# Quasi-Geodesics and Backward Orbits Under Semigroups of Holomorphic
Functions

Konstantinos Zarvalis  Department of Mathematics, Aristotle University of
Thessaloniki, 54124, Thessaloniki, Greece<EMAIL_ADDRESS>
###### Abstract.

We explore two properties of backward orbits under semigroups of holomorphic
self-maps in the unit disk. First, we prove that regular backward orbits are
quasi-geodesics for the hyperbolic distance of the unit disk. Then, we show
that backward orbits satisfy a useful property, this time in Euclidean terms.

###### Key words and phrases:

Backward orbit, semigroup of holomorphic functions, quasi-geodesic, non-
tangential convergence

###### 1991 Mathematics Subject Classification:

Primary 37F44; Secondary 30D05.

##  1\. Introduction

One-parameter continuous semigroups of holomorphic functions in the unit disk
$\mathbb{D}$, or, from now on, semigroups in $\mathbb{D}$, have been studied
extensively in recent years. They were first introduced by Berkson and Porta
in [3] and their theory is presented in several books, such as [1], [11], and
more recently in the monograph [5].

A semigroup in $\mathbb{D}$ is a family $(\phi_{t})$, $t\geq 0$, of
holomorphic self-maps of the unit disk that satisfies the following three
properties:

  1. (i)

$\phi_{0}$ is the identity in $\mathbb{D}$,

  2. (ii)

$\phi_{t+s}(z)=\phi_{t}(\phi_{s}(z))$, for all $z\in\mathbb{D}$ and all
$t,s\geq 0$,

  3. (iii)

$\lim\limits_{t\to s}\phi_{t}(z)=\phi_{s}(z)$, for all $z\in\mathbb{D}$ and
all $s\geq 0$.

All the elements of a semigroup have the same dynamic behavior ([5, Remark
12.1.2]). In particular, if, for some $t_{0}>0$, the function $\phi_{t_{0}}$
has no fixed points inside the unit disk, then the same is true for any other
element of the semigroup, other than $\phi_{0}$. Such semigroups, without any
fixed points, are called non-elliptic. For a non-elliptic semigroup
$(\phi_{t})$, there always exists a unique point $\tau\in\partial\mathbb{D}$
such that

$\lim\limits_{t\to+\infty}\phi_{t}(z)=\tau,$

for all $z\in\mathbb{D}$ ([5, Theorem 8.3.1]). This point $\tau$ is called the
Denjoy-Wolff point of the semigroup.

A very important tool, connected with the study of a semigroup $(\phi_{t})$,
is the so-called Koenigs function $h$ of the semigroup. More specifically and
for non-elliptic semigroups, this function is the unique conformal mapping
$h:\mathbb{D}\to\mathbb{C}$ satisfying $h(0)=0$, such that

$\phi_{t}(z)=h^{-1}(h(z)+t),\quad z\in\mathbb{D},\quad t\geq 0.$

The simply connected domain $\Omega:=h(\mathbb{D})$ is called the associated
planar domain of the semigroup.

A direct implication of the above relation concerning the Koenigs function is
the fact that $\Omega$ is convex in the positive direction (also known as
starlike at infinity). This means that $\\{w+t:t\geq 0\\}\subset\Omega$, for
every $w\in\Omega$.

A first distinction within the class of non-elliptic semigroups was given in
[9]. If $\Omega$ is contained in a horizontal strip, then the semigroup is
called hyperbolic. Otherwise, it is called parabolic.

Let $z\in\mathbb{D}$. Then the curve $\gamma_{z}:[0,+\infty)\to\mathbb{D}$,
with $\gamma_{z}(t)=\phi_{t}(z),t\geq 0,$ is called the trajectory of $z$.
Later on, we might refer to $\gamma_{z}$ as the forward trajectory of $z$ to
avoid confusion. It is clear, by the definition of the Denjoy-Wolff point
$\tau$, that every trajectory converges to $\tau$.

Given $z\in\mathbb{D}$, one can also study the backward trajectory of $z$
until it reaches a point of the unit circle. In particular, if such a
trajectory can be defined for all negative “times”, then we have a backward
orbit. More formally, if $(\phi_{t})$ is a semigroup in $\mathbb{D}$, then a
continuous curve $\gamma:[0,+\infty)\to\mathbb{D}$ is called a backward orbit
if for every $t\in(0,+\infty)$ and for every $s\in[0,t],$

$\phi_{s}(\gamma(t))=\gamma(t-s).$

Backward orbits for one-parameter semigroups in $\mathbb{D}$ were initially
studied in [9] and then in [12]. All the information concerning backward
orbits needed in this article will be mostly drawn from [6] (see also [5,
Chapter 13]).

Leaving aside the semigroups, for a moment, we now need to provide some
information with regard to hyperbolic geometry in order to be able to state
our first theorem. To begin with, we denote by $\lambda_{\mathbb{D}}$ the
hyperbolic density of the unit disk, which is given by the relation

$\lambda_{\mathbb{D}}(z)|dz|=\frac{|dz|}{1-|z|^{2}}.$

Then, the hyperbolic distance between any two points $z,w$ of the unit disk is
defined as

$k_{\mathbb{D}}(z,w)=\inf\limits_{\gamma}\int\limits_{\gamma}\lambda_{\mathbb{D}},$

where the infimum is taken over all smooth curves $\gamma$ that join $z$ and
$w$ inside the unit disk. Now, let $\Omega\subsetneq\mathbb{C}$ be a simply
connected domain and $f:\Omega\to\mathbb{D}$ a corresponding Riemann map. We
denote by $\lambda_{\Omega}$ the hyperbolic density of $\Omega$ given by

$\lambda_{\Omega}(z)=\lambda_{\mathbb{D}}(f(z))\cdot|f^{\prime}(z)|,\quad
z\in\Omega,$

and by $k_{\Omega}$ the hyperbolic distance in $\Omega$ given by

$k_{\Omega}(z,w)=k_{\mathbb{D}}(f(z),f(w)),\quad z,w\in\Omega.$

It can be verified that this definition is indeed independent of the choice of
the Riemann map $f$. In addition, given a simply connected domain $\Omega$ and
a smooth curve $\gamma:(a,b)\to\Omega$ (where $-\infty\leq a<b\leq+\infty$),
we denote the hyperbolic length of $\gamma$ between the points $\gamma(t_{1})$
and $\gamma(t_{2})$, where $a<t_{1}\leq t_{2}<b$, by
$l_{\Omega}(\gamma;[t_{1},t_{2}])$. The hyperbolic length is given by the
relation

$l_{\Omega}(\gamma;[t_{1},t_{2}])=\int\limits_{\gamma}\lambda_{\Omega}(z)|dz|=\int\limits_{t_{1}}^{t_{2}}\lambda_{\Omega}(\gamma(t))|\gamma^{\prime}(t)|dt.$

The definition of the hyperbolic length can be extended in a natural way in
case the domain of $\gamma$ is of the form $[a,b],[a,b)$ or $(a,b]$.

As usual, a geodesic for the hyperbolic distance is a smooth curve such that
the hyperbolic length among any two points of the curve coincides with the
hyperbolic distance between these two points. For the purposes of this
article, we need the notion of a quasi-geodesic as well. We mostly follow [5,
Chapter 6] for the necessary information concerning quasi-geodesics. For a
more detailed study on quasi-geodesics, the reader may refer to [10].

