# Invariance of Abel universality under composition and applications

Stéphane Charpentier, Myrto Manolaki, Konstantinos Maronikolakis  Stéphane
Charpentier, Institut de Mathématiques, UMR 7373, Aix-Marseille Universite, 39
rue F. Joliot Curie, 13453 Marseille Cedex 13, France
<EMAIL_ADDRESS>Myrto Manolaki, UCD School of Mathematics
and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
<EMAIL_ADDRESS>Konstantinos Maronikolakis, UCD School of Mathematics and
Statistics, University College Dublin, Belfield, Dublin 4, Ireland
<EMAIL_ADDRESS>
###### Abstract.

A holomorphic function $f$ on the unit disc ${\mathbb{D}}$ belongs to the
class $\mathcal{U}_{A}(\mathbb{D})$ of Abel universal functions if the family
$\\{f_{r}:0\leq r<1\\}$ of its dilates $f_{r}(z):=f(rz)$ is dense in the
Banach space of all continuous functions on $K$, endowed with the supremum
norm, for any proper compact subset $K$ of the unit circle. We prove that this
property is invariant under composition from the left with any non-constant
entire function. As an application, we show that $\mathcal{U}_{A}(\mathbb{D})$
is strongly-algebrable. Furthermore, we prove that Abel universality is
invariant under composition from the right with an automorphism $\Phi$ of
${\mathbb{D}}$ if and only if $\Phi$ a rotation. On the other hand, we
establish the existence of a subset of $\mathcal{U}_{A}(\mathbb{D})$ which is
residual in the space of holomorphic functions on $\mathbb{D}$ and is
invariant under composition from the right with any automorphism of
${\mathbb{D}}$.

###### Key words and phrases:

Abel universal functions, invariance under composition

###### 2020 Mathematics Subject Classification:

30K15, 30B30, 30E10

The research conducted in this paper was funded by the Irish Research Council
under the Ulysses 2022 Scheme. Konstantinos Maronikolakis acknowledges
financial support from the Irish Research Council through the grant
[GOIPG/2020/1562].

##  1\. Introduction

Let ${\mathbb{D}}=\\{z\in{\mathbb{C}}:\,|z|<1\\}$ and
${\mathbb{T}}=\partial{\mathbb{D}}$ denote the unit disc and the unit circle
respectively, and let $H(\mathbb{D})$ be the space of holomorphic functions on
$\mathbb{D}$, endowed with the topology of local uniform convergence. This
paper is concerned with a class of functions in $H(\mathbb{D})$ which have an
extremely chaotic radial behaviour.

######  Definition 1.1 (Abel universal functions).

Let $\rho=(r_{n})_{n}$ be an increasing sequence in $[0,1)$ such that
$r_{n}\to 1$. We denote by ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ the set of
all functions $f\in H({\mathbb{D}})$ that satisfy the following property: for
any compact subset $K$ of ${\mathbb{T}}$, different from ${\mathbb{T}}$, and
any continuous function $\varphi$ on $K$, there exists an increasing sequence
$(n_{k})_{k}$ in ${\mathbb{N}}$ such that

$\sup_{\zeta\in K}|f(r_{n_{k}}\zeta)-\varphi(\zeta)|\to 0,\quad k\to\infty.$

A function in $H({\mathbb{D}})$ is called an Abel universal function if it
belongs to ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for some $\rho$, and the
class of all Abel universal functions is denoted by
${\mathcal{U}}_{A}({\mathbb{D}})$.

In [18], it was shown that $\mathcal{U}_{A}(\mathbb{D},\rho)$ is a dense
$G_{\delta}$ subset of $H(\mathbb{D})$ for any $\rho$ as above, which unified
several previous results in [30], [17], [11] and [6]. A reformulation of
Definition 1.1 reads as follows: a function $f$ in $H({\mathbb{D}})$ is Abel
universal if, for any compact subset $K$ of ${\mathbb{T}}$, different from
${\mathbb{T}}$, the family $\\{f_{r}:\,0\leq r<1\\}$ of its dilates
$f_{r}(z):=f(rz)$, $z\in\overline{{\mathbb{D}}}$, is dense in the Banach space
$C(K)$ of complex-valued continuous functions on $K$, endowed with the
supremum norm. Dilation is one of the most standard techniques used in complex
function theory, and especially in the study of the boundary behaviour of
functions in $H({\mathbb{D}})$. For a detailed account of results over the
past century and the wider importance of dilation theory we refer the reader
to the recent survey [32]. The other standard technique for studying the
boundary behaviour of a function $f$ in $H({\mathbb{D}})$ involves considering
the partial sums $S_{n}(f,0)$ of its Taylor expansion about the origin. Thus,
the class $\mathcal{U}_{A}(\mathbb{D})$ of Abel universal functions is the
natural analogue of the well-studied class $\mathcal{U}_{T}(\mathbb{D})$ of
universal Taylor series, which consists of all functions $f\in
H({\mathbb{D}})$ such that the set $\\{S_{n}(f,0):n\in\mathbb{N}\\}$ is dense
in the space of functions in $C(K)$ that are holomorphic on the interior of
$K$, for any compact set $K$ in $\mathbb{C}\setminus\mathbb{D}$ with connected
complement. In [35], Nestoridis showed that $\mathcal{U}_{T}(\mathbb{D})$ is a
dense $G_{\delta}$ subset of $H(\mathbb{D})$, which opened an interesting
avenue of investigation, connecting the classical theory of overconvergence
with the more recent theory of universality.

Since the introduction of Abel universal functions in 2020, several variants
and properties have been intensively studied, such as their local and global
growth, their boundary behaviour and the behaviour of their Taylor polynomials
outside $\mathbb{D}$ (see [18, 19, 20, 21, 31, 34]). It turns out that the
notion of Abel universal functions is somehow more flexible than that of
universal Taylor series. This especially appears when we carefully look at the
construction of an Abel universal function, and when we try to build one that
satisfies additional properties. For example, such a construction can be
achieved in order to prove that the set $\mathcal{U}_{A}(\mathbb{D},\rho)$ is
not invariant under derivation [21], while the same problem is still open for
universal Taylor series. At the same time, Abel universal functions share many
boundary properties with universal Taylor series. For instance, any Abel
universal function $f$ admits infinity as an asymptotic value and satisfies
the local Picard property; that is, for every region of the form
$D_{\zeta,r}:=\\{z\in\mathbb{D}:|z-\zeta|<r\\}$, where $r>0$ and
$\zeta\in{\mathbb{T}}$, the image $f(D_{\zeta,r})$ is the entire complex plane
$\mathbb{C}$ except possibly a point [20]. (The analogues of these results for
universal Taylor series were proved in [25] and [26].) Two questions naturally
arise: (i) Can the exceptional value in the local Picard property occur? (ii)
Can Abel universal functions have finite asymptotic values?

In this paper, we will derive answers to both questions, by looking at the
bigger picture; namely, we will examine the invariance of Abel universality
under composition.

######  Main Problems.

Let $\rho=(r_{n})_{n}$ be a sequence in $[0,1)$ with $r_{n}\to 1$ as
$n\to\infty$.

  1. (1)

Given $f$ in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, describe the holomorphic
functions $g$ on $f({\mathbb{D}})$ such that $g\circ f$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$;

  2. (2)

Describe the holomorphic functions $\Phi:{\mathbb{D}}\to{\mathbb{D}}$ which
are continuous on $\overline{{\mathbb{D}}}$, such that $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, for every $f$ in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$.

As an application of one of our main results, we will deduce that if
$f\in\mathcal{U}_{A}(\mathbb{D},\rho)$ then $e^{f}$ is always in
$\mathcal{U}_{A}(\mathbb{D},\rho)$, which shows that question (i) has an
affirmative answer. Moreover, if $f$ is in $\mathcal{U}_{A}(\mathbb{D},\rho)$
and has no zeros on $\mathbb{D}$, we will derive that $1/f$ is always in
$\mathcal{U}_{A}(\mathbb{D},\rho)$, which shows that question (ii) also has an
affirmative answer. We note that although the analogues of questions (i) and
(ii) for the class $\mathcal{U}_{T}(\mathbb{D})$ have an affirmative answer as
well, the situation concerning invariance under composition is far from being
understood. The only known results about invariance from the left are due to
Costakis and Melas [22], who showed that there exists a function
$f\in\mathcal{U}_{T}(\mathbb{D})$ such that $e^{f}$ is in
$\mathcal{U}_{T}(\mathbb{D})$, and that there exists a zero-free function
$f\in\mathcal{U}_{T}(\mathbb{D})$ such that
$1/f\in\mathcal{U}_{T}(\mathbb{D})$. Thus, Problem (1) remains open for
$\mathcal{U}_{T}(\mathbb{D})$, even in simple cases. The same happens for
Problem (2). The only known result in this direction, is due to Gardiner [25],
who used a mixture of advanced tools from potential theory to show that a
certain class of universal Taylor series (defined on a strip) is not
conformally invariant. For this result, the conformal mapping was from the
disc to a strip, and this was crucial for the deep methods that were used.
Hence, even the simple case of composition from the right with a conformal map
from $\mathbb{D}$ to $\mathbb{D}$, remains unsolved for universal Taylor
series.

In this paper, we fully address Problems (1) and (2). In particular, we prove
the following:

######  Main Theorems.

Let $\rho=(r_{n})_{n}$ be a sequence in $[0,1)$ with $r_{n}\to 1$ as
$n\to\infty$.

  1. (1)

For any $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and any non-constant
holomorphic function $g$ on $f({\mathbb{D}})$ the composition $g\circ f$
belongs to ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$;

  2. (2)

Let $\Phi:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic on ${\mathbb{D}}$ and
continuous on $\overline{{\mathbb{D}}}$. Then $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for any
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, if and only if $\Phi$ is a
rotation; that is, $\Phi(z)=e^{i\theta}z$ for some $\theta\in[0,2\pi]$.

The proof of Theorem (1) relies on a variation of a path lifting theorem due
to Iversen [28], that we shall prove using an extension of the Gross Star
Theorem due to Kaplan [29]. The proof of Theorem (2) (when $\Phi$ is not a
rotation) is based on an inductive construction argument.

Theorem (1) has a nice application to the algebraic structure of the set
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. We note that
${\mathcal{U}}_{A}({\mathbb{D}},\rho)\cup\\{0\\}$ is not a linear space
(indeed for any $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and any polynomial
$P$, the function $f+P\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$). Nonetheless,
${\mathcal{U}}_{A}({\mathbb{D}},\rho)\cup\\{0\\}$ contains a linear subspace
that is dense in $H({\mathbb{D}})$ and an infinite dimensional closed subspace
[18], like the set of universal Taylor series [8, 5]. There is an extensive
literature on the topic of finding linear structures in non-linear sets; we
refer to the survey [3]. Despite recent important investigations in the
context of linear dynamical systems (see, e.g., [7, 9, 12, 15, 16, 23] and the
recent paper [2]), it is not known whether the set of universal Taylor series
contains an algebra (except the constant functions). We shall see that Theorem
(1) can be used to prove that
${\mathcal{U}}_{A}({\mathbb{D}},\rho)\cup\\{{\mathbb{D}}\ni z\mapsto
c\in{\mathbb{C}}\text{ for some constant }c\in{\mathbb{C}}\\}$ does contain a
dense infinitely generated algebra.

The paper is organized as follows: Section 2 contains the proof of Theorem
(1), and the application to the algebrability of
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Section 3 is devoted to the proof of
Theorem (2). In Section 4 we provide some further extensions of our main
results to a more general notion than Abel universality. In this case, the
origin $0$ of the radii where we have universal approximation is replaced by a
fixed point $w$ in $\mathbb{D}$. We also show that the corresponding classes
depend on the choice of $w$ (which is not the case for universal Taylor series
if we change the center of expansion). Finally, in Section 5 we establish the
existence of a subset ${\mathcal{U}}$ of
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ which is residual in $H({\mathbb{D}})$
(that is, ${\mathcal{U}}$ contains a dense $G_{\delta}$ subset of
$H({\mathbb{D}})$) and satisfies the following property: for any
$f\in{\mathcal{U}}$, any non-constant holomorphic function
$g:f({\mathbb{D}})\to{\mathbb{C}}$, and any automorphism $\Phi$ of
${\mathbb{D}}$, the function $g\circ f\circ\Phi$ belongs to ${\mathcal{U}}$.

