodesic flow invariant measures under this flow (see [KP14,
Section 6] as well as [Cra11, Chapter 4]). This is assumed in Proposition 4.6.
Continuity properties of the flow are assumed in Lemma 8.11.

  3. (3)

Ergodicity properties and the existence of an ergodic Borel geodesic flow
invariant measure of full support are used in Proposition 10.2. In this
setting, one can use the Bowen-Margulis measure, see [Cra11, 4.2].

  4. (4)

Loxodromic elements have dense set of points on the boundary, and act by
north-south dynamics, both of which are true by $\delta$-hyperbolicity (see
[CDP90, Chapter 11, Proposition 2.4]), since strictly convex real projective
Hilbert metrics are $\delta$-hyperbolic [Ben04, Theorem 4.5]. This is assumed
in Lemma 6.9.

  5. (5)

Uniqueness of geodesic paths in a given base-point homotopy class, and
uniqueness of base-point homotopy classes of loops of length at most $L$ (see
[KP14, Page 9]).

∎

From this, we obtain the following result.

######  Proposition 5.13.

The map $\mathit{dev}\circ\pi_{\mu}^{-1}$ defines a $\pi_{1}(X)$-equivariant
homeomorphism from $X_{\mu_{\rho}}$ to $\Omega$. Furthermore, if we consider
the metric on $\widetilde{X}$ induced by the convex projective structure
$\Omega_{\rho}$ on $X$, $\pi_{\mu_{\Omega}}$ induces an isometry from
$(\Omega,d_{\Omega})$ to $(X_{\mu_{\Omega}},d_{\mu_{\Omega}})$.

###### Proof.

Since $\mu_{\rho}$ has no atoms and full support, by Proposition 4.14
$\pi_{\mu_{\rho}}\colon\widetilde{X}\to X_{\mu_{\rho}}$ is a
$\pi_{1}(X)$-equivariant homeomorphism. We can thus define the
$\pi_{1}(X)$-equivariant homeomorphism
$\varphi\coloneqq\mathit{dev}\circ\pi_{\mu_{\rho}}^{-1}$.

We show that this map is, in fact, an isometry. For any points $x,y\in
X_{\mu_{\rho}}$, let $x=\pi_{\mu}(\bar{x})$ and $y=\pi_{\mu}(\bar{y})$, for
$\bar{x},\bar{y}\in\widetilde{X}$. Note that

$\displaystyle d_{\mu_{\Omega}}(x,y)$
$\displaystyle=d_{\mu_{\Omega}}(\pi_{\mu}(\bar{x}),\pi_{\mu}(\bar{y}))=\mu_{\Omega}(G[\bar{x},\bar{y}])=d_{\Omega}(\bar{x},\bar{y}).$

∎

###  5.4. Duals for hyperbolic/negatively curved Riemannian Liouville current

In this subsection, we assume that $X$ is closed. Recall that the geodesic
current $\mathcal{L}_{Y}^{X}$ has full support and has no atoms, and it is has
been defined as the Liouville current associated to
$[(Y,\varphi)]\in\operatorname{Teich}(X)$ (see Subsection 2.11 for details).

######  Lemma 5.14.

Given $[(Y,\varphi)]\in\operatorname{Teich}(X)$, the map
$\varphi\circ\pi_{\mathcal{L}_{Y}^{X}}^{-1}$ induces a
$\pi_{1}(X)$-equivariant homeomorphism between $\widetilde{Y}$ to
$\widetilde{X}_{\mathcal{L}_{Y}^{X}}$. Furthermore, if $Y=X$,
$\pi_{\mathcal{L}_{X}}$ induces an isometry between
$(\widetilde{X},d_{\widetilde{X}})$ and
$(\widetilde{X}_{\mathcal{L}_{X}},d_{\mathcal{L}_{X}})$.

###### Proof.

Note that since $\mathcal{L}_{Y}^{X}$ has no atoms and full support, it
follows from Proposition 4.14 that
$\varphi\circ\pi_{\mathcal{L}_{Y}^{X}}^{-1}$ is a $\pi_{1}(X)$-equivariant
homeomorphism. If $X=Y$, then we write $\mathcal{L}_{Y}^{X}=\mathcal{L}_{X}$,
and we have $\pi_{\mathcal{L}_{X}}(\bar{x})=x$. Let $x,y\in
X_{\mathcal{L}_{X}}$, and set $\bar{x}=\pi_{\mathcal{L}_{X}}^{-1}(x)$,
$\bar{y}=\pi_{\mathcal{L}_{X}}^{-1}(y)$.

$\displaystyle d_{\widetilde{X}}(\bar{x},\bar{y})$
$\displaystyle=\mathcal{L}_{X}(G[\pi_{\mathcal{L}_{X}}^{-1}(x),\pi_{\mathcal{L}_{X}}^{-1}(y)])=d_{\mathcal{L}_{X}}(x,y).$

∎

We end this example by remarking that the same argument for a negatively
curved Riemannian metric $Z$, and its associated geodesic current
$\mathcal{L}_{Z}$ as defined by Otal (see Subsection 2.11), yields an isometry
$\pi_{\mathcal{L}_{Z}}$ between $(\widetilde{Z},d_{\widetilde{Z}})$ and
$(\widetilde{X}_{\mathcal{L}_{Z}},d_{\mathcal{L}_{Z}})$.

For example, by [OT21, Proposition 4.2], the Blaschke metric induced by cubic
differentials is a negatively curved metric, with the Liouville current
associated to a negatively curved Riemannian metric whose dual space is
isometric to $X$ equipped with said metric. One can obtain similar
equivalences for other geodesic currents satisfying the Crofton property
associated to non-positively curved metrics, as long as the properties in
Proposition 5.12 are satisfied.

##  6\. Hyperbolicity

In this section we prove that the dual spaces $X_{\mu}$ are
$\delta$-hyperbolic metric spaces.

