# Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Filippo Bracci† F. Bracci: Dipartimento di Matematica, Università di Roma “Tor
Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italia.
<EMAIL_ADDRESS>,  Łukasz Kosiński  Ł. Kosiński: Institute of
Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian
University, Łojasiewicza 6, 30-348 Kraków, Poland<EMAIL_ADDRESS>and  Włodzimierz Zwonek†† W. Zwonek: Institute of Mathematics, Faculty of
Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6,
30-348 Kraków, Poland<EMAIL_ADDRESS>
###### Abstract.

Consider a holomorphic map $F:D\to G$ between two domains in
${\mathbb{C}}^{N}$. Let $\mathcal{F}$ denote a family of geodesics for the
Kobayashi distance, such that $F$ acts as an isometry on each element of
$\mathcal{F}$. This paper is dedicated to characterizing the scenarios in
which the aforementioned condition implies that $F$ is a biholomorphism.
Specifically, we establish this when $D$ is a complete hyperbolic domain, and
$\mathcal{F}$ comprises all geodesic segments originating from a specific
point. Another case is when $D$ and $G$ are $C^{2+\alpha}$-smooth bounded
pseudoconvex domains, and $\mathcal{F}$ consists of all geodesic rays
converging at a designated boundary point of $D$. Furthermore, we provide
examples to demonstrate that these assumptions are essentially optimal.

###### Key words and phrases:

Rigidity of holomorphic maps; invariant metrics; scaling methods

###### 2010 Mathematics Subject Classification:

Primary 32A19; Secondary 32H12

†Partially supported by PRIN Real and Complex Manifolds: Geometry and
Holomorphic Dynamics n. 2022AP8HZ9, by INdAM, and by the MUR Excellence
Department Project MatMod@TOV CUP:E83C23000330006

†† Partially supported by the Preludium bis grant no. 2021/43/O/ST1/02111 of
the National Science Centre, Poland

##  1\. Introduction

Understanding the conditions under which a holomorphic map between two complex
manifolds becomes a biholomorphism has been a longstanding and intriguing
question in the literature. The vastness of the literature makes it
challenging to provide a comprehensive overview. Notably, classical results
such as the Schwarz lemma, along with its generalizations by Carathéodory,
Cartan, Kaup, and Wu, offer examples of simple conditions that distinguish
biholomorphisms among holomorphic maps (see, e.g., [1]). These theorems also
have natural counterparts in the context of invariant metrics and
(pluri)complex Green and Poisson functions ([1, 5, 9, 3, 8]).

When dealing with bounded domains in $\mathbb{C}^{N}$, such results extend to
intriguing natural boundary counterparts. Julia’s lemma, rooted in versions of
the maximum principle for (sub)harmonic functions, and its various
generalizations (see, e.g., [5, 1]) offer geometric conditions. To some
extent, these conditions have been generalized in different ways to higher
dimensions (see [16]).

In our previous work [7], we delved into the question of when, given two
convex domains and a family of complex geodesics, a holomorphic map that is
isometric with respect to the Kobayashi distance for any complex geodesic in
the given family is indeed an automorphism. This question is meaningful
specifically within the class of convex domains, as it remains unclear outside
this class whether a domain possesses a sufficient number of complex
geodesics.

Nevertheless, in a complete Kobayashi hyperbolic domain, every two points can
be connected by a geodesic (i.e., a length-minimizing curve), and
biholomorphisms naturally preserve these geodesics, in the sense that, if
$\gamma:I\to M$ is a geodesic in $M$, and $F:M\to N$ is a biholomorphism, then
$F\circ\gamma:I\to N$ is a geodesic in $N$. In particular,

(1.1) $K_{M}(\gamma(t),\gamma(s))=K_{N}(F(\gamma(t)),F(\gamma(s))),\quad\hbox{
for all }t,s\in I,$

where, for a complex manifold $X$, $K_{X}$ denotes its Kobayashi distance.

Hence, it is pertinent to inquire about the extent to which (1.1)
characterizes biholomorphisms for a given family of geodesics.

The response proves to be remarkably broad, particularly when one examines the
family comprising all geodesic segments emanating from a specific point:

######  Theorem 1.1.

Let $D\subset\mathbb{C}^{N}$, $N\geq 1$, be a complete hyperbolic domain and
let $G$ be a domain in $\mathbb{C}^{N}$. Let $p\in D$ and let $\mathcal{F}$ be
the family of all geodesic segments in $D$ starting from $p$. Let $F:D\to G$
be holomorphic. Then $F$ is a biholomorphism if and only if $F\circ\gamma$ is
a geodesic of $G$ for all $\gamma\in\mathcal{F}$.

When addressing the same question in the context of the boundary, caution must
be exercised in selecting the framework, as geodesics may exhibit erratic
behavior near the boundary. In situations where geodesic rays consistently
”land” at a specific point (as observed in $C^{2}$ strongly pseudoconvex
domains), it is meaningful to investigate the persistence of the previous
result. Indeed, we can establish:

######  Theorem 1.2.

Let $D,G\subset\mathbb{C}^{N}$, $N\geq 1$, be $C^{2+\alpha}$-smooth bounded
strongly pseudoconvex domains, $\alpha>0$, and let $p\in\partial{D}$. Let
$\mathcal{F}$ be the family of all geodesic rays $\gamma:[0,+\infty)\to D$
such that $\lim_{t\to+\infty}\gamma(t)=p$. Let $F:D\to G$ be holomorphic. Then
$F$ is a biholomorphism if and only if $F\circ\gamma$ is a geodesic of $G$ for
all $\gamma\in\mathcal{F}$.

The key characteristic of the family $\mathcal{F}$ of geodesics examined in
the preceding theorems is that, for any $z\in D$, there exists a geodesic in
the family containing it in its image. A family of geodesics possessing this
property is termed complete.

Hence, a natural question arises: Do the earlier results apply universally to
every complete family of geodesics? The response is in the negative, and our
assumptions are found to be fundamentally optimal. We will demonstrate
examples of such scenarios in Section 4.

##  2\. Geodesics in strongly pseudoconvex domains

Let $M$ be a complex manifold and let $K_{M}$ denote its Kobayashi
pseudodistance and let $k_{M}$ be its infinitesimal Kobayashi pseudometric.
Let $I\subset\mathbb{R}$ be an interval. An absolutely continuous curve
$\gamma:I\to M$ is a geodesic for $k_{M}$ if for every $s,t\in I$, $s\leq t$,

$K_{M}(\gamma(s),\gamma(t))=\ell_{M}(\gamma;[s,t]):=\int_{s}^{t}k_{M}(\gamma(u);\gamma^{\prime}(u))du.$

Any geodesic $\gamma$ can be reparametrized in (hyperbolic) arc-length, so
that $K_{M}(\gamma(s),\gamma(t))=|t-s|$. If $\gamma$ is parametrized by arc-
length and its interval of definition $I$ is a closed interval, then $\gamma$
is called a geodesic segment. If $I=\mathbb{R}^{+}$, then $\gamma$ is a
geodesic ray, while, if $I=\mathbb{R}$, then $\gamma$ is a geodesic line. We
just say that $\gamma$ is a geodesic when there is no need to specify its
domain of definition. If $M$ is complete hyperbolic, by Hopf-Rinow’s Theorem,
given any two points $p,q\in M$ there always exists a geodesic segment
$\gamma:[a,b]\to M$ which joins $p$ and $q$, namely, such that $\gamma(a)=p$
and $\gamma(b)=q$.

If $D\subset\mathbb{C}^{N}$, we say that a geodesic ray $\gamma:[0,+\infty)\to
D$ lands at some point $p\in\partial D$ provided
$\lim_{t\to+\infty}\gamma(t)=p$.

We collect below the properties we need for geodesics in strongly pseudoconvex
domains. These are well known, and we only sketch the proof.

######  Proposition 2.1.

Let $D\subset\mathbb{C}^{N}$, $N\geq 2$, be a bounded strongly pseudoconvex
domain with $C^{2}$ boundary. Then:

  1. (1)

for every two points $p,q\in\overline{D}$ exists a geodesic segment, ray or
line $\gamma:I\to D$ such that $p,q\in\overline{\gamma(I)}$.

  2. (2)

Let $p\in\overline{D}$ and $\\{z_{n}\\}\subset D$ be a sequence converging to
$q\in\partial D\setminus\\{p\\}$. If $\\{\gamma_{n}\\}$ is a sequence of
geodesic segments/rays joining $p$ with $z_{n}$, then, up to affine
reparametrizations, $\\{\gamma_{n}\\}$ converges uniformly to a geodesic
ray/line joining $p$ with $q$.

  3. (3)

if $\gamma:[0,+\infty)\to D$ is a geodesic ray, then there exists
$p\in\partial D$ such that $\lim_{t\to+\infty}\gamma(t)=p$.

  4. (4)

If $\gamma,\eta$ are two geodesic rays in $D$ converging to $p\in\partial D$
then there exists $C>0$ such that
$\lim_{t\to+\infty}K_{D}(\gamma(t),\eta(t))\leq C$.

###### Proof.

By [2], $D$ is Gromov hyperbolic with respect to the Kobayashi distance and
the identity map extends as a homeomorphism from the Gromov closure of $D$ and
$\overline{D}$, that is, $D$ is a so-called Gromov model domain (see [6]). In
particular it is “visible” hence (1), (2) and (3) follows (for the uniform
convergence in (2) see [4, Lemma 3.1]).

In order to prove (4), let $\gamma,\gamma_{0}$ be two geodesic rays converging
to $p\in\partial D$. Since $D$ is a Gromov model domain, it has no “geodesic
loops” and hence by [4, Prop. 4.4] there exists $C>0$ such that

$K_{D}(\gamma(t),\gamma_{0}(t))\leq C,\quad\forall t\geq 0,$

and we are done. ∎

###  2.1. Application of the scaling method

Let $\Omega$ be a bounded strongly pseudoconvex domain with
$C^{2+\alpha}$-smooth boundary, $\alpha\in(0,1]$, and let
$p\in\partial\Omega$.

It follows from Lempert [15] and Fridman-Ma [11] (see also [14, Section 2] for
details) that there exists a mapping $T$ biholomorphic on $\overline{\Omega}$
sending $p$ to $e_{1}=(1,0,\ldots,0)$ such that $\Omega_{0}:=T(\Omega)$ is
contained in the ball of radius $1+\delta_{0}$ and centered at
$(-\delta_{0},0)$, where $\delta_{0}\geq 0$, and $\partial\Omega_{0}$ has a
defining function near $e_{1}$ of the form

(2.1) $\rho(z)=-1+|z|^{2}+o(|z-e_{1}|^{2+\alpha}).$

Writing $z=(z_{1},z^{\prime\prime})\in\mathbb{C}\times\mathbb{C}^{N-1}$,
define

(2.2)
$A_{t}(z)=\left(\frac{z_{1}+t}{1+tz_{1}},\sqrt{1-t^{2}}\frac{z^{\prime}}{1+tz_{1}}\right),$

and

(2.3) $\Omega_{t}:=A_{t}^{-1}(\Omega_{0}),\quad t\in(0,1).$

Then,

######  Lemma 2.2.

  1. (1)

for every $\beta>-1$, $\Omega_{t}\cap\\{\operatorname{Re}z_{1}>\beta\\}$
converges as $t\to 1$ to
$\mathbb{B}^{N}\cap\\{\operatorname{Re}z_{1}>\beta\\}$ in
$\mathcal{C}^{2+\alpha}$-topology,

  2. (2)

$\Omega_{t}$ converges to $\mathbb{B}^{N}$ in the Hausdorff topology as $t\to
1$.