##  2\. Invariance under composition from the left and algebrability

In this section, our aim is to prove that all properly defined (non-constant)
holomorphic functions preserve Abel universality under composition from the
left. Then we will use this result to derive the existence of subalgebras of
$H({\mathbb{D}})$, every non-constant element of which is an Abel universal
function.

###  2.1. Main result

Throughout this paper, the notation $\rho$ stands for an increasing sequence
$(r_{n})_{n}$ in $[0,1)$ that tends to $1$ as $n\to\infty$.

######  Theorem 2.1.

Let $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. For any
$g:f({\mathbb{D}})\to{\mathbb{C}}$ non-constant holomorphic function, we have
$g\circ f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$.

The main tool for the proof is a result similar to a theorem due to Iversen
(see [14] p.245 or [28]). We are grateful to Professor Alexandre Eremenko for
bringing this theorem to our attention. We will prove this Iversen-type
theorem (Theorem 2.3 below) with the help of a consequence of [29, Theorem 3],
which is an extension of the Gross Star Theorem, and that can be stated as
follows:

######  Theorem 2.2 (Theorem 3 in [29]).

Let $B$ be a countable closed subset of ${\mathbb{C}}$. We set
$D={\mathbb{C}}\setminus B$. Let also $g:D\to{\mathbb{C}}$ be a non-constant
holomorphic function and $w\in g(D)$ any complex number that is not a critical
value of $g$. Let $g^{-1}$ be any local inverse of $g$ defined in a
neighbourhood of $w$. Then, $g^{-1}$ can be continued indefinitely along
almost every ray starting from $w$. More precisely, for almost every
$\zeta\in{\mathbb{T}}$, if $L_{\zeta}$ is the ray starting from $w$ and
passing through $w+\zeta$ and $f:[0,\infty)\to L_{\zeta}$, given by
$f(t)=w+t\zeta$, is the natural parametrisation of $L_{\zeta}$, then there
exists a continuous path $f_{0}:[0,\infty)\to D$ such that $g\circ f_{0}\equiv
f$. In particular, $f_{0}\circ f^{-1}$ is a continuous extension of the local
inverse $g^{-1}$ of $g$ to the ray $L_{\zeta}$.

Our Iversen-type theorem reads as follows:

######  Theorem 2.3.

Let $B$ be a countable closed subset of ${\mathbb{C}}$. We set
$D={\mathbb{C}}\setminus B$. Let also $g:D\to{\mathbb{C}}$ be a non-constant
holomorphic function. Then, for every proper compact subarc $K$ of
${\mathbb{T}}$, the set of all functions $h\in C(K)$ with the property that
there exists $h_{0}\in C(K)$ such that $g\circ h_{0}=h$, is dense in $C(K)$.

###### Proof.

Let $K:=\\{e^{it}:t\in[\alpha,\beta]\\}$, $\alpha,\beta\in[0,2\pi]$ with
$\alpha<\beta$, be a proper compact subarc of ${\mathbb{T}}$ and $h\in C(K)$.
Let also $\varepsilon>0$. Then, there exist a finite number of points
$t_{1},\dots,t_{n}$ with $\alpha=t_{1}<t_{2}<\dots<t_{n}=\beta$ such that
$h(K)\subset\cup_{i=1}^{n}D\left(h(t_{n}),\frac{\varepsilon}{4}\right)$. Now,
let $w_{1}\in D\left(h(t_{1}),\frac{\varepsilon}{4}\right)$ be any point that
is not a critical value of $g$. Note that the assumption on $g$ ensures that
such a point exists, by Picard’s theorem and the fact that $g$ has at most
countably many critical values in an open set. Let $g^{-1}$ be any local
inverse of $g$ defined in a neighbourhood of $w$. By Theorem 2.2, the function
$g^{-1}$ can be continued indefinitely along almost all rays starting from
$w_{1}$. Using the same previous argument, we can then choose a point
$w_{2}\in D\left(h(t_{2}),\frac{\varepsilon}{4}\right)$ that is not a critical
value of $g$, such that $[w_{1},w_{2}]$ is contained in
$D\left(h(t_{1}),\frac{\varepsilon}{4}\right)\cup
D\left(h(t_{2}),\frac{\varepsilon}{4}\right)$ and $g^{-1}$ has a well-defined
continuation on $[w_{1},w_{2}]$. More precisely, let
$\tilde{h}:\\{e^{it}:t\in[t_{1},t_{2}]\\}\to[w_{1},w_{2}]$ be the natural
parametrisation of $[w_{1},w_{2}]$ with $\tilde{h}(t_{1})=w_{1}$ and
$\tilde{h}(t_{2})=w_{2}$. Then there exists a continuous function $h_{0}$ on
$\\{e^{it}:t\in[t_{1},t_{2}]\\}$ such that $g\circ h_{0}=\tilde{h}_{0}$ on
$\\{e^{it}:t\in[t_{1},t_{2}]\\}$.

By repeating this construction, we get the following: there exist points
$w_{1},\dots,w_{n}$ such that, for any $k=1,\dots,n$, we have $w_{k}\in
D\left(h(t_{k}),\frac{\varepsilon}{4}\right)$, $w_{k}$ is not a critical value
of $g$ and $[w_{k},w_{k+1}]\subset
D\left(h(t_{k}),\frac{\varepsilon}{4}\right)\cup
D\left(h(t_{k+1}),\frac{\varepsilon}{4}\right)$ for $k=1,\dots,n-1$. Also,
$g^{-1}$ has a well defined continuation on $\cup_{k=1}^{n-1}[w_{k},w_{k+1}]$;
that is, for every $k=1,\dots,n-1$, if
$\tilde{h}:\\{e^{it}:t\in[t_{k},t_{k+1}]\\}\to[w_{k},w_{k+1}]$ is the natural
parametrisation of $[w_{k},w_{k+1}]$ with $\tilde{h}_{0}(t_{k})=w_{k}$ and
$\tilde{h}_{0}(t_{k})=w_{k+1}$, then there exists a continuous function
$h_{0}$ on $\\{e^{it}:t\in[t_{k},t_{k+1}]\\}$ such that $g\circ
h_{0}=\tilde{h}_{0}$ on $\\{e^{it}:t\in[t_{k},t_{k+1}]\\}$.

We notice that $h_{0}$ is a well-defined continuous function on $K$. Also, for
any $t\in[\alpha,\beta]$, we have that $t\in[t_{k},t_{k+1}]$ for some
$k=1,\dots,n-1$ and then $\left|g(h_{0}(t))-h(t)\right|<\varepsilon$, since
$[w_{k},w_{k+1}]\subset D\left(h(t_{k}),\frac{\varepsilon}{4}\right)\cup
D\left(h(t_{k+1}),\frac{\varepsilon}{4}\right)$, which completes the proof. ∎

We now turn to the proof of Theorem 2.1.

###### Proof of Theorem 2.1.

Let $g:f({\mathbb{D}})\to{\mathbb{C}}$ be a fixed non-constant holomorphic
function, and let $f$ be in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$.By [20,
Corollary 3.6], the set $f({\mathbb{D}})$ is equal to ${\mathbb{C}}$ or
${\mathbb{C}}\setminus\\{\alpha\\}$ for some $\alpha\in{\mathbb{C}}$. So
Theorem 2.3 applies to $g$ and gives us, for every proper compact subset $K$
of ${\mathbb{T}}$, the existence of a dense subset $D(K)$ of $C(K)$ such that
for every $h\in D(K)$, there exists $h_{0}\in C(K)$ with $h=g\circ h_{0}$. Let
then $K$ be a proper compact subset of ${\mathbb{T}}$, $h\in D(K)$ and let
$h_{0}$ be a continuous function on $K$ such that $h=g\circ h_{0}$ on $K$.
Since $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, there exists an increasing
sequence $(\lambda_{n})_{n}$ of integers such that $(f_{r_{\lambda_{n}}})_{n}$
converges uniformly to $h_{0}$ on $K$.

$\sup_{\zeta\in K}\left\lvert
f(r_{\lambda_{n}}\zeta)-h_{0}(\zeta)\right\rvert\to 0\quad\text{as
}n\to\infty.$

Now, by compactness, there exists an open set $W\in{\mathbb{C}}$ such that
$h_{0}(K)\subset W\subset\overline{W}\subset f({\mathbb{D}})$, on which $g$ is
well-defined and uniformly continuous. Therefore, $g\circ f_{r_{\lambda_{n}}}$
is well-defined on $K$ as well, and

$g\circ f_{r_{\lambda_{n}}}\to g\circ h_{0}=h,$

uniformly on $K$. Since $D(K)$ is dense in $C(K)$, we conclude that $g\circ
f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Let
$\tilde{K}=\left\\{z\in\mathbb{C}:dist(z,h_{0}(K))\leq\frac{dist(\alpha,h_{0}(K))}{2}\right\\}$,
then, there exists $n_{1}\in\mathbb{N}$ such that for every $n\geq n_{1}$ and
$\zeta\in K$, we have that $f(r_{\lambda_{n}}\zeta)\in\tilde{K}$. Let
$\varepsilon>0$, then, by the uniform continuity of $g$ on $\tilde{K}$, there
exists $\delta>0$, such that $\left\lvert g(z)-g(w)\right\rvert<\varepsilon$
whenever $z,w\in\tilde{K}$ with $|z-w|<\delta$. There exists $n_{2}\geq
n_{1}$, such that for every for every $n\geq n_{2}$ and $\zeta\in K$, we have
that $\left\lvert f(r_{\lambda_{n}}\zeta)-h_{0}(\zeta)\right\rvert<\delta$.
Since $f(r_{\lambda_{n}}\zeta),h_{0}(\zeta)\in\tilde{K}$, we get that
$\sup_{\zeta\in K}\left\lvert
g(f(r_{\lambda_{n}}\zeta))-g(h_{0}(\zeta))\right\rvert\leq\varepsilon$ and so

$\sup_{\zeta\in K}\left\lvert
g(f(r_{\lambda_{n}}\zeta))-g(h_{0}(\zeta))\right\rvert\to 0.$

But, $D(K)$ is dense in $C(K)$ so $g\circ
f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

We immediately deduce from Theorem 2.1 the following corollary, that answers
Questions (i) and (ii) mentioned in the introduction. We recall that
$\alpha\in{\mathbb{C}}\cup\\{\infty\\}$ is an asymptotic value of $f\in
H({\mathbb{D}})$ provided that there exists a path
$\gamma:[0,1)\to{\mathbb{D}}$, with $|\gamma(t)|\to 1$ as $t\to 1$, such that
$f(\gamma(t))\to\alpha$ as $t\to 1$. By [20, Corollary 3.6], every
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ has $\infty$ as an asymptotic
value.

######  Corollary 2.4.

Let $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$.

  1. (1)

The function $e^{f}$ is Abel universal. In particular, there exist zero-free
Abel universal functions (we recall that $f({\mathbb{D}})$ is equal to
${\mathbb{C}}$ or ${\mathbb{C}}\setminus\\{\alpha\\}$ for some $\alpha$, by
[20, Corollary 3.6]).

  2. (2)

If $f$ is zero-free, then for every $\alpha\in{\mathbb{C}}$, $\alpha+1/f$ is
an Abel universal function. As a consequence, for every
$\alpha\in{\mathbb{C}}$, there exists
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ that has $\alpha$ as an asymptotic
value.

As a consequence of the second point of the previous corollary, note that, on
the one hand, the class ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is not closed
under multiplication. On the other hand, the function $P(f)$ is Abel universal
for any $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and any non-constant
polynomial $P$. In particular, the subalgebra generated by $f$ is contained in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, up to the constants. The purpose of
the rest of the section is to prove that, in fact,
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ contains, except the constants, quite
_large_ subalgebras of $H({\mathbb{D}})$.