Recall (see Definition 2.2) that $I_{a,b}$ denotes a generalized ordered
interval in $\partial\widetilde{X}$, and a box of geodesics was defined as any
subset of $\mathcal{G}(\widetilde{X})$ of the type $B=I_{a,b}\times I_{c,d}$.
Recall also $\mathcal{B}$ denotes the family of all boxes of geodesics.

######  Definition 6.1 (opposite box).

Given a box of geodesics $B=I_{a,b}\times I_{c,d}$, its _opposite box_ is
defined as $B^{\perp}=I_{d,a}\times I_{b,c}$, so that the intervals
$I_{a,b},I_{c,d},I_{d,a},I_{b,c}$ partition $\partial\widetilde{X}$. See
Figure 6.1.

$a$
---
$d$
---
$b$
---
$c$
---
$I_{a,b}$
---
$I_{c,d}$
---
$a$
---
$d$
---
$b$
---
$c$
---
$I_{d,a}$
---
$I_{b,c}$
---
$B=I_{a,b}\times I_{c,d}$
---
$B^{\perp}=I_{d,a}\times I_{b,c}$
---

Figure 6.1. Sketch of a box of geodesics and its corresponding opposite box.

The following lemma is straightforward and not new. The second equation in
Lemma 6.2 appears in work of Otal [Ota90, Page 154], without proof. We provide
a proof here, for completeness.

We introduce the following notation. Given two geodesics segments $t$ and
$t^{\prime}$, let $G(t,t^{\prime})$ be the set of geodesics intersecting both
$t$ and $t^{\prime}$ transversely. We will refer to sets of this type as
_double transversals_. Let
$\overline{x},\overline{y},\overline{z},\overline{w}\in\widetilde{X}$ appear
counter-clockwise as vertices of an embedded geodesic quadrilateral on
$\widetilde{X}$ with sides $[\overline{x},\overline{y})$,
$[\overline{z},\overline{w})$, $[\overline{y},\overline{z})$, and
$[\overline{w},\overline{x})$.

######  Lemma 6.2.

Given the setup described above, we have

$\displaystyle\mu(G([\overline{x},\overline{w}),[\overline{y},\overline{z})))+\mu(G((\overline{x},\overline{w}],(\overline{y},\overline{z}]))$
$\displaystyle=$
$\displaystyle\mu(G[\overline{x},\overline{z}))+\mu(G[\overline{w},\overline{y}))$
$\displaystyle-\mu(G([\overline{w},\overline{z}))-\mu(G([\overline{x},\overline{y}))$

and

$\displaystyle\mu(G((\overline{x},\overline{w}],(\overline{y},\overline{z}]))+\mu(G([\overline{x},\overline{w}),[\overline{y},\overline{z})))$
$\displaystyle=$
$\displaystyle\mu(G(\overline{x},\overline{z}])+\mu(G(\overline{w},\overline{y}])$
$\displaystyle-\mu(G((\overline{w},\overline{z}])-\mu(G((\overline{x},\overline{y}])$

###### Proof.

We have the following partitions:

(6.3a)
$\displaystyle\begin{split}G([\overline{x},\overline{z}))=G([\overline{x},\overline{z}),[\overline{x},\overline{y}))\cup
G([\overline{x},\overline{z}),(\overline{z},\overline{y}])\end{split}$ (6.3b)
$\displaystyle\begin{split}G([\overline{w},\overline{y}))=G([\overline{w},\overline{y}),[\overline{w},\overline{z}))\cup
G([\overline{w},\overline{y}),(\overline{y},\overline{z}])\end{split}$ (6.3c)
$\displaystyle\begin{split}G([\overline{w},\overline{z}))=G([\overline{w},\overline{z}),[\overline{w},\overline{y}))\cup
G([\overline{w},\overline{z}),(\overline{z},\overline{y}])\end{split}$ (6.3d)
$\displaystyle\begin{split}G([\overline{x},\overline{y}))=G([\overline{x},\overline{y}),[\overline{x},\overline{z}))\cup
G([\overline{x},\overline{y}),(\overline{y},\overline{z}])\end{split}$

We now applying the measure $\mu$ to all the equations, add the equations
resulting from Equation 6.3a and 6.3b, and subtract this from the equations
resulting from Equation 6.3c and 6.3d, we get the result. The other equation
follows in a similar way. ∎

######  Definition 6.4 (double transversals and boxes for 4-tuples).

In what follows, compare Figure 6.2 for illustrations. Let
$\overline{x},\overline{y},\overline{z},\overline{w}$ be four distinct points
in $\widetilde{X}$. Up to relabeling, we assume that they appear as vertices
of an embedded geodesic quadrilateral ordered counter-clockwise on
$\widetilde{X}$. Consider the oriented hyperbolic geodesic $\gamma_{1}$
connecting $\overline{x}$ to $\overline{y}$, and the oriented hyperbolic
geodesic $\gamma_{2}$ connecting $\overline{w}$ to $\overline{z}$. Let
$\delta_{1}$ be the oriented hyperbolic geodesic connecting $\overline{z}$ to
$\overline{y}$ and $\delta_{2}$ be the oriented geodesic connecting
$\overline{w}$ to $\overline{x}$. We define three sets of geodesics associated
to the tuple $(\overline{x},\overline{y},\overline{z},\overline{w})$.