In the sequel, we will need the following results.

######  Lemma 2.3.

$K_{\Omega_{t_{k}}}(\cdot,\cdot)$ converges uniformly on compacta to
$K_{\mathbb{B}^{N}}(\cdot,\cdot)$.

###### Proof.

Fix $r_{0}\in(0,1)$. For $r>0$ let $\mathbb{B}(0,r)$ be the ball of center $0$
and radius $r$. Since, clearly, $K_{\mathbb{B}(0,r)}(\cdot,\cdot)$ converges
uniformly on $\mathbb{B}(0,r_{0})$ to $K_{\mathbb{B}^{N}}(\cdot,\cdot)$ for
$r\to 1^{-}$, for every $\delta>0$ we can find $r_{1}\in(r_{0},1)$ such that
$K_{\mathbb{B}(0,r_{1})}(z,w)\leq K_{\mathbb{B}^{N}}(z,w)+\delta$ for all
$z,w\in\mathbb{B}(0,r_{0})$. But, for $k$ large enough,
$\mathbb{B}(0,r_{1})\subset\Omega_{t_{k}}$. Thus, for $k$ large enough, and
for all $z,w\in\mathbb{B}(0,r_{0})$,

$K_{\Omega_{t_{k}}}(z,w)\leq K_{\mathbb{B}(0,r_{1})}(z,w)\leq
K_{\mathbb{B}^{N}}(z,w)+\delta.$

Similarly, we show that

$k_{\mathbb{B}^{N}}(z,w)-\delta\leq K_{\mathbb{B}(0,1/r_{2})}(z,w)\leq
K_{\Omega_{t_{k}}}(z,w)$

for some $r_{2}\in(r_{0},1)$, and we are done.

∎

######  Proposition 2.4.

Let $\mathcal{F}$ be the family of all geodesic rays
$\gamma:[0,+\infty)\to\Omega_{0}$ such that
$\lim_{t\to+\infty}\gamma(t)=e_{1}$. Let $\\{t_{k}\\}\subset(0,1)$ be a
strictly increasing sequence converging to $1$. Then, for every
$w_{0}\in\mathbb{B}^{N}$ there exists a sequence of geodesics
$\\{\gamma_{k}\\}\subset\mathcal{F}$ such that
$\\{A_{t_{k}}^{-1}(\gamma_{k})\\}$ converges uniformly on compacta to a
geodesic ray $\eta:[0,+\infty)\to\mathbb{B}^{N}$ such that $\eta(0)=w_{0}$ and
$\lim_{t\to+\infty}\eta(t)=e_{1}$.

###### Proof.

Fix $w_{0}\in\mathbb{B}^{N}$. Since $A_{t_{k}}^{-1}$ is an automorphism of
$\mathbb{B}^{N}$, it follows that there exists a sequence
$\\{z_{k}\\}_{k>k_{0}}$, converging to $e_{1}$ such that
$A^{-1}_{t_{k}}(z_{k})=w_{0}$ for all $k>k_{0}$. Let
$\gamma_{k}\in\mathcal{F}$ be such that $\gamma_{k}:[0,+\infty)\to\Omega_{0}$
satisfies $\gamma_{k}(0)=z_{k}$ and $\lim_{t\to+\infty}\gamma_{k}(t)=e_{1}$.

Then $\eta_{k}:=A^{-1}_{t_{k}}\circ\gamma_{k}:[0,+\infty)\to\Omega_{t_{k}}$ is
a geodesic in $\Omega_{k}$ such that $\eta_{k}(0)=w_{0}$ and
$\lim_{t\to+\infty}\eta_{k}(t)=e_{1}$.

Take a sequence $\rho_{k}>1,$ decreasing to $1$ such that
$\Omega_{t_{k}}\subset\mathbb{B}(0,\rho_{k})$. Now, by Lemma 2.3, for every
$T>0$ and for all $k>k_{0}$ we have

$K_{\mathbb{B}(0,\rho_{k})}(\eta_{k}(T),w_{0})\leq
K_{\Omega_{t_{k}}}(\eta_{k}(T),w_{0})=T.$

Therefore, $\\{\eta_{k}\\}$ is uniformly bounded on compacta. Also, for every
$s,t\in[0,+\infty)$,

$K_{\mathbb{B}(0,\rho_{k})}(\eta_{k}(t),\eta_{k}(s))\leq
K_{\Omega_{t_{k}}}(\eta_{k}(t),\eta_{k}(s))=|t-s|,$

therefore, $\\{\eta_{k}\\}$ is equicontinuous on compacta. By Arzelà-Ascoli
theorem, we can thus extract a subsequence uniformly converging on compacta to
a continuous curve $\eta:[0,+\infty)\to\mathbb{B}^{N}$ such that
$\eta(0)=w_{0}$. Since $K_{\Omega_{t_{k}}}(\cdot,\cdot)$ converges uniformly
on compacta to $K_{\mathbb{B}^{N}}(\cdot,\cdot)$ by Lemma 2.3, it follows that
$\eta$ is a geodesic in $\mathbb{B}^{N}$. Moreover, by (1) of Lemma 2.2, it
follows that $\lim_{t\to+\infty}\eta(t)=e_{1}$.

Repeating the previous argument for every subsequence of $\\{\eta_{k}\\}$, by
the uniqueness of geodesics in $\mathbb{B}^{N}$, we have that $\\{\eta_{k}\\}$
converges uniformly on compacta to $\eta$, and we are done. ∎

We also need a sort of converse to the previous proposition. For this result
we need the following “visibility lemma” slightly extending [14, Lemma 10]:

######  Lemma 2.5.

Let $p,q\in\partial\mathbb{B}^{N}$, $p\neq q$. Then for every open
neighbourhood $U$ and $V$ of, respectively, $p$ and $q$, such that $U\cap
V=\emptyset$, there exists a compact set $L\subset\mathbb{B}^{N}$ such that
for every $t\in[0,1)$ and every geodesic $\gamma$ of $\Omega_{t}$ which joins
a point in $U$ with a point in $V$, the image of $\gamma$ intersects $L$.

###### Proof.

In case $p,q\neq-e_{1}$, the result follows immediately from [14, Lemma 5].

In case $p=-e_{1}$, we argue by contradiction. If there exists a sequence
$\\{t_{k}\\}$ converging to $1$ and geodesics $\gamma_{k}$ of $\Omega_{t_{k}}$
joining a point in $U$ to a point of $V$ such that the images of $\gamma_{k}$
escape all compact subsets of $\mathbb{B}^{N}$, then we can find a sequence
$s_{k}>0$ such that $\\{\gamma_{k}(s_{k})\\}$ converges to some point
$q^{\prime}\in\partial\mathbb{B}^{N}\setminus\\{-e_{1},q\\}$. But then, we can
apply [14, Lemma 10] to $q,q^{\prime}$ and reach a contradiction. ∎

As a matter of notation, we let $\pi_{1}:\mathbb{C}^{N}\to\mathbb{C}$ be the
projection on the first component.

######  Proposition 2.6.

Let $\mathcal{F}$ be the family of all geodesic rays
$\gamma:[0,+\infty)\to\Omega_{0}$ such that
$\lim_{t\to+\infty}\gamma(t)=e_{1}$. Let $\\{t_{k}\\}\subset(0,1)$ be a
strictly increasing sequence converging to $1$ and let
$\\{\gamma_{k}\\}\subset\mathcal{F}$ be such that there is $0<\epsilon<1$ so
that for every $k$

$0<{\sf Re}\,\pi_{1}(A_{t_{k}}^{-1}(\gamma_{k}(0)))<1-\epsilon,$

and $\\{A_{t_{k}}^{-1}\gamma_{k}(0)\\}$ is compactly divergent in
$\mathbb{B}^{N}$. Then, there exists a subsequence of
$\\{A_{t_{k}}^{-1}\circ\gamma_{k}\\}$ such that, after a suitable affine
reparametrization, converges uniformly on compacta to a geodesic line $\eta$
in $\mathbb{B}^{N}$ joining $e_{1}$ to a point $\xi\in\partial\mathbb{B}^{N}$
with $0\leq{\sf Re}\,\pi_{1}(\xi)\leq 1-\epsilon$.

###### Proof.

Since $\\{\pi_{1}\circ A_{t_{k}}^{-1}\\}$ converges uniformly on compacta of
$\overline{\mathbb{B}^{N}}\setminus\\{e_{1}\\}$ to the constant map
$\zeta\mapsto-1$, it follows that $\lim_{t\to+\infty}\gamma_{k}(0)=e_{1}$.

Up to subsequences, we can assume that $\\{A_{t_{k}}^{-1}(\gamma_{k}(0))\\}$
converges to a point $\xi\in\partial\mathbb{B}^{N}$ such that $0\leq{\sf
Re}\,\pi_{1}(\xi)\leq 1-\epsilon$, in particular, $\xi\neq e_{1}$. By Lemma
2.5, there exists a compact subset $K\subset\mathbb{B}^{N}$ such that
$A_{t_{k}}^{-1}(\gamma_{k}([0,+\infty))\cap K\neq\emptyset$ for all $k$
sufficiently large.

Let $s_{k}>0$ be such that $A_{t_{k}}^{-1}(\gamma_{k}(s_{k}))\in K$ for $k$
sufficiently large. Let $\sigma_{k}:[-s_{k},+\infty)\to\Omega_{t_{k}}$ be the
geodesic ray defined by $\sigma_{k}(t):=A_{t_{k}}^{-1}(\gamma_{k}(t-s_{k}))$.
Note that $\\{\sigma_{k}(-s_{k})\\}$ converges to $\xi$ and that
$\sigma_{k}(0)\in K$ for $k$ sufficiently large. Since for any $\rho>1$

$K_{\mathbb{B}(\rho)}(\sigma_{k}(0),\sigma_{k}(-s_{k}))\leq
K_{\Omega_{t_{k}}}(\sigma_{k}(0),\sigma_{k}(-s_{k}))=s_{k},$

providing that $k$ is big enough, it follows that
$\lim_{k\to\infty}s_{k}=+\infty$.