######  Definition 1.1.

Let $\gamma:[0,+\infty)\to\mathbb{D}$ be a Lipschitz curve that satisfies
$\lim\limits_{t\to+\infty}k_{\mathbb{D}}(\gamma(0),\gamma(t))=+\infty$ (i.e.
the curve tends to the unit circle). Then, $\gamma$ is called an
($A,B$)-quasi-geodesic for the hyperbolic distance $k_{\mathbb{D}}$ of the
unit disk provided there exist absolute positive constants $A\geq 1$ and
$B\geq 0$ such that

$l_{\mathbb{D}}(\gamma;[t_{1},t_{2}])\leq A\cdot
k_{\mathbb{D}}(\gamma(t_{1}),\gamma(t_{2}))+B,$

for all $t_{2}\geq t_{1}\geq 0$.

In case there is no possibility of confusion with the constants, we might
refer to an $(A,B)$-quasi-geodesic as just a quasi-geodesic in order to
simplify the notation. Considering, in addition, the trivial inequality

$k_{\mathbb{D}}(\gamma(t_{1}),\gamma(t_{2}))\leq
l_{\mathbb{D}}(\gamma;[t_{1},t_{2}]),\quad t_{2}\geq t_{1}\geq 0,$

it is clear that, for a quasi-geodesic, the hyperbolic length among any two
points of the curve $\gamma$ is comparable to the hyperbolic distance between
these two points. Therefore, a hyperbolic quasi-geodesic presents a behavior
that is adequately close to that of the respective hyperbolic geodesic.

Moreover, keeping in mind the conformal invariance of the hyperbolic length
and the hyperbolic distance, the definition of a quasi-geodesic extends
naturally to any other simply connected domain.

In general, a quasi-geodesic does not need to be a Lipschitz curve, but merely
a continuous curve; see [10]. However, in the study of semigroups, we usually
demand quasi-geodesics to be a priori Lipschitz. For this reason, we demand
the Lipschitz condition as well.

Now, we need to somehow connect the notion of a quasi-geodesic with our study
on semigroups of holomorphic functions. In [7], the authors prove that for a
non-elliptic semigroup $(\phi_{t})$ with Denjoy-Wolff point $\tau$, every
forward trajectory that converges non-tangentially to $\tau$ is actually a
quasi-geodesic for the hyperbolic distance $k_{\mathbb{D}}$ of the unit disk
(see also [5, Theorem 17.3.3]).

A natural question that arises from this fact is whether the same is true for
any backward orbit that converges non-tangentially to a point of the unit
circle. In fact, the answer is, at least partly for now, positive and this is
the main goal of this present article. We need the following definition in
order to make a helpful distinction within the class of backward orbits of a
semigroup; see [5, p. 346].

######  Definition 1.2.

Let $(\phi_{t})$ be a semigroup in $\mathbb{D}$ and
$\gamma:[0,+\infty)\to\mathbb{D}$ a backward orbit. Then, $\gamma$ is called
regular if

$V(\gamma):=\limsup\limits_{t\to+\infty}k_{\mathbb{D}}(\gamma(t),\gamma(t+1))<+\infty.$

The quantity $V(\gamma)$ is called the hyperbolic step of $\gamma$.

To truly understand the scope of our theorems, it would be useful to mention
all the different cases with regard to the convergence of a backward orbit. To
this goal, we require some notions concerning semigroups in general. First, a
point $\sigma\in\partial\mathbb{D}$ is called a fixed point of a semigroup
$(\phi_{t})$ if $\angle\lim\limits_{z\to\sigma}\phi_{t}(z)=\sigma$, for all
$t\geq 0$. Every non-elliptic semigroup has such a fixed point, the Denjoy-
Wolff point. Any other fixed point that also has a finite angular derivative
$\phi_{t}^{{}^{\prime}}(\sigma)$ is characterized as a repelling fixed point
of the semigroup. For a repelling fixed point $\sigma$, there exists
$\mu\in(-\infty,0)$ such that $\phi_{t}^{{}^{\prime}}(\sigma)=e^{-\mu t}$, for
all $t\geq 0$. This $\mu$ is called the repelling spectral value of $\sigma$.
Finally, any fixed point $\sigma$ of the semigroup satisfying
$\phi_{t}^{{}^{\prime}}(\sigma)=\infty$, for some (and equivalently all)
$t>0$, is characterized as a super-repelling fixed point of the semigroup. The
Denjoy-Wolff point along with any repelling fixed point of the semigroup are
collectively known as the regular fixed points of the semigroup. For more
information on the fixed points of a semigroup, the reader may refer to [5,
Chapter 12].

As for the convergence, a regular backward orbit converges non-tangentially to
a point of the unit circle if and only if it converges to a repelling fixed
point of the semigroup. On the other hand, a non-regular backward orbit
converges non-tangentially only if it converges to a super-repelling fixed
point, while the converse does not hold. More details with respect to the
convergence of a backward orbit follow in Section 2.5.

We will prove the following:

######  Theorem 1.1.

Let $(\phi_{t})$ be a non-elliptic semigroup of holomorphic functions in the
unit disk. Suppose that $\gamma:[0,+\infty)\to\mathbb{D}$ is a regular
backward orbit for $(\phi_{t})$ that converges non-tangentially to a point of
the unit circle. Then, $\gamma$ is a quasi-geodesic for the hyperbolic
distance $k_{\mathbb{D}}$ of the unit disk.

Next, we are going to study backward orbits from a Euclidean perspective. We
are in need of some terminology first. Let $\Omega\subsetneq\mathbb{C}$ be a
simply connected domain. For $z\in\Omega$, we denote by $\delta_{\Omega}(z)$
the distance of $z$ from the boundary $\partial\Omega$ (i.e.
$\delta_{\Omega}(z)=\text{dist}(z,\partial\Omega))$. It can be proved (see
Section 2) that the image of a backward orbit $\gamma$ through the Koenigs
function $h$ is a horizontal half-line. If, in addition, the backward orbit
converges non-tangentially to a point of the unit circle, the carrier of the
half-line separates $\partial\Omega$, where $\Omega=h(\mathbb{D})$, into two
disjoint sets $\partial\Omega^{+}$ and $\partial\Omega^{-}$. We denote by
$\partial\Omega^{+}$ the component that is “over” the half-line $h\circ\gamma$
and by $\partial\Omega^{-}$ the one “below”. Then, for $z\in\Omega$, we set
$\delta_{\Omega}^{+}(z)=\text{dist}(z,\partial\Omega^{+})$ and
$\delta_{\Omega}^{-}(z)=\text{dist}(z,\partial\Omega^{-})$. More information
and stricter definitions follow in the next sections. We will prove the
following:

######  Theorem 1.2.

Let $(\phi_{t})$ be a non-elliptic semigroup of holomorphic functions in the
unit disk with Koenigs function $h$. Suppose that
$\gamma:[0,+\infty)\to\mathbb{D}$ is a backward orbit for $(\phi_{t})$ that
converges non-tangentially to a point of the unit circle. Set
$\Omega=h(\mathbb{D})$. Then, there exists a constant $c\geq 1$ such that

$\frac{1}{c}\cdot\delta_{\Omega}^{-}(h(\gamma(t)))\leq\delta_{\Omega}^{+}(h(\gamma(t)))\leq
c\cdot\delta_{\Omega}^{-}(h(\gamma(t))),$

for all $t\geq 0$.

In fact, the above inequality is true for every forward trajectory of a
semigroup that converges non-tangentially to the Denjoy-Wolff point (see [7]).
Therefore, it is once again only natural to assume that the same should be
true for the respective backward orbits as well.

A first interesting observation into the statement of Theorem 1.2 is the fact
that it is not limited only to regular backward orbits. Indeed, the proof that
will be provided in Section 4 holds for non-regular backward orbits as well.
Theorem 1.2 can be actually expressed in a more general statement that applies
to a wider family of curves, but for the purposes of this present article, we
will limit ourselves to backward orbits. After the proof, we will give some
final insight on backward orbits with respect to the boundary distance.