By applying Theorem 2.1 with different holomorphic functions $g$ we can get
new results about the class ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Indeed, we
first notice that $e^{f}\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ whenever
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, which shows that we can have zero-
free Abel universal functions (this complements Theorem 3.6.4 in [20]). Taking
the last result into account, by considering the function
${\mathbb{C}}\setminus\\{0\\}\ni z\mapsto\frac{1}{z}\in{\mathbb{C}}$ we get
that $\frac{1}{f}\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for every zero-free
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. This clearly gives that the class
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is not closed under multiplication.

###  2.2. Algebrability of ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$

By Theorem 2.1, we get that $p\circ f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$
for every $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and any non-constant
polynomial $p$. So, the subalgebra generated by any function in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is contained in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, except the constants. It is natural to
ask if we can have algebras contained in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, except the constants, that are
generated by more than one function.

Before answering this question, we give the appropriate definitions. Let us
start with some definitions.

######  Definition 2.5.

Let $X$ be a unital algebra over ${\mathbb{C}}$. For a subset $B$ of $X$ we
denote by $\mathcal{A}(B)$ the subalgebra generated by $B$, that is

$\mathcal{A}(B)=\\{P(b_{1},\dots,b_{n}):b_{1},\dots,b_{n}\in
B,P\in{\mathbb{C}}[t_{1},\dots,t_{n}],n\in{\mathbb{N}}\\}.$

If the set $B$ is algebraically independent, we will say that $B$ freely
generates the algebra $\mathcal{A}(B)$.

With the assumptions of the previous definition, we say that a subset $A$ of
$X$ is _algebrable_ if $A\cup\\{c\cdot\mathbf{1}:c\in{\mathbb{C}}\\}$ contains
a subalgebra that is not generated by a finite set. In our case, we will use a
stronger form of algebrability. We give the appropriate definition introduced
in [4].

######  Definition 2.6.

Let $X$ be a unital commutative algebra with unit $\mathbf{1}$ over
${\mathbb{C}}$ that is also a topological space. A subset $A$ of $X$ is
_strongly-algebrable_ if $A\cup\\{c\cdot\mathbf{1}:c\in{\mathbb{C}}\\}$
contains a subalgebra that is freely generated by an infinite set. If the
subalgebra is also dense in $X$, we say that $A$ is _densely strongly-
algebrable_.

From Theorem 2.1, we get the following:

######  Theorem 2.7.

The class ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is strongly-algebrable.

###### Proof.

Let $B$ be an infinite set that freely generates $H({\mathbb{C}})$ and fix any
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. We define the set
$\tilde{B}:=\\{g\circ f:g\in B\\}$. By Theorem 2.1, the algebra generated by
$\tilde{B}$ is contained in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, except the
constants. Moreover, the set $\tilde{B}$ is algebraically independent. Indeed,
assume on the contrary that there exist $g_{1},\dots,g_{N}$ and a polynomial
$P$ in $N$ variables such that $P(g_{1},\dots,g_{N})\circ f\equiv 0$. Then, by
the Identity Principle, we get that $P(g_{1},\dots,g_{N})\equiv 0$, hence
$P\equiv 0$. This finishes the proof. ∎

We note that the idea of composing a fixed function in our class with an
appropriate set of functions has already been used in [24], [1] and [13].

To complete the discussion, we shall see that one can actually prove that the
set ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is densely strongly-algebrable, by
using an extension of a criterion given by Bès and Papathanasiou in [16], for
the algebrability of the set of hypercyclic vectors for multiplicative
operators. Indeed, straightforward modifications to the proof of [16, Theorem
1.6] (the details are left to the reader) yields to the following
generalisation of their result to general sequences of operators (instead of
iterates of a single operator). We refer to [27] for the relevant concepts
from the theory of universality.

######  Theorem 2.8.

Let $X,Y$ be separable, commutative unital algebras over the real or complex
scalar field ${\mathbb{C}}$ that are also Fréchet spaces. Assume also that $X$
supports a dense freely generated subalgebra. Let also $(T_{n})_{n}$ be a
sequence of multiplicative and continuous operators from $X$ to $Y$ with the
property that $(\underbrace{T_{n}\times\dots\times T_{n}}_{N\text{
times}})_{n}$ is topologically transitive for every $N\in{\mathbb{N}}$. Then,
the following are equivalent:

  1. (1)

The sequence $(T_{n})_{n}$ supports a universal algebra (except the
constants).

  2. (2)

For each non-constant polynomial $P$ in one variable with coefficients in
${\mathbb{C}}$, the map $\hat{P}:Y\to Y$ with $f\to P(f)$ has dense range.

  3. (3)

For each $N\in{\mathbb{N}}$ and each non-constant polynomial $P$ in $N$
variables with coefficients in ${\mathbb{C}}$, the map $\hat{P}:Y^{N}\to Y$
with $f\to P(f)$ has dense range.

  4. (4)

The set of all sequences $(f_{j})_{j}\in X^{{\mathbb{N}}}$ that freely
generate a dense universal algebra (except the constants) is residual in
$X^{{\mathbb{N}}}$.

The previous criterion leads us to the following strengthening of Theorem 2.7.

######  Theorem 2.9.

The set of all sequences $(f_{j})_{j}\in H({\mathbb{D}})^{{\mathbb{N}}}$ that
freely generate a dense subalgebra of $H({\mathbb{D}})$, every non-constant
element of which is an Abel universal function, is residual in
$H({\mathbb{D}})^{{\mathbb{N}}}$. In particular,
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is densely strongly-algebrable.

###### Proof.

For a proper compact subset $K$ of $\mathbb{T}$ we denote by
$\mathcal{U}_{A}^{K}(\mathbb{D},\rho)$ the set of all holomorphic functions
$f$ on $\mathbb{D}$ with the property that
$\\{f_{r_{n}}:K\to\mathbb{C}:n\in\mathbb{N}\\}$ is dense in $C(K)$. Clearly,

$\mathcal{U}_{A}(\mathbb{D},\rho)=\bigcap_{n\in{\mathbb{N}}}\mathcal{U}_{A}^{K_{n}}(\mathbb{D},\rho),$

where $(K_{n})_{n}$ a sequence of compact proper subsets of ${\mathbb{T}}$,
with the property that for every proper compact subset $K$ of ${\mathbb{T}}$,
there exists $n\in{\mathbb{N}}$ such that $K\subset K_{n}$. So, by Baire
Category Theorem theorem, it is enough to show that, for a fixed proper
compact subset $K$ of $\mathbb{T}$, the set of all $(f_{j})_{j}\in
H({\mathbb{D}})^{{\mathbb{N}}}$ that freely generate a dense subalgebra
contained in ${\mathcal{U}}_{A}^{K}({\mathbb{D}},\rho)$ (except the
constants), is residual in $H({\mathbb{D}})^{{\mathbb{N}}}$.

We will apply Theorem 2.8 with $X=H({\mathbb{D}}),Y=C(K)$ and
$T_{n}(f)=f_{r_{n}}$, $f\in H({\mathbb{D}})$, $n\in{\mathbb{N}}$. Using the
standard terminology from the universality theory, the set
$\mathcal{U}_{A}^{K}(\mathbb{D},\rho)$ is the set of all universal elements
for $(T_{n})_{n}$. Let $N\in{\mathbb{N}}$, we will check that
$(\underbrace{T_{n}\times\dots\times T_{n}}_{N\text{ times}})_{n}$ is
topologically transitive. Indeed, let $L$ be a compact subset of
${\mathbb{D}},\varepsilon>0,f_{1},\dots,f_{N}\in H({\mathbb{D}})$ and
$h_{1},\dots,h_{N}\in C(K)$. We intend to find $g_{1},\dots,g_{N}\in
H({\mathbb{D}})$ with

$\sup_{z\in L}|g_{i}(z)-f_{i}(z)|<\varepsilon,\ \ \ i=1,\dots,N$

and $n_{0}\in{\mathbb{N}}$ such that

$\sup_{z\in K}|T_{n_{0}}(g_{i})(z)-h_{i}(z)|<\varepsilon,\ \ \ i=1,\dots,N.$

Let $n_{0}\in{\mathbb{N}}$ be such that $L\cap r_{n_{0}}K=\emptyset$, where
$r_{n_{0}}K:=\\{r_{n_{0}}z:\,z\in K\\}$. Then, for $i=1,\dots,N$, by
Mergelyan’s theorem there exists a polynomial $g_{i}$ such that

$\sup_{z\in L}|g_{i}(z)-f_{i}(z)|<\varepsilon\quad\text{and}\quad\sup_{z\in
K}|g_{i}(r_{n_{0}}z)-h_{i}(z)|<\varepsilon.$

Then the functions $g_{1},\dots,g_{N}$ have the desired properties.

Thus, we proved that (1) in Theorem 2.8 holds, and so (4) is true as well, as
desired. ∎

##  3\. Invariance under composition from the right

We recall that by $\rho$ we will denote a sequence $(r_{n})_{n}$ in $[0,1)$
that tends to $1$ as $n\to\infty$. Our aim in this section is to prove the
following theorem.

######  Theorem 3.1.

Let $\Phi:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic on ${\mathbb{D}}$ and
continuous on $\overline{{\mathbb{D}}}$. Then $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}})$ for any $f$ in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ if and only if $\Phi$ is a rotation.

It is clear that ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is invariant under
composition from the right by any rotation. So, using the fact that
${\mathcal{U}}_{A}({\mathbb{D}})=\cup_{\rho}{\mathcal{U}}_{A}({\mathbb{D}},\rho)$,
Theorem 3.1 yields the following:

######  Corollary 3.2.

For any $\Phi:{\mathbb{D}}\to{\mathbb{D}}$ holomorphic and continuous on
$\overline{{\mathbb{D}}}$, the following statements are equivalent:

  1. (1)

${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is invariant under composition from the
right by $\Phi$;

  2. (2)

${\mathcal{U}}_{A}({\mathbb{D}})$ is invariant under composition from the
right by $\Phi$;

  3. (3)

$\Phi$ is a rotation.

The proof of Theorem 3.1 will be divided into two steps. The first one
consists in reducing the problem, using the following proposition.

######  Proposition 3.3.

Let $\Phi:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic and continuous on
$\overline{{\mathbb{D}}}$ and $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. If
$f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}})$, then $\Phi$ is an
automorphism.

###### Proof.

Let $\Phi$ and $f$ be as in the statement. Let us recall that given any path
$\gamma:[0,1)\to{\mathbb{D}}$ such that $\gamma(r)\to\zeta$ as $r\to 1$ for
some $\zeta\in{\mathbb{T}}$, the set $f(\gamma([0,1))$ is dense in
${\mathbb{C}}$ (see [18]). Therefore, the continuity of $\Phi$ on
$\overline{{\mathbb{D}}}$ implies that, if
$f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, then
$|\Phi(\gamma(r))|\to 1$ as $r\to 1$ for any path $\gamma$ as above. Now, a
standard argument shows that a function in $H({\mathbb{D}})$ that enjoys the
latter property must be a finite Blaschke product.

Let us assume that $\Phi$ is a finite Blaschke product that is not an
automorphism. It is then $l$-valent on $\overline{{\mathbb{D}}}$ for some
integer $l\geq 2$. We claim that for any compact set $L$ in ${\mathbb{D}}$,
there exists $r_{0}\in[0,1)$ and a compact arc $I$ in ${\mathbb{T}}$,
different from ${\mathbb{T}}$, such that for any $r\in[r_{0},1)$,
${\mathbb{C}}\setminus\Phi(rI)$ has a bounded connected component $L_{r}$,
contained in ${\mathbb{D}}$, and such that $L\subset L_{r}$. Indeed, by
continuity of $\Phi$ on $\overline{{\mathbb{D}}}$ and since
$\Phi({\mathbb{T}})={\mathbb{T}}$, the set $\Phi(C(0,r))$ is a Jordan curve
with at least one point of self-intersection and such that the index of $0$
with respect to it is equal to $l$, that is also the index of $0$ with respect
to $\Phi({\mathbb{T}})$. Now the claim follows from the fact that $l\geq 2$
and that $\Phi$ is injective in some neighbourhood of any point of
${\mathbb{T}}$ (since $\Phi^{\prime}(\zeta)\neq 0$ for any
$\zeta\in{\mathbb{T}}$).