Arguing as in the proof of Proposition 2.4 it is easy to see that, up to
subsequences, $\\{\sigma_{k}\\}$ converges uniformly on compacta to a geodesic
line $\eta:(-\infty,+\infty)\to\mathbb{B}^{N}$ such that
$\lim_{t\to+\infty}\eta(t)=e_{1}$.

We are left to show that $\lim_{t\to-\infty}\eta(t)=\xi$. Suppose this is not
the case and that
$\lim_{t\to-\infty}\eta(t)=q\in\partial\mathbb{B}^{N}\setminus\\{\xi\\}$. Let
$V$ and $U$ be open neighborhoods of $\xi$ and $q$ respectively such that
$V\cap U=\emptyset$.

Fix $M\in\mathbb{N}$, such that $\eta(-M)\in U$. Since $\\{\sigma_{k}\\}$
converges uniformly on compacta to $\eta$, there exists $k(M)$ so that, for
$k\geq k(M)$ we have $-s_{k}<-M$, $\sigma_{k}(-M)\in U$ and
$\sigma_{k}(-s_{k})\in V$.

Therefore, for all $k\geq k(M)$, $\sigma_{k}:[-s_{k},-M]\to\Omega_{t_{k}}$ is
a geodesic which joins a point of $U$ to a point of $V$. By Lemma 2.5, there
exists a compact subset $L\subset\mathbb{B}^{N}$ and $-s_{M,k}\in(-s_{k},-M)$
such that $\sigma_{k}(-s_{M,k})\in L$ for all $k\geq k(M)$. Hence,

$M\leq
K_{\Omega_{t_{k}}}(\sigma_{k}(0),\sigma_{k}(-M))+K_{\Omega_{t_{k}}}(\sigma_{k}(-M),\sigma_{k}(-s_{M,k}))=K_{\Omega_{t_{k}}}(\sigma_{k}(-s_{M,k}),\sigma_{k}(0))\leq
C,$

for some constant $C>0$ independent of $M$ and $k$, which gives a
contradiction for $M\to\infty$. ∎

A similar argument, that we leave to the reader (see also [14, Lemma 6])
gives:

######  Proposition 2.7.

Let $\mathcal{F}$ be the family of all geodesic rays
$\gamma:[0,+\infty)\to\Omega_{0}$ such that
$\lim_{t\to+\infty}\gamma(t)=e_{1}$. Let $\\{t_{k}\\}\subset(0,1)$ be a
strictly increasing sequence converging to $1$ and let
$\\{\gamma_{k}\\}\subset\mathcal{F}$ such that $\\{\gamma_{k}(0)\\}$ is
relatively compact in $\mathbb{B}^{N}$. Then, there exist a subsequence
$\\{k_{m}\\}$ and $s_{m}\geq 0$ with $\lim_{m\to\infty}s_{m}=+\infty$ such
that $\\{A_{t_{k_{m}}}^{-1}(\gamma_{k_{m}}(t+s_{m}))\\}$ converges uniformly
on compacta to a geodesic line $\eta:\mathbb{R}\to\mathbb{B}^{N}$ whose image
is $(-1,1)e_{1}$.

Conversely, if $s_{k}>0$ is chosen so that
$\\{A_{t_{k}}^{-1}(\gamma_{k}(s_{k}))\\}$ is relatively compact in
$\mathbb{B}^{N}$, then, up to subsequence,
$\\{A_{t_{k}}^{-1}(\gamma_{k}(s_{k}+t))\\}$ converges uniformly on compacta to
a geodesic line $\eta:\mathbb{R}\to\mathbb{B}^{N}$ whose image is
$(-1,1)e_{1}$.

##  3\. The proof of Theorem 1.1 and Theorem 1.2

###### Proof of Theorem 1.1.

We first show that $F$ is proper.

For each $\gamma\in\mathcal{F}$ we can assume that $\gamma(0)=p$. We need to
show that if $\\{z_{k}\\}$ is a sequence in $D$ converging to some
$q\in\partial D$ then $\\{F(z_{k})\\}$ accumulates to $\partial G$.

To this aim, let $\gamma_{k}\in\mathcal{F}$ be such that
$\gamma_{k}(T_{k})=z_{k}$ for some $T_{k}\geq 0$. Note that
$\lim_{k\to+\infty}T_{k}=+\infty$. Indeed,

$T_{k}=K_{D}(\gamma_{k}(0),\gamma_{k}(T_{k}))=K_{D}(p,z_{k}),$

and $\lim_{k\to\infty}K_{D}(p,z_{k})=+\infty$ since $D$ is complete
hyperbolic. By hypothesis,

$K_{G}(F(p),F(z_{k}))=K_{G}(F(\gamma_{k}(0)),F(\gamma_{k}(T_{k})))=K_{D}(\gamma_{k}(0),\gamma_{k}(T_{k}))=T_{k},$

hence $\lim_{k\to\infty}K_{G}(F(p),F(z_{k}))=+\infty$, which, since $G$ is a
domain (hence connected), implies that $\\{F(z_{k})\\}$ accumulates on
$\partial G$.

Now we show that $F$ is actually a biholomorphism onto $G$.

If there were $q\in D$, $q\neq p$ such that $F(p)=F(q)$ then a geodesic
$\gamma\in\mathcal{F}$ joining $p=\gamma(0)$ with $q=\gamma(t)$, for some
$t>0$, could not be mapped onto a geodesic as $F\circ\gamma$ were not
injective since $F(\gamma(0))=F(\gamma(t))$. Thus $F^{-1}(F(p))=\\{p\\}$.

We claim that the differential $dF_{p}$ is non degenerate. Indeed, if $dF_{p}$
is degenerate there is a vector $0\neq X\in\mathbb{C}^{n}$ with $dF_{p}(X)=0$.
Expanding, we have

(3.1) $F(p+tX)=F(p)+O(t^{2}).$

But our assumption on preserving geodesics of $\mathcal{F}$ implies that

(3.2) $K_{D}(p,p+tX)=K_{G}(F(p),F(p+tX)),$

for $t>0$ small so that $p+tX\in D$.

Now, since $D$ is complete hyperbolic and then the Euclidean topology of $D$
is equivalent to the topology induced by $K_{D}$, there exists $c>0$ such that
$K_{D}(p,p+tX)\geq c\|p-(p+tX)\|$. Also, by definition of Kobayashi pseudo-
distance, there exists $C>0$ such that $K_{G}(F(p),w)\leq C\|F(p)-w\|$ for $w$
close to $F(p)$. Therefore, dividing (3.2) by $t>0$, and using (3.1), we have

$c\|X\|\leq\frac{C}{t}\|F(p)-F(p+tX)\|=O(t),$

which, for $t\to 0^{+}$ gives a contradiction, and hence $dF_{p}$ is non
degenerate.

Thus, the argument principle (see, e.g., [10, Prop. 2 p.48]) implies that, for
every open set $W\subset\subset D$ such that $\overline{W}$ is diffeomorphic
to a ball and such that $\partial W$ has the same orientation of the
$2N-1$-dimensional sphere $\mathbb{S}^{2N-1}$, the map
$\frac{F(z)-F(p)}{\|F(z)-F(p)\|}:\partial W\to\mathbb{S}^{2N-1}$ has degree
$1$.

Now, let $w_{0}\in G$. Let $\eta:[0,1]\to G$ be a continuous injective curve
such that $\eta(0)=F(p)$ and $\eta(1)=w_{0}$. Since $F$ is proper,
$F^{-1}(\eta([0,1]))$ is contained in a compact set $K\subset\subset D$. Let
$W$ be an open set as before such that $K\subset W$. By construction,
$\|F(z)-\eta(t)\|>0$ for all $z\in\partial W$ and $t\in[0,1]$. Hence,
$S_{t}:=\frac{F(z)-\eta(t)}{\|F(z)-\eta(t)\|}:\partial W\to\mathbb{S}^{2N-1}$
is well defined for $t\in[0,1]$ and homotopic to $S_{0}$. Therefore, the
degree of $S_{t}$ is $1$ for all $t\in[0,1]$. Hence, again by the argument
principle, $F^{-1}(w_{0})$ contains exactly only one point. Thus, $F$ is a
biholomorphism onto $G$. ∎

In order to prove Theorem 1.2 we need a lemma:

######  Lemma 3.1.

Let $\Phi:\mathbb{B}^{N}\to\mathbb{B}^{N}$. Let $\mathcal{G}$ be the family of
all geodesic rays landing at $e_{1}$. If $\Phi\circ\gamma$ is a geodesic of
$\mathbb{B}^{N}$ for all $\gamma\in\mathcal{G}$ then $\Phi$ is an automorphism
of $\mathbb{B}^{N}$.

###### Proof.

Let $\varphi:\mathbb{D}\to\mathbb{B}^{N}$ be a complex geodesic containing
$e_{1}$ in its closure—that is, $\varphi$ is an injective holomorphic map such
that its image is the intersection of a complex affine line with
$\mathbb{B}^{N}$ and the closure of its image contains $e_{1}$. Hence,
$\varphi:\mathbb{D}\to\mathbb{B}^{N}$ is an affine map. It is well known that
all geodesics rays of $\mathbb{B}^{N}$ landing at $e_{1}$ are contained in one
such a complex geodesic. We are going to show that $\Phi\circ\varphi$ is a
complex geodesic of $\mathbb{B}^{N}$ for every $\varphi\in\mathcal{G}$. Then
the result follows from [7, Theorem 1.2].

To this aim, let $\varphi$ be a complex geodesic in $\mathbb{B}^{N}$ whose
closure contains $e_{1}$. Up to post-composing $\varphi$ with an automorphism
of $\mathbb{B}^{N}$ fixing $e_{1}$, we can assume that $\varphi(\zeta)=\zeta
e_{1}$. Since $[0,1)\ni r\mapsto re_{1}$ is a geodesic of $\mathbb{B}^{N}$, it
follows that $[0,1)\ni r\mapsto F(re_{1})$ is a geodesic in $\mathbb{B}^{N}$.
Up to composing with an automorphism of $\mathbb{B}^{N}$, we can assume that
$F([0,1)e_{1})=[0,1)e_{1}$. Let $F_{1}$ be the first component of $F$. Thus
$F_{1}:\mathbb{D}\to\mathbb{D}$ is a holomorphic map which maps a geodesic
into a geodesic of $\mathbb{D}$. By the Schwarz-Pick lemma, $F_{1}(\zeta
e_{1})=\zeta$ for all $\zeta\in\mathbb{D}$. Since $|F_{1}(\zeta
e_{1})|^{2}+|F^{\prime}(\zeta e_{1})|^{2}<1$ for all $\zeta\in\mathbb{D}$
(here $F^{\prime}=(F_{2},\ldots,F_{N})$), we have that
$\lim_{\mathbb{D}\ni\zeta\to e^{i\theta}}|F^{\prime}(\zeta e_{1})|=0$ for all
$\theta\in\mathbb{R}$. By the maximum principle, $F^{\prime}(\zeta
e_{1})\equiv 0$. Therefore, $F(\zeta e_{1})=\zeta e_{1}$ for all
$\zeta\in\mathbb{D}$ and we are done. ∎

###### Proof of Theorem 1.2.

We divide the proof into some Steps:

Step 1. There exists $p^{\prime}\in\partial{G}$ such that $\\{F(\gamma(t))\\}$
converges to $p^{\prime}$ for all $\gamma\in\mathcal{F}$.