The structure of the article is as follows: in Section 2, we will provide all
the necessary information that will deem useful for the proofs of our
theorems. Then, in Section 3, we will deal with our first theorem. Finally, in
Section 4, we will present our second theorem along with some relevant
corollaries.

##  2\. Preliminary Results

###  2.1. Hyperbolic Distance

Having already introduced the notion of the hyperbolic distance, we now need
to mention some of its properties. First of all, by the definition of the
hyperbolic distance with regard to an arbitrary simply connected domain, it
becomes clear that the hyperbolic distance is conformally invariant. In
addition, the hyperbolic distance has the following monotonicity property: let
$\Omega_{1},\Omega_{2}$ be two simply connected domains, other than the
complex plane, such that $\Omega_{1}\subset\Omega_{2}$. Then, for all
$z,w\in\Omega_{1}$ we have

$k_{\Omega_{2}}(z,w)\leq k_{\Omega_{1}}(z,w).$

###  2.2. Quasi-hyperbolic Distance

Let $\Omega\subsetneq\mathbb{C}$ be a simply connected domain. Then, for any
$z_{1},z_{2}\in\Omega$, the quasi-hyperbolic distance is defined as

$d_{\Omega}(z_{1},z_{2})=\inf\limits_{\gamma}\int\limits_{\gamma}\frac{|dz|}{\delta_{\Omega}(z)},$

where the infimum is taken over all curves $\gamma$ that join $z_{1}$ and
$z_{2}$ inside $\Omega$. The following inequality ([5, Theorem 5.2.1])
correlates the hyperbolic and quasi-hyperbolic metrics:

$\frac{1}{4\delta_{\Omega}(z)}\leq\lambda_{\Omega}(z)\leq\frac{1}{\delta_{\Omega}(z)},$

for all $z\in\Omega$. An immediate corollary of the above inequality is the
fact that

$\frac{d_{\Omega}(z_{1},z_{2})}{4}\leq k_{\Omega}(z_{1},z_{2})\leq
d_{\Omega}(z_{1},z_{2}),$

for all $z_{1},z_{2}\in\Omega$. These two inequalities will play a principal
role in the proof of our theorems. In particular, this next Distance Lemma
provides a useful correlation between the hyperbolic distance and the
Euclidean distance from the boundary.

######  Lemma 2.1.

[5, Theorem 5.3.1] Let $\Omega\subsetneq\mathbb{C}$ be a simply connected
domain. Then, for every $z_{1},z_{2}\in\Omega,$

$\frac{1}{4}\log\left(1+\frac{|z_{1}-z_{2}|}{\min\\{\delta_{\Omega}(z_{1}),\delta_{\Omega}(z_{2})\\}}\right)\leq
k_{\Omega}(z_{1},z_{2})\leq\int\limits_{\Gamma}\frac{|dw|}{\delta_{\Omega}(w)},$

where $\Gamma$ is any piecewise $C^{1}$-smooth curve joining $z_{1}$ to
$z_{2}$ inside $\Omega$.

###  2.3. Harmonic Measure

We provide here some basic facts concerning a second conformal invariant, the
harmonic measure. For a comprehensive presentation of its theory, the reader
may refer to [5, Chapter 7], [13], [15, Chapter 1]. Let
$\Omega\subsetneq\mathbb{C}$ be a simply connected domain with non-polar
boundary and $B$ a Borel subset of $\partial\Omega$. Then, the harmonic
measure of $B$ with respect to $\Omega$ is exactly the solution of the
generalized Dirichlet problem for the Laplacian in $\Omega$ with boundary
function equal to $1$ on $B$ and to $0$ on $\partial\Omega\setminus B$. For
the harmonic measure of $B$ with respect to $\Omega$ and for $z\in\Omega$, we
use the notation $\omega(z,B,\Omega)$. It is known that for a fixed
$z\in\Omega$, $\omega(z,\cdot,\Omega)$ is a Borel probability measure on
$\partial\Omega$.

######  Remark 2.1.

Let $B\subset\partial\mathbb{D}$ be a circular arc with endpoints
$\sigma,\tau\in\partial\mathbb{D}$. Then (see e.g. [8, p.155]), the level set

$L_{k}=\\{z\in\mathbb{D}:\omega(z,B,\mathbb{D})=k\\},\quad 0<k<1,$

is a circular arc (or a diameter in case $B$ is a half-circle and
$k=\frac{1}{2}$) with endpoints $\sigma$ and $\tau$ that meets the unit circle
with angle $k\cdot\pi$. As a result, a sequence $\\{z_{n}\\}\subset\mathbb{D}$
that converges to $\sigma$ or $\tau$, converges non-tangentially if and only
if

$0<\liminf\limits_{n\to+\infty}\omega(z_{n},B,\mathbb{D})\leq\limsup\limits_{n\to+\infty}\omega(z_{n},B,\mathbb{D})<1.$

If either the liminf is $0$ or the limsup is $1$, then $\\{z_{n}\\}$ contains
a subsequence that converges tangentially and thus, we cannot say that the
convergence of $\\{z_{n}\\}$ is non-tangential.

###  2.4. Extremal Length

Another conformally invariant quantity we are going to need through the course
of the article is the extremal length. For a detailed presentation of the
theory of the extremal length, the interested reader may refer to [15].

Let $\Gamma$ be a family of locally rectifiable curves inside a domain
$\Omega\subsetneq\mathbb{C}$. We consider all the non-negative Borel
measurable functions $\rho$ defined on $\Omega$ such that the $\rho$-area

$A(\rho,\Omega)=\iint\limits_{\Omega}\rho(z)^{2}dxdy,\quad z=x+iy,$

is a positive real number. We define the $\rho$-length of $\Gamma$ as

$L(\rho,\Gamma)=\inf\limits_{\gamma\in\Gamma}\int\limits_{\gamma}\rho(z)|dz|.$

Then, the extremal length of $\Gamma$ with respect to $\Omega$ is the quantity

$\lambda_{\Omega}(\Gamma)=\sup\limits_{\rho}\frac{L(\rho,\Gamma)^{2}}{A(\rho,\Omega)},$

where the supremum is taken over all non-negative Borel measurable functions
$\rho$ such that $A(\rho,\Omega)\in(0,+\infty)$.

Next, we consider two families of locally rectifiable curves
$\Gamma_{1},\Gamma_{2}$ defined on two domains
$\Omega_{1},\Omega_{2}\subsetneq\mathbb{C}$ respectively. If the family
$\Gamma_{1}$ is a subfamily of the family $\Gamma_{2}$, then by the definition
of extremal length, it is easily understood that
$\lambda_{\Omega_{1}}(\Gamma_{1})\geq\lambda_{\Omega_{2}}(\Gamma_{2})$.
Moreover, we are going to need the following lemma.

######  Lemma 2.2.

[2, Lemma 8.4] If every curve of the family $\Gamma_{1}$ contains a subcurve
that belongs to the family $\Gamma_{2}$, then
$\lambda_{\Omega_{1}}(\Gamma_{1})\geq\lambda_{\Omega_{2}}(\Gamma_{2})$.

In this present article, along with the above two properties, we will mostly
need one specific instance of the extremal length. Once again, let
$\Omega\subsetneq\mathbb{C}$ be a domain and let
$E,F\subset\overline{\Omega}$. Then, the extremal distance between $E$ and $F$
inside $\Omega$ is defined as the quantity

$\lambda(E,F,\Omega)=\lambda_{\Omega}(\Gamma),$

where $\Gamma$ is the family of all locally rectifiable curves that join $E$
to $F$ inside $\Omega$. By extension, the extremal distance is also
conformally invariant.

