$\sigma^{\prime}\neq\sigma$) such that
$\lim_{t\to-\infty}h^{-1}(\gamma(t))=\sigma^{\prime}$. Also, by construction,
$\lim_{t\to+\infty}h^{-1}(\gamma(t))=\sigma$. Thus, by $(\star)$, both
connected components of $\mathbb{C}_{\infty}\setminus\Gamma$ are not contained
in $\Omega$.

Let $c^{\prime\prime}\in(a_{-}(\sigma),c^{\prime})$. By construction, there
exists $T_{0}\in\mathbb{R}$ such that
$((-\infty,T_{0}]+ic^{\prime\prime})\cap\Gamma=\emptyset$. Let $U$ be the
connected component of $\mathbb{C}_{\infty}\setminus\Gamma$ such that
$T_{0}+ic^{\prime\prime}\in U$.

Since $U\not\subset\Omega$, there exists $w\in U\setminus\Omega$. Hence
$w-t\not\in\Omega$ for all $t>0$ and, since
$\Gamma\cap\mathbb{C}\subset\Omega$, we actually have $w-t\in
U\setminus\Omega$ for all $t>0$.

Taking into account that $\mathbb{R}+i(a_{-}(\sigma),c)\subset\Omega$ it
follows ${\sf Im}\,w\leq a_{-}(\sigma)$. Moreover, since by construction for
every $\varepsilon>0$ there exists $R\in\mathbb{R}$ such that

$U\cap\\{w\in\mathbb{C}:{\sf
Re}\,w<R\\}\subset\\{w\in\mathbb{C}:a_{-}(\sigma)-\varepsilon<{\sf
Im}\,w<c^{\prime}\\},$

the only possibility is ${\sf Im}\,w=a_{-}(\sigma)$. In particular,
$\psi(a_{-}(\sigma))>-\infty$. Taking into account that $w-t\in
U\setminus\Omega$ for all $t>0$, we also see that ${\sf
Im}\,h(t\sigma)<a_{-}(\sigma)$ for $t$ close to $1$. It follows that
$\lim_{y\to a_{-}(\sigma)}\psi(y)=-\infty<\psi(a_{-}(\sigma))$. By Theorem
6.14 we have $\hbox{Int}(\overline{\Omega})\neq\Omega$, contradicting our
hypothesis. Therefore, $a_{-}(\sigma)\not\in\overline{J_{\infty}}$.

Next, we show that $\mathcal{B}_{N}\ni\sigma\mapsto a_{-}(\sigma)\in I_{N}$ is
injective. Assume by contradiction that there exist
$\sigma_{1},\sigma_{2}\in\mathcal{B}_{N}$, $\sigma_{1}\neq\sigma_{2}$, such
that $a_{-}(\sigma_{1})=a_{-}(\sigma_{2})$. Let $\gamma_{j}(t):=t\sigma_{j}$,
$t\in[0,1)$ and $j=1,2$. Let $\Gamma$ be the closure in $\mathbb{C}_{\infty}$
of $\gamma_{1}([0,1))\cup\gamma_{2}([0,1))$. Then by Lemma 7.3, $\Gamma$ is a
Jordan curve in $\mathbb{C}_{\infty}$. By the same lemma, it follows that
there exists $R\in\mathbb{R}$ such that $\Gamma\cap\mathbb{C}$ is contained in
$\\{w\in\mathbb{C}:{\sf Re}\,w<R\\}$. Let $U$ be the connected component of
$\mathbb{C}_{\infty}\setminus\Gamma$ which does not contain $h(0)+R$.

Since we are assuming that $\sigma_{1}\neq\sigma_{2}$, by $(\star)$ we have
$U\not\subset\Omega$. We show that, if this is the case, then
$\hbox{Int}(\overline{\Omega})\neq\Omega$, reaching a contradiction.

Indeed, assume $w\in U\setminus\Omega$. We have that $w-t\in U\setminus\Omega$
for all $t>0$, since $\Omega$ is starlike at infinity and $\partial
U=\Gamma\cap\mathbb{C}\subset\Omega$.

Taking into account Lemma 7.3, we see that for every $\varepsilon>0$ there
exists $R_{\varepsilon}\in\mathbb{R}$ so that

$U\cap\\{w\in\mathbb{C}:{\sf
Re}\,w<R_{\varepsilon}\\}\subset\\{w\in\mathbb{C}:|{\sf
Im}\,w-a_{-}(\sigma)|<\varepsilon\\}.$

Hence, if ${\sf Im}\,w\neq a_{-}(\sigma)$, it follows that there exists
$t_{0}>0$ such that $w-t_{0}\in\Gamma\cap\mathbb{C}\subset\Omega$, which is a
contradiction. Therefore, ${\sf Im}\,w=a_{-}(\sigma)$. In particular
$\psi(a_{-}(\sigma))>-\infty$. Also, since ${\sf Re}\,w+ia_{-}(\sigma)-t\in U$
for all $t>0$, we have that either ${\sf Im}\,\gamma_{1}(t)>a_{-}(\sigma)$ and
${\sf Im}\,\gamma_{2}(t)<a_{-}(\sigma)$ or ${\sf
Im}\,\gamma_{1}(t)<a_{-}(\sigma)$ and ${\sf Im}\,\gamma_{2}(t)>a_{-}(\sigma)$,
for $t$ close to $1$. In both cases, $\lim_{y\to
a_{-}(\sigma)}\psi(y)=-\infty<\psi(a_{-}(\sigma))$. Thus, by Theorem 6.14 we
have $\hbox{Int}(\overline{\Omega})\neq\Omega$, contradicting our hypothesis.
Hence the map is injective.

Finally, we prove that $\mathcal{B}_{N}\ni\sigma\mapsto a_{-}(\sigma)\in
I_{N}$ is surjective. Let $a\in I_{N}$. We assume $\lim_{y\to
a^{+}}\psi(y)=-\infty$ (the other case is similar). Therefore, for every
$n=1,2,\dots$ there exists $\varepsilon_{n}>0$ such that $\psi(y)<-n$ for all
$y\in I$ such that $0<y-a<2\varepsilon_{n}$. We can choose
$\\{\varepsilon_{n}\\}$ to be decreasing and such that
$(a,a+2\varepsilon_{1})\subset I$. In particular, for every $n$,

$B_{n}:=\\{w\in\mathbb{C}:{\sf Re}\,w\geq-n,0<{\sf
Im}\,w-a<2\varepsilon_{n}\\}\subset\Omega.$

Now, let
$\theta(t):=(-n+(a+\varepsilon_{n})i)(1-(t-n))+(-n-1+(a+\varepsilon_{n+1})i)(t-n)$,
for $n\leq t<n+1$, $n=1,2,\ldots$. Clearly, $\theta:[1,+\infty)\to\Omega$ is a
continuous injective curve. By construction, $\lim_{t\to+\infty}{\sf
Re}\,\theta(t)=-\infty$ and $\lim_{t\to+\infty}{\sf Im}\,\theta(t)=a$. Hence
(see, e.g., [8, Prop. 3.3.3]), there exists $\sigma\in\partial\mathbb{D}$ such
that $\lim_{t\to+\infty}h^{-1}(\theta(t))=\sigma$ and
$h^{\ast}(\sigma)=\infty$. Therefore, $\sigma$ is a boundary fixed point of
$(\varphi_{t})_{t\geq 0}$.

Now, we first prove that $\sigma\neq\tau$. Indeed, if this were the case, we
can define the injective continuous curve $\eta:(-\infty,+\infty)\to\Omega$ by
setting $\eta(t):=\theta(t)$ for $t\geq 1$ and $\eta(t):=\theta(1)-(t-1)$ for
$t<1$. By construction, and since we are assuming $\sigma=\tau$, it follows
that $L:=\overline{h^{-1}(\theta((-\infty,+\infty))}$ is a Jordan curve such
that $L\setminus\\{\tau\\}\subset\mathbb{D}$ and
$L\cap\partial\mathbb{D}=\\{\tau\\}$. Let $U\subset\mathbb{D}$ be the bounded
connected component of $\mathbb{C}\setminus L$. Thus $h(U)\subset\Omega$ is
one of the connected components of
$\mathbb{C}\setminus\overline{\eta(-\infty,+\infty)}$. But then $a$ belongs to
the closure of one open unbounded connected component of $\psi^{-1}(-\infty)$,
against the hypothesis $a\in I_{N}$. Thus, $\sigma\neq\tau$.

Finally, $\lim_{t\to+\infty}h(h^{-1}(\theta(t)))=a$, and
$\lim_{t\to+\infty}h^{-1}(\theta(t))=\sigma$. Therefore,

$\liminf_{z\to\sigma}{\sf Im}\,h(z)\leq a\leq\limsup_{z\to\sigma}{\sf
Im}\,h(z).$

Thus, by Lemma 7.3 and since $a\in I_{N}$, we have $\lim_{z\to\sigma}{\sf
Im}\,h(z)=a$. This implies that $\sigma\in\mathcal{B}\mathcal{F}_{N}$,
$a=a_{-}(\sigma)$, and we are done. ∎

###  7.2. Unbounded discontinuities

######  Definition 7.7.

Let $h$ be the Koenigs function of a non-elliptic semigroup in $\mathbb{D}$.
We say that $\sigma\in\mathcal{S}(h)$ is bounded, provided there exists
$R\in\mathbb{R}$ such that $\\{w\in\mathbb{C}:{\sf Re}\,w<R\\}\cap
I(\hat{h}(\underline{\sigma}))=\emptyset$. If $\sigma\in\mathcal{S}(h)$ is not
bounded, we say it is unbounded. We let $\mathcal{S}_{u}(h)$ denote the set of
all unbounded discontinuities.

Except for the Denjoy-Wolff point, unbounded discontinuities can be deduced by
analytic properties of the semigroup:

######  Proposition 7.8.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be its Koenigs
function, and $\Omega=h(\mathbb{D})$. Let
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$. Then the following are
equivalent.

  1. (1)

$\lim_{t\to+\infty}\varphi_{t}^{\ast}(\sigma)=\tau$, and for every $t_{0}>0$
there exists a sequence $\\{w_{n}\\}\subset\mathbb{D}$ converging to $\sigma$
such that $\lim_{n\to\infty}\varphi_{t_{0}}(w_{n})=\sigma$.