If $\gamma\in\mathcal{F}$, by Proposition 2.1.(3) there exists
$p^{\prime}\in\partial G$ such that $\gamma(t)$ converges to $p^{\prime}$ as
$t\to+\infty$. If $\eta\in\mathcal{F}$ then by Proposition 2.1.(4) there
exists $C>0$ such that $\lim_{t\to+\infty}K_{D}(\gamma(t),\eta(t))\leq C$.
Hence, $\lim_{t\to+\infty}K_{G}(F(\gamma(t)),F(\eta(t)))\leq C$, which, by the
standard estimates on the Kobayashi distance in strongly pseudoconvex domains
(see also D’Addezio’s Lemma [6, Lemma A.2]), implies that $F(\eta(t))$
converges to $p^{\prime}$ as $t\to+\infty$.

Step 2. $F$ is proper (and hence surjective).

Arguing as in the proof of Theorem 1.1, we see that if $\\{z_{k}\\}\subset D$
converges to some $\xi\in\partial D\setminus\\{p\\}$, then $\\{F(z_{k})\\}$
accumulates at $\partial G$.

In case $\\{z_{k}\\}\subset D$ converges to $p$, let
$\gamma_{0}:[0,+\infty)\to D$ be a geodesic such that
$\lim_{t\to+\infty}\gamma(t)=p$.

If there exists $C>0$ such that for every $k$ there exists $s_{k}>0$ with
$K_{D}(\gamma(s_{k}),z_{k})<C$ (in particular, $\\{s_{k}\\}$ converges to
$+\infty$), then

$K_{G}(F(\gamma(s_{k})),F(z_{k}))\leq K_{D}(\gamma(s_{k}),z_{k})<C,$

and, as before, it follows that $\\{F(z_{k})\\}$ converges to $p^{\prime}$.

Therefore, we can assume that

(3.3) $\lim_{k\to+\infty}K_{D}(\gamma_{0}([0,+\infty)),z_{k})=+\infty.$

We argue by contradiction and assume that $\\{F(z_{k})\\}$ is relatively
compact in $G$.

Let $T_{D}:D\to D_{0}$ and $T_{G}:G\to G_{0}$ be the maps defined in Section
2.1, mapping, respectively, $p$ to $e_{1}$ and $p^{\prime}$ to $e_{1}$, such
that $G_{0},D_{0}$ are contained in some balls and the defining functions of
$D_{0},G_{0}$ at $e_{1}$ are as in (2.1). Let $F_{0}:=T_{G}\circ F\circ
T^{-1}_{D}$. Since $T_{D}$ extends biholomorphically through the boundary of
$D$, the family $T(\mathcal{F})$ coincides with the family
$\mathcal{F}^{\prime}$ of geodesic rays $\gamma:[0,+\infty)\to D_{0}$ such
that $\lim_{t\to+\infty}\gamma(t)=e_{1}$, and $F_{0}\circ\gamma$ is a geodesic
in $G_{0}$ for all $\gamma\in\mathcal{F}^{\prime}$.

We let $\tilde{z}_{k}:=T_{D}(z_{k})$ and
$\tilde{\gamma}_{0}=T_{D}\circ\gamma_{0}$. For $t\in(-1,1)$ let $A_{t}$ be as
in (2.2).

Since $\\{\tilde{z}_{k}\\}$ converges to $e_{1}$, we can find a strictly
increasing sequence $\\{t_{k}\\}$ converging to $1$ such that for all $k$,

(3.4) $0<{\sf Re}\,\pi_{1}(A_{t_{k}}^{-1}(\tilde{z}_{k}))<\frac{1}{2},$

where $\pi_{1}$ is the projection on the first component.

Let $D_{k}:=A^{-1}_{t_{k}}(D_{0})$. Also, let
$\tau_{k}:=A_{t_{k}}^{-1}\circ\tilde{\gamma}_{0}$. Finally, if
$\eta_{k}\in\mathcal{F}^{\prime}$ is such that $\eta_{k}(0)=\tilde{z}_{k}$, we
let $\sigma_{k}:=A_{t_{k}}^{-1}\circ\eta_{k}$.

Note that $\tau_{k}$ and $\sigma_{k}$ are geodesic rays in $D_{k}$ landing at
$e_{1}$.

By Proposition 2.7, up to subsequences and a reparametrization $t\mapsto
t+s_{k}$, for some $s_{k}>0$ such that $\lim_{k\to\infty}s_{k}=+\infty$,
$\\{\tau_{k}(t+s_{k})\\}$ converges to a geodesic line
$\tau:(-\infty,+\infty)\to\mathbb{B}^{N}$ whose image is $(-1,1)e_{1}$.

Now, we claim that the sequence formed by
$\sigma_{k}(0)=A_{t_{k}}^{-1}(\eta_{k}(0))=A_{t_{k}}^{-1}(T_{D}(z_{k}))$ is
compactly divergent in $\mathbb{B}^{N}$. Indeed, assume that there is a
subsequence, still denoted by $\\{\sigma_{k}(0)\\}$, which converges to some
$\xi_{0}\in\mathbb{B}^{N}$. Hence, there is $C>0$ such that
$K_{\mathbb{B}^{N}}(\tau(0),\xi_{0})<C$. Therefore, for $k$ sufficiently
large,

$K_{D_{k}}(\tau_{k}(s_{k}),\sigma_{k}(0))\leq 2C.$

Hence,

$K_{D}(\gamma_{0}(s_{k}),z_{k})=K_{A^{-1}_{t_{k}}(T_{D}(D))}(A_{t_{k}}^{-1}(T_{D}(\gamma_{0}(s_{k}))),A_{t_{k}}^{-1}(T_{D}(z_{k})))=K_{D_{k}}(\tau_{k}(s_{k}),\sigma_{k}(0))\leq
2C,$

contradicting (3.3).

Thus, $\\{\sigma_{k}(0)\\}$ is compactly divergent in $\mathbb{B}^{N}$ and by
(3.4) we can apply Proposition 2.6, and, up to subsequences and
reparametrization, we see that $\\{\sigma_{k}\\}$ converges to a geodesic line
$\sigma:(-\infty,+\infty)\to\mathbb{B}^{N}$ such that
$\lim_{t\to+\infty}\sigma(t)=e_{1}$ and $\lim_{t\to-\infty}\sigma(t)=q$ for
some $q\neq-e_{1}$.

In particular, by the uniqueness of geodesic lines in $\mathbb{B}^{N}$ landing
at $e_{1}$, we have

(3.5) $\tau((-\infty,+\infty))\cap\sigma((-\infty,+\infty))=\emptyset.$

By (3.3),

$\lim_{k\to+\infty}K_{D_{k}}(\tau_{k}([0,+\infty)),\sigma_{k}(0))=+\infty.$

Now, since $\\{s_{k}\\}$ converges to $+\infty$, it follows that
$F_{0}(\tilde{\gamma}_{0}(s_{k}))$ converges to $e_{1}$. Hence, for every $k$
we can find $a_{k}\in[0,1)$ such that
$\\{A_{a_{k}}^{-1}(F_{0}(\tilde{\gamma}_{0}(s_{k})))\\}$ is contained in a
compact set $L\subset\mathbb{B}^{N}$.

Let $G_{k}:=A^{-1}_{a_{k}}(G_{0})$ and

$F_{k}:=A^{-1}_{a_{k}}\circ F_{0}\circ A_{t_{k}}:D_{k}\to G_{k}.$

By construction, $\\{\tau_{k}(s_{k})\\}$ converges to $\tau(0)$ and
$F_{k}(\tau_{k}(s_{k}))=A_{a_{k}}^{-1}(F_{0}(\tilde{\gamma}_{0}(s_{k}))\in L$.

Hence, again up to subsequences, we can assume that $\\{F_{k}\\}$ converges
uniformly on compacta to a holomorphic self-map
$\Phi:\mathbb{B}^{N}\to\mathbb{B}^{N}$ with the property that $\Phi$ maps
$\tau(0)$ in $L$.

We claim that, for every geodesic ray $\eta:[0,+\infty)\to\mathbb{B}^{N}$ such
that $\lim_{t\to+\infty}\eta(t)=e_{1}$ then $\Phi\circ\eta$ is a geodesic in
$\mathbb{B}^{N}$.

Indeed, fix such a $\eta$ and let $w_{0}=\eta(0)$. By Proposition 2.4, there
exists a sequence $\\{\gamma_{k}\\}\subset\mathcal{F}$ such that
$\\{A_{t_{k}}^{-1}\circ\gamma_{k}\\}$ converges uniformly on compacta to a
geodesic ray starting at $w_{0}$ and landing at $e_{1}$. By the uniqueness of
geodesics in $\mathbb{B}^{N}$, such a geodesic ray is $\eta$ itself.

Therefore, by Lemma 2.3, for every fixed $t,s\in[0,+\infty)$, we have

$\begin{split}K_{\mathbb{B}^{N}}(\Phi(\eta(t)),\Phi(\eta(s)))&=\lim_{k\to\infty}K_{G_{k}}(F_{k}(A_{t_{k}}^{-1}(\gamma_{k}(t))),F_{k}(A_{t_{k}}^{-1}(\gamma_{k}(s))))\\\
&=\lim_{k\to\infty}K_{A^{-1}_{a_{k}}(G_{0})}(A^{-1}_{a_{k}}(F_{0}(\gamma_{k}(t))),A^{-1}_{a_{k}}(F_{0}(\gamma_{k}(s))))\\\
&=\lim_{k\to\infty}K_{G_{0}}(F_{0}(\gamma_{k}(t)),F_{0}(\gamma_{k}(s)))=\lim_{k\to\infty}|t-s|=|t-s|.\end{split}$

It follows by Lemma 3.1 that $\Phi$ is an automorphism of $\mathbb{B}^{N}$. In
particular, by (3.5),

(3.6)
$\Phi(\tau((-\infty,+\infty)))\cap\Phi(\sigma((-\infty,+\infty)))=\emptyset.$

Now, $\\{F_{k}(\tau_{k}(t+s_{k}))\\}$ converges uniformly on compacta to
$\Phi(\tau((-\infty,+\infty)))$. Since

$F_{k}\circ\tau_{k}(t+s_{k})=(A^{-1}_{a_{k}}\circ F_{0}\circ
A_{t_{k}})\circ(A_{t_{k}}^{-1}\circ\tilde{\gamma}_{0})(t+s_{k})=A^{-1}_{a_{k}}\circ
F_{0}\circ\tilde{\gamma}_{0}(t+s_{k}),$

and $t\mapsto F_{0}\circ\tilde{\gamma}_{0}(t+s_{k})$ is a geodesic ray in
$G_{0}$ landing at $e_{1}$, by Proposition 2.7 we have

(3.7) $\Phi(\tau((-\infty,+\infty)))=(-1,1)e_{1}.$

On the other hand, since $\\{F(z_{k})\\}$ is relatively compact in $G$—hence
$\\{F_{0}(\tilde{z}_{k})\\}$ is relatively compact in $G_{0}$, it follows as
before and by Proposition 2.7 that

(3.8) $\Phi(\sigma((-\infty,+\infty)))=(-1,1)e_{1}.$

But (3.7) and (3.8) contradict (3.6). Therefore, $F$ is proper. In particular,
$\det(dF_{z})\not\equiv 0$.