A very useful tool that connects the extremal distance with the harmonic
measure is the following inequality, the so-called Beurling’s estimate.

######  Lemma 2.3.

[13, Theorem H.7] Let $\Omega$ be a simply connected domain and let $B$ be a
finite union of arcs lying on $\partial\Omega$. Let $z\in\Omega$. Then,

$\omega(z,B,\Omega)\leq C\cdot\exp(-\pi\cdot\lambda(\gamma,B,\Omega)),$

for every Jordan arc $\gamma$ joining $z$ to $\partial\Omega\setminus B$
inside $\Omega$, where $C>0$ is an absolute constant.

Finally, for $r\in(0,1)$, we call the doubly connected domain
$\mathbb{D}\setminus[0,r]$ a Grötzsch ring (see Figure 1). Through the study
of Grötzsch rings (see [15]), a lot of information has arisen about the
extremal distance between the line segment $[0,r]$ and the unit circle inside
the unit disk. In particular, it has been proved that $\lim\limits_{r\to
0^{+}}\lambda([0,r],\partial\mathbb{D},\mathbb{D})=+\infty$ (see [15, Theorem
2.80]), a fact that we will need in the final stages of the proof of Theorem
1.2.

Figure 1. A Grötzsch ring

###  2.5. Backward Orbits

Let $\gamma:[0,+\infty)\to\mathbb{D}$ be a backward orbit for a non-elliptic
semigroup $(\phi_{t})$ with Denjoy-Wolff point $\tau$ and Koenigs function
$h$. Set $\Omega=h(\mathbb{D})$. By the definition of backward orbits, we have

$\phi_{t}(\gamma(t))=\gamma(t-t)=\gamma(0),$

for all $t\geq 0$. Remembering the basic property of the Koenigs function, we
get $h(\gamma(t))=h(\gamma(0))-t,t\geq 0$. Therefore, we see that $h$
linearizes backward orbits, turning them to half-lines that converge to
$\infty$ through the negative direction (i.e. with strictly decreasing real
part and constant imaginary part).

Next, since $\gamma(0)\in\mathbb{D}$, we can find the forward trajectory of
$\gamma(0)$ under the semigroup along with its image through $h$. More
precisely, we can find the image of the set $\\{\phi_{t}(\gamma(0)):t\geq
0\\}$. In a similar fashion, its image in $\Omega$ is the half-line
$\\{h(\gamma(0))+t:t\geq 0\\}$ which converges to $\infty$ through the
positive direction. As a result, the union of the images of the backward orbit
and the forward trajectory that emanate from $\gamma(0)$ is the horizontal
line $\\{h(\gamma(0))+t:t\in\mathbb{R}\\}$. Clearly, this line separates
$\partial\Omega$ into two disjoint sets
$\partial\Omega^{+}=\\{w\in\partial\Omega:\Im w>\Im h(\gamma(0))\\}$ and
$\partial\Omega^{-}=\\{w\in\partial\Omega:\Im w<\Im h(\gamma(0))\\}$.

Having already mentioned some details in the Introduction, we need some
information concerning the convergence of a backward orbit. All the following
results are drawn from [5, Chapter 13]. If $\gamma$ is regular, then either
$\lim\limits_{t\to+\infty}\gamma(t)=\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$,
where $\sigma$ is a repelling fixed point of the semigroup, and the
convergence is non-tangential or $\lim\limits_{t\to+\infty}\gamma(t)=\tau$ and
the convergence is tangential. In the former case, the associated planar
domain $\Omega$ must contain a maximal horizontal strip and the image of the
backward orbit through $h$ is contained in this strip. In the latter case,
$\Omega$ must contain a maximal horizontal half-plane and the image of the
backward orbit through $h$ is contained in this half-plane.

Here, it seems useful to make some comments concerning prime ends. Since $h$
maps the unit disk $\mathbb{D}$ conformally onto the Koenigs domain $\Omega$,
by Carathédory’s Theorem ([5, Theorem 4.2.3]), the points of the unit circle
$\partial\mathbb{D}$ correspond one-to-one to the prime ends of $\Omega$.
However, multiple prime ends of $\Omega$ might correspond to the same point of
$\partial\Omega$. For example, if for some backward orbit
$\gamma:[0,+\infty)\to\mathbb{D}$ both $\partial\Omega^{+}$ and
$\partial\Omega^{-}$ are non-empty, then there are at least two prime ends of
$\Omega$ corresponding to $\infty$. This is because
$\lim\limits_{t\to+\infty}h(\gamma(t))=\lim\limits_{t\to+\infty}h(\phi_{t}(\gamma(0)))=\infty$,
while $\lim\limits_{t\to+\infty}\gamma(t)\neq\tau$. Combining all the above,
we can say that $\partial\Omega^{+}$ corresponds to a set of prime ends of
$\Omega$ which in turn corresponds through $h^{-1}$ to a set of points
$\partial\mathbb{D}^{+}$ on the unit circle. Similarly, $\partial\Omega^{-}$
corresponds to a set $\partial\mathbb{D}^{-}$. It is easy to check that
$\partial\mathbb{D}^{+}$ and $\partial\mathbb{D}^{-}$ are disjoint circular
arcs with endpoints $\lim\limits_{t\to+\infty}\gamma(t)$ and $\tau$, whose
union is the whole unit circle. For more information on prime ends and how
they connect to semigroups of holomorphic functions, we refer to [5, Chapter
11].

Using a previous remark that the image of a backward orbit in $\Omega$ must
tend to $\infty$ through the negative direction, we can understand that there
are only three categories left for the image through $h$ of a non-regular
backward orbit (see Figure 2):

(a) $\gamma_{1}$ is non-regular, $\gamma_{2}$ is regular and they both
converge to a repelling point

(b) $\gamma_{1}$ is non-regular, $\gamma_{2}$ is regular and they both
converge to the Denjoy-Wolff point

(c) $\gamma$ is non-regular and converges to a super-repelling point

Figure 2. Images of backward orbits through the Koenigs function

  1. (i)

either it is part of the boundary of a maximal horizontal strip, in which case
the non-regular backward orbit itself converges tangentially to a repelling
fixed point,

  2. (ii)

or it is part of the boundary of a maximal horizontal half-plane, in which
case the non-regular backward orbit itself converges tangentially to the
Denjoy-Wolff point,

  3. (iii)

or the image gets arbitrarily close to both $\partial\Omega^{+}$ and
$\partial\Omega^{-}$, in which case the non-regular backward orbit itself
converges to a super-repelling fixed point either tangentially or non-
tangentially, or can even contain sequences of points that converge
tangentially and other sequences that converge non-tangentially.

It can be easily proved that any backward orbit that converges tangentially to
a point of the unit circle cannot be a quasi-geodesic for the hyperbolic
distance $k_{\mathbb{D}}$ of the unit disk. Therefore, we limit our search for
quasi-geodesics, only in the case of a backward orbit converging non-
tangentially.

###  2.6. Infinitesimal Generators

Let $(\phi_{t})$ be a semigroup in $\mathbb{D}$. Then, there exists a unique
holomorphic function $G:\mathbb{D}\to\mathbb{C}$ such that

$\frac{\partial\phi_{t}(z)}{\partial t}=G(\phi_{t}(z)),\quad\quad
z\in\mathbb{D},\quad t\geq 0.$

The function $G$ is called the infinitesimal generator of $(\phi_{t})$; see
[5, Chapter 10]. We need the notion of the infinitesimal generators in order
to prove that certain backward orbits are actually Lipschitz curves. We are
going to mainly need the following lemma:

######  Lemma 2.4.

[5, Proposition 12.2.2] Let $G$ be the infinitesimal generator of a semigroup
in $\mathbb{D}$ and let $\sigma\in\partial\mathbb{D}$ be a repelling fixed
point of the semigroup. Then, $\angle\lim\limits_{z\to\sigma}G(z)=0$.