  2. (2)

$\sigma\in\mathcal{S}_{u}(h)$.

###### Proof.

(1) implies (2). It is clear that $\sigma$ is not a boundary fixed point.
Moreover, $\varphi_{t}$ does not extend continuously at $\sigma$ for $t>0$ by
hypothesis. Hence, as we already remarked, it follows that $h$ is not
continuous at $\sigma$, thus $\sigma\in\mathcal{S}(h)$. The cluster set of $h$
at $\sigma$ is the impression of $\hat{h}(\underline{\sigma})$—which is either
a segment or a half-line parallel to the real axis (see Lemma 6.5). We have to
prove that $\sigma$ is an unbounded discontinuity, that is,
$I(\hat{h}(\underline{\sigma}))$ is a half-line.

Suppose by contradiction that
$K:=I(\hat{h}(\underline{\sigma}))=\\{w\in\mathbb{C}:a\leq{\sf Re}\,w\leq
b,{\sf Im}\,w=y\\}$ for some $-\infty<a<b={\sf Re}\,h^{\ast}(\sigma)<+\infty$.
Fix $t>b-a$. Recall that $\varphi_{t}(z)=h^{-1}(h(z)+t)$ for all
$z\in\mathbb{D}$. Let $\\{w_{n}\\}\subset\mathbb{D}$ be the sequence
converging to $\sigma$ such that $\lim_{n\to\infty}\varphi_{t}(w_{n})=\sigma$
given by hypothesis (1). Up to subsequences, we can assume that
$\\{h(w_{n})\\}$ converges to some $p\in K$. Let $\xi_{n}:=h(w_{n})+t$. Hence,
$\\{\xi_{n}\\}$ converges to $p+t$. Since the identity map
$\sf{id}_{\mathbb{D}}$ extends as a homeomorphism between
$\widehat{\mathbb{D}}$ (the Carathéodory closure of $\mathbb{D}$) and
$\overline{\mathbb{D}}$, it follows that $\\{\varphi_{t}(w_{n})\\}$ converges
to $\sigma$ if and only if $\\{h^{-1}(\xi_{n})\\}$ converges to
$\underline{\sigma}\in\partial_{C}\mathbb{D}$ in the Carathéodory topology.
Moreover, taking into account that $\hat{h}$ is a homeomorphism between
$\widehat{\mathbb{D}}$ and $\widehat{\Omega}$, it follows that
$\\{h^{-1}(\xi_{n})\\}$ converges to
$\underline{\sigma}\in\partial_{C}\mathbb{D}$ if and only if $\\{\xi_{n}\\}$
converges to $\hat{h}(\underline{\sigma})$ in the Carathèodory topology. Let
$(C_{n})$, $n\geq 0$ be a null-chain representing
$\hat{h}(\underline{\sigma})$ and let $V_{n}$, $n\geq 1$ denotes the interior
part of $C_{n}$. Fix $s\in(b,t)$. Since $\bigcap_{n\geq 1}\overline{V_{n}}=K$
and $V_{n+1}\subset V_{n}$ it follows that there exists $m$ such that
$\overline{V_{m}}\subset\\{w\in\mathbb{C}:{\sf Re}\,w<s\\}$. But this implies
that $\\{\xi_{n}\\}$ is not eventually contained in $V_{m}$, and hence (see,
e.g., [8, Remark 4.2.2]), $\\{\xi_{n}\\}$ does not converge to
$\hat{h}(\underline{\sigma})$ in the Carathéodory topology, contradiction.
Therefore (1) implies (2).

(2) implies (1). Recall that, by the so-called “boundary Denjoy-Wolff Theorem”
(see, e.g., [8, Theorem 14.1.1]), for every $\sigma\in\partial\mathbb{D}$
which is not a boundary fixed point of $(\varphi_{t})_{t\geq 0}$, the curve
$[0,+\infty)\ni t\mapsto\phi_{t}^{\ast}(\sigma)$ is a continuous injective
curve converging to $\tau$. Thus
$\lim_{t\to+\infty}\varphi_{t}^{\ast}(\sigma)=\tau$. Finally, since
$I(\hat{h}(\underline{\sigma}))$ is a half-line, using again that
$\varphi_{t}(z)=h^{-1}(h(z)+t)$ for all $z\in\mathbb{D}$ and $t\geq 0$, that
$I(\hat{h}(\underline{\sigma}))$ is the cluster set of $h$ at $\sigma$ and a
similar argument as before, we have the conclusion. ∎

Now we relate unbounded discontinuities to the defining function of the
Koenigs domain.

######  Proposition 7.9.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be its Koenigs
function, $\Omega=h(\mathbb{D})$ the associated Koenigs domain . Let
$\psi:I\to\mathbb{R}$ be the defining function of $\Omega$. Let $D_{U}$ be the
set of points $y_{0}\in\overline{I}$ such that either

$\liminf_{y\to y_{0}^{-}}\psi(y)=-\infty,\quad-\infty<\limsup_{y\to
y_{0}^{-}}\psi(y)<+\infty,$

or

$\liminf_{y\to y_{0}^{+}}\psi(y)=-\infty,\quad-\infty<\limsup_{y\to
y_{0}^{+}}\psi(y)<+\infty.$

Then the map $\mathcal{U}:\mathcal{S}_{u}(h)\setminus\\{\tau\\}\to D_{U}$
defined by $\mathcal{U}(\sigma)={\sf Im}\,h^{\ast}(\sigma)$ is surjective.
Moreover, for each $y_{0}\in I\cap D_{U}$ the fibre $\mathcal{U}^{-1}(y_{0})$
contains at most two points, while, if $y_{0}\in\partial I\cap D_{U}$ the
fibre $\mathcal{U}^{-1}(y_{0})$ contains only one point.

Finally, assume $\hbox{Int}(\overline{\Omega})=\Omega$ and let $y_{0}\in I\cap
D_{U}$. Then, there exist $\sigma_{1}\neq\sigma_{2}$ such that
$\mathcal{U}^{-1}(y_{0})=\\{\sigma_{1},\sigma_{2}\\}$ if and only if either
$\sigma_{1}\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{-}(h)$ and
$\sigma_{2}\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{+}(h)$ or
$\sigma_{1}\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{+}(h)$ and
$\sigma_{2}\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{-}(h)$ and
$\sigma_{1},\sigma_{2}$ are the end points of a non-exceptional maximal
contact arc of $(\varphi_{t})_{t\geq 0}$.

###### Proof.

Given $\sigma\in\mathcal{S}_{u}(h)$, it follows easily from Lemma 6.5 that
${\sf Im}\,h^{\ast}(\sigma)\in D_{U}$. Hence, the map $\mathcal{U}$ is well
defined.

In order to see that $\mathcal{U}$ is surjective, take $y_{0}\in D_{U}$. We
can assume without loss of generality that $\liminf_{y\to
y_{0}^{-}}\psi(y)=-\infty$ and $-\infty<M:=\limsup_{y\to
y_{0}^{-}}\psi(y)<+\infty$. By definition of $\psi$ and since $\psi$ is upper
semicontinuous, it is clear from the reverse implication of Lemma 6.5 that
$\\{w\in\mathbb{C}:{\sf Re}\,w\leq M,{\sf Im}\,w=y_{0}\\}$ is the impression
of a prime end corresponding to a point
$\sigma\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ such that
$\mathcal{U}(\sigma)=y_{0}$.

By Proposition 6.7, the fibre $\mathcal{U}^{-1}(y_{0})$ contains at most two
points for every $y_{0}\in D_{U}$ and only one if $y_{0}\in\partial I\cap
D_{U}$,

Assume now $\hbox{Int}(\overline{\Omega})=\Omega$. By Theorem 6.14, every two
non-exceptional maximal contact arcs of $(\varphi_{t})_{t\geq 0}$ have
different final points. Thus, by Proposition 6.7, if
$\sigma_{1},\sigma_{2}\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ are such that
$\sigma_{1}\neq\sigma_{2}$ and
$y_{0}:=\mathcal{U}(\sigma_{1})=\mathcal{U}(\sigma_{1})$ then $y_{0}\in I$,
either $\sigma_{1}\in\mathcal{S}^{+}(h)$ and $\sigma_{2}\in\mathcal{S}^{-}(h)$
or $\sigma_{1}\in\mathcal{S}^{-}(h)$ and $\sigma_{2}\in\mathcal{S}^{+}(h)$ and
either ${\sf Re}\,h^{\ast}(\sigma_{1})<{\sf Re}\,h^{\ast}(\sigma_{2})$ or
${\sf Re}\,h^{\ast}(\sigma_{2})<{\sf Re}\,h^{\ast}(\sigma_{1})$.

We assume $x_{1}:={\sf Re}\,h^{\ast}(\sigma_{1})<x_{2}:={\sf
Re}\,h^{\ast}(\sigma_{2})$ and $\sigma_{1}\in\mathcal{S}^{-}(h)$,
$\sigma_{2}\in\mathcal{S}^{+}(h)$ (the other cases are similar).

We need to show that there exists a maximal contact arc for
$(\varphi_{t})_{t\geq 0}$ starting at $\sigma_{1}$ and ending at $\sigma_{2}$.
The proof is very similar to the one for the proof of the existence of the two
maximal contact arcs in Proposition 6.7, so we only sketch it.

Since $\sigma_{1}\in\mathcal{S}^{-}(h)$, using (1.b) of Lemma 6.5, by Schwarz
reflection principle, $h^{-1}$ restricted to $\Omega\cap\\{w\in\mathbb{C}:{\sf
Im}\,w<y_{0}\\}$ extends holomorphically through
$B:=\\{w\in\mathbb{C}:x_{1}<{\sf Re}\,w<x_{2},{\sf Im}\,w=y_{0}\\}$. Hence,
there exists an open arc $A\subset\partial\mathbb{D}$ such that $h$ extends
holomorphically through $A$ and $h(A)=B$. By Lemma 6.2, $A$ is a contact arc.