Step 3. $F$ is injective.

As in Step. 2, we let $D_{0}:=T_{D}(D)$, $G_{0}:=T_{G}(G)$ and
$F_{0}:=T_{G}\circ F\circ T_{D}^{-1}:D_{0}\to G_{0}$. The statement is thus
equivalent to show that $F_{0}$ is injective.

By Step. 3, $\mathcal{Z}:=\\{z\in D_{0}:\det(dF_{0})_{z}=0\\}$ is a (possibly
empty) analytic subvariety of $D_{0}$—if $D_{0}$ and $G_{0}$ are sufficiently
smooth (see, e.g., [12]) then $\mathcal{Z}=\emptyset$ and $F_{0}$ is a
covering map, but we do not need this in our argument. Since $F_{0}$ is proper
by Step. 2, $F_{0}(\mathcal{Z})$ is an analytic subvariety of $G_{0}$ by
Remmert’s Theorem. In particular, $G_{0}^{\prime}:=G_{0}\setminus
F_{0}(\mathcal{Z})$ is a connected open and dense set in $G_{0}$ and
$F:D_{0}\setminus\mathcal{Z}\to G_{0}^{\prime}$ is a finite covering map.

For $w\in G_{0}$, let $\nu(w)$ be the number of points of $F^{-1}(w)$. Thus,
$\nu(w)=\nu(w^{\prime})=:\nu$ for all $w,w^{\prime}\in G^{\prime}_{0}$, and
$1\leq\nu(w)\leq\nu$ for all $w\in F_{0}(\mathcal{Z})$. Hence, it is enough to
show that $\nu=1$.

Arguing by contradiction, we assume that $\nu\geq 2$. Fix $\tilde{w}\in
G^{\prime}_{0}$ such that the vector $\tilde{w}-e_{1}$ is almost complex
tangential. Then by [14, Corollary 4], for $w$ sufficiently close to
$\tilde{w}$ there exists a unique geodesic ray $\eta_{w}:[0,+\infty)\to G_{0}$
such that $\eta_{w}(0)=w$ and $\lim_{t\to+\infty}\eta_{w}(t)=e_{1}$. Moreover,
if two such $\eta_{w}$ and $\eta_{w^{\prime}}$ intersect at one point, then
the range of one is contained in the other. Also $\eta_{w}(t)$ varies smoothly
with respect to $w$ and $t$. Take an open subset $\Delta$ in a real hyperplane
such that any $w\in\Delta$ gives rise to exactly one $\eta_{w}$. Then,
$\Delta\times[0,\infty)\to D$, $(w,t)\mapsto\eta_{w}(t)$ is a
$\mathcal{C}^{1}$ smooth diffeomorphism onto its image, consequently it is
locally bi-Lipschitz. The set $F(\mathcal{Z})$ has real codimension at least
$2$, while the real codimension of $\Delta$ is clearly 1. Thus, the set of
$w\in\Delta$ such that the range of $\eta_{w}$ does not intersect
$F(\mathcal{Z})$ has real Hausdorff codimension at least 1. In particular, it
is non-empty. Fix one such a $w_{0}$ and let $\eta:=\eta_{w_{0}}$. By
construction, $\eta([0,+\infty))\subset G_{0}^{\prime}$.

Since we are assuming that $\nu\geq 2$, there exists $z_{1},z_{2}\in D_{0}$,
$z_{1}\neq z_{2}$ such that $F(z_{1})=F(z_{2})=w_{0}$.

Let $\gamma_{j}:[0,+\infty)\to D_{0}$ be geodesic rays such that
$\gamma_{j}(0)=z_{j}$ and $\lim_{t\to+\infty}\gamma_{j}(t)=e_{1}$, $j=1,2$. By
hypothesis, $F_{0}\circ\gamma_{j}$ is a geodesic in $G_{0}$, and, by Step. 1,
$\lim_{t\to+\infty}F_{0}(\gamma_{j}(t))=e_{1}$ for $j=1,2$. Moreover, by
construction, $F_{0}(\gamma_{1}(0))=F_{0}(\gamma_{2}(0))=w_{0}$. Therefore, by
the uniqueness of $\eta$,

$F_{0}(\gamma_{j}(t))=\eta(t),\quad\hbox{for $t\geq 0$ and $j=1,2$}.$

It follows from $\eta([0,+\infty))\subset G_{0}^{\prime}$ that there are no
$s,t>0$ such that $z=\gamma_{1}(s)=\gamma_{2}(t)$. Take any sequence
$\\{s_{k}\\}$ converging to $+\infty$. Then
$\gamma_{1}(s_{k})\neq\gamma_{2}(s_{k})$.

Let $\\{a_{k}\\}$ be a strictly increasing sequence of positive numbers
converging to $1$ such that $\\{A_{a_{k}}^{-1}(\gamma_{1}(s_{k}))\\}$ is
relatively compact in $\mathbb{B}^{N}$, where $A_{a_{k}}$ is given by (2.2).

Likewise, we can find a sequence $\\{b_{k}\\}$, $0<b_{k}<1$ converging to $1$
such that $\\{A_{b_{k}}^{-1}(\eta(s_{k}))\\}$ is relatively compact in
$\mathbb{B}^{N}$.

Let $D_{k}:=A_{a_{k}}^{-1}(D_{0})$, $G_{k}:=A_{b_{k}}^{-1}(G_{0})$ and
$F_{k}:=A^{-1}_{b_{k}}\circ F_{0}\circ A_{a_{k}}:D_{k}\to G_{k}$.

By Proposition 2.7, up to subsequences and reparametrization, we can assume
that $\\{A_{a_{k}}^{-1}\circ\gamma_{1}\\}$ converges uniformly on compacta to
a geodesic $\sigma_{1}$ of $\mathbb{B}^{N}$ whose image is $(-1,1)e_{1}$.

By Proposition 2.1.(4), there exists $C>0$ such that for every $s_{k}$ we can
find $u_{k}>0$ such that $K_{D_{0}}(\gamma_{1}(s_{k}),\gamma_{2}(u_{k}))<C$.
It follows, again by Proposition 2.7, that, up to subsequences and
reparametrization, also $\\{A_{a_{k}}^{-1}\circ\gamma_{2}\\}$ converges
uniformly on compacta to a geodesic $\sigma_{2}$ of $\mathbb{B}^{N}$ whose
image is $(-1,1)e_{1}$. In particular,

$\sigma_{1}((-\infty,+\infty))=\sigma_{2}((-\infty,+\infty)).$

A similar argument shows that, up to subsequences and reparametrization,
$\\{A_{b_{k}}^{-1}\circ\eta\\}$ converges uniformly on compacta to a geodesic
$\tau$ of $\mathbb{B}^{N}$ whose image is $(-1,1)e_{1}$.

Arguing as in Step. 2, and up to subsequences, we can also assume that
$\\{F_{k}\\}$ converges uniformly on compacta of $\mathbb{B}^{N}$ to an
automorphism $\Phi$ of $\mathbb{B}^{N}$.

Since
$F_{k}(A_{a_{k}}^{-1}(\gamma_{j}((0+\infty))))=A_{b_{k}}^{-1}(\eta(0,+\infty))$
for $j=1,2$ and for all $k$, it follows that
$\Phi(\sigma_{j}((-\infty,+\infty)))=\tau((-\infty,+\infty))$, $j=1,2$.

Let $\zeta_{k}^{1}:=A_{a_{k}}^{-1}(\gamma_{1}(s_{k}))$ and let
$x_{0}\in(-1,1)e_{1}$ be the limit of $\\{\zeta_{k}^{1}\\}$. Let
$\zeta_{k}^{2}\in A_{a_{k}}^{-1}(\gamma_{2}([0,+\infty)))$ be such that
$F_{k}(\zeta_{k}^{2})=F_{k}(\zeta_{k}^{1})$ for all $k$ (it exists since
$F_{0}(\gamma_{1}([0,+\infty))=F_{0}(\gamma_{2}([0,+\infty))$).

We claim that $\\{\zeta_{k}^{2}\\}$ converges to $x_{0}$. To this aim, it is
enough to show that $\\{\zeta_{k}^{2}\\}$ is relatively compact in
$\mathbb{B}^{N}$, because, if so and, up to subsequences,
$x_{1}\in\mathbb{B}^{N}$ is the limit of $\\{\zeta_{k}^{2}\\}$ then

$\Phi(x_{1})=\lim_{k\to+\infty}F_{k}(\zeta^{2}_{k})=\lim_{k\to+\infty}F_{k}(\zeta^{1}_{k})=\Phi(x_{0}),$

and, since $\Phi$ is an automorphism, $x_{0}=x_{1}$.

In order to show that $\\{\zeta_{k}^{2}\\}$ is relatively compact in
$\mathbb{B}^{N}$, since $\\{A_{a_{k}}^{-1}\circ\gamma_{2}\\}$ converges, up to
reparametrization, to $\sigma_{2}$ whose image is equal to that of
$\sigma_{1}$, there exists a sequence $\\{\xi_{k}\\}$ converging to $x_{0}$
and such that $\xi_{k}\in A_{a_{k}}^{-1}(\gamma_{2}([0,+\infty))$ for all $k$.
Thus,

$K_{D_{k}}(\xi_{k},\zeta_{k}^{2})=K_{G_{k}}(F_{k}(\xi_{k}),F_{k}(\zeta_{k}^{2}))=K_{G_{k}}(F_{k}(\xi_{k})),F_{k}(\zeta_{k}^{1})).$

Since $\\{\xi_{k}\\}$ and $\\{\zeta_{k}^{1}\\}$ converge to $x_{0}$, it
follows by Lemma 2.3 that the last quantity is bounded. Hence, $\\{\xi_{k}\\}$
is relatively compact in $\mathbb{B}^{N}$ and hence it converges to $x_{0}$.

Since $\gamma_{1}(s_{k})\neq\gamma_{2}(s_{k})$, we see that
$\zeta_{k}^{1}\neq\zeta_{k}^{2}$ for all $k$.