  1. (1)

Let $b^{+}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ be the box
of geodesics defined by
$(\gamma_{2}^{-},\gamma_{1}^{-}]\times[\gamma_{1}^{+},\gamma_{2}^{+})$ and
$b^{-}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ the box defined
by $[\gamma_{2}^{-},\gamma_{1}^{-})\times(\gamma_{1}^{+},\gamma_{2}^{+}]$

  2. (2)

Let ${G^{+}}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ denote a
_double transversal_ , defined as the set of geodesics intersecting both
$[\overline{x},\overline{w})$ and $[\overline{y},\overline{z})$. Let
$G^{-}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ denote the set
of geodesics intersecting $(\overline{x},\overline{w}]$ and
$(\overline{y},\overline{z}]$ transversely. Let, also $(G^{+})^{\perp}$ denote
the set of geodesics intersecting $[\overline{x},\overline{y})$ and
$[\overline{w},\overline{z})$, and $(G^{-})^{\perp}$ the set of geodesics
intersecting $(\overline{x},\overline{y}]$ and $(\overline{w},\overline{z}]$

  3. (3)

Let $B^{+}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ be the box
of geodesics defined by
$(\delta_{1}^{-},\delta_{2}^{-}]\times[\delta_{2}^{+},\delta_{1}^{+})$, and
$B^{-}_{\overline{x},\overline{y},\overline{z},\overline{w}}$ be the box of
geodesics defined by
$[\delta_{1}^{-},\delta_{2}^{-})\times(\delta_{2}^{+},\delta_{1}^{+}]$.

The following result follows directly from Definition 6.4.

######  Proposition 6.5.

Given the setting as described above, dropping the subscripts, we have

$b^{\pm}\subseteq G^{\pm}\subseteq B^{\pm}.$

Moreover, we have

$(B^{\perp})^{\pm}\subseteq(G^{\perp})^{\pm}\subseteq(b^{\perp})^{\pm}.$

We define the following two quantities.

######  Definition 6.6 ($\delta_{\mu}$ with boxes).

For a given geodesic current $\mu$, define

$\delta^{B}_{\mu}=\frac{1}{2}\sup_{B\in\mathcal{B}}\min\\{\mu(B^{+})+\mu(B^{-}),\mu((B^{\perp})^{+})+\mu((B^{\perp})^{-})\\}$

We observe that $B\times B^{\perp}\subset\mathfrak{I}$ where $\mathfrak{I}$ is
the subset of $\mathcal{G}(\widetilde{X})\times\mathcal{G}(\widetilde{X})$
consisting of transversely intersecting geodesics, used in the definition of
intersection number of geodesic currents (see Definition 2.4). Thus,
$\delta_{\mu}$ is giving another measure, related to intersection number, of
‘how far is $\mu$ from being a measured lamination’.

######  Definition 6.7 ($\delta_{\mu}$ with double transversals).

For a given geodesic current $\mu$, define

$\delta^{G}_{\mu}=\frac{1}{2}\sup_{G}\min\\{\mu(G^{+})+\mu(G^{-}),\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})\\}$

where $G$ ranges over all $G_{x,y,z,w}$ with $x,y,z,w$ all distinct.

######  Lemma 6.8.

For any geodesic current $\mu$,

$\delta_{\mu}^{B}=\delta_{\mu}^{G}.$

###### Proof.

By the inequalities above we have that, for fixed $x,y,z,w$,

$b^{\perp}\supseteq G^{\perp}\supseteq B^{\perp}.$

Since ranging over all $x,y,z,w$, $B$ and $b$ exhaust all possible boxes (and
same for $B^{\perp}$,$b^{\perp}$), we have, taking measure $\mu$ and supremum
over all distinct $x,y,z,w$, that

$\delta_{\mu}^{B}\geq\delta_{\mu}^{G}\geq\delta_{\mu}^{B}.$

as we wanted to show. ∎

$w$
---
$x$
---
$\gamma_{1}$
---
$\delta_{1}$
---
$\delta_{2}$
---
$z$
---
$\gamma_{2}^{+}$
---
$\gamma_{2}^{-}$
---
$[\gamma_{2}^{-},\gamma_{1}^{-})$
---
$[\gamma_{1}^{+},\gamma_{2}^{+})$
---
$\gamma_{2}$
---
$\gamma_{1}^{-}$
---
$\gamma_{1}^{-}$
---
$\gamma_{1}^{+}$
---
$\gamma_{1}^{+}$
---
$B=[\gamma_{2}^{-},\gamma_{1}^{-})\times[\gamma_{1}^{+},\gamma_{2}^{+})$
---
$B^{\prime}=[\delta_{1}^{-},\delta_{2}^{-})\times[\delta_{2}^{+},\delta_{1}^{+})$
---
$w$
---
$x$
---
$y$
---
$y$
---
$x$
---
$z$
---
$\gamma_{2}^{-}$
---
$\gamma_{2}$
---
$B^{\perp}$
---
$\gamma_{2}^{+}$
---

Figure 6.2.

Since $\delta^{G}_{\mu}$ and $\delta^{B}_{\mu}$ are the same quantity, we will
simply refer to it as $\delta_{\mu}$. For some proofs it will be easier to use
one viewpoint or the other.

######  Proposition 6.9.

Let $\mu$ be any geodesic current on $X$. Then $\delta_{\mu}$ is finite.

###### Proof.

Assume $\delta_{\mu}$ is not finite. Hence there exists a sequence of boxes
$B^{\prime}_{n}$ such that $\mu(B^{\prime}_{n})$ and
$\mu({B^{\prime}_{n}}^{\perp})$ both diverge. Suppose that
$B^{\prime}_{n}=[a^{\prime}_{n},b^{\prime}_{n})\times[c^{\prime}_{n},d^{\prime}_{n})$
and thus
$B^{\prime\perp}=[b^{\prime}_{n},c^{\prime}_{n})\times[d^{\prime}_{n},a^{\prime}_{n})$,
with $a^{\prime}_{n},b^{\prime}_{n},c^{\prime}_{n},d^{\prime}_{n}$ are ordered
counter clockwise for every $n$. Consider a compact subset $K$ in
$\widetilde{X}$ containing a fundamental region for the $\pi_{1}(X)$-action on
$\widetilde{X}$. For every box
$B^{\prime}_{n}=[a^{\prime}_{n},b^{\prime}_{n}]\times[c^{\prime}_{n},d^{\prime}_{n}]$
let $m(B^{\prime}_{n})$ be the _center_ of $B^{\prime}_{n}$, i.e. the
intersection between the two ‘diagonal’ geodesics joining $a^{\prime}_{n}$ to
$c^{\prime}_{n}$ and $b^{\prime}_{n}$ to $d^{\prime}_{n}$. Since the
$\pi_{1}(X)$ action on $\widetilde{X}$ is cocompact, for each box
$B^{\prime}_{n}$ in the sequence, there exists $g_{n}\in\pi_{1}(X)$ such that
$g_{n}m(B^{\prime}_{n})\in K$. By $\pi_{1}(X)$ invariance of the measure $\mu$
we have that since $\mu(B^{\prime}_{n})$ and $\mu({B^{\prime}_{n}}^{\perp})$
diverge, then also $\mu(g_{n}B^{\prime}_{n})$ and
$\mu((g_{n}{B^{\prime}_{n}})^{\perp})$ diverge. Let
$B_{n}:=g_{n}B^{\prime}_{n}$.