Since $\sigma_{1}\in\mathcal{S}^{-}(h)$ and $\sigma_{2}\in\mathcal{S}^{+}(h)$,
properties (1.a) and (1.b) of Lemma 6.2 show that for every $x<x_{1}$ the
point $x+iy_{0}$ is not accessible with injective continuous curves in
$\Omega$ and thus cannot belong to any contact arc. Since
$x_{1}+iy_{0}\in\overline{B}$, it follows then that the starting point
$\xi_{1}$ of $A$ satisfies $h^{\ast}(\xi_{1})=h^{\ast}(\sigma_{1})$.

Now, arguing exactly as in the proof of Proposition 6.7 one can show that
$\xi_{1}=\sigma_{1}$.

In order to show that $\sigma_{2}$ is the final point of $A$, it is enough to
prove that $\psi(y_{0})=x_{2}$. Indeed, by construction, $\psi(y_{0})\geq
x_{2}$. Since $\sigma_{1}\in\mathcal{S}^{-}(h)$ and
$\sigma_{2}\in\mathcal{S}^{+}(h)$ (here we use (2.a) and (2.b) of Lemma 6.5),
for every $\varepsilon>0$ we can find $\delta>0$ such that $\liminf_{t\to
y}\psi(y)\leq x_{2}+\varepsilon$ for all $|y-y_{0}|<\delta$. This clearly
implies that $\tilde{\psi}(y_{0})\leq x_{2}$ and hence, by Theorem 6.14,
$\psi(y_{0})=\tilde{\psi}(y_{0})=x_{2}$.

Finally, if $\sigma_{1},\sigma_{2}$ are two unbounded discontinuities which
are the end points of a non-exceptional maximal contact arc $A$ of
$(\varphi_{t})_{t\geq 0}$ then clearly
$\mathcal{U}(\sigma_{1})=\mathcal{U}(\sigma_{2})$ since $h^{\ast}(A)$ is
contained in a line parallel to the real axis. ∎

###  7.3. Boundary fixed points and unbounded discontinuities

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$,
with Koenigs’ function $h$, Koenigs’ domain $\Omega$ and Denjoy-Wolff point
$\tau\in\partial\mathbb{D}$. Let $\psi:I\to\mathbb{R}$ be the defining
function of $\Omega$. In the previous sections, we introduced the following
sets:

  * •

$\mathcal{B}\mathcal{F}_{R}\subset\partial\mathbb{D}$, the set of all boundary
regular fixed points of $(\varphi_{t})_{t\geq 0}$ different from $\tau$,

  * •

$\mathcal{B}\mathcal{F}_{N}\subset\partial\mathbb{D}$, the set of all boundary
super-repelling fixed points of $(\varphi_{t})_{t\geq 0}$,

  * •

$\mathcal{S}^{+}(h),\mathcal{S}^{-}(h)\subset\partial\mathbb{D}\setminus\\{\tau\\}$
the set of “upper”/“lower” discontinuities,

  * •

$\mathcal{S}_{u}(h)\subset\partial\mathbb{D}$ the set of all unbounded
discontinuities,

  * •

$I_{R}\subset I$, the set of all bounded open connected components of
$\psi^{-1}(-\infty)$,

  * •

$J_{\infty}\subset I$, the union of all open connected components of
$\psi^{-1}(-\infty)$,

  * •

$I_{N}\subset\overline{I}$, the set of all points
$y_{0}\in\overline{I}\setminus\overline{J_{\infty}}$ such that either
$\lim_{y\to y_{0}^{+}}\psi(y)=-\infty$ or $\lim_{y\to
y_{0}^{-}}\psi(y)=-\infty$ (or both),

  * •

$D_{U}\subset\overline{I}$, the set of all points $y_{0}\in\overline{I}$ such
that either

$\liminf_{y\to y_{0}^{+}}\psi(y)=-\infty<\limsup_{y\to
y_{0}^{+}}\psi(y)<+\infty,$

or

$\liminf_{y\to y_{0}^{-}}\psi(y)=-\infty<\limsup_{y\to
y_{0}^{-}}\psi(y)<+\infty.$

Assuming $\hbox{Int}(\overline{\Omega})=\Omega$, we proved that there is a
one-to-one correspondence between $\mathcal{B}\mathcal{F}_{R}$ and $I_{R}$; a
one-to-one correspondence between $\mathcal{B}\mathcal{F}_{N}$ and $I_{R}$;
and a surjective map from $\mathcal{S}_{u}(h)$ to $D_{U}$, so that, if the
fibre above $y_{0}$ contains two points $\sigma_{1},\sigma_{2}$, then
$y_{0}\in I$ and there is a maximal contact arc starting from $\sigma_{1}$ and
ending at $\sigma_{2}$.

Let us now explore what happens at those points of intersection between
$D_{U}$ and $I_{N}$ or those points of $D_{U}$ which belong to the closure of
an element of $I_{R}$.

Note that, by definition, if $(a,b)\in I_{R}$ then $x\not\in D_{U}$ for every
$x\in(a,b)$. However, it might happen that $a$ or $b$ belongs to $D_{U}$.

######  Proposition 7.10.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be its Koenigs
function, $\Omega=h(\mathbb{D})$ the associated Koenigs domain and assume that
$\hbox{Int}(\overline{\Omega})=\Omega$. Let $\psi:I\to\mathbb{R}$ be the
defining function of $\Omega$.

  1. (1)

$y_{0}\in D_{U}\cap I_{N}$ if and only there exist a unique
$\sigma\in\mathcal{B}\mathcal{F}_{N}$ and a unique
$\xi\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ such that ${\sf
Im}\,h^{\ast}(\sigma)={\sf Im}\,h^{\ast}(\xi)=y_{0}$.

  2. (2)

Let $(y_{0},b)\in I_{R}$. Then $y_{0}\in D_{U}$ if and only if there exist a
unique $\sigma\in\mathcal{B}\mathcal{F}_{R}$ and a unique
$\xi\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{-}(h)$ such that
$\liminf_{z\to\sigma}{\sf Im}\,h^{\ast}(z)={\sf Im}\,h^{\ast}(\xi)=y_{0}$.

  3. (3)

Let $(a,y_{0})\in I_{R}$. Then $y_{0}\in D_{U}$ if and only if there exist a
unique $\sigma\in\mathcal{B}\mathcal{F}_{R}$ and a unique
$\xi\in\mathcal{S}_{u}(h)\cap\mathcal{S}^{+}(h)$ such that
$\limsup_{z\to\sigma}{\sf Im}\,h^{\ast}(z)={\sf Im}\,h^{\ast}(\xi)=y_{0}$.

Moreover, in the hypothesis of (1), (2) or (3), $y_{0}\in I$ and there exists
a maximal contact arc $A$ starting from $\sigma$ and ending at $\xi$ such that
${\sf Im}\,h^{\ast}(A)=y_{0}$.

###### Proof.

(1) Assume $y_{0}\in D_{U}\cap I_{N}$. Then, by Proposition 7.6 there exists a
unique $\sigma\in\mathcal{B}\mathcal{F}_{N}$ such that ${\sf
Im}\,h^{\ast}(\sigma)=y_{0}$. By Proposition 7.9, either there exist a unique
$\xi\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ such that ${\sf
Im}\,h^{\ast}(\xi)=y_{0}$ or there are
$\xi^{+},\xi^{-}\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ such that ${\sf
Im}\,h^{\ast}(\xi^{\pm})=y_{0}$ and $\xi^{+}\in S^{+}(h)$, $\xi^{-}\in
S^{-}(h)$. By (1.a) and (1.b) of Lemma 6.5, the existence of such $\xi^{+}$
and $\xi^{-}$ implies that there exist two sequences $\\{z_{n}^{+}\\}$ and
$\\{z_{n}^{-}\\}$ such that ${\sf Im}\,z_{n}^{+}>y_{0}$, ${\sf
Im}\,z_{n}^{+}<y_{0}$, $\\{z_{n}^{\pm}\\}$ converges to $h^{\ast}(\xi^{\pm})$
and $z_{n}^{\pm}-t\in\mathbb{C}\setminus\Omega$ for all $t\geq 0$. Since the
continuous curve $[0,1)\ni t\mapsto h(t\sigma)$ has the property that
$\lim_{t\to 1^{-}}{\sf Re}\,h(t\sigma)=-\infty$ and $\lim_{t\to 1^{-}}{\sf
Im}\,h(t\sigma)=y_{0}$, this immediately leads to a contradiction.

Therefore, there exists a unique $\xi\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$
such that ${\sf Im}\,h^{\ast}(\xi)=y_{0}$. By the same argument as before, it
follows that either $\xi\in\mathcal{S}^{-}(h)\setminus\mathcal{S}^{+}(h)$ and
${\sf Im}\,h(t\sigma)>y_{0}$ for all $t$ close to $1$ or
$\xi\in\mathcal{S}^{+}(h)\setminus\mathcal{S}^{-}(h)$ and ${\sf
Im}\,h(t\sigma)<y_{0}$ for all $t$ close to $1$. In both cases, $y_{0}\in I$.

The converse implication of (1) is straightforward from Proposition 7.6 and
Proposition 7.9.

(2) If $(y_{0},b)\in I_{R}$ then $\psi(y)=-\infty$ for all $y\in(y_{0},b)$.
Hence, one can argue as in the proof of (1) showing the existence of a unique
$\sigma\in\mathcal{B}\mathcal{F}_{R}$ and a unique $\xi\in S^{-}(h)$ such that
$\liminf_{z\to\sigma}{\sf Im}\,h^{\ast}(z)={\sf Im}\,h^{\ast}(\xi)=y_{0}$, and
that $y_{0}\in I$. The reverse implication follows again by Proposition 7.6
and Proposition 7.9.

(3) is similar to (2).