Now, $\\{F_{k}\\}$ converges uniformly on compact to $\Phi$, which is
injective. Hence, for $\epsilon>0$ so that
$B(x_{0},\epsilon):=\\{z\in\mathbb{C}^{N}:\|z-x_{0}\|<\epsilon\\}\subset\mathbb{B}^{N}$,
there exists $k_{0}$ such that $F_{k}$ is injective on $B(x_{0},\epsilon)$ for
$k\geq k_{0}$. However, for $k$ large, $\zeta_{k}^{1},\zeta_{k}^{2}\in
B(x_{0},\epsilon)$, $\zeta_{k}^{1}\neq\zeta_{k}^{2}$ and
$F_{k}(\zeta_{k}^{1})=F_{k}(\zeta_{k}^{2})$, a contradiction. ∎

##  4\. Special cases: holomorphic coverings and Reinhardt domains

It is natural to explore the possibility of relaxing assumptions on the
mapping $F$ (and the domains $D$ and $G$) in results like those in Theorem 1.1
and Theorem 1.2, while preserving the same conclusion. However, as we will
demonstrate, such assumptions cannot be omitted in general. Below, we provide
examples of non-trivial proper holomorphic mappings, which are also
holomorphic coverings, between Reinhardt domains (that can be chosen to be
strongly pseudoconvex). These mappings do not constitute biholomorphisms,
despite being Kobayashi isometries along a complete family of geodesic lines.

First, we present a general result on the behaviour of geodesics under
holomorphic coverings.

######  Proposition 4.1.

Lef $\pi:D\to G$ be a holomorphic covering between complete hyperbolic domains
in $\mathbb{C}^{n}$, $w,z\in G$. Then $\gamma$ is a geodesic segment in $G$
joining $w$ and $z$ if and only if there exist points $\tilde{w},\tilde{z}\in
D$ with $\pi(\tilde{w})=w$, $\pi(\tilde{z})=z$, and a geodesic segment
$\tilde{\gamma}$ in $D$ such that $\pi\circ\tilde{\gamma}=\gamma$ and
$K_{D}(w,z)=K_{G}(\tilde{w},\tilde{z})$.

In particular, given $w_{0}\in G$ and $\tilde{w}_{0}\in D$ such that
$\pi(\tilde{w}_{0})=w_{0}$, any geodesic passing through $w_{0}\in G$ can be
lifted to a geodesic of $D$ passing through $\tilde{w}_{0}$.

###### Proof.

Recall that the holomorphic coverings preserve the Kobayashi metric. Denote by
$\gamma:[t_{0},t_{1}]\to G$ a geodesic segment joining $w$ and $z$ and denote
by $\tilde{\gamma}$ its lifting to $D$. Let $\tilde{\gamma}(t_{0})=\tilde{w}$,
$\tilde{\gamma}(t_{1})=\tilde{z}$. Then we have the following inequalities

$K_{G}(w,z)\leq
K_{D}(\tilde{w},\tilde{z})\leq\int_{t_{0}}^{t_{1}}k_{D}(\tilde{\gamma}(t);\tilde{\gamma}^{\prime}(t))dt=\int_{t_{0}}^{t_{1}}k_{G}(\gamma(t);\gamma^{\prime}(t))dt,$

which straightforwardly establishes the desired equivalence. ∎

######  Remark 4.2.

Let us consider the following holomorphic covering

$\\{\lambda\in\mathbb{C}:-\log R<\operatorname{Re}(\lambda)<\log
R\\}=:H_{R}\owns\lambda\mapsto\exp(\lambda)\in
A_{R}:=\\{\lambda\in\mathbb{C}:1/R<|\lambda|<R\\}.$

Note that the geodesic lines in $H_{R}$ given by $\mathbb{R}\ni s\mapsto
t_{0}+is$ for $-\log R<t_{0}<\log R$, a complete collection of geodesic lines;
they are mapped isometrically onto geodesic lines $\\{t\exp(is):1/R<t<R\\}$,
$s\in\mathbb{R}$ though the mapping itself is not even proper.

######  Remark 4.3.

More generally, in view of the results above we see that under the assumption
that the mapping $F:D\mapsto G$ is a holomorphic covering and $G$ admits a
complete collection $\mathcal{F}$ of geodesic lines then we may lift the
collection $\mathcal{F}$ to a complete family $\widetilde{\mathcal{F}}$ of
geodesic lines so that $F$ maps isometrically elements of
$\widetilde{\mathcal{F}}$ preserving the Kobayashi distance along them.
Consequently, when the mapping is not biholomorphic we get a kind of
counterexample to the desired biholomorphcity of the mapping $F$. We shall
make it very concrete in the next subsection.

###  4.1. Examples in Reinhardt domains

In this part we are dealing with Reinhardt pseudoconvex domains
$D\subset\mathbb{C}_{*}^{n}$, $n\geq 1$. In such a situation the domain

$\log D:=\\{\log|z|:=(\log|z_{1}|,\ldots,\log|z_{n}|):z\in D\\}$

is convex. For simplicity we impose a stronger condition on $D$, namely, we
assume that the following is satisfied: $\log D$ is bounded, which means that
$D$ is additionally bounded and ’separated’ from the axes.

Recall that the mapping

$T_{\log D}\owns u\mapsto\exp u:=(\exp(u_{1}),\ldots,\exp(u_{n}))\in D,$

where $T_{\log D}:=\log D+i\mathbb{R}^{n}$ is a tube domain with the basis
$\log D$, is a holomorphic covering. A formula describing the behaviour of the
Kobayashi distance under holomorphic coverings gives the following

$K_{D}(w,z)=\inf\\{K_{T_{\log D}}(u,v+i2\pi\nu):\nu\in\mathbb{Z}^{n}\\},$

where $\exp(u)=w$, $\exp(v)=z$. Note that in our situation the infimum is
always attained.

This leads us to the next example.

######  Example 4.4.

The rays $(0,1)\owns t\mapsto t\omega\in\mathbb{D}_{*}$, $|\omega|=1$ are
geodesic lines such that we have the equality
$K_{\mathbb{D}_{*}}(t^{n}\omega^{n},s^{n}\omega^{n})=K_{\mathbb{D}_{*}}(t\omega,s\omega)$,
$0<t,s<1$ showing that the mapping
$\mathbb{D}_{*}\owns\lambda\mapsto\lambda^{n}\in\mathbb{D}_{*}$ maps
isometrically a complete collection of geodesic lines and is not a
biholomorphism (if $n\geq 2$).

The analogous result may be proven for the annuli $A(0,r,1)$ and
$A(0,r^{n},1)$ – we shall give the details below in a more general setting (in
arbitrary dimension).

One may also easily see that in the case $x,y\in\log D$ we have the following
equality

(4.1) $K_{D}(\exp(x),\exp(y))=K_{{}_{T_{\log D}}}(x,y).$

In fact to see the last property it is sufficient to see that any competing
analytic disc $f:\mathbb{D}\to T_{\log D}$ for $x$ and $y+i2\pi\nu$ for some
$\nu\in\mathbb{Z}^{n}$, i.e., the one with $f(0)=x$, $f(t)=y+i2\pi\nu$, $t>0$
gives rise to a competitor
$g(\lambda):=(f(\lambda)+\overline{f(\overline{\lambda})})/2$ with $g(0)=x$,
$g(t)=y$, from which the formula follows at once.

Adopting the definition from [19] we call two points $x,y\in\partial\log D$
antipodal if there is a hyperplane $H$ such that $x+H$ and $y+H$ are not
identical and they are supporting hyperplanes of $\log D$. Let us make some
comments. First note that the segment $(-1,1)$ is a geodesic line (up to a
parametrization) in the strip
$H:=\\{\lambda\in\mathbb{C}:-1<\operatorname{Re}\lambda<1\\}$. Consequently,
the open segment joining antipodal points $x,y\in\partial\log D$ is a geodesic
line in $T_{\log D}$ and thus by the fact that the map defined by the formula
(13) in [19] is a complex geodesic we easily conclude that (up to a
parametrization) the curve

$(-1,1)\owns t\mapsto\exp((x+y)/2)\exp(t(x-y)/2)$

is a geodesic line in $D$.

Now we recall the definition of the monomial mapping $\Phi_{A}$ (see [17],
[18]) where $A$ is $n\times n$ matrix with integer elements. First for
$\alpha\in\mathbb{C}^{n}$ we define

(4.2) $z^{\alpha}:=z_{1}^{\alpha_{1}}\cdot\ldots\cdot z_{n}^{\alpha_{n}}$

for all $z$ with $z_{j}\neq 0$ if $\alpha_{j}<0$ (this is always satisfied in
our situation). Then we define

(4.3) $\Phi_{A}(z):=(\Phi_{A^{1}}(z),\ldots,\Phi_{A^{n}}(z)),$

where $A^{j}$ denotes the $j$-th row of $A$. It is known that $\Phi_{A}$ is a
proper holomorphic mapping of $\mathbb{C}_{*}^{n}$ if and only if $\det A\neq
0$; additionally, its multiplicity equals $|\det A|$ (see e.g. Theorem 2.1 in
[17]).

Below we consider a non-singular $A$. Note that $\log(\Phi_{A}(D))=A(\log D)$,
so $\Phi_{A}$ when restricted to $D$ is a proper holomorphic mapping onto the
pseudoconvex Reinhardt domain $\Phi_{A}(D)$. It is also a holomorphic
covering.

Recall that Proposition 4.1 lets us to lift any geodesic segment/ray/line in
$\Phi_{A}(D)$ to a geodesic segment/ray/line in $D$ such that $\Phi$,
restricted to it, is a Kobayashi isometry.

To make the next example concrete let $\log D$ be the unit Euclidean ball and
$A=2\mathbb{I}_{n}$. We then construct a holomorphic map
$\exp(T_{\mathbb{B}_{n}^{\mathbb{R}}})\owns
z\mapsto(z_{1}^{2},\ldots,z_{n}^{2})\in\exp(T_{2\mathbb{B}_{n}^{\mathbb{R}}})$
that maps isometrically a complete collection of geodesic lines in $D$ that is
obtained by coordinate-wise rotations of geodesics lying in
$\mathbb{B}_{n}^{\mathbb{R}}$ and is proper, with multiplicity equal to
$2^{n}$.

Note that $A$ maps antipodal points of $\log D$ to antipodal points in $A(\log
D)$. Moreover, we get the following equality for geodesic lines $\gamma$
joining the antipodal points (such that the graph is the segment)

(4.4)
$K_{D}(\gamma(t),\gamma(s))=K_{\Phi_{A}(D)}(\Phi_{A}(\gamma(t)),\Phi_{A}(\gamma(s))),$

which easily shows that the above construction may be easily transferred to
Reinhardt domains with $\log D$ satisfying the following property: For any
$u\in\log D$ there are two antipodal points such that $u$ lies on the segment
joining $x$ and $y$.