To conclude, if we assume that $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}({\mathbb{D}})$, then there exists an increasing sequence
$(r_{n})_{n}\in[0,1)$, tending to $1$, such that $|f\circ\Phi(r_{n}\zeta)|\leq
1$ for any $\zeta\in I$. By the maximum modulus principle, this implies that,
for any $n$ large enough, $f$ is bounded by $1$ on the set $L_{r_{n}}$ given
by the claim. Since, $L_{r_{n}}$ contains any compact subset of ${\mathbb{D}}$
for $n$ large enough, it follows that $f$ is bounded in ${\mathbb{D}}$, which
contradicts the fact that $f$ is in ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. ∎

We recall that we are interested in the following problem: for which
holomorphic self-map $\Phi$ of ${\mathbb{D}}$, does $f\circ\Phi$ belong to
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for any
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$?

First of all, if $\Phi$ is continuous on $\overline{{\mathbb{D}}}$ and
$f\circ\Phi$ belongs to $\mathcal{U}_{A}({\mathbb{D}},\rho)$ for some $f\in
H({\mathbb{D}})$, then $\Phi$ is a proper holomorphic selfmap of
${\mathbb{D}}$, i.e. $|\Phi(z)|\to 1$ as $|z|\to 1$. Indeed, if not, upon
taking a subsequence and by continuity of $\Phi$ on $\overline{{\mathbb{D}}}$,
there exists a sequence $(z_{n})_{n}\in{\mathbb{D}}$ such that
$z_{n}\to\zeta\in{\mathbb{T}}$ and $\Phi(\zeta)\in{\mathbb{D}}$. Again by
continuity of $\Phi$ on $\overline{{\mathbb{D}}}$, hence in particular at
$\zeta$, this implies that $\Phi(r\zeta)$ is contained in a compact subset of
${\mathbb{D}}$, $r\in[0,1]$. Then $f\circ\Phi([0,1)\zeta)$ is bounded and thus
is not in $\mathcal{U}_{A}({\mathbb{D}},\rho)$, a contradiction.

Now, a standard characterisation of finite Blaschke products asserts that if
$\Phi$ is a proper holomorphic selfmap of ${\mathbb{D}}$ then it is a finite
Blaschke product. Therefore, if it is not an automorphism, then
$\Phi(z)=\zeta$ has more than two distinct solutions in ${\mathbb{T}}$ for any
$\zeta\in{\mathbb{T}}$. Now it is clear that for such $\Phi$, the function
$f\circ\Phi$ does not belong to $\mathcal{U}_{A}({\mathbb{D}},\rho)$, whatever
$f\in H({\mathbb{D}})$.

All in all, if we restrict ourselves to holomorphic selfmaps $\Phi$ that are
continuous on $\overline{{\mathbb{D}}}$, and for the problem we are interested
in, it is enough to consider that $\Phi$ is a Möbius transform. For
$a\in{\mathbb{D}}$, we denote by $\Phi_{a}$ the Möbius transformation given by
$\Phi_{a}(z)=\frac{a-z}{1-\bar{a}z}$. We will prove the following result.

By the previous proposition, in order to prove Theorem 3.1, we are reduced to
proving the following one.

######  Theorem 3.4.

Let $\Phi$ be an automorphism of ${\mathbb{D}}$. Then, the function
$f\circ\Phi$ belongs to ${\mathcal{U}}_{A}({\mathbb{D}})$ for any $f$ in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, if and only if $\Phi$ is a rotation.

For the proof of Theorem 3.4 we will use the following lemma.

######  Lemma 3.5.

Let $\Phi$ be an automorphism of ${\mathbb{D}}$.

  1. (1)

$\Phi$ is _not_ a rotation if and only if for any non-empty open arc
$I\subset{\mathbb{T}}$, there exist $\zeta_{1}$ and $\zeta_{2}$ in $I$ such
that for any $r\in(0,1)$, $|\Phi(r\zeta_{1})|\neq|\Phi(r\zeta_{2})|$.

  2. (2)

For every $\zeta\in{\mathbb{T}}$, there exists $r_{0}\in(0,1)$ such that the
function $r\mapsto|\Phi(r\zeta)|$ is increasing on $(r_{0},1)$.

###### Proof of Lemma 3.5.

For the first part, we note that if $\Phi$ is a rotation we obviously have
$|\Phi(r\zeta_{1})|=|\Phi(r\zeta_{2})|$ for all $r\in(0,1)$ and $\zeta_{1}$,
$\zeta_{2}\in{\mathbb{T}}$ . Let us check the non-trivial direction. Without
loss of generality we can suppose that $\Phi$ is a Möbius transformation,
namely $\Phi=\Phi_{a}$ for some $a\in{\mathbb{D}}$, where

(1) $\Phi_{a}(z)=\frac{a-z}{1-\bar{a}z}.$

Let $r\in(0,1)$ and $I$ an open arc in ${\mathbb{T}}$. Using the formula

(2) $1-|\Phi(z)|^{2}=\frac{(1-|a|^{2})(1-|z|^{2})}{|1-\bar{a}z|^{2}},$

we easily get that for any $\zeta_{1},\zeta_{2}\in{\mathbb{T}}$, the equality
$1-|\Phi(r\zeta_{1})|^{2}=1-|\Phi(r\zeta_{2})|^{2}$ is equivalent to

(3) $\Re{(\bar{a}\zeta_{1})}=\Re{(\bar{a}\zeta_{2})},$

which is also equivalent to
$1-|\Phi(R\zeta_{1})|^{2}=1-|\Phi(R\zeta_{2})|^{2}$ for any $R\in(0,1)$. Now,
it is clear that (3) is true for any $\zeta_{1},\zeta_{2}$ in the non-empty
open arc $I$ if and only if $a=0$, i.e. $\Phi$ is a rotation.

The second part can be easily seen by using (2) and studying the function
$r\mapsto\frac{1-r^{2}}{r^{2}|a|^{2}-2r\Re{(\bar{a}\zeta)}+1}$. ∎

Now we will use the above lemma to prove Theorem 3.4.

###### Proof of Theorem 3.4.

We have already noticed that, if $f$ is any function in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and $\Phi$ is a rotation, then
$f\circ\Phi$ belongs to ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$. So it is
enough to show that if $\Phi$ is any Möbius transformation that is not a
rotation, then there exists $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such
that $f\circ\Phi$ does not belong to ${\mathcal{U}}_{A}({\mathbb{D}})$. Let us
then fix $\Phi=\Phi_{a}$ with $a\neq 0$ (see (1) for the definition of
$\Phi_{a}$). Since $\Phi^{-1}=\Phi$, we shall always denote also by $\Phi$ the
function $\Phi^{-1}$.

By assumption and Lemma 3.5, there exist $\zeta_{1}\neq\zeta_{2}$ in
${\mathbb{T}}$ such that for any $r\in(0,1)$,
$|\Phi(r\zeta_{1})|\neq|\Phi(r\zeta_{2})|$. Let $r_{-1}\in(0,1)$ be such that
both functions $r\mapsto|\Phi(r\zeta_{1})|$ and $r\mapsto|\Phi(r\zeta_{2})|$
are increasing on $(r_{-1},1)$ (using (2) of Lemma 3.5). Note that for any
$r,r^{\prime}\in(r_{-1},1)$, $\Phi(r\zeta_{1})\neq\Phi(r^{\prime}\zeta_{2})$,
by injectivity of $\Phi$ on $\overline{{\mathbb{D}}}$.

Let us write $\rho=(r_{n})_{n}$. Without loss of generality we may and shall
assume that $r_{n}\in(r_{-1},1)$ for any $n\in{\mathbb{N}}$. By all the
previous, together with the continuity of $\Phi$ and the fact that
$|\Phi(r\zeta)|\to 1$ as $r\to 1$ for any $\zeta\in{\mathbb{T}}$, the sets
$\cup_{n}\\{R\in(0,1):\,|\Phi(R\zeta_{1})|=r_{n}\\}$ and
$\cup_{n}\\{R\in(0,1):\,|\Phi(R\zeta_{2})|=r_{n}\\}$ are disjoint and can be
enumerated as, respectively, two increasing sequences $(R_{n}^{1})_{n}$ and
$(R_{n}^{2})_{n}$ in $(0,1)$, tending to $1$ as $n\to\infty$. Then we have
$|\Phi(R_{n}^{i}\zeta_{i})|=r_{n}$ for any $n$ and any $i\in\\{1,2\\}$. For
technical reasons, we will also make use of the two increasing enumerations
$(s_{n}^{1})_{n\geq-1}$ and $(s_{n}^{2})_{n\geq-1}$ in $(0,1)$ of the sets
$\cup_{n\geq-1}\\{s\in(0,1):\,|\Phi(s\zeta_{1})|=(r_{n}+r_{n+1})/2\\}$ and
$\cup_{n\geq-1}\\{s\in(0,1):\,|\Phi(s\zeta_{2})|=(r_{n}+r_{n+1})/2\\}$,
respectively. Again, the lastly considered four sets are disjoint, and we have
$R_{n}^{i}<s_{n}^{i}<R_{n+1}^{i}$ for any $n\geq-1$ and any $i\in\\{1,2\\}$.
For $n\geq-1$, we denote $L_{n}=\overline{D(0,r_{n})}$.

Finally, let us fix technical notations that we will used in the construction
below:

  * •

$(\varepsilon_{n})_{n}$ is a sequence of positive real numbers decreasing to
$0$ with $\sum_{n=1}^{\infty}\varepsilon_{n}\leq 1/2$;

  * •

$(\varphi_{n})_{n}$ is an enumeration of the polynomials with coefficients in
${\mathbb{Q}}+i{\mathbb{Q}}$;

  * •

$(K_{n})_{n}$ is a sequence of compact subsets of ${\mathbb{T}}$, different
from ${\mathbb{T}}$, such that for any compact set $K\subset{\mathbb{T}}$
different from ${\mathbb{T}}$, there exists $n\in{\mathbb{N}}$ such that
$K\subset K_{n}$;

  * •

$\alpha,\beta:{\mathbb{N}}\to{\mathbb{N}}$ are two functions such that for any
$l,m\in{\mathbb{N}}$, there exists infinitely many $n\in{\mathbb{N}}$ for
which $(\alpha(n),\beta(n))=(l,m)$.

We now proceed to the construction of the desired function. To do so, we will
build by induction a sequence $(P_{n})_{n}$ of polynomials with specific
approximation properties. We set $P_{0}=0$. Assume now that
$P_{0},\ldots,P_{n-1}$ have been defined. To specify $P_{n}$ we distinguish
three cases:

Case I: if $r_{n}K_{\beta(n)}\cap\Phi([r_{-1},1)\zeta_{i})=\emptyset$ for each
$i\in\\{1,2\\}$.

We choose $P_{n}$ by applying Mergelyan’s theorem in order to have for any
$i\in\\{1,2\\}$,

  1. (a)

$\sup_{z\in
L_{n-1}\cup\Phi([r_{-1},1]\zeta_{i})}|P_{n}(z)|\leq\varepsilon_{n}$;

  2. (b)

$\sup_{z\in
r_{n}K_{\beta(n)}}|P_{n}(z)+\left(\sum_{k=0}^{n-1}P_{k}(z)-\varphi_{\alpha(n)}(z)\right)|\leq\varepsilon_{n}$.

Note that this choice is possible by assumption and since the set
$\cup_{i\in\\{1,2\\}}\Phi([r_{-1},1]\zeta_{i})\cup L_{n-1}\cup
r_{n}K_{\beta(n)}$ is compact and has connected complement.