In order for $\mu(B_{n})$ to diverge, $B_{n}$ has to leave every compact set
of the space of geodesics. Hence either
$d_{\partial\widetilde{X}}(d_{n},a_{n})\to 0$ or
$d_{\partial\widetilde{X}}(b_{n},c_{n})\to 0$ Let’s assume without loss of
generality that distance between $d_{n}$ and $a_{n}$ in
$\partial\widetilde{X}$ tends to $0$ as $n$ goes to infinity. By compactness
of $S^{1}$, each of the sequences $(a_{n}),(b_{n}),(c_{n}),(d_{n})$ admits a
convergent subsequence. Therefore, up to subsequences, we can assume that
$a_{n},d_{n}\to x$, $b_{n}\to b$ and $c_{n}\to c$.

Now we make some case distinctions

  1. (1)

$b\neq c$ and $b,c\neq x$;

  2. (2)

$x\neq c$ and $x\neq b$ and $b=c$;

  3. (3)

$x$ is equal to $b$ or $c$ (or both).

In cases (1) and (2) we immediately reach a contradiction because we can find
a compact set $C\subseteq\mathcal{G}(\widetilde{X})$ in which $B_{n}^{\perp}$
is included for all $n$ large enough. This is in contradiction with the fact
that both $\mu(B_{n})$ and $\mu(B_{n}^{\perp})$ diverge.

Finally, let $B_{n}$ a sequence of boxes falling into case (3), i.e. three
among the vertices $a_{n},b_{n},c_{n},d_{n}$ converge to the same point on
$\partial\widetilde{X}$. In this case we also get a contradiction because
$m(B_{n})$ would escape all compact sets of $\widetilde{X}$, but the center of
the boxes $B_{n}$ belongs to the compact set $K$ for all $n$. This completes
the proof.

∎

Given a geodesic current $\mu$, we say a box of geodesics $B$ is _$\mu$
-generic_ if $\mu(\partial B)=0$. Let $\mathcal{B}^{\mu}\subset\mathcal{B}$
denote the subset of $\mu$-generic geodesic boxes.

The following is an easy but crucial observation.

######  Lemma 6.10.

In the definition of $\delta_{\mu}$ we can restrict to $\mu$-generic boxes,
i.e.,

$\delta_{\mu}=\sup_{B\in\mathcal{B}^{\mu}}\min\\{\mu(B),\mu(B^{\perp})\\}$

where $\mathcal{B}^{\mu}\subset\mathcal{B}$ consists of boxes $B$ so that $B$
and $B^{\perp}$ are $\mu$-generic.

###### Proof.

Suppose that $B=[a,b)\times[c,d)$, and $\mu(P(a))>0$. By Lemma 2.5, there
exists a lift of a closed geodesic in the support. For every $\varepsilon>0$,
there exists a point $a^{\prime}\in\partial\widetilde{X}$, so that the box
$K=[a^{\prime},a)\times[c,d)$ has no atoms and $\mu(B^{\prime})<\varepsilon$.
Indeed, it has no atoms, since atoms are concentrated on the set $C_{\mu}$ of
lifts of closed geodesics in the support of $\mu$, and this set $C_{\mu}$ is
discrete, by local finiteness of $\mu$. By choosing $a^{\prime}$ close enough
to $a$, we can ensure $K$ has no atoms. Otherwise, some $K$ would contain
infinitely many elements in $C_{\mu}$, but this would violate local finiteness
of $\mu$. Note that then the box $B^{\prime}\coloneqq K\cup B$ is
$\mu$-generic, and by taking $a^{\prime}$ closer to $a$, we can guarantee
$\mu(B^{\prime})-\mu(B)<\varepsilon$. Moreover, $\partial B^{\prime\perp}$ has
the same atoms as $\partial B^{\perp}$. By repeating the same argument with
$B^{\perp}$, we can guarantee that $B^{\perp}$ is also $\mu$-generic.

∎

The following is a restatement of the Gromov 4-point condition for
$\delta$-hyperbolic spaces [BH11, Page 410].

######  Definition 6.11 ($\delta$-hyperbolicity).

A metric space $(X,d)$ is _$\delta$ -hyperbolic_ if and only if for any
4-tuple of points $x,y,z,w\in X$, among the following three quantities

  * •

$d(x,y)+d(z,w)$

  * •

$d(x,z)+d(y,w)$

  * •

$d(y,z)+d(x,w)$

the two largest of them are within $2\delta$ of each other. If $\delta=0$,
then it means that the maximum appears at least twice.

######  Theorem 6.12.

If $\mu$ is a geodesic current then $X_{\mu}$ is a $\delta_{\mu}$-hyperbolic
space in the sense of Definition 6.11, and $\delta_{\mu}$ is the optimal
$\delta$-hyperbolicity constant.