##  3\. Backward Orbits and Quasi-Geodesics

The first goal of this article is to prove that any regular backward orbit
that converges non-tangentially to a point of the unit circle is a quasi-
geodesic for the hyperbolic distance of the unit disk. By the definition of
quasi-geodesics, we first need to show that any such backward orbit is a
Lipschitz curve. The proof of the following proposition follows almost exactly
the same steps as the proof of Proposition 10.1.7(4) in [5].

######  Proposition 3.1.

Let $(\phi_{t})$ be a non-elliptic semigroup in $\mathbb{D}$ and let
$\gamma:[0,+\infty)\to\mathbb{D}$ be a regular backward orbit for $(\phi_{t})$
that converges to a point of the unit circle non-tangentially. Then $\gamma$
is a Lipschitz curve.

###### Proof.

Let $\lim\limits_{t\to+\infty}\gamma(t)=\sigma$. By Section 2.5, the fact that
$\gamma$ converges non-tangentially to $\sigma$, forces $\sigma$ to be a
repelling fixed point of $(\phi_{t})$. Let $s\geq t_{2}\geq t_{1}\geq 0$ and
set $s_{1}=s-t_{1}$ and $s_{2}=s-t_{2}$. Obviously, $s_{1}\geq s_{2}$. Then,

$|\gamma(t_{2})-\gamma(t_{1})|=|\gamma(s-s_{2})-\gamma(s-s_{1})|=|\phi_{s_{2}}(\gamma(s))-\phi_{s_{1}}(\gamma(s))|,$

where the last equality follows from the definition of a backward orbit. Let
$G$ be the infinitesimal generator of $(\phi_{t})$. Then,

$\displaystyle|\gamma(t_{2})-\gamma(t_{1})|$ $\displaystyle=$
$\displaystyle|\phi_{s_{1}}(\gamma(s))-\phi_{s_{2}}(\gamma(s))|$
$\displaystyle\leq$
$\displaystyle\int\limits_{s_{2}}^{s_{1}}\left|\frac{\partial\phi_{t}(\gamma(s))}{\partial
t}\right|dt$ $\displaystyle=$
$\displaystyle\int\limits_{s_{2}}^{s_{1}}|G(\phi_{t}(\gamma(s)))|dt$
$\displaystyle=$ $\displaystyle\int\limits_{s_{2}}^{s_{1}}|G(\gamma(s-t))|dt.$

However, $\lim\limits_{t\to+\infty}\gamma(t)=\sigma$ and
$\angle\lim\limits_{z\to\sigma}G(z)=0$. As a consequence, $G$ is bounded on
every Stolz angle with vertex $\sigma$. Since $\gamma$ converges to $\sigma$
non-tangentially, there must exist a Stolz angle $S$ with vertex $\sigma$ such
that $\gamma([0,+\infty))\subset S$. Therefore, there exists an absolute
constant $C>0$, depending only on $S$, such that $|G(\gamma(t))|<C$, for every
$t\geq 0$. Combining with the above relations, we get
$|\gamma(t_{2})-\gamma(t_{1})|\leq C(s_{1}-s_{2})=C(t_{2}-t_{1})$. This last
relation yields the desired result. ∎

Now, we are ready for the proof of Theorem 1.1.

Proof of Theorem 1.1 First of all, since $\gamma(0)\in\mathbb{D}$ and $\gamma$
converges to a point of the unit circle, it is trivial that
$\lim\limits_{t\to+\infty}k_{\mathbb{D}}(\gamma(0),\gamma(t))=+\infty$. As a
result, $\gamma$ satisfies the elementary necessary condition to be considered
as a candidate for a quasi-geodesic. Moreover, as we showed before, if
$\gamma$ converged to the Denjoy-Wolff point $\tau$, the convergence would be
tangential. Therefore, $\gamma$ converges to a point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ and $\sigma$ must be a
repelling fixed point of $(\phi_{t})$, because $\gamma$ is regular. For the
remainder of the proof, we denote by $\mu\in(-\infty,0)$ the repelling
spectral value of $\sigma$.

Set $\Omega=h(\mathbb{D})$, where $h$ is the Koenigs function of $(\phi_{t})$.
According to [4], $\Omega$ must contain a maximal horizontal strip of
amplitude $-\frac{\pi}{\mu}$. We denote by $S$ this maximal strip. Then, we
can write $S=\\{z\in\mathbb{C}:|\Im z-a^{*}|<-\frac{\pi}{2\mu}\\}$, where
$a^{*}=\lim\limits_{r\to 1^{-}}\Im h(r\sigma)$. We know that the set
$\Delta=h^{-1}(S)$ is called a petal of $(\phi_{t})$ and every regular
backward orbit converging to $\sigma$ is contained inside $\Delta$. Therefore,
$\gamma([0,+\infty))\subset\Delta$ or equivalently
$h\circ\gamma([0,+\infty))\subset S$. In addition, since $\gamma$ is a
backward orbit, it is true that

$\phi_{t}(\gamma(t))=\gamma(t-t)=\gamma(0),$

for all $t\geq 0$. Turning to $\Omega$ through $h$,
$h(\phi_{t}(\gamma(t)))=h(\gamma(t))+t$, which yields

$h(\gamma(t))=h(\gamma(0))-t,$

for all $t\geq 0$. On the other hand, a similar computation shows that the
image of the forward trajectory of $\gamma(0)$ through $h$ is the horizontal
half-line $\\{h(\gamma(0))+t:t\geq 0\\}$. Thus, the union of the images of the
backward and forward trajectories of $\gamma(0)$ through $h$ is a horizontal
line which separates $\partial\Omega$ into two disjoint sets. We write
$\partial\Omega^{+}=\\{w\in\partial\Omega:\Im w>\Im h(\gamma(0))\\}$ and
$\partial\Omega^{-}=\\{w\in\partial\Omega:\Im w<\Im h(\gamma(0))\\}$. It is
easy to see that $\partial\Omega^{+}\cap\partial\Omega^{-}=\emptyset$ and
$\partial\Omega^{+}\cup\partial\Omega^{-}=\partial\Omega$. For $z\in\Omega$,
we have already set $\delta_{\Omega}(z)=\text{dist}(z,\partial\Omega)$. For
the remainder of the article, we also set
$\delta_{\Omega}^{+}(z)=\text{dist}(z,\partial\Omega^{+})$ and
$\delta_{\Omega}^{-}(z)=\text{dist}(z,\partial\Omega^{-})$.

We find $p^{+}=x^{+}+iy^{+}\in\partial\Omega^{+}$ such that
$\delta_{\Omega}^{+}(h(\gamma(0)))=|h(\gamma(0))-p^{+}|$. Likewise, we find
$p^{-}=x^{-}+iy^{-}\in\partial\Omega^{-}$ such that
$\delta_{\Omega}^{-}(h(\gamma(0)))=|h(\gamma(0))-p^{-}|$. Setting
$x=\min\\{x^{+},x^{-}\\}$ and by the convexity in the positive direction of
$\Omega$, we get that the horizontal half-lines $L^{+}=\\{w:\Re w\leq x,\Im
w=y^{+}\\}$ and $L^{-}=\\{w:\Re w\leq x,\Im w=y^{-}\\}$ are both subsets of
$\mathbb{C}\setminus\Omega$. Therefore, $S\subset\Omega\subset
B:=\mathbb{C}\setminus(L^{+}\cup L^{-})$ (see Figure 3). Moreover, since
$\lim\limits_{t\to+\infty}\Re h(\gamma(t))=-\infty$, there exists a $t_{0}\geq
0$ such that $\Re h(\gamma(t))\leq x$, for all $t\geq t_{0}$.