Case II: if $r_{n}K_{\beta(n)}\cap\Phi([r_{-1},1)\zeta_{i})=\emptyset$ and
$r_{n}K_{\beta(n)}\cap\Phi([r_{-1},1)\zeta_{j})\neq\emptyset$ for some
$i,j\in\\{1,2\\}$.

Since both cases $(i,j)=(1,2)$ and $(i,j)=(2,1)$ are treated in the same way,
we assume that $r_{n}K_{\beta(n)}\cap\Phi([r_{-1},1)\zeta_{2})=\emptyset$. Let
$\psi_{n}(z)$ be any function continuous on the curve
$\Phi([r_{-1},1]\zeta_{1})$ that is equal to $0$ on
$\Phi([r_{-1},s_{n-1}^{1}]\zeta_{1}\cup[s_{n}^{1},1]\zeta_{1})$ and to
$\sum_{k=0}^{n}P_{k}(z)-\varphi_{\alpha(n)}(z)$ at
$z=\Phi(R_{n}^{1}\zeta_{1})$. Then we define the function

$\varphi_{n}^{\text{II}}(z)=\left\\{\begin{array}[]{ll}\sum_{k=0}^{n-1}P_{k}(z)-\varphi_{\alpha(n)}(z)&\text{if
}z\in r_{n}K_{\beta(n)}\\\ \psi_{n}(z)&\text{if
}z\in\Phi([r_{-1},1]\zeta_{1}).\end{array}\right.$

Note that $\varphi_{n}^{\text{II}}(z)$ is continuous on
$r_{n}K_{\beta(n)}\cup\Phi([r_{-1},1]\zeta_{1})$ and holomorphic in its
interior. Moreover, by assumption, the set
$\cup_{i\in\\{1,2\\}}\Phi([r_{-1},1]\zeta_{i})\cup L_{n-1}\cup
r_{n}K_{\beta(n)}$ has connected complement. Then we choose $P_{n}$ by
applying Mergelyan’s theorem in order to have

  1. (c)

$\sup_{z\in
L_{n-1}\cup\Phi([r_{-1},1]\zeta_{2})}|P_{n}(z)|\leq\varepsilon_{n}$;

  2. (d)

$\sup_{z\in
r_{n}K_{\beta(n)}\cup\Phi([r_{-1},1]\zeta_{1})}|P_{n}(z)+\varphi_{n}^{\text{II}}|\leq\varepsilon_{n}$.

Case III: if $r_{n}K_{\beta(n)}\cap\Phi([r_{-1},1)\zeta_{i})\neq\emptyset$ for
each $i\in\\{1,2\\}$. There are two subcases that correspond to
$R_{n}^{1}<R_{n}^{2}$ and $R_{n}^{1}>R_{n}^{2}$. Since both are treated
similarly, we only consider the case where $R_{n}^{1}<R_{n}^{2}$, i.e.
$|\Phi(R_{n}^{2}\zeta_{1})|>r_{n}$ and $|\Phi(R_{n}^{1}\zeta_{2})|<r_{n}$. By
continuity, there exists $\eta_{n}>0$ such that the following inequalities
hold:

$\max\left(|\Phi(s_{n-1}^{2}\zeta_{2})|,|\Phi((R_{n}^{1}+\eta_{n})\zeta_{2})|\right)<|\Phi((R_{n}^{2}-\eta_{n})\zeta_{2})|<r_{n}\\\
<|\Phi((R_{n}^{1}+\eta_{n})\zeta_{1})|<\min\left(|\Phi(s_{n}^{1}\zeta_{1})|,|\Phi((R_{n}^{2}-\eta_{n})\zeta_{1})|\right)$

and

$|\Phi((R_{n}^{1}-\eta_{n})\zeta_{1}|\geq|\Phi(s_{n-1}^{1}\zeta_{1})|\quad\text{and}\quad|\Phi((R_{n}^{2}+\eta_{n})\zeta_{2})|\leq|\Phi(s_{n}^{2}\zeta_{2})|.$

Now consider any function $\kappa_{n}$ continuous on the disjoint union of the
two curves $\Phi([r_{-1},1]\zeta_{1})$ and $\Phi([r_{-1},1]\zeta_{2})$ such
that

  * •

$\kappa_{n}(z)=0$ if
$z\in\Phi([r_{-1},(R_{n}^{2}-\eta_{n})]\zeta_{2})\cup\Phi((R_{n}^{2}+\eta_{n}),1]\zeta_{2})$
and if
$z\in\Phi([r_{-1},(R_{n}^{1}-\eta_{n})]\zeta_{1})\cup\Phi((R_{n}^{1}+\eta_{n}),1]\zeta_{1})$;

  * •

$\kappa_{n}(z)$ is equal to $\sum_{k=0}^{n-1}P_{k}(z)-\varphi_{\alpha(n)}(z)$
at $z=\Phi(R_{n}^{1}\zeta_{1})$ and $z=\Phi(R_{n}^{2}\zeta_{2})$.

Finally, we define the function

$\varphi_{n}^{\text{III}}(z)=\left\\{\begin{array}[]{ll}\sum_{k=0}^{n-1}P_{k}(z)-\varphi_{\alpha(n)}(z)&\text{if
}z\in r_{n}K_{\beta(n)}\\\ \kappa_{n}(z)&\text{if
}z\in\Phi([r_{-1},1]\zeta_{1})\cup\Phi([r_{-1},1]\zeta_{2}).\end{array}\right.$

By definition $\varphi_{n}^{\text{III}}(z)$ is continuous on the compact set
$r_{n}K_{\beta(n)}\cup\Phi([r_{-1},1]\zeta_{1})\cup\Phi([r_{-1},1]\zeta_{2})$
and holomorphic in its interior, hence also on the smaller compact set

$G_{n}:=r_{n}K_{\beta(n)}\cup\Phi([s_{n-1}^{1},1]\zeta_{1})\cup\Phi([r_{-1},1]\zeta_{2}).$

Note that the $L_{n-1}\cup G_{n}$ is compact and has connected complement,
since $\Phi(s_{n-1}^{1}\zeta_{1})>r_{n-1}$ by definition. Thus we can choose
$P_{n}$ by applying Mergelyan’s theorem in such a way that

  1. (e)

$\sup_{z\in L_{n-1}}|P_{n}(z)|\leq\varepsilon_{n}$;

  2. (f)

$\sup_{z\in G_{n}}|P_{n}(z)+\varphi_{n}^{\text{III}}(z)|\leq\varepsilon_{n}$.

This concludes the induction. We set $f=\sum_{n}P_{n}$ which defines a
function in $H({\mathbb{D}})$ (by (a), (c) and (e)). Let us first check that
$f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$. Fix $n,m\in{\mathbb{N}}$ and let
$(n_{l})_{l}\subset{\mathbb{N}}$ be such that for any $l\in{\mathbb{N}}$,
$\alpha(n_{l})=n$ and $\beta(n_{l})=m$. By (b), (d) or (f) we have, for any
$z\in K_{m}$,

(4)
$\left|f(r_{n_{l}}z)-\varphi_{n}(z)\right|\leq\left|\sum_{n=0}^{n_{l}}P_{n}(z)-\varphi_{n}(z)\right|+\sum_{n>n_{l}}|P_{n}(z)|\leq\varepsilon_{n_{l}}+\sum_{n>n_{l}}\varepsilon_{n}\to
0$

as $l\to\infty$. To see that $f\circ\Phi$ does not belong to
${\mathcal{U}}_{A}({\mathbb{D}})$, it is enough to see that the construction
yields, for any $r\in[r_{-1},1)$,

$\min\left(|f\circ\Phi(r\zeta_{1})|,|f\circ\Phi(r\zeta_{2})|\right)\leq\sum_{n}\varepsilon_{n}.$

In particular, it is not possible to uniformly approximate on
$\\{\zeta_{1},\zeta_{2}\\}$ any constant greater than
$\sum_{n}\varepsilon_{n}$ by some dilates $(f\circ\Phi)_{r}$,
$r\in[r_{-1},1)$, of $f\circ\Phi$.

∎

######  Remark 3.6.

If $\Phi$ is not continuous on $\overline{{\mathbb{D}}}$, it is not clear what
necessary and sufficient condition $\Phi$ should satisfy for the conclusion of
the previous corollary to hold.

##  4\. Further extensions and applications

In [18], a notion slightly more general than that of Abel universal functions
was introduced. As usual, we denote by $\rho=(r_{n})_{n}$ an increasing
sequence in $[0,1)$ that converges to $1$ as $n\to\infty$.

######  Definition 4.1.

Let $\rho$ and $w\in{\mathbb{D}}$ be fixed. We will say that $f\in
H({\mathbb{D}})$ belongs to the class
${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$ of $w$-Abel universal functions if,
for any proper compact subset $K$ of ${\mathbb{T}}$, and any function
$\varphi\in C(K)$,

$\sup_{\zeta\in K}\left|f(w+r_{n}(\zeta-w))-\varphi(\zeta)\right|\to
0\quad\text{as }n\to\infty.$

We set

$\mathcal{U}_{A}^{w}({\mathbb{D}})=\bigcup_{\rho}{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho),$

where $\rho$ varies over the set of all increasing sequences in $[0,1)$
tending to $1$.

Note that if $w=0$, then ${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$ coincides
with ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, and
$\mathcal{U}_{A}^{w}({\mathbb{D}})$ with ${\mathcal{U}}_{A}({\mathbb{D}})$.
Roughly speaking, the definition of a $w$-Abel universal function is that of
an Abel universal function for which the origin $0$ of the usual radii is
moved to another point $w$ in ${\mathbb{D}}$. It was proven in [18] that the
set

$\bigcap_{w\in{\mathbb{D}}}{\mathcal{U}}_{A}({\mathbb{D}},\rho,w)$

is a residual subset of $H({\mathbb{D}})$.

Since ${\mathcal{U}}_{A}({\mathbb{D}})$ does not coincide with any of the sets
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ (see [21]), and in view of the result
of the previous parts, it is natural to ask the following:

######  Questions.

  1. (1)

Do the classes ${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$ depend on the choice
of the “origin” $w$?

  2. (2)

How do Theorems 2.1 and 3.1 extend to the classes
${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$ and
${\mathcal{U}}_{A}^{w}({\mathbb{D}})$ for $w\in{\mathbb{D}}$ different from
$0$.

First of all, we underline that, up to superfluous modifications that are left
to the reader, the proof of Theorem 2.1 works for
${\mathcal{U}}_{A}^{w}({\mathbb{D}})$, for any $w\in{\mathbb{D}}$. Thus we
have:

######  Theorem 4.2.

Let $f\in{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$. For any non-constant
holomorphic function $g:f({\mathbb{D}})\to{\mathbb{C}}$, the function $g\circ
f$ belongs to ${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$.

For Question 1 and the second part of Question 2, we shall see that the main
ideas of the proof of Theorem 3.1 are general enough to give complete answers.

Let $\Psi$ be a homeomorphism from $\overline{{\mathbb{D}}}$ onto
$\overline{{\mathbb{D}}}$ and let $(P_{\Psi})$ be the property that is
satisfied by a function $f$ in $H({\mathbb{D}})$ whenever, for every proper
compact set $K\subset{\mathbb{T}}$, the set
$\\{(f\circ\Psi)_{r}:K\to{\mathbb{C}};\,r\in[0,1)\\}$ is dense in $C(K)$.
Analogously, if $\rho=(r_{n})_{n}\subset[0,1)$, tending to $1$ as $n\to\infty$
is given, then a function $f$ in $H({\mathbb{D}})$ satisfies the property
$(P_{\Psi}^{\rho})$ whenever the set
$\\{(f\circ\Psi)_{r_{n}}:K\to{\mathbb{C}};\,n\in{\mathbb{N}}\\}$ is dense in
$C(K)$.