## References

  * [1] M. Abate, Iteration theory of holomorphic maps on taut manifolds. Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989.
  * [2] Z. M. Balogh, M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75 (2000), 504-533.
  * [3] J. Bland, T. Duchamp, M. Kalka, On the automorphism group of strictly convex domains in $\mathbb{C}^{n}$. Contemp. Math. 49, 19–30 (1986).
  * [4] F. Bracci, Frames of quasi-geodesics, visibility and geodesic loops, arXiv:2304.11025.
  * [5] F. Bracci, M. D. Contreras, S. Díaz-Madrigal, Continuous semigroups of holomorphic self-maps of the unit disc. Springer Monograph in Mathematics 2020
  * [6] F. Bracci, H. Gaussier, Abstract boundaries and continuous extension of biholomorphisms, Anal. Math. 48 (2022), no. 2, 393–409.
  * [7] F. Bracci, L. Kosiński, W. Zwonek, Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics. J. Geom. Anal., 31, 11292–11311 (2021).
  * [8] F. Bracci, D. Kraus, O. Roth, A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps. Adv. Math., 432, (2023), paper N. 109262.
  * [9] F. Bracci, G. Patrizio, Monge-Ampère foliations with singularities at the boundary of strongly convex domains. Math. Ann. 332, 3, (2005) 499–522.
  * [10] J. P. D’Angelo, Several complex variables and the geometry of real hypersurfaces. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1993. xiv+272 pp.
  * [11] B. L. Fridman, D. Ma, On Exhaustion of Domains, Indiana University Mathematics Journal, Vol. 44, No. 2 (Summer, 1995), pp. 385–395.
  * [12] F. Forstnerič, Proper holomorphic mappings: a survey.Several complex variables (Stockholm, 1987/1988), 297-363. Math. Notes, 38 Princeton University Press, Princeton, NJ, 1993.
  * [13] Ł. Kosiński, N. Nikolov, Lower estimates of the Kobayashi distance and limits of complex geodesics, https://arxiv.org/abs/2208.13239, Math. Ann., to appear.
  * [14] Ł. Kosiński, N. Nikolov, P.J. Thomas, A Gehring-Hayman inequality for strongly pseudoconvex domains, https://arxiv.org/abs/2303.04071.
  * [15] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. Fr. 109 (1981), 427–474.
  * [16] W. Rudin, Function theory in the Unit Ball of $\mathbb{C}^{n}$, Springer-Verlag, Berlin 1980.
  * [17] W. Zwonek, On hyperbolicity of pseudoconvex Reinhardt domains, Arch. Math. (Basel) 72 (1999), 304–314.
  * [18] W. Zwonek, Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions, Diss. Math., 388 (2000), 103 pp.
  * [19] W. Zwonek, Complex geodesics in tube domains and their role in the study of harmonic mappings in the disc, Anal Math 48, 601–617 (2022).


# UAlberta at SemEval 2022 Task 2: Leveraging Glosses and Translations for
Multilingual Idiomaticity Detection

Bradley Hauer, Seeratpal Jaura, Talgat Omarov, Grzegorz Kondrak
Alberta Machine Intelligence Institute, Department of Computing Science
University of Alberta, Edmonton, Canada
<EMAIL_ADDRESS>
###### Abstract

We describe the University of Alberta systems for the SemEval-2022 Task 2 on
multilingual idiomaticity detection. Working under the assumption that
idiomatic expressions are non-compositional, our first method integrates
information on the meanings of the individual words of an expression into a
binary classifier. Further hypothesizing that literal and idiomatic
expressions translate differently, our second method translates an expression
in context, and uses a lexical knowledge base to determine if the translation
is literal. Our approaches are grounded in linguistic phenomena, and leverage
existing sources of lexical knowledge. Our results offer support for both
approaches, particularly the former.

UAlberta at SemEval 2022 Task 2: Leveraging Glosses and Translations for
Multilingual Idiomaticity Detection

Bradley Hauer, Seeratpal Jaura, Talgat Omarov, Grzegorz Kondrak Alberta
Machine Intelligence Institute, Department of Computing Science University of
Alberta, Edmonton, Canada<EMAIL_ADDRESS>

##  1 Introduction

In this paper, we describe the University of Alberta systems for the task of
classifying multi-word expressions (MWEs) in context as either _idiomatic_ or
_literal_ Tayyar Madabushi et al. (2022). Each instance in the data includes a
MWE (e.g., closed book), its language, and its context, composed of the three
surrounding sentences. We participate in both the zero-shot and one-shot
settings.

While the exact definitions of the two key terms are not stated explicitly in
the task
description111https://sites.google.com/view/semeval2022task2-idiomaticity, it
is suggested that idiomatic is synonymous with non-compositional. The Pocket
Oxford Dictionary defines idiomatic as “not immediately comprehensible from
the words used,” and literal as “taking words in their basic sense.”
Therefore, we adopt the following MWE compositionality criterion

$\mbox{literal}\equiv\mbox{compositional}\equiv\neg\,\mbox{idiomatic}$

where the three terms are considered to be Boolean variables. In addition, the
shared task considers all proper noun MWEs (e.g., Eager Beaver) as literal.

Our goal is to explore the idea that glosses and translations of word senses
can help decide whether the meaning of a given MWE occurrence is
compositional. Based on the above-stated compositionality criterion, this in
turn could facilitate idiomaticity detection. In particular, we hypothesize
that at least one of the words in any idiomatic expression is used in a non-
standard sense. Following the intuition that a traditional word sense
disambiguation (WSD) system can only identify senses that are included in a
given sense inventory, we propose two methods that indirectly detect non-
standard senses by leveraging either glosses or translations of senses from
such an inventory.

Figure 1: An example of defBERT input.

Our gloss-based method follows from the intuition that the meaning of a given
MWE occurrence is related to any of the existing sense glosses of its
component words only if the expression is compositional. Therefore, the
addition of the glosses to the context of the expression should help the
classifier in deciding whether the MWE is used in a literal or idiomatic
sense. We implement this method by adding the glosses of each sense of each
individual word, retrieved from a lexical knowledge base, to the input to a
neural classifier which fine-tunes multilingual BERT (mBERT; Devlin et al.,
2019) for the idiomaticity detection task. We refer to this method as defBERT
(Figure 1).

Our translation-based method follows from the observation that compositional
expressions are typically translated word-for-word (“literally”), which
implies that each content word and its translation should have the same
meaning. Therefore, each such multilingual word pair should share a multi-
synset in a multi-wordnet Hauer and Kondrak (2020b). The procedure is as
follows: (1) translate the MWE in context; (2) word-align the source and
target sentences; (3) lemmatize and POS-tag the source MWE; and (4) for each
lemma in the MWE, search for a multi-synset that contains both the lemma and
its translation. This method is unsupervised, and we refer to it as MT.

Our results provide evidence that leveraging lexical resources is beneficial
for idiomaticity detection. In particular, our gloss-based method, when
combined with a type-based UNATT heuristic, is among the top-scoring
submissions in the one-shot setting. The heuristic is based on the observation
that some MWEs are inherently idiomatic or literal, regardless of their
context, which is confirmed by our analysis of the development set
annotations.

##  2 Related Work

Early attempts to represent idiomatic MWEs involve treating idiomatic phrases
as individual tokens and learning corresponding static embeddings (Mikolov et
al., 2013). However, Cordeiro et al. (2016) show that the effectiveness of
this method is limited by data sparsity for longer idiomatic expressions.
Furthermore, Shwartz and Dagan (2019) and Garcia et al. (2021) conclude that
idiomaticity is not yet accurately represented even by contextual embedding
models. Tayyar Madabushi et al. (2021) create a new manually labeled dataset
containing idiomatic and literal MWEs, and propose a method based on a pre-
trained neural language model.

Regarding using lexical translations for idiomaticity detection, Moirón and
Tiedemann (2006) measure semantic entropy in bitext alignment statistics,
while Salehi et al. (2014) predict compositionality by presenting an
unsupervised method that uses Wiktionary translation, synonyms, and definition
information. We extend these ideas by applying machine translation, and
consulting a multilingual lexical knowledge base.

Our prior work has already demonstrated the utility of lexical translations
for various semantic tasks, including prior SemEval tasks on predicting cross-
lingual entailment Hauer et al. (2020) and contextual synonymy detection Hauer
et al. (2021), as well as word sense disambiguation Luan et al. (2020), and
homonymy detection Hauer and Kondrak (2020a); Habibi et al. (2021).

##  3 Methods

In this section, we describe our methods for idiomaticity detection.

###  3.1 Baseline mBERT

We re-implemented the mBERT classifier baseline (Devlin et al., 2019)
following the methodology of Tayyar Madabushi et al. (2021). The model takes
the context sentence and the relevant MWE as an input, and outputs a binary
label indicating the idiomaticity of the target MWE. The input sequence is
constructed by concatenating the MWE to the end of the context sentence after
the special [SEP] token.

It is important to note the differences between our re-implementation and the
official baseline provided by the task organizers. In the official baseline,
the organizers add the target MWE as an additional feature in the one-shot
setting but not in the zero-shot setting. Furthermore, the organizers include
the sentences preceding and succeeding the target sentence only in the zero-
shot setting. In our re-implementation, we add the target MWE and exclude the
preceding and succeeding sentences in both zero-shot and one-shot settings.

###  3.2 Gloss-based Method

Our first method, defBERT, extends the baseline model by adding the glosses of
all possible senses of each individual word in the target MWE to the
classifier’s input. The intuition is that the addition of the glosses to the
input should help the classifier decide if the meaning of the target MWE can
be deduced from the definitions of the individual words, i.e., if it is
compositional. In the example in Figure 1, the disparity between the context
in which _fish story_ appears, and the glosses of the various senses of the
words _fish_ and _story_ indicates that the MWE is idiomatic in this context.

The intuition for this method is that non-native speakers can identify
idiomatic expressions, provided they understand the standard meanings of the
words which comprise them. Suppose that the vocabulary of a non-native speaker
covers most of the essential words necessary to understand a language, but not
idiomatic expressions. Even if the speaker cannot deduce the meaning of an
idiomatic expression in context, they can guess that the expression was used
in an idiomatic sense because individual words of this expression do not make
sense in the given context.

###  3.3 Translation-based Method

Our MT method is based on translating the target MWE in context, and leverages
multilingual semantic resources. The intuition behind this method is that
idioms are generally specific to a particular language, and, being non-
compositional, their meanings cannot be conveyed simply by translating the
individual words.

Under this hypothesis, to classify an MWE as literal or idiomatic, we need
only determine whether the words in the MWE are translated literally. We do
this by first identifying the translation of each word via alignment. We then
consult a multilingual wordnet, or _multi-wordnet_ , a lexical knowledge-base
which organizes words in two or more languages into multilingual synonym sets,
or _multi-synsets_. Each multi-synset corresponds to a unique concept, and
contains the words which express that concept. Given a word in context, and a
translation of that word in that context, we consider the word to be literally
translated if it shares at least one multi-synset with its translation.