###### Proof.

We prove that $X_{\mu}$ satisfies the $\delta_{\mu}$-hyperbolic 4-point
condition. Let $x,y,z,w\in X_{\mu}$ be four arbitrary points, and, up to
relabeling, assume that the geodesic segments $[x,z]$ and $[y,w]$ intersect.
Then $d(x,z)+d(y,w)\geq d(x,y)+d(z,w)$ or $d(x,z)+d(y,w)\geq d(x,w)+d(y,z)$.
Assume first that $d(x,z)+d(y,w)>d(x,y)+d(z,w)+2\delta$. We want to show that
$d(x,z)+d(y,w)\leq d(y,z)+d(x,w)+2\delta$. Notice how the first inequality is
equivalent to

$\displaystyle\frac{1}{2}$
$\displaystyle[\mu(G[\overline{x},\overline{z}))+\mu(G[\overline{y},\overline{w}))-\mu(G[\overline{x},\overline{y}))-\mu(G[\overline{z},\overline{w}))]+$
$\displaystyle\frac{1}{2}[\mu(G(\overline{x},\overline{z}])+\mu(G(\overline{y},\overline{w}])-\mu(G(\overline{x},\overline{y}])-\mu(G(\overline{z},\overline{w}])]>2\delta$

which is equivalent to

$\frac{1}{2}[\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})]>\delta$

where $G=G_{x,w,y,z}$, by Lemma 6.2. On the other hand, $d(x,z)+d(y,w)\leq
d(y,z)+d(x,w)+2\delta$, is equivalent to

$\displaystyle\frac{1}{2}[\mu(G[\overline{x},\overline{z}))+\mu(G[\overline{y},\overline{w}))-\mu(G[\overline{y},\overline{z}))-\mu(G[\overline{x},\overline{w}))]+$
$\displaystyle\frac{1}{2}[\mu(G(\overline{x},\overline{z})]+\mu(G(\overline{y},\overline{w}])-\mu(G(\overline{y},\overline{z}])-\mu(G(\overline{x},\overline{w}])]\leq
2\delta$

i.e., $\frac{1}{2}[\mu(G^{+}),\mu(G^{-})]\leq\delta$, by Lemma 6.2.

Suppose, for the sake of contradiction, that
$\frac{1}{2}[\mu(G^{+}),\mu(G^{-})]>\delta$. Then,

$\displaystyle\min\\{\mu(G^{+})+\mu(G^{-}),\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})\\}$
$\displaystyle>\delta$
$\displaystyle=\sup_{G}\min\\{\mu(G^{+})+\mu(G^{-}),\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})\\}$

which is a contradiction. If, instead, we had
$d(x,z)+d(y,w)>d(y,z)+d(x,w)+2\delta$, then, similarly as above, this would
mean $\frac{1}{2}[\mu(G^{+}),\mu(G^{-})]>\delta$, and if
$\frac{1}{2}[\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})]>\delta$, we would
again contradict that $\delta$ is a supremum, so we must have
$\frac{1}{2}[\mu((G^{\perp})^{+})+\mu((G^{\perp})^{-})]\leq\delta$, and thus
$d(x,z)+d(y,w)\leq d(x,y)+d(z,w)+2\delta$, so the 4-point condition is proven.
Since $\delta_{\mu}$ is defined in terms of a supremum, it follows it is the
optimal $\delta$-hyperbolicity constant. ∎

######  Corollary 6.13.

$X_{\mu}$ is $0$-hyperbolic if and only if $\mu$ is a measured lamination.

###### Proof.

By [BIPP21b, Proposition 2.1], $\mu$ is a measured lamination if and only if,
for every box $B\subset G(X)$, we have $\mu(B)\mu(B^{\perp})=0$. This last
equality is true if and only if $\min\\{\mu(B)\mu(B^{\perp})\\}=0$. If $\mu$
is a measured lamination, then $\delta_{\mu}=0$, and thus by Theorem 6.12,
$X_{\mu}$ is $0$-hyperbolic. If $X_{\mu}$ is $0$-hyperbolic, since
$\delta_{\mu}$ is the smallest hyperbolicity constant, we must have
$\delta_{\mu}=0$. Since $\delta_{\mu}$ is defined in terms of a supremum, this
implies that for all boxes $B\subset\mathcal{G}(\widetilde{X})$,
$\mu(B)\mu(B^{\perp})=0$, and hence $\mu$ must be a lamination. ∎

As a consequence, we recover the following well-known result.

######  Corollary 6.14.

If $\mu$ is a Liouville current $\mathcal{L}_{Y}$, for
$Y\in\operatorname{Teich}(S)$, then its optimal hyperbolicity constant is
$\delta_{\mu}=\log(2)$.

###### Proof.

By [Bon88, Theorem 13], a geodesic current $\mu$ is a Liouville current
$\mathcal{L}_{X}$ for some hyperbolic structure $Y$, if and only if
$e^{-\mu(B)}+e^{-\mu(B^{\perp})}=1$. Maximizing the function $\delta_{\mu}$
subject to this relation, yields $\log(2)$. Thus, by Theorem 6.12, the result
follows. ∎

######  Remark 6.15.

Recall that when $\mu$ is the hyperbolic Liouville current of $X$, then
$X_{\mu}$ is isometric to the hyperbolic plane by Example 5.4. Observe that
$\log(2)$ is the optimal $\delta$-hyperbolicity constant which was computed
for example in [Nv16, Corollary 5.4]. Corollary 6.14 recovers this result.

######  Proposition 6.16.

$\delta_{\mu}$ is a lower semi-continuous function on geodesic currents.