Note that $f$ belongs to ${\mathcal{U}}_{A}^{w}({\mathbb{D}})$ (respectively
${\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$) if and only if it has
$(P_{\Psi_{w}})$ (respectively $(P_{\Psi_{w}}^{\rho})$), with

(5) $\Psi_{w}(re^{i\theta})=w+r(e^{i\theta}-w),\quad
re^{i\theta}\in{\mathbb{D}}.$

Similarly, if $f\in H({\mathbb{D}})$ and $\Phi$ is an automorphism of the
disc, then $f\circ\Phi$ belongs to ${\mathcal{U}}_{A}({\mathbb{D}})$
(respectively ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$) if and only if $f$
satisfies the property $(P_{\Phi})$ (respectively $(P_{\Phi}^{\rho})$).

Now, the crucial technical point that made the construction in the proof of
Theorem 3.4 possible was the fact that if $\Phi$ is an automorphism of
${\mathbb{D}}$ that is not a rotation, then there exist $r_{0}\in[0,1)$ and
$\zeta_{1},\zeta_{2}\in{\mathbb{T}}$ such that
$|\Phi(r\zeta_{1})|\neq|\Phi(r\zeta_{2})|$ for any $r\in[r_{0},1)$. With this
condition, it was possible to construct a function $f$ in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that the minimum of
$|f\circ\Phi(r\zeta_{1})|$ and $|f\circ\Phi(r\zeta_{2})|$ is less than a
constant, for any $r$ large enough. Up to minor technical details, it is not
difficult to see that this condition can be replaced by the slightly weaker
one: there exist $r_{0}\in[0,1)$ and
$\zeta_{1},\zeta_{2},\zeta_{3}\in{\mathbb{T}}$ such that for any
$r\in[r_{0},1)$, the points $\Phi(r\zeta_{1})$, $\Phi(r\zeta_{2})$ and
$\Phi(r\zeta_{3})$ do not belong to the same circle centred at $0$ (i.e.,
there exists $i,j\in\\{1,2,3\\}$ such that
$|\Phi(r\zeta_{i})|\neq|\Phi(r\zeta_{j})|$). Under this condition, we can
build a function $f\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ such that for any
$r\in[r_{0},1)$, the quantity

$\min\left(|f\circ\Phi(r\zeta_{1})|,|f\circ\Phi(r\zeta_{2})|,|f\circ\Phi(r\zeta_{3})|\right)$

is less than a finite constant independent of $r$. Thus the function
$f\circ\Phi$ cannot belong to ${\mathcal{U}}_{A}({\mathbb{D}})$.

More generally, if $\Psi_{1}$ and $\Psi_{2}$ are two homeomorphisms from
$\overline{{\mathbb{D}}}$ onto $\overline{{\mathbb{D}}}$, the same type of
construction can be made in order to prove the following proposition:

######  Proposition 4.3.

Let $\Psi_{1}$ and $\Psi_{2}$ be two homeomorphisms from
$\overline{{\mathbb{D}}}$ onto $\overline{{\mathbb{D}}}$. If there exist
$r_{0}\in[0,1)$ and $\zeta_{1},\zeta_{2},\zeta_{3}\in{\mathbb{T}}$ such that
for any $r\in[r_{0},1)$, there is no $R\in[r_{0},1)$ such that the points
$\Psi_{2}(r\zeta_{1})$, $\Psi_{2}(r\zeta_{2})$ and $\Psi_{2}(r\zeta_{3})$ all
belong to the same closed curve $\Psi_{1}(C(0,R))$, then there exists $f\in
H({\mathbb{D}})$ that satisfy $(P^{\rho}_{\Psi_{1}})$, but not
$(P_{\Psi_{2}})$.

The proof of this proposition can be written following the same lines as that
of Theorem 3.4, up to technical details, and so is omitted. Now, we will
explain how this proposition can be used in order to answer the above
Questions 1 and 2.

###  4.1. Answer to Question 1

Let $w_{1}\neq w_{2}$ in ${\mathbb{D}}$. To answer Question 1, one can
directly apply Proposition 4.3 to the homeomorphisms $\Psi_{w_{1}}$ and
$\Psi_{w_{2}}$ defined by (5). Indeed, since $\Psi_{1}(C(0,R))$ and
$\Psi_{2}(C(0,r))$ are different circles for any values of $r,R\in[0,1)$, and
since there is only one circle passing through $3$ pairwise distinct points,
it is true that, for any $3$-tuple $\zeta_{1}$, $\zeta_{2}$ and $\zeta_{3}$ in
${\mathbb{T}}$, the points $\Psi_{2}(r\zeta_{1})$, $\Psi_{2}(r\zeta_{2})$ and
$\Psi_{2}(r\zeta_{3})$ cannot all belong to $\Psi_{1}(C(0,R))$, whatever
$R\in[0,1)$. Thus, by Proposition 4.3 we get that all classes introduced above
are different:

######  Proposition 4.4.

Let $\rho_{1},\rho_{2}$ be two increasing sequences in $[0,1)$ tending to $1$
and let $w_{1}\neq w_{2}$ in ${\mathbb{D}}$. Then

$\mathcal{U}_{A}^{w_{1}}({\mathbb{D}})\setminus\mathcal{U}_{A}^{w_{2}}({\mathbb{D}})\neq\emptyset\quad\text{and}\quad{\mathcal{U}}_{A}^{w_{1}}({\mathbb{D}},\rho_{1})\setminus{\mathcal{U}}_{A}^{w_{2}}({\mathbb{D}},\rho_{2})\neq\emptyset.$

######  Remark 4.5.

One might be tempted to see a contrast with the fact that the universality of
the partial sums of universal Taylor series on ${\mathbb{D}}$ does not depend
on the choice of the center of their Taylor expansions, see [33].

###  4.2. Anwser to Question 2

Let now $w\in{\mathbb{D}}$ be given, and let
$\Phi:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic and continuous on
$\overline{{\mathbb{D}}}$. First of all, the same proof as that of Proposition
3.3 makes it clear that if $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}^{w}({\mathbb{D}})$ for any
$f\in{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$, then $\Phi$ is an
automorphism. Thus we may and shall assume that $\Phi$ is an automorphism of
${\mathbb{D}}$.

Now, Proposition 4.3 tells us that if $\Phi$ is an automorphism of
${\mathbb{D}}$ such that $f\circ\Phi\in{\mathcal{U}}_{A}^{w}({\mathbb{D}})$
for any $f\in{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$, then there exist at
least two (in fact infinitely many) distinct numbers $r_{1},r_{2}$ in $[0,1)$
and two (again, in fact infinitely many) $3$-tuples
$(\zeta_{1},\zeta_{2},\zeta_{3})$ and $(\xi_{1},\xi_{2},\xi_{3})$ of pairwise
distinct points in ${\mathbb{T}}^{3}$, such that there exist
$R_{1},R_{2}\in[0,1)$ for which

  * •

the points
$\Phi\circ\Psi_{w}(r_{1}\zeta_{1}),\Phi\circ\Psi_{w}(r_{1}\zeta_{2}),\Phi\circ\Psi_{w}(r_{1}\zeta_{3})$
belong to $\Psi_{w}(C(0,R_{1}))$;

  * •

the points
$\Phi\circ\Psi_{w}(r_{2}\xi_{1}),\Phi\circ\Psi_{w}(r_{2}\xi_{2}),\Phi\circ\Psi_{w}(r_{2}\xi_{3})$
belong to $\Psi_{w}(C(0,R_{2}))$.

Observe that the pairwise distinct points $r_{1}\zeta_{i}$ (respectively
$r_{2}\xi_{i}$), $i\in\\{1,2,3\\}$, belong to the circle $C(0,r_{1})$
(respectively $C(0,r_{2})$), and that $\Psi_{w}(C(0,r))$ is the circle
$C((1-r)w,r)$. Since automorphisms of ${\mathbb{D}}$ map circles onto circles
and since there is only one circle passing through $3$ pairwise distinct
points, we deduce from the previous that the property
“$f\circ\Phi\in{\mathcal{U}}_{A}^{w}({\mathbb{D}})$ for any
$f\in{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$” holds only if, for at least
two pairs $(r_{1},r_{1}^{\prime}),(r_{2},r_{2}^{\prime})\in(0,1)^{2}$, with
$r_{i}\neq r_{j}$ and $r_{i}^{\prime}\neq r_{j}^{\prime}$, $\Phi$ maps the
circle $C((1-r_{i})w,r_{i})$ onto $C((1-r^{\prime}_{i})w,r^{\prime}_{i})$,
$i=1,2$.

In the case $w=0$, the proof of Lemma 3.5 tells us that the latter condition
can only happen if $\Phi$ is a rotation (or it can also be recalled as a well-
known fact that the only automorphisms of ${\mathbb{D}}$ that map a circle
centred at $0$ onto a circle centred at $0$ are the rotations). The following
lemma is a generalization of this fact, and tells us in particular that, if
$w\neq 0$, this condition holds only if $\Phi$ is the identity map. For
simplicity, we will use in the sequel the notation $C_{w}(r)=C((1-r)w,r)$.

######  Lemma 4.6.

Let $\Phi$ be an automorphism of ${\mathbb{D}}$. Let
$(r_{1},r_{1}^{\prime}),(r_{2},r_{2}^{\prime})\in(0,1)^{2}$, with $r_{i}\neq
r_{j}$ and $r_{i}^{\prime}\neq r_{j}^{\prime}$, and assume that $\Phi$ maps
$C_{w}(r_{i})$ onto $C_{w}(r_{i}^{\prime})$ for $i=1,2$. Then
$r_{i}=r_{i}^{\prime}$ for $i=1,2$ and, either $w=0$, in which case $\Phi$ is
a rotation, or $w\neq 0$ and $\Phi(z)=z$ for any $z\in{\mathbb{D}}$.

###### Proof.

Let $\Phi$ be an automorphism of ${\mathbb{D}}$ satisfying the condition of
the lemma. Then, necessarily, $\Phi$ maps $\overline{D((1-r_{i})w,r_{i})}$
onto $\overline{D((1-r^{\prime}_{i})w,r^{\prime}_{i})}$, $i=1,2$. Now, it is
easily seen that $D((1-r)w,r)$ is contained in $D((1-r^{\prime})w,r^{\prime})$
whenever $r\leq r^{\prime}$. As a consequence, $\Phi$ must have a fixed point
in one of the above discs, and so in ${\mathbb{D}}$. Let us denote by $a$ this
fixed point. Let us also denote by $\varphi_{a}$ an automorphism of
${\mathbb{D}}$ that maps $0$ to $a$. Then
$\Psi:=\varphi_{a}^{-1}\circ\Phi\circ\varphi_{a}$ is an automorphism of
${\mathbb{D}}$ that fixes $0$, hence $\Psi$ is a rotation. Moreover, since
$\Phi$ maps $C_{w}(r_{i})$ onto $C_{w}(r_{i}^{\prime})$ for $i=1,2$, we get

$\Psi\left(\varphi_{a}^{-1}\left(C_{w}(r_{i})\right)\right)=\varphi_{a}^{-1}\left(C_{w}(r^{\prime}_{i})\right).$

Moreover, using again the fact that $D((1-r)w,r)$ is contained in
$D((1-r^{\prime})w,r^{\prime})$ whenever $r\leq r^{\prime}$, and since
automorphisms of ${\mathbb{D}}$ map circles onto circles,
$\varphi_{a}^{-1}\left(C_{w}(r_{i})\right)$ and
$\varphi_{a}^{-1}\left(C_{w}(r^{\prime}_{i})\right)$ are circles and one of
them is contained in the open disc bounded by the other. Since $\Psi$ is a
rotation, this implies that $r_{i}=r_{i}^{\prime}$ and that, either $\Psi$ is
the identity map, or that $\varphi_{a}^{-1}\left(C_{w}(r_{i})\right)$ is a
circle $C(0,R_{i})$ centred at $0$, for some $R_{i}\in(0,1)$, $i=1,2$, with
$R_{1}\neq R_{2}$. If $\Psi$ is the identity map, then so is $\Phi$. Let us
then assume that $\Psi$ is a rotation that is not the identity map.