Figure 3. The sets of Theorem 1.1

Given the geometrical properties of $S$ and $B$ and the fact that the set
$h(\gamma([0,+\infty)))$ is a horizontal half-line, there exist positive
constants $c,d$ such that $\delta_{S}(h(\gamma(t)))=c$ and
$\delta_{B}(h(\gamma(t)))=d$, for all $t\geq t_{0}$. Combining all the above
and making use of the monotonicity and the conformal invariance of the
hyperbolic metric, we have for $t_{2}\geq t_{1}\geq t_{0}$:

$\displaystyle
l_{\mathbb{D}}(\gamma;[t_{1},t_{2}])=l_{\Omega}(h\circ\gamma;[t_{1},t_{2}])$
$\displaystyle\leq$ $\displaystyle l_{S}(h\circ\gamma;[t_{1},t_{2}])$
$\displaystyle=$
$\displaystyle\int\limits_{t_{1}}^{t_{2}}\lambda_{S}(h(\gamma(t)))|(h\circ\gamma)^{\prime}(t)|dt$
$\displaystyle\leq$
$\displaystyle\int\limits_{t_{1}}^{t_{2}}\frac{|(h\circ\gamma)^{\prime}(t)|dt}{\delta_{S}(h(\gamma(t)))}$
$\displaystyle=$
$\displaystyle\frac{4d}{c}\int\limits_{t_{1}}^{t_{2}}\frac{|(h\circ\gamma)^{\prime}(t)|dt}{4\delta_{B}(h(\gamma(t)))}.$

The horizontal line $\\{w:\Im w=\frac{y^{+}+y^{-}}{2}\\}$ is an axis of
symmetry for $B$. Therefore, the curve $\eta:[t_{0},+\infty)\to B$ with
$\eta(t)=\Re h(\gamma(t))+i\frac{y^{+}+y^{-}}{2}$ is a geodesic arc for the
hyperbolic distance $k_{B}$ of $B$. Setting $\epsilon=\frac{y^{+}-y^{-}}{2}$,
it is evident that $\delta_{B}(\eta(t))=\epsilon$, for all $t\geq t_{0}$.
Furthermore, an easy direct computation shows that
$|(h\circ\gamma)^{\prime}(t)|=|\eta^{\prime}(t)|=1$, for all $t\geq t_{0}$.
Continuing the above relations, we get

$\displaystyle l_{\mathbb{D}}(\gamma;[t_{1},t_{2}])$ $\displaystyle\leq$
$\displaystyle\frac{4\epsilon}{c}\int\limits_{t_{1}}^{t_{2}}\frac{|(h\circ\gamma)^{\prime}(t)|dt}{4\delta_{B}(\eta(t))}$
$\displaystyle=$
$\displaystyle\frac{4\epsilon}{c}\int\limits_{t_{1}}^{t_{2}}\frac{|\eta^{\prime}(t)|dt}{4\delta_{B}(\eta(t))}$
$\displaystyle\leq$
$\displaystyle\frac{4\epsilon}{c}l_{B}(\eta;[t_{1},t_{2}])$ $\displaystyle=$
$\displaystyle\frac{4\epsilon}{c}k_{B}(\eta(t_{1}),\eta(t_{2})),$

for all $t_{2}\geq t_{1}\geq t_{0}$, where the last equality is due to the
fact that $\eta$ is a geodesic arc. However, by the triangle inequality, we
get

$k_{B}(\eta(t_{1}),\eta(t_{2}))\leq
k_{B}(h(\gamma(t_{1})),h(\gamma(t_{2})))+\sum\limits_{i=1}^{2}k_{B}(h(\gamma(t_{i})),\eta(t_{i})).$

For $t\geq t_{0}$, we denote by $\Gamma_{t}$ the vertical line segment joining
$\eta(t)$ to $h(\gamma(t))$. We can parametrize it as follows:
$\Gamma_{t}:[0,1]\to B$, with $\Gamma_{t}(s)=(1-s)\eta(t)+s\cdot
h(\gamma(t))$. Then, by Lemma 2.1,

$k_{B}(\eta(t),h(\gamma(t)))\leq\int\limits_{0}^{1}\frac{|\Gamma_{t}^{\prime}(s)|ds}{\delta_{B}(\Gamma_{t}(s))}\leq\int\limits_{0}^{1}\frac{|h(\gamma(t))-\eta(t)|ds}{\min\limits_{s\in[0,1]}\delta_{B}(\Gamma_{t}(s))}\leq\int\limits_{0}^{1}\frac{\epsilon
ds}{d}=\frac{\epsilon}{d}.$

Combining this last result with the triangle inequality and returning to the
previous relations, we find that

$l_{\mathbb{D}}(\gamma;[t_{1},t_{2}])\leq\frac{4\epsilon}{c}k_{B}(h(\gamma(t_{1})),h(\gamma(t_{2})))+\frac{8\epsilon^{2}}{cd},$

for all $t_{2}\geq t_{1}\geq t_{0}$. But $\Omega\subset B$, so

$k_{B}(h(\gamma(t_{1})),h(\gamma(t_{2})))\leq
k_{\Omega}(h(\gamma(t_{1})),h(\gamma(t_{2})))=k_{\mathbb{D}}(\gamma(t_{1}),\gamma(t_{2})).$

Setting $A=\frac{4\epsilon}{c}$ and
$B=\frac{8\epsilon^{2}}{cd}+l_{\mathbb{D}}(\gamma;[0,t_{0}])$, we finally get

$l_{\mathbb{D}}(\gamma;[t_{1},t_{2}])\leq A\cdot
k_{\mathbb{D}}(\gamma(t_{1}),\gamma(t_{2}))+B,$

for all $t_{2}\geq t_{1}\geq 0$, and this yields the desired result. ∎

######  Remark 3.1.

Perhaps the condition of regularity for Theorem 1.1 is actually not necessary.
Indeed, even if $\gamma$ is non-regular, it can still be a quasi-geodesic and
even a geodesic. This happens, for instance, when $\Omega$ contains no
horizontal strips and is symmetric with respect to a horizontal line and
$h(\gamma([0,+\infty)))$ lies on this line. So, a natural conjecture is that
Theorem 1.1 is true for non-regular backward orbits that converge non-
tangentially as well.

##  4\. Backward Orbits and Boundary Distance

In this section, we will give a property of backward orbits in terms of
Euclidean geometry. We will first prove Theorem 1.2 and in the end we will
state some corollaries.

Proof of Theorem 1.2 We aim for a contradiction. To this goal, suppose that
$\limsup\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t))}=+\infty$.
Then, there exists a strictly increasing sequence
$\\{t_{n}\\}\subset[0,+\infty)$ with $\lim\limits_{n\to+\infty}t_{n}=+\infty$
such that
$\lim\limits_{n\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t_{n})))}{\delta_{\Omega}^{-}(h(\gamma(t_{n})))}=+\infty$.
For the sake of convenience in notation, we set
$\delta^{+}(n)=\delta_{\Omega}^{+}(h(\gamma(t_{n})))$ and
$\delta^{-}(n)=\delta_{\Omega}^{-}(h(\gamma(t_{n})))$. Without loss of
generality, by extracting a subsequence, we can assume that
$\delta^{+}(n)>\delta^{-}(n)$, for all $n\in\mathbb{N}$. In addition, we
denote by $p_{n}^{+}$ a point of $\partial\Omega^{+}$ such that
$\delta^{+}(n)=|h(\gamma(t_{n}))-p_{n}^{+}|$ and similarly, by $p_{n}^{-}$ a
point of $\partial\Omega^{-}$ such that
$\delta^{-}(n)=|h(\gamma(t_{n}))-p_{n}^{-}|$. Moreover, let $\gamma_{n}^{+}$
be the line segment $[h(\gamma(t_{n})),p_{n}^{+})$ and $\gamma_{n}^{-}$ the
line segment $[h(\gamma(t_{n})),p_{n}^{-})$. Clearly,
$\gamma_{n}^{+},\gamma_{n}^{-}\subset\Omega$, for all $n\in\mathbb{N}$ (see
Figure 4). Finally, we denote by $\Gamma_{n}^{-}$ the family of all Jordan
arcs that join $h(\gamma(t_{n}))$ to $\partial\Omega^{-}$ inside $\Omega$.
Obviously, $\gamma_{n}^{-}\in\Gamma_{n}^{-}$, for all $n\in\mathbb{N}$.