###### Proof.

Let $\mu_{i}\to\mu$ in the weak∗-topology. Recall that, by Lemma 6.10, in the
definition of $\delta_{\mu}$, we can restrict to $\mu$-generic boxes $B\subset
G(X)$ without affecting the supremum. For any such $B$, let

$f_{B}(\mu)\coloneqq\min\\{\mu(B),\mu(B^{\perp})\\}.$

By the Portmanteau theorem [Bau01, Theorem 30.12], $f_{B}(\mu_{i})\to
f_{B}(\mu)$, so $f_{B}$ is a continuous function on geodesic currents. Since
$\delta_{\mu}=\sup_{B}f_{B}(\mu)$ is a supremum of continuous functions, by
[vRS82, Theorem 10.3] it must be lower semi-continuous. ∎

######  Example 6.17.

We discuss an example. Take a sequence $\mu_{i}$ of scaled hyperbolic
Liouville currents $\mu_{i}\coloneqq a_{i}\mathcal{L}_{X_{i}}$ converging to a
measured lamination $\lambda$ in the weak∗-topology. Then, we must have
$a_{i}\to 0$ (see argument at the bottom of [Bon88, Page 152]). By Corollary
6.14, $\delta_{\mu_{i}}=a_{i}\log(2)$ for all $i\in\mathbb{N}$, and thus
$\lim_{i}a_{i}\log(2)=0$. On the other hand $\delta_{\lambda}=0$, also.

To finish this section, we show a few inequalities between the
$\delta$-hyperbolicity constants of $X_{\mu}$ and the ones of its subcurrents
according to the decomposition theorem for geodesic currents Theorem 7.2. In
Section 7 we will see that $X_{\mu}$ decomposes as a graph of spaces with
vertices the duals of its subcurrents $\mu_{i}$. The following inequalities
relate the $\delta$-hyperbolicity constants of the components of the space to
those of its pieces. The proof is straightforward.

######  Lemma 6.18.

Let $\mu,\mu_{1},\mu_{2}$ be geodesic currents so that $\mu=\mu_{1}+\mu_{2}$,
and let $B\subseteq\mathcal{G}(\widetilde{X})$ be a box of geodesics.

  1. (1)

We have

$\min\\{\mu(B),\mu(B^{\perp})\\}\geq\min\\{\mu_{1}(B),\mu_{1}(B^{\perp})\\}+\min\\{\mu_{2}(B),\mu_{2}(B^{\perp})\\}.$

  2. (2)

If, furthermore $\mu_{1}\perp\mu_{2}$, then

$\min\\{\mu(B),\mu(B^{\perp})\\}=\min\\{\mu_{1}(B),\mu_{1}(B^{\perp})\\}+\min\\{\mu_{2}(B),\mu_{2}(B^{\perp})\\}.$

######  Proposition 6.19.

Let $\mu$ be a geodesic current which decomposes according to the structural
Theorem 7.2, $\mu=\sum_{i}\nu_{i}+\sum_{j}\lambda_{j}+\sum_{k}a_{k}s_{k}$,
where $s_{k}$ is a geodesic current supported on a simple closed curve,
$\lambda_{j}$ is a non-discrete measured lamination, and $\nu_{i}$ is a
geodesic current which is filling in a subsurface, and all the currents in the
decomposition have orthogonal supports (in the sense that
$i(\operatorname{supp}(\alpha),\operatorname{supp}(\beta))=0$, according to
[BIPP21a, Proposition 3.2]). Then, we have

(6.20)
$\min\\{\mu(B),\mu(B^{\perp})\\}=\sum_{i}\min\\{\nu_{i}(B),\nu_{i}(B^{\perp})\\},$

and, thus, for every $i$, we have

(6.21) $\delta_{\nu_{i}}\leq\delta_{\mu}\leq\sum_{i}\delta_{\nu_{i}}.$

###### Proof.

By the structural theorem [BIPP21a, Proposition 3.2] all the currents in the
decomposition are pairwise orthogonal. Thus, equation 6.20 follows by
Proposition 6.18, and considering [BIPP21a, Proposition 2.1], it follows that

$\min\\{\lambda_{j}(B),\lambda_{j}(B^{\perp})\\}=0$

and

$\min\\{s_{k}(B),s_{k}(B^{\perp})\\}=0.$

The first inequality follows from the fact that, by Equation 6.20,
$\min\\{\mu(B),\mu(B^{\perp})\\}\geq\min\\{\nu_{i}(B),\nu_{i}(B^{\perp})\\}$
for every $i$, and for any two real valued functions $f,g$ so that $f\leq g$,
we have $\sup f\leq\sup g$. The second inequality in Equation 6.21 follows by
Equation 6.20 and the fact that for two real valued functions $f,g$, we have
$\sup f+g\leq\sup f+\sup g$. ∎

##  7\. Decomposition theorem

We begin by defining the set of _special geodesics_ , as introduced in
[BIPP21a]. Given a geodesic current $\mu$ on a compact hyperbolic surface $X$,
let

$\displaystyle\mathcal{E}_{\mu}:=\\{$ $\displaystyle c\subset X:c\text{ closed
geodesic such that }i(\mu,c)=0$ $\displaystyle\text{ and for every
}c^{\prime}\subset X\text{ closed geodesic, }i(\mu,c^{\prime})>0\text{
whenever }i(c,c^{\prime})>0\\}$

The set $\mathcal{E}_{\mu}=\\{s_{1},\dots,s_{n}\\}$ is a finite set of
pairwise disjoint simple closed geodesics, called _special curves_ , which
decomposes $X$ in subsurfaces with geodesic boundary

(7.1) $X=\bigcup_{i}X_{i}.$

Given a current $\mu$ on $X$, recall that the systole of $\mu$ relative to
$X_{i}$ is

$\mathrm{sys}_{X_{i}}(\mu)\coloneqq\inf\\{i(\mu,c):c\in\mathit{Curves}(X_{i}-\partial
X_{i})\\}.$

We now state the decomposition theorem for geodesic currents, as proven in
[BIPP21a, Theorem 1.2].

######  Theorem 7.2 (Decomposition Theorem for Geodesic Currents).

Any current $\mu$ on $X$ decomposes as

(7.3) $\mu=\sum_{i=1}^{n}\mu_{i}+\sum_{j=1}^{m}a_{j}s_{j}$

where each non-zero $\mu_{i}$ is a geodesic current supported on $X_{i}$ and
$\sum_{j=1}^{m}a_{j}s_{j}$ is a weighted simple multi-curve, and the weights
$a_{j}$ need not be positive.