We are left to show that there exists a maximal contact arc starting from
$\sigma$ and ending at $\xi$. Since the argument is quite similar to the one
for proving the existence of the two maximal contact arcs in Proposition 6.7,
we omit it and leave details to the reader. ∎

###  7.4. Discontinuities and the Denjoy-Wolff point

For the Denjoy-Wolff point, recalling the notation introduced in (6.3) and
setting $L_{y}:=\\{w\in\mathbb{C}:{\sf Im}\,w=y\\}$ for $y\in\mathbb{R}$, we
have (see [8, Thm.11.1.4]),

######  Lemma 7.11.

Let $h$ be the Koenigs function of a non-elliptic semigroup in $\mathbb{D}$
with Denjoy-Wolff point $\tau$ and let $\Omega=h(\mathbb{D})$ the Koenigs
domain. If $\tau\in\mathcal{S}(h)$ then

  1. (1)

either there exists $y\in\mathbb{R}$ such that
$\Omega\subset\\{w\in\mathbb{C}:{\sf Im}\,w>y\\}$ or
$\Omega\subset\\{w\in\mathbb{C}:{\sf Im}\,w<y\\}$ and
$I(\hat{h}(\underline{\tau}))=\overline{L_{a,+\infty;y}}^{\infty}$, where
$a\in[-\infty,\infty)$,

  2. (2)

or, there exist real numbers $y_{1}<y_{2}$ such that
$\Omega\subset\\{w\in\mathbb{C}:y_{1}<{\sf Im}\,w<y_{2}\\}$ and
$I(\hat{h}(\underline{\tau}))=\overline{L_{a_{1},+\infty;y_{1}}}^{\infty}\cup\overline{L_{a_{2},+\infty;y_{2}}}^{\infty}$,
where $a_{1},a_{2}\in[-\infty,\infty)$.

Moreover, in the first case, there are no $\sigma\in\partial\mathbb{D}$ such
that $h^{\ast}(\sigma)\in L_{y}$, while in the second case there are no
$\sigma\in\partial\mathbb{D}$ such that $h^{\ast}(\sigma)\in L_{y_{1}}\cup
L_{y_{2}}$.

Such lemma suggests the following definition:

######  Definition 7.12.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup with Denjoy-Wolff
point $\tau\in\partial\mathbb{D}$ and Koenigs function $h$. Suppose
$\tau\in\mathcal{S}_{u}(h)$. If $I(\hat{h}(\underline{\tau}))=L_{y_{1}}\cup
L_{y_{1}}\cup\\{\infty\\}$, for some $-\infty<y_{1}<y_{2}<+\infty$, we say
that $\tau$ is a double unbounded discontinuity. Otherwise, we say that $\tau$
is a simple unbounded discontinuity.

######  Remark 7.13.

By Lemma 7.11 it follows that if $\tau\in\mathcal{S}(h)$ then the semigroup
$(\varphi_{t})_{t\geq 0}$ is either hyperbolic or parabolic with positive
hyperbolic step. Moreover, if $\tau\in\mathcal{S}(h)$ is a double unbounded
discontinuity then $(\varphi_{t})_{t\geq 0}$ is necessarily hyperbolic and, if
$\psi:(a_{0},a_{1})\to\mathbb{R}$ is the defining function of its Koenigs
domain, for some $-\infty<a_{0}<a_{1}<+\infty$ then

$\liminf_{y\to a_{0}^{+}}\psi(y)=-\infty,\quad\limsup_{y\to
a_{0}^{+}}\psi(y)=+\infty,$

and

$\liminf_{y\to a_{1}^{-}}\psi(y)=-\infty,\quad\limsup_{y\to
a_{1}^{-}}\psi(y)=+\infty.$

For future aims, we need the following proposition (recall the definition of
$a_{\pm}(\sigma)$ from (7.3)):

######  Proposition 7.14.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function and $\Omega=h(\mathbb{D})$ the Koenigs domain. Assume that the
defining function $\psi$ of $\Omega$ is defined in $(a_{0},a_{1})$ for some
$-\infty<a_{0}<a_{1}<+\infty$. Then the following are equivalent.

  1. (1)

$\tau$ is a simple unbounded discontinuity,

  2. (2)

one and only one of the following two conditions holds:

     * (2.a)

either

$\liminf_{y\to a_{0}^{+}}\psi(y)=-\infty,\quad\limsup_{y\to
a_{0}^{+}}\psi(y)=+\infty,\quad\limsup_{y\to a_{1}^{-}}\psi(y)<+\infty$

     * (2.b)

or,

$\liminf_{y\to a_{1}^{-}}\psi(y)=-\infty,\quad\limsup_{y\to
a_{1}^{-}}\psi(y)=+\infty,\quad\limsup_{y\to a_{0}^{+}}\psi(y)<+\infty.$

Moreover, if (2.a) holds, then

  * (a)

for every boundary fixed point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ we have
$a_{0}\not\in[a_{-}(\sigma),a_{+}(\sigma)]$,

  * (b)

for every $\xi\in\mathcal{S}_{u}(h)$ we have ${\sf Im}\,h^{\ast}(\xi)\neq
a_{0}$.

While, if (2.b) holds, then

  * (a’)

for every boundary fixed point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ we have
$a_{1}\not\in[a_{-}(\sigma),a_{+}(\sigma)]$,

  * (b’)

for every $\xi\in\mathcal{S}_{u}(h)$ we have ${\sf Im}\,h^{\ast}(\xi)\neq
a_{1}$.

###### Proof.

The equivalence between (1) and (2) and the statements (b), (b’) follows
immediately from Lemma 7.11.

As for (a) (and (a’)), if $\sigma$ is a boundary fixed point different from
$\tau$, and $a_{0}=a_{-}(\sigma)$, then for every $x\in\mathbb{R}$ there
exists $\delta>0$ such that $\\{w\in\mathbb{C}:{\sf Re}\,w>x-\delta,a_{0}<{\sf
Im}\,w<a_{0}+\delta\\}\subset\Omega$. This is immediate if
$a_{+}(\sigma)>a_{0}$, and it follows from the fact that $\Omega$ is starlike
at infinity and the continuous injective curve $[0,1)\ni t\mapsto h(t\sigma)$
has the property that $\lim_{t\to 1^{-}}{\sf Re}\,h(t\sigma)=-\infty$ and
$\lim_{t\to 1^{-}}{\sf Im}\,h(t\sigma)=a_{0}$ in case $a_{+}(\sigma)=a_{0}$.
But this implies that there for every $x\in\mathbb{R}$ exists
$\zeta\in\partial\mathbb{D}$ such that $h^{\ast}(\zeta)=x+ia_{0}$ (the point
$\zeta$ being the limit of $h^{-1}(x+i(a_{0}+s))$ for $s\to 0^{+}$), against
Lemma 7.11. ∎

For our aims, we need also the following (see Proposition 7.9 for the
definition of $\mathcal{U}$):

######  Lemma 7.15.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function and $\Omega=h(\mathbb{D})$ the Koenigs domain. Assume that the
defining function $\psi$ of $\Omega$ is defined in $(a_{0},a_{1})$ for some
$-\infty<a_{0}<a_{1}\leq+\infty$. Then the following are equivalent.

  1. (1)

There exists a maximal contact arc for $(\varphi_{t})_{t\geq 0}$ with starting
point $\tau$ and final point $\sigma\in\mathcal{S}_{u}(h)$.

  2. (2)

$a_{1}=+\infty$ and there exists $a\in(a_{0},+\infty)$ such that
$\psi(y)=-\infty$ for all $y>a$ and

$-\infty=\liminf_{y\to a^{-}}\psi(y)<\limsup_{y\to a^{-}}\psi(y)=\psi(a).$

Moreover, if either (1) or (2)—and hence both—holds, then
$\mathcal{U}(\sigma)=a$, $\mathcal{U}^{-1}(a)=\\{\sigma\\}$ and
$h^{\ast}(\sigma)=\psi(a)+ia$.

###### Proof.

(1) implies (2). Let $A\subset\partial\mathbb{D}$ be a maximal contact arc
starting at $\tau$ and ending at $\sigma\in\mathcal{S}_{u}(h)$. If
$\sigma=\tau$, then $(\varphi_{t})_{t\geq 0}$ is a parabolic group (see [8,
Proposition 14.2.7]), in particular, $\sigma\not\in\mathcal{S}(h)$. Hence,
$\sigma\neq\tau$ and, since it is not a boundary fixed point of
$(\varphi_{t})_{t\geq 0}$, it follows that $h^{\ast}(\sigma)\in\mathbb{C}$.
Since $x_{0}(A)=\tau$, a fixed point of $(\varphi_{t})_{t\geq 0}$, by Lemma
6.2 and (6.2), there exists $a\geq a_{0}$ and $b\in(-\infty,+\infty]$ such
that $h(A)=\\{w\in\mathbb{C}:{\sf Im}\,w=a,{\sf Re}\,w<b\\}$ and
$\lim_{A\ni\xi\to\sigma}h(\xi)=h^{\ast}(\sigma)$. Thus $a={\sf
Im}\,h^{\ast}(\sigma)$ and $b={\sf Re}\,h^{\ast}(\sigma)\in\mathbb{R}$. In
particular, $\mathcal{U}(\sigma)=a$.

Moreover, we already recalled (see the discussion on maximal contact arcs)
that $h^{\ast}(\sigma)+t\in\Omega$ for all $t>0$.

Let $\Gamma_{1}$ be the closure of the image of the continuous injective curve
given by $(0,+\infty)\ni t\mapsto h^{-1}(h^{\ast}(\sigma)+t)$. Since
$\varphi_{t}(w)=h^{-1}(h(w)+t)$ for all $w\in\mathbb{D}$ and $t\geq 0$, it
follows that $\Gamma_{1}$ is a Jordan arc with end points $\sigma$ and $\tau$
and $\Gamma_{1}\setminus\\{\sigma,\tau\\}\subset\mathbb{D}$. Hence,
$\Gamma:=\overline{A}\cup\Gamma_{1}$ is a Jordan curve. Let
$V\subset\mathbb{D}$ be the bounded connected component of
$\mathbb{C}\setminus\Gamma$. Note that $V$ is a Jordan domain and hence $h$
extends as a homeomorphism to $\partial V$. Since by construction
$h(\Gamma\setminus\\{\sigma,\tau\\})=\\{w\in\mathbb{C}:{\sf
Im}\,w=a\\}\setminus\\{b+ia\\}$, it follows that $h(\partial
V)=\\{w\in\mathbb{C}:{\sf Im}\,w=a\\}\cup\\{\infty\\}$. Thus,
$h(V)=\\{w\in\mathbb{C}:{\sf Im}\,w<a\\}$ or $h(V)=\\{w\in\mathbb{C}:{\sf
Im}\,w>a\\}$. But $h(V)\subset\Omega$ and $\Omega\subset\\{w\in\mathbb{C}:{\sf
Im}\,w>a_{0}\\}$, therefore, $\\{w\in\mathbb{C}:{\sf
Im}\,w>a\\}=h(V)\subset\Omega$. This implies that $\psi$ is defined on
$(a_{0},+\infty)$ and $\psi(y)=-\infty$ for all $y>a$.