Then $\varphi_{a}\left(C(0,R_{i})\right)=C_{w}(r_{i})$ for $i=1,2$. Let
$R\in(0,1)$ be fixed. By Schwartz-Pick lemma, $\varphi_{a}(C(0,R))$ is the
pseudo-hyberbolic disc centred at $a$ with radius $R$ (for the pseudo-
hyperbolic distance), or equivalently the hyperbolic disc centred at $a$ with
radius $2\tanh^{-1}(R)$ (with respect to the hyperbolic distance), see for
e.g. [10, Theorem 2.2]. This disc is a Euclidean one with center $c$ and
(euclidean) radius $r$ given by the formulae ([10, Exercise 3, Page 16])

$c=\frac{a\left(1-R^{2}\right)}{1-|a|^{2}R^{2}}\quad\text{and}\quad
r=\frac{\left(1-|a|^{2}\right)R}{1-|a|^{2}R^{2}}.$

Thus, given $r\in(0,1)$, the equality
$\varphi_{a}\left(C(0,R)\right)=C_{w}(r)$ for some $R\in(0,1)$ imposes the
conditions

(6) $(1-r)w=\frac{a\left(1-R^{2}\right)}{1-|a|^{2}R^{2}}\quad\text{and}\quad
r=\frac{\left(1-|a|^{2}\right)R}{1-|a|^{2}R^{2}}.$

Replacing the value of $r$ given by the second equality in the first one, a
direct computation leads to the conditions $a=0$ or

$R=\frac{w-a}{a-|a|^{2}w}.$

If $a\neq 0$ then the above condition imposes that
$R_{i}=\frac{w-a}{a-|a|^{2}w}$, $i=1,2$, which contradicts the fact that
$R_{1}\neq R_{2}$. Therefore $a=0$, that is equivalent to $w=0$ by (6). Now
$w=0$ implies that $\Phi$ is rotation by Lemma 3.5, which concludes the proof
of the lemma. ∎

Altogether, we have proved the following complement to Theorem 3.4:

######  Theorem 4.7.

Let $w\in{\mathbb{D}}\setminus\\{0\\}$ and let
$\Phi:{\mathbb{D}}\to{\mathbb{D}}$ be holomorphic and continuous on
$\overline{{\mathbb{D}}}$. Then $f\circ\Phi$ belongs to
${\mathcal{U}}_{A}^{w}({\mathbb{D}})$ for any
$f\in{\mathcal{U}}_{A}^{w}({\mathbb{D}},\rho)$ if and only if $\Phi$ is the
identity map.

The explanations that lead us to this result suggest the following remark.

######  Remark 4.8.

In view of the above explanations, the definition of the class
${\mathcal{U}}^{w}_{A}({\mathbb{D}},\rho)$ may not seem natural. Rather, it
might be more natural to define the class
${\mathcal{V}}_{A}^{w}({\mathbb{D}},\rho)$ of those $f\in H({\mathbb{D}})$
such that $f\circ\varphi_{w}\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$, where
$\varphi_{w}$ is an automorphism of ${\mathbb{D}}$ that maps $0$ to $w$. In
this case, $\varphi_{w}$ sends circles centered at $0$ to hyperbolic circles
centered at $w$ and it is clear from the proof of Theorem 4.7 that if
$\Phi:{\mathbb{D}}\to{\mathbb{D}}$ is holomorphic and continuous on
$\overline{{\mathbb{D}}}$, then
$f\circ\Phi\in{\mathcal{V}}_{A}^{w}({\mathbb{D}},\rho)$ for any
$f\in{\mathcal{V}}_{A}^{w}({\mathbb{D}},\rho)$ if and only if
$\varphi_{w}^{-1}\circ\Phi\circ\varphi_{w}$ is a rotation.

The class ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is contained in another class
of holomorphic functions whose existence was first proved in [17] (see also
[11] and [BernalGonzalezBonillaCalderonMorenoPradoBassas2007]). Firstly, we
give some relevant context. Let $f\in H({\mathbb{D}})$ and
$\gamma:[0,1)\to{\mathbb{D}}$ be a continuous path in ${\mathbb{D}}$ such that
$\lim_{t\to 1^{-}}\gamma(t)=\zeta$ for some $\zeta\in{\mathbb{T}}$, then the
cluster set of $f$ along $\gamma$ is the set

$\displaystyle C_{\gamma}(f)=\Big{\\{}w\in\mathbb{C}:$ $\displaystyle\
w=\lim_{n\to\infty}f\left(\gamma(t_{n})\right)\text{ where
}(t_{n})_{n\in\mathbb{N}}\text{ is a sequence in }[0,1)$ $\displaystyle\text{
that converges to }1\Big{\\}}.$

We say that a cluster set is maximal if it is equal to $\mathbb{C}$. Let
${\mathcal{U}}_{c}({\mathbb{D}})$ to be the set of all functions in
$H({\mathbb{D}})$ with the property that the cluster set of $f$ along any
continuous path that converges to a point of ${\mathbb{T}}$ is maximal. It was
shown in [21] that ${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ is a strict subset
of ${\mathcal{U}}_{c}({\mathbb{D}})$.

We have the following analogue of the main results of Section 2:

######  Proposition 4.9.

Let $f\in\mathcal{U}_{c}({\mathbb{D}})$. Then

  1. (1)

For any non-constant entire function $g$, the composition $g\circ f$ belongs
to $\mathcal{U}_{c}({\mathbb{D}})$.

  2. (2)

If $\Phi$ is a finite Blaschke product then $f\circ\Phi$ belongs to
$\mathcal{U}_{c}({\mathbb{D}})$. If we assume that $\Phi$ belongs to
$A({\mathbb{D}})$, then the converse holds true.

###### Proof.

(1) Let $\gamma:[0,1)\to{\mathbb{D}}$ be a continuous path in ${\mathbb{D}}$
such that $\lim_{t\to 1^{-}}\gamma(t)=\zeta$ for some $\zeta\in{\mathbb{T}}$.
Let also $w\in\mathbb{C}$. Since $g$ is non-constant, there exists a sequence
$z_{n}\in\mathbb{C}$ such that $|g(z_{n})-w|<\varepsilon_{n}$. Then, for any
$n\in{\mathbb{N}}$, there exists $\delta_{n}>0$ such that
$|g(z)-g(z_{n})|<\varepsilon$ whenever $|z-z_{n}|<\delta_{n}$. Since
$f\in\mathcal{U}_{c}({\mathbb{D}})$ there exists a sequence $t_{n}\in[0,1)$
converging to 1 such that
$\left|f\left(\gamma(t_{n})\right)-z_{n}\right|<\delta_{n}$ for any
$n\in{\mathbb{N}}$. This gives us that
$\left|g\left(f\left(\gamma(t_{n})\right)\right)-w\right|<2\varepsilon$ for
any $n\in{\mathbb{N}}$ which completes the proof.

(2) We notice that a holomorphic selfmap $\Phi$ of ${\mathbb{D}}$ is a finite
Blaschke product if and only if for any continuous path
$\gamma:[0,1)\to{\mathbb{D}}$ such that $\lim_{t\to 1^{-}}\gamma(t)=\zeta$ for
some $\zeta\in{\mathbb{T}}$, we have that
$\Phi\circ\gamma:[0,1)\to{\mathbb{D}}$ is a continuous path such that
$\lim_{t\to 1^{-}}\gamma(t)=\zeta^{\prime}$ for some
$\zeta^{\prime}\in{\mathbb{T}}$. This clearly implies the desired result. ∎

##  5\. A residual set of Abel universal functions that is invariant under
composition by any automorphism

We conclude the paper by establishing the existence of a class of functions
that is residual in $H({\mathbb{D}})$, contained in
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$, and that is invariant under
composition from the right by any automorphism of ${\mathbb{D}}$.

######  Theorem 5.1.

Let $\rho=(r_{n})_{n}$ be an increasing sequence converging to $1$ as
$n\to\infty$. There exists a residual set of functions $f\in H({\mathbb{D}})$
such that $f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for every
automorphism $\Phi$ of the unit disc.

###### Proof.

According, for e.g., to Theorem 3.4, it is enough to prove the existence of a
residual set of functions $f\in H({\mathbb{D}})$ such that
$f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for every automorphism of
the form $\Phi_{w}(z)=\frac{z+w}{1+\overline{w}z}$, $w\in{\mathbb{D}}$.

Let $(\varphi_{n})_{n}$ be an enumeration of the polynomials with coefficients
in ${\mathbb{Q}}+i{\mathbb{Q}}$ and let $(K_{n})_{n}$ be a sequence of compact
subsets of ${\mathbb{T}}$, different from ${\mathbb{T}}$, such that for any
compact set $K\subset{\mathbb{T}}$ different from ${\mathbb{T}}$, there exists
$n\in{\mathbb{N}}$ such that $K\subset K_{n}$.

We claim that, for any fixed integers $k,m,n$, there exists a sequence
$(w_{l}(k,m,n),\delta_{l}(k,m,n))_{l}$ in ${\mathbb{D}}\times(0,+\infty)$ such
that the following conditions hold:

  1. (i)

for every $w\in{\mathbb{D}}$, there exists $l\in{\mathbb{N}}$ such that $w\in
D(w_{l}(k,m,n),\delta_{l}(k,m,n))$;

  2. (ii)

for every $l\in{\mathbb{N}}$, the set

$F(k,m,n,l):=\bigcup_{w\in\overline{D(w_{l}(k,m,n),\delta_{l}(k,m,n))}}\Phi_{w}(r_{k}K_{n}).$

is contained in ${\mathbb{D}}$ and has connected complement in ${\mathbb{C}}$;

  3. (iii)

for every $l\in{\mathbb{N}}$ and $\tau\in D(w_{l}(k,m,n),\delta_{l}(k,m,n))$,

$\sup_{z\in\overline{D}}|\varphi_{m}(\Phi_{w_{l}(k,m,n)}^{-1}(\Phi_{\tau}(r_{k}z)))-\varphi_{m}(r_{k}z)|<\frac{1}{m}.$

Let us check the claim first. We fix $k,m,n\in{\mathbb{N}}$. If $L$ is a
compact set in ${\mathbb{D}}$ and $w\in L$ is fixed, the uniform continuity of
the maps $(\tau,z)\in L\times\overline{{\mathbb{D}}}\mapsto\Phi_{\tau}(z)$,
$z\in\overline{{\mathbb{D}}}\mapsto\Phi_{w}^{-1}(z)$ and
$z\in\overline{{\mathbb{D}}}\mapsto\varphi_{m}(z)$ implies the existence of
$\delta(k,m,n)\in(0,+\infty)$ such that (ii) and (iii) hold (with $w$ instead
of $w_{l}(k,m,n)$ and $\delta(k,m,n)$ instead of $\delta_{l}(k,m,n)$). To
obtain the desired sequence it is then enough to cover $L$ by finitely many
discs of the form $D(w,\delta(k,m,n))$ and to exhaust ${\mathbb{D}}$ by a
sequence of compact sets.