Figure 4. The sets of Theorem 1.2

We need to return, for a bit, to the unit disk $\mathbb{D}$. The line
$\\{w:\Im w=\Im h(\gamma(0))\\}$ separates $\partial\Omega$ into
$\partial\Omega^{+}$ and $\partial\Omega^{-}$. We know that this line is the
image through $h$ of the set
$\gamma([0,+\infty))\cup\\{\phi_{t}(\gamma(0)):t\geq 0\\}$, where
$\\{\phi_{t}(\gamma(0)):t\geq 0\\}$ is the forward trajectory of $\gamma(0)$
under the semigroup $(\phi_{t})$. The above union is a maximal invariant curve
of $(\phi_{t})$ (see [5, Section 13.3]) and separates the unit circle
$\partial\mathbb{D}$ into two circular arcs $\partial\mathbb{D}^{+}$ and
$\partial\mathbb{D}^{-}$ with endpoints the Denjoy-Wolff point $\tau$ and the
point $\sigma$ where $\gamma$ converges. Let’s say that
$\partial\mathbb{D}^{+}$ corresponds through $h$ to $\partial\Omega^{+}$ and
$\partial\mathbb{D}^{-}$ to $\partial\Omega^{-}$ respectively. Then, by the
bijective nature of the Koenigs function $h$, the family of all Jordan arcs
that join $\gamma(t_{n})$ to $\partial\mathbb{D}^{-}$ inside $\mathbb{D}$
certainly contains the family $h^{-1}(\Gamma_{n}^{-})$.

For $n\in\mathbb{N}$, Beurling’s estimate in Lemma 2.3 yields

$\omega(\gamma(t_{n}),\partial\mathbb{D}^{+},\mathbb{D})\leq
C\cdot\exp(-\pi\cdot\lambda(\xi,\partial\mathbb{D}^{+},\mathbb{D})),$

for all Jordan arcs $\xi\in h^{-1}(\Gamma_{n}^{-})$. Using the conformal
invariance of both the harmonic measure and the extremal distance, we get

$\omega(h(\gamma(t_{n})),\partial\Omega^{+},\Omega)\leq
C\cdot\exp(-\pi\cdot\lambda(\xi^{\prime},\partial\Omega^{+},\Omega)),$

for all Jordan arcs $\xi^{\prime}\in\Gamma_{n}^{-}$. Since
$\gamma_{n}^{-}\in\Gamma_{n}^{-}$, taking limits as $n\to+\infty$, we have

(1)
$\limsup\limits_{n\to+\infty}\omega(h(\gamma(t_{n})),\partial\Omega^{+},\Omega)\leq
C\cdot\exp(-\pi\cdot\liminf\limits_{n\to+\infty}\lambda(\gamma_{n}^{-},\partial\Omega^{+},\Omega)).$

We set $D_{n}^{-}=D(h(\gamma(t_{n})),\delta^{-}(n))$, where by $D(z,r)$,
$z\in\mathbb{C},r>0$, we denote the Euclidean disk of center $z$ and radius
$r$. Then, the line segment $\gamma_{n}^{-}$ is a radius of $D_{n}^{-}$. In a
similar manner, we consider the disks
$D_{n}^{+}=D(h(\gamma(t_{n})),\delta^{+}(n))$, each one having the line
segment $\gamma_{n}^{+}$ as its radius. Bearing in mind the assumption
$\delta^{+}(n)>\delta^{-}(n)$, by construction, $D_{n}^{-}\subset\Omega$, for
all $n\in\mathbb{N}$, while the same is not true for the disks $D_{n}^{+}$.
Nevertheless, it is evident that $\gamma_{n}^{-}\subset D_{n}^{+}$, for all
$n\in\mathbb{N}$ (see Figure 4).

We denote by $\Omega_{n}$ the connected component of $\Omega\cap D_{n}^{+}$
that contains $\gamma_{n}^{-}$ and set
$E_{n}=\partial\Omega_{n}\setminus\partial\Omega^{-}$. Every curve that joins
$\gamma_{n}^{-}$ to $\partial\Omega^{+}$ inside $\Omega$ contains a subcurve
joining $\gamma_{n}^{-}$ to $E_{n}$ inside $\Omega_{n}$. By Lemma 2.2,

$\lambda(\gamma_{n}^{-},\partial\Omega^{+},\Omega)\geq\lambda(\gamma_{n}^{-},E_{n},\Omega_{n}),$

for every $n\in\mathbb{N}$. Moreover, the family of curves joining
$\gamma_{n}^{-}$ to $E_{n}$ inside $\Omega_{n}$ is a subfamily of the family
of curves joining $\gamma_{n}^{-}$ to $\partial D_{n}^{+}$ inside $D_{n}^{+}$.
Therefore,

$\lambda(\gamma_{n}^{-},E_{n},\Omega_{n})\geq\lambda(\gamma_{n}^{-},\partial
D_{n}^{+},D_{n}^{+}),$

for every $n\in\mathbb{N}$. Combining, we get

(2)
$\lambda(\gamma_{n}^{-},\partial\Omega^{+},\Omega)\geq\lambda(\gamma_{n}^{-},\partial
D_{n}^{+},D_{n}^{+}),$

for all $n\in\mathbb{N}$. By the conformal invariance of the extremal
distance, we have

$\displaystyle\lambda(\gamma_{n}^{-},\partial D_{n}^{+},D_{n}^{+})$
$\displaystyle=$
$\displaystyle\lambda(\gamma_{n}^{-}-h(\gamma(t_{n})),\partial
D_{n}^{+}-h(\gamma(t_{n})),D(0,\delta^{+}(n)))$ $\displaystyle=$
$\displaystyle\lambda\left(\frac{\gamma_{n}^{-}-h(\gamma(t_{n}))}{\delta^{+}(n)},\partial\mathbb{D},\mathbb{D}\right).$

The line segment $\gamma_{n}^{-}-h(\gamma(t_{n}))$ starts at $0$, while its
length is $\delta^{-}(n)$, so by a suitable rotation we can map the line
segment $\frac{\gamma_{n}^{-}-h(\gamma(t_{n}))}{\delta^{+}(n)}$ conformally
onto the line segment $\left[0,\frac{\delta^{-}(n)}{\delta^{+}(n)}\right).$
Therefore, through this chain of conformal mappings, we get

$\lambda(\gamma_{n}^{-},\partial
D_{n}^{+},D_{n}^{+})=\lambda\left(\left[0,\frac{\delta^{-}(n)}{\delta^{+}(n)}\right],\partial\mathbb{D},\mathbb{D}\right).$

By our hypothesis,
$\lim\limits_{n\to+\infty}\frac{\delta^{-}(n)}{\delta^{+}(n)}=0$. Through the
theory of Grötzsch domains, this implies that
$\lim\limits_{n\to+\infty}\lambda(\gamma_{n}^{-},\partial
D_{n}^{+},D_{n}^{+})=+\infty$. Returning to (2), this yields
$\lim\limits_{n\to+\infty}\lambda(\gamma_{n}^{-},\partial\Omega^{+},\Omega)=+\infty$.
Finally, going back to (1) and Beurling’s estimate, we find that
$\lim\limits_{n\to+\infty}\omega(h(\gamma(t_{n})),\partial\Omega^{+},\Omega)=0$.
By Remark 2.1, this signifies that the sequence $\\{\gamma(t_{n})\\}$
converges to $\sigma$ tangentially, as $n\to+\infty$. Contradiction!