This also implies that $\sigma\in\mathcal{S}^{-}(h)$ and there cannot be
$\xi\in\mathcal{S}^{+}(h)$ such that ${\sf Im}\,h^{\ast}(\xi)=a$. Therefore,
Proposition 7.9 implies that $\mathcal{U}^{-1}(a)=\\{\sigma\\}$. Finally,
Lemma 6.5 shows that

$-\infty<{\sf Re}\,h^{\ast}(\sigma)=\limsup_{y\to a^{-}}\psi(y)\leq\psi(a).$

Since ${\sf Re}\,h^{\ast}(\sigma)+t\in\Omega$ for all $t>0$, we have actually
$\limsup_{y\to a^{-}}\psi(y)=\psi(a)$ and hence $h^{\ast}(\sigma)=\psi(a)+ia$.

Thus (2) and the last part of the proposition are proved.

(2) implies (1). By Proposition 7.9, and arguing as before one can see that
there exists a unique $\sigma\in\mathcal{S}_{u}(h)\setminus\\{\tau\\}$ such
that $h^{\ast}(\sigma)=\psi(a)+ia$.

Next, the hypothesis on $\psi$ implies that $H:=\\{w\in\mathbb{C}:{\sf
Im}\,w>a\\}\subset\Omega$, $\\{w\in\mathbb{C}:{\sf Im}\,w=a,{\sf
Re}\,w\leq\psi(a)\\}\cap\Omega=\emptyset$ and $\\{w\in\mathbb{C}:{\sf
Im}\,w=a,{\sf Re}\,w>\psi(a)\\}\subset\Omega$. Since for every sequence
$\\{w_{n}\\}\subset H$ converging to any point of $\\{w\in\mathbb{C}:{\sf
Im}\,w=a,{\sf Re}\,w\leq\psi(a)\\}\cup\\{\infty\\}$ the cluster set of
$h^{-1}(w_{n})$ is contained in $\partial\mathbb{D}$, by Schwarz reflection
principle it follows that $h^{-1}$ extends holomorphically through
$\partial_{\infty}H$. Let $V:=h^{-1}(H)$. Therefore, $V\subset\mathbb{D}$ is a
Jordan domain and $\partial V$ is the union of an open arc
$A\subset\partial\mathbb{D}$ and a Jordan arc $\Gamma$ with the same end
points of $A$ and such that $\Gamma$ minus its end points is contained in
$\mathbb{D}$. By construction, $h^{\ast}(A)=\\{w\in\mathbb{C}:{\sf
Im}\,w=a,{\sf Re}\,w<\psi(a)\\}$. Therefore, $A$ is a contact arc for
$(\varphi_{t})_{t\geq 0}$ (see Lemma 6.2). The order on $A$ is given by the
direction of the flow $(h|_{\partial H})^{-1}(w+t)$, $t>0$, $w\in\mathbb{C}$
with ${\sf Im}\,w=a,{\sf Re}\,w<\psi(a)$. Hence, if $\zeta\in\partial A$ is
such that $h^{\ast}(\zeta)=\psi(a)+ia$, it follows that $\zeta$ is the final
point of $A$. Using an argument similar to the one in the proof of Proposition
6.7 one can easily show that $\zeta=\sigma$.

We are left to prove that $\tau$ is the initial point of $A$. Denote by
$\tilde{h}$ the homeomorphic extension of $h|_{V}$ to $\overline{V}$. By
construction, $\tilde{h}(\overline{V}\setminus\\{x_{0}(A)\\})=\overline{H}$.
Since $\tilde{h}$ is a homeomorphism from $\overline{V}$ onto
$\overline{H}^{\infty}$, it follows that $\tilde{h}(x_{0}(A))=\infty$. Thus
$h^{\ast}(x_{0}(A))=\lim_{A\ni\xi\to x_{0}(A)}h(\xi)=\infty$. Therefore,
$x_{0}(A)=\lim_{t\to+\infty}h^{-1}(\gamma(t))$ for any continuous curve
$\gamma:[0,+\infty)\to H$ such that $\lim_{t\to+\infty}\gamma(t)=\infty$ (see,
e.g., [8, Proposition 3.3.3]). Fix $w_{0}\in H$ and let $\gamma(t):=w_{0}+t$.
Hence

$\lim_{t\to+\infty}h^{-1}(\gamma(t))=\lim_{t\to+\infty}\varphi_{t}(h^{-1}(w_{0}))=\tau,$

and therefore $\tau=x_{0}(A)$. Thus (1) holds. ∎

##  8\. Koenigs domains $\Omega$ such that such that
$\hbox{Int}(\overline{\Omega})=\Omega$ and
$\mathbb{C}\setminus\overline{\Omega}$ has at most two connected components

In this section we characterize Koenigs domains whose complementary in
$\mathbb{C}$ is connected or doubly connected (and contained in a strip).

######  Remark 8.1.

Suppose $\Omega$ is the Koenigs domain of a non-elliptic semigroup
$(\varphi_{t})_{t\geq 0}$. Let $\psi:I\to[-\infty,+\infty)$ be the defining
function of $\Omega$. If $\Omega\subset H$, where $H$ is a half-plane not
parallel to $\\{w\in\mathbb{C}:{\sf Im}\,w>0\\}$, then
$\mathbb{C}\setminus\overline{\Omega}$ is connected. This follows from the
fact that for any $p\in\mathbb{C}\setminus\overline{\Omega}$ there exists
$t_{0}\geq 0$ such that $p-t_{0}\in\mathbb{C}\setminus\overline{H}$ and
$p-t.\in\mathbb{C}\setminus\overline{\Omega}$ for all $t\geq 0$.

Moreover, it is clear that in such a case $(\varphi_{t})_{t\geq 0}$ has
neither boundary fixed points other than the Denjoy-Wolff point nor unbounded
discontinuities. Also $\liminf_{y\to y_{0}}\psi(y)>-\infty$ for all $y_{0}\in
I$.

######  Theorem 8.2.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function, $\Omega=h(\mathbb{D})$ the Koenigs domain and let
$\psi:I\to\mathbb{R}$ be the defining function of $\Omega$. Suppose
$\hbox{Int}(\overline{\Omega})=\Omega$ and
$\Omega\subset\\{w\in\mathbb{C}:{\sf Im}\,w>0\\}$. Let $I_{\infty}$ denotes
the unbounded connected component of $\psi^{-1}(-\infty)$ (possibly empty).
Then the following are equivalent.

  1. (1)

$\mathbb{C}\setminus\overline{\Omega}$ is connected.

  2. (2)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$ and
$\mathcal{S}(h)$ contains only bounded discontinuities except, at most, one
unbounded discontinuity $\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ which
is the final point of a maximal contact arc starting at $\tau$.

  3. (3)

$\liminf_{I\ni y\to y_{0}}\psi(y)>-\infty$ for every
$y_{0}\in\overline{I}\setminus\overline{I_{\infty}}$.

###### Proof.

(2) implies (3) by Proposition 7.6, Proposition 7.9 and Lemma 7.15.

(3) implies (2). Indeed, if either $I_{\infty}=\emptyset$ or there exists
$a>0$ such that $I_{\infty}=[a,+\infty)$, it follows from Proposition 7.6 and
Proposition 7.9 that (2) holds. Similarly, if $I_{\infty}=(a,+\infty)$ for
some $a>0$ and $\liminf_{y\to a^{-}}\psi(y)>-\infty$.

Thus, we assume that (3) holds, that $I_{\infty}=(a,+\infty)$ for some $a>0$,
$\psi(a)>-\infty$ and $\liminf_{y\to a^{-}}\psi(y)=-\infty$. Notice that
$\limsup_{y\to a^{-}}\psi(y)=\limsup_{y\to a}\psi(y)\leq\psi(a)$. If
$M:=\limsup_{y\to a^{-}}\psi(y)<\psi(a)$, then the interval $(M,\psi(a)]+ia$
is contained in $\hbox{Int}(\overline{\Omega})\setminus\Omega$, which
contradicts our hypothesis. Therefore $\limsup_{y\to a^{-}}\psi(y)=\psi(a)$.
By Lemma 7.15, (2) holds.

(1) implies (3). Arguing by contradiction we assume that there exists
$y_{0}\in\overline{I}\setminus\overline{I_{\infty}}$ such that $\liminf_{I\ni
y\to y_{0}}\psi(y)=-\infty$.

By the same argument as in the proof of Lemma 5.7,
$L_{y_{0}}\subset\overline{\Omega}$.

Since by hypothesis $\\{w\in\mathbb{C}:{\sf
Im}\,w<0\\}\subset\mathbb{C}\setminus\overline{\Omega}$ and we are assuming
that $\mathbb{C}\setminus\overline{\Omega}$ is connected, it follows that
$\\{w\in\mathbb{C}:{\sf Im}\,w>y_{0}\\}\subset\overline{\Omega}$. But
$\\{w\in\mathbb{C}:{\sf Im}\,w>y_{0}\\}$ is open, thus $\\{w\in\mathbb{C}:{\sf
Im}\,w>y_{0}\\}\subset\hbox{Int}(\overline{\Omega})=\Omega$, by hypothesis.
Therefore, $(y_{0},+\infty)\subset I$ and $\psi(y)=-\infty$ for all $y>y_{0}$.
In other words, $y_{0}\in\overline{I_{\infty}}$, a contradiction. Thus (3)
holds.