Moreover, for each $\mu_{i}$ we either have

  * •

(type 1) $\mathrm{sys}_{X_{i}}(\mu_{i})>0$;

  * •

(type 2) $\mu_{i}$ is a measured lamination compactly supported on the
interior of ${X}_{i}$ and intersecting every curve in $X_{i}$.

######  Remark 7.4.

For the remaining of this paper we will refer to currents which fall into the
first case as _subcurrents of type 1_ , the ones falling into the second case
will be referred as _subcurrents of type 2_ , and the weighted simple curves
in the special simple multi-curve will be referred as _subcurrents of type 3_.

Figure 7.1. The support of a current on a surface, and its corresponding lift
in the universal cover.

Figure 7.1 shows a sketch of a genus $3$ surface with a geodesic current $\mu$
whose special multi-curve $m$ consists of one single geodesic (red curve,
separating the left two handles from the right handle), and yields two
subsurfaces. The left one, genus 2, supports a filling geodesic current
$\mu_{1}$ (in blue) within that subsurface. The right one, of genus 1,
supports a non-discrete measured lamination $\mu_{2}$ (in green). The lower
figure shows a part of the support of the geodesic current $\mu$ in the
universal cover. The lifts of $m$ separate $\widetilde{X}$ into countably many
regions. In the figure, three are depicted, the central one is a region
corresponding to the support of $\mu_{2}$, whereas the upper and lower ones
correspond to the support of $\mu_{1}$.

######  Definition 7.5 (subdual).

Each component $\mu_{i}$ of $\mu$ is a geodesic current itself, supported on
the subsurface $X_{i}$. On $\widetilde{X_{i}}\subseteq\widetilde{X}$ we can
define the pseudo-distance $d_{\mu_{i}}$ in the same way as for $d_{\mu}$. The
_sub-dual space_ $X_{\mu_{i}}$ is the quotient space
$\widetilde{X_{i}}/{\\{d_{\mu_{i}}=0\\}}$ endowed with the
$\pi_{1}(X_{i})$-action.

In order to precisely describe the dual $X_{\mu}$ in terms of the sub-duals
$X_{\mu_{i}}$ we use the notion of _tree-graded space_.

For the standard definition of tree-graded space when $(X,d)$ is a geodesic
metric space we refer to the work by Drutu-Sapir [DS05].

######  Definition 7.6 (Tree-graded space).

A geodesic metric space $(X,d)$ is said to be _tree-graded_ with respect to a
collection of geodesic subspaces $\mathcal{P}$, called _pieces_ , if

  1. (1)

_axiom pieces_. Given two distinct pieces $P_{1},P_{2}\in\mathcal{P}$ the
intersection $P_{1}\cap P_{2}$ contains at most one point;

  2. (2)

_axiom triangles_. Any simple geodesic triangle in $(X,d)$ is contained in a
piece.

The following can be thought as a local to global principle for geodesics in a
tree graded space.

######  Definition 7.7 (Piece-wise geodesic).

Let $(X,\mathcal{P})$ be a tree graded space. Suppose that the pieces
$P_{k}\in\mathcal{P}$ are geodesics with respect to the restricted metric. Let
$\gamma=\gamma_{1}\gamma_{2}\cdots\gamma_{2m}$ be a curve in the tree-graded
space $(X,\mathcal{P})$ which is a composition of geodesics $\gamma_{k}$ in
$X$. Suppose that all geodesics $\gamma_{2k}$ with $k\in\\{1,\cdots,m-1\\}$
are non-trivial and for every $k\in\\{1,\cdots,m\\}$ the geodesic
$\gamma_{2k}$ is contained in a piece $P_{k}$ while for every
$k\in\\{0,\cdots,m-1\\}$ the geodesic $\gamma_{2k+1}$ intersects $P_{k}$ and
$P_{k+1}$ only in its respective endpoints. In addition assume that if
$\gamma_{2k+1}$ is empty then $M_{k}\subset M_{k+1}$. We call this $\gamma$ a
_piece-wise geodesic_.

The next proposition is Lemma [DS05, Lemma 2.28].

######  Proposition 7.8.

A curve $\gamma$ in a tree graded space $(X,\mathcal{P})$ is a geodesic if and
only if it is a piece-wise geodesic.

Geodesics in a tree graded space can be then thought of as concatenations of
geodesics within pieces and geodesics in the _transversal trees_ $T_{x}$. Here
$T_{x}$ denotes the set of points $y\in X$ that can be connected to $x$ by a
geodesic intersecting each piece at most once (see [DS05, Lemma 2.14] to see
why these are trees).

Our goal is to show that a dual space $X_{\mu}$ is a tree graded space where
its pieces can be isometrically identified with the duals $X_{\mu_{i}}$ of the
subcurrents of $\mu$.

However, the dual spaces have not been endowed with a canonical geodesic
structure in general (see Subsection 3.2). Because of that, we will give a
more general definition of tree-graded space that is not assumed to be
geodesic, but coincides with the usual definition of tree graded space when
the underlying space $X$ is assumed to be geodesic.

######  Definition 7.9 ((metric) tree-graded space).

A metric space $(X,d)$ is said to be a _(metric) tree-graded_ space with
respect to a collection of subspaces $\mathcal{P}$, called _pieces_ , if

  1. (1)

_axiom pieces_. Given two distinct pieces $P_{1},P_{2}\in\mathcal{P}$ the
intersection $P_{1}\cap P_{2}$ contains at most one point;

  2. (2)

_axiom transversals_. For every $x\in X$, there exists a $0$-hyperbolic metric
subspace $T_{x}\subset X$ containing $x$ and intersecting each piece at most
in one point with the following properties:

    1. (a)

for every $y\in X$, if $y\in T_{x}$, then $T_{x}=T_{y}$. Every intersection of
a piece $P$ with a $T_{x}$ is called _a point of contact_ of $P$.