For example, consider an instance in which the MWE _wedding anniversary_ is
translated into Italian as anniversario di matrimonio. Our method checks if
either of the translation pairs (_wedding_ , _matrimonio_) and (_anniversary_
, _anniversario_) share a multi-synset in a multi-wordnet. We test two
versions of this method: in MT(all), this condition must be satisfied for all
content words in the MWE; in MT(one), detecting a literal translation for one
word is sufficient to classify the MWE as literal. In addition, multiple
languages of translation may be considered.

###  3.4 Additional Heuristics

The annotation methodology for this shared task includes proper nouns in the
literal class. We therefore use a part-of-speech tagger to detect proper
nouns; if any word in the MWE is tagged as a proper noun, MT automatically
classifies it as literal without further consideration.

In the one-shot setting, we also use a type-based heuristic which we refer to
as UNATT. The intuition behind this heuristic is that certain MWEs are
inherently idiomatic or literal, regardless of the context that they appear
in. If the training data has no example of an MWE in a particular class, the
heuristic exploits this fact as evidence that the MWE should always be
classified as the opposite, attested class. For example, this heuristic always
classifies life vest as idiomatic and economic aid as literal, as these are
the only classes in which these MWEs appear in the training data. In practice,
since UNATT returns no classification if the training set contains instances
that belong to either class, this heuristic must be used in combination with
another method.

###  3.5 Combination

Our defBERT and MT methods take different views of the data, with the former
using a neural language model and gloss information, and the latter using
translation and a lexical knowledge base. We therefore consider combining the
two methods. In this approach, we independently apply defBERT and MT to a
given instance. If the two methods agree, we return the agreed-upon
classification; if they disagree, we return a default class, which is a
tunable parameter. As with the other methods, we can combine this method with
the UNATT heuristic in the one-shot setting.

##  4 Experiments

We now describe our experiments, including the tools and resources, the
experimental setup, the results, and a discussion of our findings.

###  4.1 Lexical Resources

As lexical resources for sense translations and glosses, we use two different
multi-wordnets: BabelNet (BN; Navigli and Ponzetto, 2010, 2012), and Open
Multilingual WordNet (OMW; Bond and Foster, 2013). The defBERT method and the
alignment tool access BN 4.0 via the provided Java
API222https://babelnet.org/guide. For the MT method, we access the BN 5.0 via
the HTTP API. We access OMW via the NLTK interface Bird et al. (2009). For the
MT method, we consider the translation of a word to be literal if it shares a
multi-synset with the word in either BN or OMW. For lemmatization and POS
tagging, we use TreeTagger333We use the pre-trained models for English,
Portuguese, and Galician from https://cis.uni-
muenchen.de/~schmid/tools/TreeTagger. Schmid (2013).

Both BN and OMW contain English glosses for most concepts, but the
availability of glosses in other languages varies. In particular, OMW contains
no Portuguese or Galician glosses. With BabelNet, we experimented with two
techniques: using English glosses for all languages, and using glosses from
the language of the instance, i.e. the source language, when available. We
refer to these variants as defBERT-BN-en and defBERT-BN-src, respectively.
Since defBERT uses a multilingual pre-trained language model, it can
seamlessly handle input from multiple languages. Furthermore, because of the
relatively poor coverage of Galician in the lexical resources (only 54% of
glosses are available in this language), we attempt to leverage its close
relationship to Portuguese by processing Galician as if it was Portuguese.

###  4.2 Translation and Word Alignment

We translate the context sentence of each MWE with Google Translate
API444https://cloud.google.com/translate. We translated English instances into
Italian, and Portuguese/Galician instances into English, because of the good
coverage of these languages in our resources. We also conducted development
experiments with translation into less related languages, as well as with
combining translation information from multiple languages, but we observed no
consistent improvements.

We align each input sentence with its translation using BabAlign Luan et al.
(2020), which consults BabelNet to refine the alignments generated by a base
aligner, FastAlign Dyer et al. (2013). To further improve the alignment
quality, we augment the set of sentence-translation pairs with additional
parallel data from the OpenSubtitles parallel corpus Lison and Tiedemann
(2016). We note that the English-Galician bitext is less than 1% of the size
of the other two bitexts.

###  4.3 mBERT and defBERT

We fine-tune the mBERT-based models using the binary classification objective
on the labeled training dataset. In the zero-shot setting, the MWEs in the
training data are disjoint from those in the development and test splits,
while in the one-shot setting, all MWEs in the development and test splits
have at least one example in the training data. In the zero-shot setting, we
trained the models only on the zero-shot training set, while in the one-shot
setting, we trained the models on both training sets. In particular, we fine-
tuned the models for 20 epochs with a maximum sequence length of 256, a
learning rate of 2e-5, and a per device batch size of 16, using the
HuggingFace Transformers library.555https://huggingface.co

|  | Development results | Test results
---|---|---|---
|  | Zero-Shot | One-Shot | Zero-Shot | One-Shot
|  | EN | PT | EN | PT | EN | PT | GL | ALL | EN | PT | GL | ALL
0 | Baseline | 66.2 | 63.9 | 87.0 | 86.7 | 70.7 | 68.0 | 50.7 | 65.4 | 88.6 | 86.4 | 81.6 | 86.5
1 | mBERT | 74.6 | 62.5 | 85.7 | 85.9 | 75.1 | 63.3 | 61.1 | 68.2 | 90.0 | 83.6 | 86.6 | 87.7
2 | defBERT-BN-src | 75.5 | 64.8 | 85.4 | 86.7 | 72.0 | 66.4 | 57.8 | 67.2 | 95.7 | 88.5 | 88.9 | 92.2
3 | defBERT-BN-en | 75.3 | 66.4 | 87.6 | 86.6 | 73.4 | 68.4 | 59.7 | 69.5 | 95.0 | 89.3 | 87.9 | 91.8
4 | defBERT-OMW-en | 74.8 | 64.5 | 87.1 | 84.5 | 71.0 | 65.6 | 56.5 | 66.5 | 92.4 | 86.7 | 88.5 | 90.1
5 | UNATT + defBERT | - | - | 92.0 | 87.7 | - | - | - | - | 94.5 | 89.2 | 91.2 | 92.4
6 | MT(one) + defBERT | 77.3 | 64.9 | 84.5 | 78.0 | 68.2 | 54.6 | 56.3 | 62.7 | 85.9 | 70.6 | 78.2 | 80.6
7 | MT(all) + defBERT | 66.4 | 69.2 | 73.7 | 78.0 | 65.4 | 62.5 | 54.3 | 62.1 | 80.3 | 73.8 | 73.9 | 77.3
Table 1:  The macro F1 scores on the development and test datasets. Our
official submissions are in rows 4-7. Where not otherwise specified, defBERT
is in the OMW-en configuration.

###  4.4 Development experiments

Table 1 contains the results of the following models: the official mBERT-based
baseline (row 0) as reported by the shared task organizers, our re-
implementation of the official baseline (row 1), three variants of defBERT
method which is based on mBERT (rows 2-4), defBERT combined with the UNATT
heuristic (row 5), and the MT method combined with defBERT (rows 6-7)666After
the test output submission deadline, we discovered errors in our
implementation of the MT methods. We report our original results for
consistency with the official results.. For rows 1-5 we average the macro F1
score obtained over five runs with random initializations.

Our experiments with defBERT explored the impact of adding glosses to the
mBERT model, including the source and language of the glosses. With English
glosses retrieved from BabelNet, defBERT improves the total score over the
mBERT model in the zero-shot setting, especially on Portuguese. The results
also suggest that the English glosses may be preferable to glosses in the
source language, a finding which could simplify work on lower-resourced
languages, where glosses may not be available.

Combining the predictions of the mBERT-based models with the UNATT heuristic
improves the one-shot F1 scores in all cases (row 5 vs. row 4).

The MT methods achieve the best results when combined with defBERT on the
development set in the zero-shot setting: MT(one) for English (row 6), and
MT(all) for Portuguese (row 7). This demonstrates the utility of using lexical
translation information for idiomaticity detection when annotated training
data is not available.

###  4.5 Error Analysis

We found that the defBERT method performs slightly better, by about 1% F1, on
literal instances as compared to idiomatic instances in the one-shot setting.
In other words, the method is less likely to make an error when given a
literal instance. We speculate that this is explained by the model’s
consistent classification of proper nouns as literal expressions. Indeed, a
proper noun is identified incorrectly in only one instance. The fraction of
idiomatic vs. literal instances is 39% in English and 56% in Portuguese.

For the MT method, a large number of of errors were caused by a literal
translation of an idiomatic expression by Google Translate, even though the
corresponding expression is not meaningful in the target language. For
example, “she was different, like a closed book” is translated into Italian as
“era diversa, come un libro chiuso” even though the Italian translation does
not carry the meaning of a person being secretive. In a few cases, the
translation would simply copy the source language expression, yielding output
which is not fully translated. In addition, some correct lexical translations
are not in our lexical resources. Finally, a number of incorrect idiomatic
predictions could be traced to word alignment errors, especially in cases of
many-to-one alignments (e.g., bow tie correctly translated as papillon).

Manual analysis performed on the development set corroborates our hypothesis
that most multi-word expressions are inherently idiomatic (e.g., home run) or
literal (e.g., insurance company). Only about one-third of the expressions are
ambiguous in the sense that they can be classified as either class depending
on the context (e.g. closed book). Our judgements are generally corroborated
by the gold labels, with the exception of proper nouns, which are consistently
marked as literal. The UNATT heuristic (Section 3.4), which is based on this
observation, obtains a remarkable 98.3% precision and 55.8% recall on the set
of 739 instances in the development set.

###  4.6 Test set results

The results on the test set are shown in Table 1. Our best results are
produced by defBERT-BN-en in the zero-shot setting, and the combination of
defBERT with the UNATT heuristic in the one-shot setting. The latter also
obtains the best result on Galician, which demonstrates its applicability to
low-resource languages, as this method only requires English glosses.

The results of combining defBERT with MT are well below the baseline, which
may be due to a different balance of classes in the test set, omissions in
lexical resources, and/or errors in our initial implementation. Another
possible reason is that modern idiomatic expressions are often translated
word-for-word (“calqued”), especially from English into other European
languages. Examples from the development set include flower child, banana
republic, and sex bomb.

##  5 Conclusion

Our top result ranks third overall in the one-shot setting. The corresponding
method is applicable to a wide variety of languages. It takes advantage of the
ability of neural language models to seamlessly incorporate textual
information such as glosses, even if it is expressed in a different language.
These results strongly support our hypothesis that the gloss information of
individual words can improve idiomaticity detection. Moreover, our development
results support the hypothesis that non-compositional expressions can be
identified through their translations. These findings conform with our prior
work on leveraging translation for various semantic tasks (Section 2). We hope
that this work will motivate further investigation into the role of
multilinguality in semantics.

## Acknowledgments

This research was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC), and the Alberta Machine Intelligence Institute
(Amii).

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