(3) implies (1). Notice that $\Pi=\\{w\in\mathbb{C}:{\sf Im}\,w<0\\}$ is
contained in $\mathbb{C}\setminus\overline{\Omega}$. The condition (3) implies
that $\inf_{y\in J}\psi(y)>-\infty$ for any compact interval
$J\subset\overline{I}\setminus\overline{I_{\infty}}$. It follows that any
point in $\mathbb{C}\setminus\overline{\Omega}$ can be connected with a point
in $\Pi$ in $\mathbb{C}\setminus\overline{\Omega}$ by a union of a horizontal
interval (going to the left) and a vertical one (going down). Therefore
$\mathbb{C}\setminus\overline{\Omega}$ is connected.

∎

As a direct corollary of Lemma 7.15 and Theorem 8.2, we have:

######  Corollary 8.3.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function, $\Omega=h(\mathbb{D})$ the Koenigs domain and let
$\psi:I\to\mathbb{R}$ be the defining function of $\Omega$. Suppose
$\hbox{Int}(\overline{\Omega})=\Omega$ and $\Omega\subset\mathbb{S}$, for some
strip $\mathbb{S}$. Then the following are equivalent.

  1. (1)

$\mathbb{C}\setminus\overline{\Omega}$ is connected.

  2. (2)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$ and
$\mathcal{S}(h)$ contains only bounded discontinuities.

  3. (3)

$\liminf_{I\ni y\to y_{0}}\psi(y)>-\infty$ for every $y_{0}\in\overline{I}$.

Now we examine the case $\Omega$ is contained in a strip and
$\mathbb{C}\setminus\overline{\Omega}$ has two connected components.

######  Theorem 8.4.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function, $\Omega=h(\mathbb{D})$ the Koenigs domain and let
$\psi:I\to\mathbb{R}$ be the defining function of $\Omega$. Suppose
$\hbox{Int}(\overline{\Omega})=\Omega$ and $\Omega\subset\mathbb{S}$, for some
strip $\mathbb{S}$. Then the following are equivalent.

  1. (1)

$\mathbb{C}\setminus\overline{\Omega}$ has two connected components,

  2. (2)

there exist $a\leq b$ such that $[a,b]\subset\overline{I}$, $\liminf_{y\to
y_{0}}\psi(y)=-\infty$ for all $y_{0}\in[a,b]$ and $\liminf_{I\ni y\to
y_{1}}\psi(y)>-\infty$ for all $y_{1}\in\overline{I}\setminus[a,b]$,

  3. (3)

$(\varphi_{t})_{t\geq 0}$ satisfies one, and only one, of the following:

     * (3.1)

$(\varphi_{t})_{t\geq 0}$ has one and only one boundary fixed point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ and
$\mathcal{S}_{u}(h)=\emptyset$,

     * (3.2)

$(\varphi_{t})_{t\geq 0}$ has one and only one boundary fixed point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$,
$\mathcal{S}_{u}(h)=\\{\xi\\}$ with $\xi\neq\tau$, and there exists a maximal
contact arc starting from $\sigma$ and ending at $\xi$,

     * (3.3)

$(\varphi_{t})_{t\geq 0}$ has one and only one boundary fixed point
$\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ which is a boundary regular
fixed point, $\mathcal{S}_{u}(h)=\\{\xi_{1},\xi_{1}\\}$ with
$\xi_{j}\neq\tau$, and there exists a maximal contact arc starting from
$\sigma$ and ending at $\xi_{j}$, $j=1,2$,

     * (3.4)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$ and
$\mathcal{S}_{u}(h)=\\{\xi\\}$, with $\xi\neq\tau$,

     * (3.5)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$ and
$\mathcal{S}_{u}(h)=\\{\xi_{1},\xi_{2}\\}$, with $\xi_{j}\neq\tau$, $j=1,2$
and there exists a maximal contact arc starting from $\xi_{1}$ and ending at
$\xi_{2}$,

     * (3.6)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$,
$\mathcal{S}_{u}(h)=\\{\tau\\}$ and $\tau$ is a simple unbounded
discontinuity.

###### Proof.

(1) is equivalent to (2) by Lemma 5.7.

(2) implies (3). We consider three cases:

Case 1: $a<b$. Since by hypothesis $\hbox{Int}(\overline{\Omega})=\Omega$ and
$\liminf_{y\to y_{0}}\psi(y)=-\infty$ for all $y_{0}\in[a,b]$, it follows from
Theorem 6.14 it follows that $\psi(y)=-\infty$ for all $y\in(a,b)$. By
Proposition 7.6 there exists a unique boundary fixed point $\sigma\neq\tau$
such $a_{-}(\sigma)=a$ and $a_{+}(\sigma)=b$ so that $\sigma$ is a boundary
regular fixed point. Since $\liminf_{I\ni y\to y_{1}}\psi(y)>-\infty$ for all
$y_{1}\in\overline{I}\setminus[a,b]$, the same Proposition 7.6 implies that
there are no other boundary fixed points different from $\sigma$ and $\tau$.
By Proposition 7.14 and since $\liminf_{I\ni y\to y_{1}}\psi(y)>-\infty$ for
all $y_{1}\in\overline{I}\setminus[a,b]$, $\tau\not\in\mathcal{S}_{u}(h)$.
Finally, Proposition 7.9 and Proposition 7.10 show at once that either (3.1)
or (3.2) or (3.3) holds.

Case 2: $a=b\not\in\partial I$. By Proposition 7.14,
$\tau\not\in\mathcal{S}_{u}(h)$.

Since $a\in I$ and $\psi$ is upper semicontinuous, it follows that
$\limsup_{y\to a}\psi(y)\leq\psi(a)<+\infty$.

If either $\lim_{y\to a^{-}}\psi(y)=-\infty$ or $\lim_{y\to
a^{+}}\psi(y)=-\infty$ and since $\liminf_{I\ni y\to y_{1}}\psi(y)>-\infty$
for all $y_{1}\in\overline{I}\setminus\\{a\\}$, it follows from Proposition
7.6 that there exists a unique boundary fixed point $\sigma$ different from
$\tau$. Moreover, ${\sf Im}\,h^{\ast}(\sigma)=a$. Proposition 7.9 and
Proposition 7.10 show at once that either (3.1) or (3.2) holds.

If $\limsup_{y\to a^{\pm}}\psi(y)>-\infty$, then by Proposition 7.6, there are
no boundary fixed points other than $\tau$. Proposition 7.9 shows that there
exists an unbounded discontinuity $\xi$, and Proposition 7.10 implies that
either (3.4) or (3.5) holds.

Case 3: $a=b\in\partial I$. We can assume that $I=(a,c)$ for some $c>a$.

If $\limsup_{y\to a^{+}}\psi(y)=+\infty$, it follows from Proposition 7.14,
Proposition 7.6 and Proposition 7.9 that $\tau$ is a simple unbounded
discontinuity and there are no other boundary fixed points nor unbounded
discontinuities, thus (3.6) holds.

If $\limsup_{y\to a^{+}}\psi(y)<+\infty$, then $\tau\not\in\mathcal{S}_{u}(h)$
and we can argue as in Case 2 depending on whether $\lim_{y\to
a^{+}}\psi(y)=-\infty$ or not, and we see that in this case either (3.1) or
(3.4) holds. Note here that in this case, (3.2) and (3.5) cannot hold because
Proposition 7.10 asserts that in such cases $a\in I$.

All possible cases are covered by the previous arguments, and hence (2)
implies (3).

(3) implies (2). It follows at once examining the various cases and applying
Proposition 7.6, Proposition 7.9, Proposition 7.10 and Proposition 7.14. ∎

##  9\. Proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3

The proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3 follows by gathering
together the results we proved in the previous sections.

###### Proof of Theorem 1.1.

The semigroup $(\varphi_{t})_{t\geq 0}$ is hyperbolic if and only if $\Omega$
is contained in a strip $\mathbb{S}$. Hence, by Theorem 5.8 and Corollary 5.9,
(1) and (2) are equivalent. Then (2) is equivalent to (3) and (4) by Theorem
6.14, Corollary 8.3 and Theorem 8.4. The last assertion follows from (1) and
Lemma 2.4. ∎

###### Proof of Theorem 1.2.

The semigroup $(\varphi_{t})_{t\geq 0}$ is parabolic of positive hyperbolic
step if and only if $\Omega$ is contained in a half-plane parallel to the real
axes. Hence, by Theorem 5.5 and Corollary 5.9, (1) and (2) are equivalent.
Then (2) is equivalent to (3) and (4) by Theorem 6.14 and Theorem 8.2. The
last assertion follows from (1) and Lemma 2.4. ∎

###### Proof of Theorem 1.3.

The semigroup $(\varphi_{t})_{t\geq 0}$ is parabolic of zero hyperbolic step
if and only if $\Omega$ is not contained in any half-plane parallel to the
real axes. By Proposition 3.3.(4), if there is not a half-plane $H$ such that
$\Omega\subset H$, then $\Lambda_{\infty}(\Omega)=\\{0\\}$, and the first
assertion follows.

If there is a half-plane $H$ such that $\Omega\subset H$, by Theorem 5.5 and
Corollary 5.9, (1) and (2) are equivalent. Since $H$ cannot be parallel to the
real axis, by Remark 8.1 and Theorem 6.14 we have the equivalence among (2),
(3) and (4). The last assertion follows from (1) and Lemma 2.4. ∎

##  10\. Frequencies for $p\in[1,\infty)$

Let $\Omega$ be the Koenigs domain of a non-elliptic semigroup. The aim of
this section is to discuss completeness of $\mathcal{E}_{p}(\Omega)$ for
$p\in[1,\infty)$, giving the proof of Theorem 1.4 and discuss some necessary
conditions for non-completeness of linear combinations of exponentials.

###  10.1. Logarithmic starlike domains at infinity with frequencies and the
proof of Theorem 1.4

We start with the following proposition:

######  Proposition 10.1.

Let $(\varphi_{t})_{t\geq 0}$ be a non-elliptic semigroup in $\mathbb{D}$ with
Denjoy-Wolff point $\tau\in\partial\mathbb{D}$. Let $h$ be the Koenigs
function, $\Omega=h(\mathbb{D})$ the Koenigs domain and let
$\psi:I\to\mathbb{R}$ be the defining function of $\Omega$. Suppose
$\hbox{Int}(\overline{\Omega})=\Omega$. Assume $\Omega$ is contained in the
rotation of a logarithmic starlike at infinity domain. Let $I_{\infty}$
denotes the unbounded connected component of $\psi^{-1}(-\infty)$ (possibly
empty). Then the following are equivalent.