    2. (b)

All points of $T_{x}$ are cut points except (possibly) the points of contact.

  3. (3)

_axiom contact chain triangle_. We say that two contact points are _adjacent_
if they are contained in the same $T_{x}$ or in the same piece. Let a
_straight contact chain_ be an ordered sequence of adjacent contact points
$(y_{1},\cdots,y_{n})$ where no two consecutive points in the sequence are
equal. We say that two points $x,y\in X$ are _connected by a straight contact
chain_ if there is a straight contact chain $(x,\cdots,y)$. An _interior
point_ of that chain is any point of the chain different from $x$ and $y$. A
_contact chain triangle_ $\Delta=xyz$ is a set of three points in $X$ and
three straight contact chains connecting the points, one per side. The axiom
contact chain triangle says that if a straight contact chain triangle is
contained in more than one piece, an interior point of one of the side chains
must be shared with another chain.

To motivate the role of the $0$-hyperbolic subspaces $T_{x}$, note that these
will be the transversal trees when $X$ is geodesic. In any case, the following
Remark gives the intuitive picture the reader should probably keep in mind.

######  Remark 7.10 (Geometric realization of the $T_{x}$).

Let $\mu=\sum_{i=1}^{n}\mu_{i}+\sum_{j=1}^{m}a_{j}s_{j}$ as in theorem 7.2,
with set of special geodesics $\\{s_{1},\dots,s_{n}\\}$. A _region_ is the
closure of a connected component of the complement of the lifts of special
geodesics in $\widetilde{X}$. A _piece_ $P=R/\sim$ is the restriction of the
quotient $\widetilde{X}/{d_{\mu}=0}$ to a region $R$. Two regions $R_{i}$ and
$R_{j}$ bounded by the lift of a special geodesic $c$ correspond in the dual
to two pieces $P_{i}$ and $P_{j}$ joined in a single point $x:=[c]$. Let
$a_{i,j}$ be the weight of $c$. Hence every arc joining the piece $P_{i}$ to
the piece $P_{j}$ will pass through $x$ and since there is an $a_{i,j}$
measure accumulated at the point $x_{i,j}$, every such arc will gain an
$a_{i,j}$ contribution to its length as soon as it passes by through the point
$x$. It follows that we may imagine, loosely speaking, that the pieces $P_{i}$
and $P_{j}$ are joined by an edge of length $a_{i,j}$, and any arc joining
$P_{i}$ to $P_{j}$ passes through such arc. Let us now make this idea precise.
Compare Figure 7.2 for what follows.

Given two adjacent pieces $P_{i}$ and $P_{j}$ as above, consider their
disjoint sum $P_{i}\amalg P_{j}$, and attach an edge $e_{i,j}$ of length
$a_{i,j}$ joining the point $x_{i}\coloneqq[c]\in P_{i}$ and
$x_{j}\coloneqq[c]\in P_{j}$. At the level of the surface, this corresponds to
pinching the special geodesics to points, obtaining a noded surface, and then
replacing the nodes by edges of length equal to the weight of the component of
the especial multi-curve.

More generally, for a metric tree graded space, one can define a projection
map $p\colon X\to T$, where $T$ is a $0$-hyperbolic space obtained by
collapsing each piece to a point. This space is $0$-hyperbolic because it
consists of the union of the subspaces $T_{x}$ equipped with the restriction
of the distance on $X$. By Section 5.1, this $0$-hyperbolic space can be
isometrically embedded into an $\mathbb{R}$-tree. This induces a
$\mathbb{R}$-tree structure on the $T_{x}$.

Figure 7.2.  The figure shows a sketch of the result of collapsing the special
multi-curve $m$ of the geodesic current of Figure 7.1. That is how one can go
about putting a geodesic structure on the transversals $T_{x}$: by pinching
$m$ to obtain nodal surfaces, and add edges between them.

Under the geodesic hypothesis, both definitions above are equivalent. See
Proposition A.1.

$\alpha$
---
$P_{1}$
---
$P_{2}$
---
$P_{3}$
---
$y$
---
$x$
---
$z$
---
$\beta$
---
$y_{1}^{1}$
---
$y_{1}^{\ell_{1}}$
---
$y_{3}^{1}$
---
$y_{3}^{\ell_{3}}$
---
$y_{2}^{1}$
---
$y_{2}^{\ell_{2}}$
---

Figure 7.3. This figure shows a schematic of a triangle composed of segments
of type $\alpha$ (green, dashed) and $\beta$ (red, solid), and the
corresponding straight chains for its sides, labelled by $y^{i}_{j}$ (red
solid dots). It is useful to visualize _axiom straight chain triangle_ , as
well as the proof of implication from Definition 7.6 to Definition 7.9 (in
Appendix A).

######  Theorem 7.11.

The current dual $X_{\mu}$ is a (metric) tree-graded space with respect to
$\mathcal{P}$.

###### Proof.

  1. (1)

_axiom pieces_. We firstly recall that $\mathrm{supp}(m)$ is a pairwise
disjoint union of geodesics in $\widetilde{X}$. The fact that two pieces
$P_{1}$ and $P_{2}$ have at most one point in common will follow from the fact
that the intersection of two (closed) complementary regions $R_{1},R_{2}$ is
either empty, or consists of a single geodesic line. If $R_{1}\cap
R_{2}=\emptyset$ the condition is trivially satisfied. Hence assume that
$R_{1}\cap R_{2}$ is non-empty. The pieces are defined as quotients of
complementary regions of special geodesics. Let
$P_{1}=R_{1}/{\\{d_{\mu}=0\\}}$ and $P_{2}=R_{2}/{\\{d_{\mu}=0\\}}$, then by
construction if $R_{1}\cap R_{2}\neq\emptyset$ it means they are adjacent,
i.e. there is a special geodesic $r\in R_{1}\cap R_{2}$ bounding both. Notice