Consequently,
$\limsup\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}<+\infty$.
Following a similar procedure, we can prove that
$\liminf\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}>0$.
The combination of these last two relations certifies that there exists indeed
an absolute constant $c\geq 1$ such that

$\frac{1}{c}\cdot\delta_{\Omega}^{-}(h(\gamma(t)))\leq\delta_{\Omega}^{+}(h(\gamma(t)))\leq
c\cdot\delta_{\Omega}^{-}(h(\gamma(t))),$

for all $t\geq 0$. Thus, Theorem 1.2 is proved. ∎

######  Remark 4.1.

As we can see from the proof, the curve $\gamma$ needed to just be a backward
orbit. There was no added condition of regularity, contrary to Theorem 1.1.
Furthermore, the proof itself shows that if either
$\liminf\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}=0$
or
$\limsup\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}=+\infty$
(or both), then $\gamma$ converges to $\sigma$ by a set of angles that
contains either $\\{0\\}$ or $\\{\pi\\}$ (or both).

The preceding remark leads to a partial converse of Theorem 1.2. Before that,
we need to return to boundary distances again. In some cases, the image of a
backward orbit $\gamma$ through the Koenigs function $h$ might not separate
the boundary of $\Omega=h(\mathbb{D})$ into two components. For example, if
$\gamma$ is a regular backward orbit that converges to the Denjoy-Wolff point
$\tau$ of a semigroup $(\phi_{t})$, then $\Omega$ contains a maximal
horizontal half-plane $H$ and $h(\gamma([0,+\infty)))\subset H$. Suppose that
the half-plane is of the form $H=\\{w\in\mathbb{C}:\Im w>a\\}$, for some
$a\in\mathbb{R}$ (see Figure 2). Then, maintaining the previous symbolisms,
$\partial\Omega^{+}=\emptyset$, while
$\lim\limits_{t\to+\infty}\delta_{\Omega}^{-}(h(\gamma(t)))=\Im
h(\gamma(0))-a$ due to the maximality of $H$. In such a case, we can write
$\delta_{\Omega}^{+}(h(\gamma(t)))=+\infty$, for all $t\in[0+\infty)$. With
this in mind, we have the following corollary.

######  Corollary 4.1.

Let $(\phi_{t})$ be a non-elliptic semigroup in $\mathbb{D}$ with Koenigs
function $h$. Suppose that $\gamma:[0,+\infty)\to\mathbb{D}$ is a backward
orbit for $(\phi_{t})$ that converges tangentially to a regular fixed point of
the semigroup. Then, the limit
$\lim\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}$
is equal to 0 or $+\infty$.

###### Proof.

The type of convergence in the hypothesis can happen in three distinct cases,
as we have already seen in Section 2.5: either $\gamma$ is regular and
converges to the Denjoy-Wolff point, or $\gamma$ is non-regular and converges
to the Denjoy-Wolff point, or finally $\gamma$ is non-regular and converges to
a repelling fixed point of $(\phi_{t})$. In the first case,
$h\circ\gamma([0,+\infty))$ is contained in a maximal horizontal half-plane
contained in $\Omega=h(\mathbb{D})$. In the second case,
$h\circ\gamma([0,+\infty))$ lies on the boundary of such a half-plane.
Finally, in the third case, $h\circ\gamma([0,+\infty))$ lies on the boundary
of a maximal horizontal strip contained in $\Omega$. It is easy to see that in
any case, the desired result is true.

∎

Studying again the proof of Theorem 1.2, we can also make a conclusion about a
non-regular backward orbit $\gamma:[0,+\infty)\to\mathbb{D}$ that lands at a
super-repelling fixed point $\sigma$. In this case, as we mentioned before,
the convergence can be of any kind. More specifically, the cluster set of
$\arg(1-\bar{\sigma}\gamma(t))$ can be any compact and connected subset of
$[-\frac{\pi}{2},\frac{\pi}{2}]$ (see [14]). Keeping in mind that conformal
mappings preserve orientations and angles, it is directly deduced that:

  1. (i)

$\liminf\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}=0$
if and only if
$\limsup\limits_{t\to+\infty}\arg(1-\bar{\sigma}\gamma(t))=\frac{\pi}{2}$
(i.e. there exists a strictly increasing sequence
$\\{t_{n}\\}\subset[0,+\infty)$ with $\lim\limits_{n\to+\infty}t_{n}=+\infty$
such that $\\{\gamma(t_{n})\\}$ converges to $\sigma$ by angle $0$),

  2. (ii)

$\limsup\limits_{t\to+\infty}\frac{\delta_{\Omega}^{+}(h(\gamma(t)))}{\delta_{\Omega}^{-}(h(\gamma(t)))}=+\infty$
if and only if
$\liminf\limits_{t\to+\infty}\arg(1-\bar{\sigma}\gamma(t))=-\frac{\pi}{2}$
(i.e. there exists a strictly increasing sequence
$\\{t_{n}\\}\subset[0,+\infty)$ with $\lim\limits_{n\to+\infty}t_{n}=+\infty$
such that $\\{\gamma(t_{n})\\}$ converges to $\sigma$ by angle $\pi$).

Combining Theorems 1.1 and 1.2 with Corollary 4.1, we directly get one final
corollary.

######  Corollary 4.2.

Let $(\phi_{t})$ be a non-elliptic semigroup in $\mathbb{D}$ with Koenigs
function $h$. Suppose that $\gamma:[0,+\infty)\to\mathbb{D}$ is a regular
backward orbit for $(\phi_{t})$. Then, the following are equivalent:

  1. (i)

$\gamma$ converges non-tangentially to a point of the unit circle,

  2. (ii)

$\gamma$ is a quasi-geodesic for the hyperbolic distance $k_{\mathbb{D}}$ of
the unit disk,

  3. (iii)

there exists a constant $c\geq 1$ such that

$\frac{1}{c}\cdot\delta_{\Omega}^{-}(h(\gamma(t)))\leq\delta_{\Omega}^{+}(h(\gamma(t)))\leq
c\cdot\delta_{\Omega}^{-}(h(\gamma(t))),$

for all $t\geq 0$.

Analogous results with the above hold for forward trajectories as well (see
[5, Corollary 17.3.2, Theorem 17.3.3]).

######  Remark 4.2.

Observing all the proofs of all the aforementioned theorems and corollaries,
it is easily understood that any difficulty is mostly due to super-repelling
fixed points. For example, Corollary 4.1 (or Corollary 4.2) might seize to
hold if we let $\gamma$ converge tangentially (or non-tangentially
respectively) to a super-repelling fixed point. A natural conjecture is that
both corollaries can be generalized for such backward orbits as well.

## Acknowledgments

I thank Professor D. Betsakos for the helpful conversations during the
preparation of this work.

I also thank Professor M. D. Contreras for the helpful correspondence
concerning quasi-geodesics.

## References

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  * [2] G. D. Anderson, M. K. Vamanamurthy, M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley and Sons, Inc., 1997
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