  1. (1)

$\mathbb{C}\setminus\overline{\Omega}$ is connected.

  2. (2)

$(\varphi_{t})_{t\geq 0}$ has no boundary fixed points other than $\tau$ and
$\mathcal{S}(h)$ contains only bounded discontinuities except, at most, one
unbounded discontinuity $\sigma\in\partial\mathbb{D}\setminus\\{\tau\\}$ which
is the final point of a maximal contact arc starting at $\tau$.

  3. (3)

$\liminf_{I\ni y\to y_{0}}\psi(y)>-\infty$ for every
$y_{0}\in\overline{I}\setminus\overline{I_{\infty}}$.

###### Proof.

Observe that if $\Omega$ is contained in $D$, the rotation of a logarithmic
starlike at infinity domain, then there is $v\in\mathbb{C}$, ${\sf Re}\,v\leq
0$ and $|v|=1$, such that for all $w\in D$ there exists $t>0$ so that
$w+tv\in\mathbb{C}\setminus\overline{D}$. If $v$ is not pure imaginary, then
it follows that for all $w\in D$ there exists $t^{\prime}>0$ such that
$w-t^{\prime}\in\mathbb{C}\setminus\overline{D}$. Hence, arguing as in Remark
8.1 we have the result.

If $v$ is pure imaginary, then $D$ is the $\pi/2$ rotation of a logarithmic
starlike at infinity domain. Hence, there is a half-plane $H$ parallel to the
real axis which is contained in $D$ and such that $\partial H\cap\partial
D\neq\emptyset$. We can assume $H=\\{w\in\mathbb{C}:{\sf Im}\,w>0\\}$ and
$x_{0}\in\mathbb{R}\cap\partial D$ is the maximum of the points in $\partial
D\cap\mathbb{R}$.

Since $\Omega$ is starlike at infinity, then

$\Omega\subset\\{w\in\mathbb{C}:{\sf Im}\,w>0\\}\cup\\{w\in\mathbb{C}:{\sf
Re}\,w>x_{0}\\}.$

Then the same argument as in the proof of Theorem 8.2 gives the result. ∎

With this result at hand, we can give the

###### Proof of Theorem 1.4.

It follows at once by Theorem 5.10, Theorem 6.14 and Proposition 10.1. ∎

Now, we provide examples of logarithmic starlike at infinity domains with
$p$-frequencies, which are not contained in any half-plane:

######  Proposition 10.2.

Let $\eta:\mathbb{C}_{+}\to\mathbb{C}$ be defined by
$\eta(w):=w-(\log(w+3))^{a}$, where $a\in(0,1)$ is a constant. Then $\eta$ is
univalent on $\mathbb{C}_{+}$, and the domain
$\Omega_{0}=\eta(\mathbb{C}_{+})$ is a logarithmic starlike at infinity domain
with $p$-frequencies, which is not contained in any half-plane.

###### Proof.

It is easy to see that $|\eta^{\prime}-1|<(\log 3)^{-1}<1$ on
$\mathbb{C}_{+}$, hence ${\sf Re}\,\eta^{\prime}>0$ on $\mathbb{C}_{+}$. Since
$\mathbb{C}_{+}$ is convex, it follows by the Noshiro-Warschawski’s theorem
(see, e.g. [8, Theorem 3.1.3]) that $\eta$ is univalent on $\mathbb{C}_{+}$.

Moreover, the boundary curve of the domain $\eta(\mathbb{C}_{+})$ is
parametrized as $\mathbb{R}\ni t\mapsto\alpha(t)+i\beta(t)$ where,
$\alpha(t):={\sf Re}\,\eta(it)$ and $\beta(t):={\sf Im}\,\eta(it)$. Since
$\beta^{\prime}(t)={\sf Re}\,\eta^{\prime}(it)>0$ for all $t$ and $\beta(t)$
tends to $\pm\infty$ when $t\to\pm\infty$, it follows that $\beta$ is an
increasing diffeomorphism of $\mathbb{R}$ to $\mathbb{R}$.

Hence the function $\psi=\alpha\circ\beta^{-1}$ is differentiable, and
$\Omega_{0}=\eta(\mathbb{C}_{+})$ has the form
$\\{x>\psi(y):y\in\mathbb{R}\\}$. Therefore, $\Omega_{0}$ is a starlike at
infinity domain and clearly it is not contained in any half-plane, since
$\psi(y)\to-\infty$ as $y\to\pm\infty$.

Since $\beta^{\prime}(t)={\sf Re}\,\eta^{\prime}(t)\in(1-a,1+a)$ for all $t$,
it follows that $\psi$ satisfies (1) of Definition 4.4. Also,
$|\beta^{-1}(t)|\leq|t|/(1-a)$ for all $t$, implies that $\psi$ satisfies (2)
of Definition 4.4. Therefore, $\Omega_{0}$ is a logarithmic starlike at
infinity domain.

Finally, let $\lambda\leq 0$. By the conformal invariance of the Hardy spaces,
$e^{\lambda z}\in H^{p}(\Omega_{0})$ if and only if $e^{\lambda\eta(z)}\in
H^{p}(\mathbb{C}_{+})$. Take any $b\in(0,1)$. It is easy to see that for some
$C>0$,

$p\lambda{\sf Re}\,\eta(w)\leq C+\log{\sf Re}\,(w+3)^{b},\quad
w\in\mathbb{C}_{+},$

which implies that $e^{p\lambda{\sf Re}\,\eta(w)}\leq e^{C}\cdot{\sf
Re}\,(w+3)^{b}$ in $\mathbb{C}_{+}$. Since the function ${\sf Re}\,(w+3)^{b}$
is harmonic in $\mathbb{C}_{+}$, $|e^{\lambda\eta}(z)|^{p}$ has a harmonic
majorant in $\mathbb{C}_{+}$, which proves that $e^{\lambda z}\in
H^{p}(\Omega_{0})$. Thus, $\Omega_{0}$ has $p$-frequencies, and we are done. ∎

As a direct application of Theorem 1.4 and Proposition 10.2, we have:

######  Corollary 10.3.

Let $\psi$ be any continuous function on $\mathbb{R}$ such that
$\psi(y)>C_{1}-\big{(}\log(|y|+3)\big{)}^{a}$ for all $y$, where $C_{1}$ and
$a\in(0,1)$ are constants. Then $\Omega:=\\{x+iy:x>\psi(y)\\}$ is a Koenigs
domain of a non-elliptic semigroup, and $\mathcal{E}_{p}(\Omega)$ is dense in
$H^{p}(\Omega)$ for all $p\in[1,\infty)$.

###  10.2. Necessary conditions for completeness of $p$-frequencies

The aim of this section is to give some necessary conditions for completeness
of frequencies in $H^{p}(\Omega)$. We first show that a necessary condition is
that the frequencies are not reduced to an interval:

######  Proposition 10.4.

Suppose $\Omega$ is a Koenigs domain of a non-elliptic semigroup such that
$\mathbb{C}_{+}+c\subset\Omega$ for some $c\in\mathbb{R}$. Suppose that for
some $p_{0}\in[1,+\infty)$ and $M>0$,

$\Lambda_{p_{0}}(\Omega)\subset[-M,0].$

Then $\mathcal{E}_{p}(\Omega)$ is not dense in $H^{p}(\Omega)$ for every
$p\in[1,\infty)$.

###### Proof.

Notice that by Proposition 3.3.(1), the condition
$\Lambda_{p_{0}}(\Omega)\subset[-M,0]$ implies that either
$\Lambda_{p_{0}}(\Omega)=[-N,0]$ for some $N\geq 0$ or
$\Lambda_{p_{0}}(\Omega)=(-N,0]$ for some $N>0$. Then, by Proposition 3.3.(2),
for any $q\in[1,\infty)$, $\Lambda_{q}(\Omega)$ is also a finite interval.

We show that, if $\Lambda_{p}(\Omega)$ is a finite interval then
$\mathcal{E}_{p}(\Omega)$ is not dense in $H^{p}(\Omega)$.

We may assume that $c=0$. Arguing by contradiction, we assume that
$\mathcal{E}_{p}(\Omega)$ is dense in $H^{p}(\Omega)$.

Let $h:\mathbb{D}\to\Omega$ be a Riemann map. We claim that $h^{-1}$ extends
continuously and injectively on $\overline{\mathbb{C}_{+}}^{\infty}$. Indeed,
if $ir\in\Omega$ for some $r\mathbb{R}$ then clearly $h^{-1}$ extends
continuously at $ir$. If $ir\in\partial\Omega$, then (see, e.g., [8,
Proposition 3.3.3]) there exists a point $\sigma\in\partial\mathbb{D}$ such
that $\lim_{t\to 0^{+}}h^{-1}(t+ir)=\sigma$ and $h^{\ast}(\sigma)=ir$. Now,
let $\\{z_{n}\\}\subset\mathbb{C}_{+}$ be a sequence converging to $ir$. We
can assume that $\\{{\sf Re}\,z_{n}\\}$ is strictly decreasing. Let
$\Gamma_{n}$ be the segment joining $z_{n}$ to $z_{n+1}$. Hence,
$\Gamma=\cup_{n}\Gamma_{n}$ is a Jordan arc contained in $\mathbb{C}_{+}$
(since it is convex) converging to $ir$. It is easy to show (see, e.g., [8,
Proposition 3.3.5]) that the preimage of $\Gamma$ via $h^{-1}$ converges to
$\sigma$. Thus $h^{-1}$ extends univocaly at every $ir$, $r\in\mathbb{R}$. It
is also easy to show (by the same tokens) that there exists
$\tau\in\partial\mathbb{D}$ such that $\lim_{\mathbb{C}_{+}\ni
z\to\infty}h^{-1}(z)=\tau$. We set $h^{-1}(\infty)=\tau$. In order to show
that $h^{-1}$ is continuous on $\overline{\mathbb{C}_{+}}$, it is enough to
show that if $\\{ir_{n}\\}$ is a sequence converging to $ir$ (with
$r\in\mathbb{R}\cup\\{\infty\\}$) then $\\{h^{-1}(ir_{n})\\}$ converges to
$h^{-1}(ir)$. Indeed, let $\Gamma_{n}$ be a Jordan arc in
$\overline{\mathbb{C}_{+}}^{\infty}$ such that $ir_{n},ir\in\Gamma_{n}$ and
$\Gamma_{n}\setminus\\{ir_{n},ir\\}\subset\mathbb{C}_{+}$. We can take
$\Gamma_{n}$ in such a way that its spherical diameter tends to $0$ as
$n\to\infty$. Thus, if $\\{h^{-1}(ir_{n})\\}$ does not converge to
$h^{-1}(ir)$, there is a subsequence $\\{n_{k}\\}$ such that
$h^{-1}(\Gamma_{n_{k}})\cap\mathbb{D}$ has diameter bounded from below by a
positive constant. By this violates the no Koebe arcs theorem (see, e.g., [8,
Theorem 3.2.4]). Thus $h^{-1}$ extends continuously on
$\overline{\mathbb{C}_{+}}$. Finally, it is easy to see that $h^{-1}$ is
injective on $\overline{\mathbb{C}_{+}}$ (see, e.g., [8, Proposition 3.3.4]).