Then, let a sequence $(w_{l}(k,m,n),\delta_{l}(k,m,n))_{l}$ in
${\mathbb{D}}\times(0,+\infty)$ be given by the claim. Denoting by
${\mathcal{U}}$ the set of all functions $f\in H({\mathbb{D}})$ such that
$f\circ\Phi_{w}\in{\mathcal{U}}_{A}({\mathbb{D}})$ for any $w\in{\mathbb{D}}$,
we also claim that

(7)
${\mathcal{U}}\supset\bigcap_{k,m,n,l\in{\mathbb{N}}}\bigcup_{k^{\prime}\geq
k}\left\\{f\in H({\mathbb{D}}):\,\sup_{z\in
F(k^{\prime},m,n,l)}|f(z)-\varphi_{m}(\Phi_{w_{l}(k^{\prime},m,n)}^{-1}(z))|<\frac{1}{m}\right\\}.$

Indeed, let $f$ be in the right-hand side of the above inclusion and fix
$\varepsilon>0$, a compact set $K$ in ${\mathbb{T}}$ different from
${\mathbb{T}}$, $\varphi\in C(K)$ and $w\in{\mathbb{D}}$. First let
$k,n,m\in{\mathbb{N}}$ be such that $K\subset K_{n}$,
$|\varphi(r_{k^{\prime}}z)-\varphi(z)|<\varepsilon/3$ for any $z\in K_{n}$ and
any $k^{\prime}\geq k$, $1/m<\varepsilon/3$ and $\sup_{z\in
K_{n}}|\varphi_{m}(r_{k^{\prime}}z)-\varphi(r_{k^{\prime}}z)|<\varepsilon/3$
for every $k^{\prime}\geq k$. Note that the choice of $m$ can be done by an
application of Mergelyan’s theorem. Now, let $k^{\prime}\geq k$ be given by
the fact that $f$ belongs to the right-hand side of $\eqref{eq-all-auto}$. By
(i), there exists $l^{\prime}\in{\mathbb{N}}$ such that $w\in
D(w_{l^{\prime}}(k^{\prime},m,n),\delta_{l^{\prime}}(k^{\prime},m,n))$. Thus
we get for every $z\in K$,

$\begin{array}[]{ccc}\left|f\circ\Phi_{w}(r_{k^{\prime}}z)-\varphi(r_{k^{\prime}}z)\right|&\leq&\left|f\circ\Phi_{w}(r_{k^{\prime}}z)-\varphi_{m}(r_{k^{\prime}}z)\right|+\left|\varphi(r_{k^{\prime}}z)-\varphi_{m}(r_{k^{\prime}}z)\right|\\\
&\leq&\left|f\circ\Phi_{w}(r_{k^{\prime}}z)-\varphi_{m}(\Phi_{w_{l^{\prime}}(k^{\prime},m,n)}^{-1}(\Phi_{w}(r_{k^{\prime}}z)))\right|\\\
&&\hfill+\left|\varphi_{m}(\Phi_{w_{l^{\prime}}(k^{\prime},m,n)}^{-1}(\Phi_{w}(r_{k^{\prime}}z)))-\varphi_{m}(r_{k^{\prime}}z)\right|+\frac{\varepsilon}{3}\\\
&\overset{\eqref{eq-all-auto}\text{ and
}(iii)}{\leq}&\frac{\varepsilon}{3}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}.\end{array}$

To conclude by the Baire Category Theorem, it is enough to check that the set

$\bigcup_{k^{\prime}\geq k}\left\\{f\in H({\mathbb{D}}):\,\sup_{z\in
F(k^{\prime},m,n,l)}|f(z)-\varphi_{m}(\Phi_{w_{l}(k^{\prime},m,n)}^{-1}(z))|<\frac{1}{m}\right\\}$

is open and dense in $H({\mathbb{D}})$ for any $k\in{\mathbb{N}}$. It is clear
that it is open, so let us focus on the density. Let us then fix
$k\in{\mathbb{N}}$, a closed disc $L$ contained in ${\mathbb{D}}$, $g\in
H({\mathbb{D}})$ and $\varepsilon>0$. By uniform continuity of the map
$(\tau,z)\in K\times\overline{{\mathbb{D}}}\mapsto\Phi_{\tau}(z)$ for any
compact set $K$ in ${\mathbb{D}}$, and since automorphisms of ${\mathbb{D}}$
maps ${\mathbb{T}}$ into ${\mathbb{T}}$, there exists $k^{\prime}\geq k$ such
that the set $L\cap F(k^{\prime},m,n,l)=\emptyset$. Then, by (ii) the set
$L\cup F(k^{\prime},m,n,l)$ is compact and has connected complement. Therefore
we can conclude by applying Mergelyan’s theorem that gives us a polynomial $P$
that uniformly approximates $g$ in $L$ up to $\varepsilon$, and
$\varphi_{m}\circ\Phi_{w_{l}(k^{\prime},m,n)}^{-1}$ on $F(k^{\prime},m,n,l)$
up to $1/m$. ∎

######  Remark 5.2.

The previous theorem can be seen as a result on uncountable common
universality. Indeed, let $\rho$ be fixed and, for $k\in{\mathbb{N}}$, $K$ a
compact set in ${\mathbb{T}}$ different from ${\mathbb{T}}$, and
$\Phi\in\text{Aut}({\mathbb{D}})$, set $T_{\Phi_{r_{k}}}:H({\mathbb{D}})\to
C(K)$ defined by $T_{\Phi}:f\mapsto(z\mapsto f\circ\Phi_{r_{k}}(z))$. Then
$f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for any
$\Phi\in\text{Aut}({\mathbb{D}})$ if and only if $f$ is universal for
$(T_{\Phi_{r_{k}}})_{k}$ for any $\Phi$ (and any $K$).

Let us denote by ${\mathcal{U}}$ the residual set of all functions in
$H({\mathbb{D}})$ such that
$f\circ\Phi\in{\mathcal{U}}_{A}({\mathbb{D}},\rho)$ for any automorphism
$\Phi$ of ${\mathbb{D}}$. By definition, ${\mathcal{U}}$ is a subset of
${\mathcal{U}}_{A}({\mathbb{D}},\rho)$ and is invariant under composition from
the right by any automorphism of the unit disc. Thus, as a corollary of
Theorem 2.1 and Theorem 5.1, we get the following:

######  Corollary 5.3.

For any $f\in{\mathcal{U}}$, any non-constant holomorphic function
$g:f({\mathbb{D}})\to{\mathbb{C}}$, and any automorphism $\Phi$ of
${\mathbb{D}}$, the function $g\circ f\circ\Phi$ belongs to ${\mathcal{U}}$.

## Acknowledgment

We are thankful to Prof. Alexander Eremenko for bringing our attention to
Iversen’s theorem. We would also like to thank Dr Dimitris Papathanasiou who
suggested us an adaptation of [16, Theorem 1.6] in order to prove Theorem 2.9.

## References

  * [1] N. Albuquerque, L. Bernal-González, D. Pellegrino, and J. B. Seoane-Sepúlveda, _Peano curves on topological vector spaces_ , Linear Algebra Appl. 460 (2014), 81–96.
  * [2] W. Alexandre, C. Gilmore, and S. Grivaux, _Typicality of operators on Fréchet algebras admitting a hypercyclic algebra_ , Preprint, arXiv:2312.08321v1.
  * [3] R. M. Aron, L. Bernal González, D. M. Pellegrino, and J. B. Seoane Sepúlveda, _Lineability: The search for linearity in mathematics_ , Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2016.
  * [4] A. Bartoszewicz and S. Głąb, _Strong algebrability of sets of sequences and functions_ , Proc. Amer. Math. Soc. 141 (2013), no. 3, 827–835.
  * [5] F. Bayart, _Linearity of sets of strange functions_ , Michigan Math. J. 53 (2005), no. 2, 291–303.
  * [6] by same author, _Universal radial limits of holomorphic functions_ , Glas. Math. J. 47 (2005), no. 2, 261–267.
  * [7] by same author, _Hypercyclic algebras_ , J. Funct. Anal. (2019), no. 11, 3441–3467.
  * [8] F. Bayart, K.-G. Grosse-Erdmann, V. Nestoridis, and C. Papadimitropoulos, _Abstract theory of universal series and applications_ , Proc. London Math. Soc. (3) 96 (2008), 417–463.
  * [9] F. Bayart, F. Costa Júnior, and D. Papathanasiou, _Baire theorem and hypercyclic algebras_ , Adv. Math. 376 (2021), Paper No. 107419, 58pp.
  * [10] A. F. Beardon and D. Minda, _The hyperbolic metric and geometric function theory_ , Quasiconformal mappings and their applications, Narosa, New Delhi, 2007, pp. 9–56.
  * [11] L. Bernal-González, M. C. Calderón-Moreno, and J. A. Prado-Bassas, _Maximal cluster sets along arbitrary curves_ , J. Approx. Theory 129 (2004), 207–216.
  * [12] L. Bernal-González and M. d. C. Calderón-Moreno, _Hypercyclic algebras for $D$-multiples of convolution operators_, Proc. Amer. Math. Soc. 147 (2019), no. 2, 647–653.
  * [13] L. Bernal-González, J. López-Salazar, and J. B. Seoane-Sepúlveda, _Holomorphic surjectivity near the boundary: Topological and algebraic genericity_ , Linear Multilinear Algebra 68 (2020), no. 7, 1434–1456.
  * [14] W. Berweiler and A. Eremenko, _Direct singularities and completely invariant domains of entire functions_ , Illinois J. Math 52 (2008), no. 1, 243–259.
  * [15] J. Bès, J.A. Conejero, and D. Papathanasiou, _Hypercyclic algebras for convolution and composition operators_ , J. Funct. Anal. 274 (2018), no. 10, 2884–2905.
  * [16] J. Bès and D. Papathanasiou, _Algebrable sets of hypercyclic vectors for convolution operators_ , Israel J. Math. 238 (2020), no. 1, 91–119.
  * [17] A. Boivin, P. M. Gauthier, and P. V. Paramonov, _Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications_ , Canad. J. Math. 54 (2002), no. 5, 945–969.
  * [18] S. Charpentier, _Holomorphic functions with universal boundary behaviour_ , J. Approx. Theory 254 (2020), 105391.
  * [19] S. Charpentier and L. Kosiński, _Wild boundary behaviour of holomorphic functions in domains of $\mathbb{C}^{N}$_, Indiana Univ. Math. J. 70 (2021), no. 6, 2351–2367.
  * [20] S. Charpentier, M. Manolaki, and K. Maronikolakis, _Abel universal functions: Boundary behaviour and Taylor polynomials_ , Preprint, arXiv:2310.05611.
  * [21] S. Charpentier and A. Mouze, _Abel universal functions_ , Canad. J. Math. 75 (2023), no. 6, 1957–1985.
  * [22] G. Costakis and A. Melas, _On the range of universal functions_ , Bull. London Math. Soc. 32 (2000), no. 4, 458–464.
  * [23] J. Falcó and K.-G. Grosse-Erdmann, _Algebrability of the set of hypercyclic vectors for backward shift operators_ , Advances in Mathematics 366 (2020), 107082.
  * [24] D. García, B. C. Grecu, M. Maestre, and J. B. Seoane-Sepúlveda, _Infinite dimensional Banach spaces of functions with nonlinear properties_ , Math. Nachr. 283 (2010), no. 5, 712–720.
  * [25] S. J. Gardiner, _Universal Taylor series, conformal mappings and boundary behaviour_ , Ann. Inst. Fourier 64 (2014), 327–339.
  * [26] S. J. Gardiner and D. Khavinson, _Boundary behaviour of universal Taylor series_ , C. R. Math. Acad. Sci. Paris 352 (2014), no. 2, 99–103.
  * [27] K.-G. Grosse-Erdmann and A. Peris Manguillot, _Linear chaos_ , Universitext, Springer, London, 2011.
  * [28] F. Iversen, _Recherches sur les fonctions inverses des fonctions méromorphes_ , Thèse, Helsingfors, 1914.
  * [29] W. Kaplan, _Extensions of the Gross star theorem_ , Michigan Math. J. 2 (1954), 105–108.
  * [30] S. Kierst and E. Szpilrajn, _Sur certaines singularités des fonctions analytiques uniformes_ , Fundam. Math. 21 (1933), 276–294.
  * [31] K. Maronikolakis, _Universal radial approximation in spaces of analytic functions_ , J. Math. Anal. Appl. 512 (2022), no. 1, 126102.
  * [32] J. Mashreghi, _Dilation theory. A century of distancing from the frontiers_ , Acta Sci. Math. (Szeged) 88 (2022), 321–370.
  * [33] A. Melas and V. Nestoridis, _Universality of Taylor series as a generic property of holomorphic functions_ , Adv. Math. 157 (2001), no. 2, 138–176.
  * [34] T. Meyrath, _Universal radial limits of meromorphic functions in the unit disk_ , C. R. Math. Acad. Sci. Paris 360 (2022), 893–898.
  * [35] V. Nestoridis, _Universal Taylor series_ , Ann. Inst. Fourier 46 (1996), 1293–1306.