Therefore, $G:=h^{-1}(\mathbb{C}_{+})$ is a Jordan domain in $\mathbb{D}$. By
Mergelyan’s approximation theorem, polynomials are dense in the algebra of
holomorphic functions which are continuous up to $\overline{G}$, since this
last space is dense in $H^{p}(G)$, it follows that $H^{p}(\mathbb{D})$ is
dense in $H^{p}(G)$ and thus $H^{p}(\Omega)$ is dense in
$H^{p}(\mathbb{C}_{+})$.

By Lemma 5.1, the density of $H^{p}(\Omega)$ in $H^{p}(\mathbb{C}_{+})$
implies that $\mathcal{E}_{p}(\Omega)$ is dense in $H^{p}(\mathbb{C}_{+})$.

So to finish the proof it suffices to check that the last statement is false.
To this end, choose a nonzero $C^{\infty}$ function $s$ on $\mathbb{R}_{+}$,
which vanishes on $[0,M]$. Let $\mathcal{H}^{2}(\mathbb{C}_{+})$ be the Hardy-
Smirnov space of the semiplane $\mathbb{C}_{+}$. By the Paley-Wiener theorem,
$(2\pi)^{-1/2}\,\mathcal{L}$ is an isometric isomorphism of
$L^{2}(\mathbb{R}_{+},dx)$ onto $\mathcal{H}^{2}(\mathbb{C}_{+})$ (here
$\mathcal{L}$ is the Laplace transform). Set $u=\mathcal{L}s$. Then
$u\in\mathcal{H}^{2}(\mathbb{C}_{+})$, and, moreover, $(z+1)u\in
H^{\infty}(\mathbb{C}_{+})$. The function $u$ gives rise to a bounded linear
functional on $H^{p}(\mathbb{C}_{+})$, defined by

$\tilde{u}(v)=\int_{i\mathbb{R}}u(-z)v(z)|dz|,\quad v\in
H^{p}(\mathbb{C}_{+}).$

By the Fourier inversion formula,

$\tilde{u}(e^{\lambda z})=2\pi s(-\lambda),\quad\lambda\leq 0.$

Since $s$ is nonzero, $\tilde{u}$ is a nonzero functional on
$H^{p}(\mathbb{C}_{+})$. On the other hand, $s(-\lambda)=0$ for all
$\lambda\in\Lambda_{p}(\Omega)$, which implies that $\tilde{u}$ vanishes on
exponentials $e^{\lambda z}$ for $\lambda\in\Lambda_{p}(\Omega)$. Hence the
space of linear combinations of these exponentials are not dense in
$H^{p}(\mathbb{C}_{+})$. ∎

######  Example 10.5.

Let $\eta(w)=w-\log(w+3)$. We claim that $\eta$ is univalent on
$\mathbb{C}_{+}$, and the domain $\Omega_{0}=\eta(\mathbb{C}_{+})$ has the
form $\Omega=\\{x+iy\in\mathbb{C}:x>\psi_{0}(y)\\}$, and it is a logarithmic
starlike at infinity domain. Moreover, the functions $e^{\lambda z}$, for
$\lambda\leq 0$, belong to $H^{p}(\Omega_{0})$ if $p|\lambda|<1$ and do not
belong to $H^{p}(\Omega_{0})$ if $p|\lambda|\geq 1$.

Indeed, one has only to check the last statement on the membership in
$H^{p}(\Omega_{0})$, because the arguments given in Proposition 10.2 apply to
all other statements. Suppose first that $p|\lambda|<1$. Then the estimate

$p\lambda{\sf Re}\,\eta(w)\leq p|\lambda|\log|w+3|$

implies that

$|e^{\lambda\eta}|^{p}\leq|w+3|^{p|\lambda|}\leq K{\sf
Re}\,(w+3)^{p|\lambda|}$

in $\mathbb{C}_{+}$. Hence ${\sf Re}\,(w+3)^{p|\lambda|}$ is a harmonic
majorant of $|e^{\lambda\eta}|^{p}$ in $\mathbb{C}_{+}$.

Now let $p|\lambda|\geq 1$. Notice that
$|e^{\lambda\eta(w)}|^{p}\asymp|w|^{p|\lambda|}$ as
$w\in\partial\mathbb{C}_{+}$ goes to infinity. Since the harmonic measure
$d\omega_{\mathbb{C}_{+}}(w)=(|w|^{2}+1)^{-1}/\pi$, one gets that
$e^{\lambda\eta}\notin L^{p}(d\omega_{\mathbb{C}_{+}})$, and therefore the
exponential function $e^{\lambda z}$ is not in $H^{p}(\Omega_{0})$.

Next, we see that the existence of two non-exceptional maximal contact arcs
with the same final point is an obstruction to $p$-completeness of frequencies
(and not only to $\infty$-weak-star completeness):

######  Proposition 10.6.

Assume $(\varphi_{t})_{t\geq 0}$ is a non-elliptic semigroup with Denjoy-Wolff
point $\tau\in\partial\mathbb{D}$. If there exist two contact arcs $A,B$ with
final point $\sigma=x_{1}(A)=x_{1}(B)\in\partial\mathbb{D}\setminus\\{\tau\\}$
then $(\varphi_{t})_{t\geq 0}$ has no $p$-complete frequencies for every
$p\in[1,\infty)$.

###### Proof.

Let $h$ be the Koenigs function of $(\varphi_{t})_{t\geq 0}$. By Lemma 6.2, we
can find $\varepsilon>0$, $z_{0}\in\mathbb{C}$ and two non-empty open arcs
$A^{\prime}\subset A$ and $B^{\prime}\subset B$ such that $h$ extends
holomorphically through $A^{\prime}$ and $B^{\prime}$,

$\Omega\cap\\{\zeta\in\mathbb{C}:|\zeta-
z_{0}|<\varepsilon\\}=\\{\zeta\in\mathbb{C}:|\zeta-
z_{0}|<\varepsilon\\}\setminus L,$

where

$L:=\\{w\in\mathbb{C}:|{\sf Re}\,w-{\sf Re}\,z_{0}|<\varepsilon,{\sf
Im}\,w={\sf Im}\,z_{0}\\},$

and $h(A^{\prime})=h(B^{\prime})=L$.

We claim that $h^{-1}\not\in\overline{\mathcal{E}(\Omega)}^{H^{p}(\Omega)}$.
Since $h^{-1}\in H^{\infty}(\Omega)\subset H^{p}(\Omega)$, this proves the
statement.

The claim, using conformal invariance, is equivalent to
$z\not\in\overline{\hbox{span}\\{e^{\lambda
h(z)}:\lambda\in\Lambda_{p}(\Omega)\\}}^{H^{p}(\mathbb{D})}$.

Indeed, for every $\lambda\in\Lambda_{p}(\Omega)$, it follows that $e^{\lambda
h(A^{\prime})}=e^{\lambda h(B^{\prime})}$. Hence, any function $f$ in
$\mathcal{E}_{p}(\Omega)$ extends holomorphically through $A^{\prime}$ and
$B^{\prime}$ and $f(A^{\prime})=f(B^{\prime})$.

Now, if $\\{P_{n}\\}\subset\mathcal{E}_{p}(\Omega)$ converges in
$H^{p}(\mathbb{D})$ to $z$, since the sequence of boundary values
$\\{P_{n}^{\ast}\\}$ converges in $L^{p}(\partial\mathbb{D})$ to the boundary
values of $z$, it follows that $\\{P_{n}^{\ast}(e^{i\theta})\\}$ converges to
$e^{i\theta}$ for almost every $\theta\in[0,2\pi)$.

Recall that $P_{n}$ extends holomorphically through $A^{\prime}\cup
B^{\prime}$ and let $A^{\prime\prime}\subset A^{\prime}$ be the set of full
measure such that $\\{P_{n}(e^{i\theta})\\}$ converges to $e^{i\theta}$ and
similarly define $B^{\prime\prime}\subset B^{\prime}$.

Let $V$ be the (bounded) Jordan domain whose boundary is given by $A^{\prime}$
and the segment joining the two end points of $A^{\prime}$. By the Riesz-
Privalov theorem, applied to $h|_{V}$ (see, e.g., [21, Theorem 6.8]), we see
that $h(A^{\prime\prime})$ has full measure in $h(A^{\prime})=L$. Similarly,
$h(B^{\prime\prime})$ has full measure in $h(B^{\prime})=L$. Therefore, there
exist $q\in A^{\prime\prime}$ and $p\in B^{\prime\prime}$ such that
$P_{n}(q)=P_{n}(p)$ for all $n$, but $\\{P_{n}(p)\\}$ converges to $p$ and
$\\{P_{n}(q)\\}$ converges to $q$, a contradiction. Therefore,
$z\not\in\overline{\hbox{span}\\{e^{\lambda
h(z)}:\lambda\in\Lambda_{p}(\Omega)\\}}^{H^{p}(\mathbb{D})}$, and we are done.
∎

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