# Dynamical zeta functions for differentiable parabolic maps of the interval

Claudio Bonanno  Dipartimento di Matematica, Università di Pisa, Largo Bruno
Pontecorvo 5, 56127 Pisa, Italy<EMAIL_ADDRESS>and  Roberto
Castorrini  Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa,
Italy<EMAIL_ADDRESS>
###### Abstract.

This paper explores the domain of meromorphic extension for the dynamical zeta
function associated to a class of one-dimensional differentiable parabolic
maps featuring an indifferent fixed point. We establish the connection between
this domain and the spectrum of the weighted transfer operators of the induced
map. Furthermore, we discuss scenarios where meromorphic extensions occur
beyond the confines of the natural disc of convergence of the dynamical zeta
function.

##  1\. Introduction

In the last few years, the _dynamical zeta function_ introduced by D. Ruelle
in the late 1970s has proved to play a crucial role in the study of the
statistical properties of hyperbolic dynamical systems, providing a powerful
tool for understanding the underlying structure and behavior of such systems.
Its definition is based on the distribution of the periodic orbits of the
system, therefore the dynamical zeta functions can be considered a bridge
between the microscopic and macroscopic aspects of the system.

One of the most interesting properties of the dynamical zeta function is that
its poles are related with the eigenvalues of the associated transfer
operator, another important tool in the thermodynamic formalism approach to
dynamical systems. The eigenvalues of the transfer operator, in turn, shed
light on the stability and mixing properties of the system, offering a deeper
understanding of its overall dynamics. The basic idea is that the smaller is
the essential spectrum of the transfer operator, the broader is the domain
over which the dynamical zeta function can be meromorphically extended. We
refer the reader to [3] for a detailed discussion on the subject and for an
exhaustive list of the literature on the first steps of the development of
this theory.

It is clear from the basic idea sketched above that it is important to find a
good space of functions on which the essential spectral radius of the transfer
operator becomes smaller. This is typically obtained by looking at functions
of increasing regularity111In the presence of discontinuities, there exist
example for which this can not be done, as shown in [8] and [7]. Nevertheless,
estimates on the essential spectral radius close to optimal do exist also in
this case, as shown in [4]. if the dynamical system is uniformly hyperbolic.
However, in not uniformly hyperbolic systems, capturing information on the
dynamical zeta function becomes more challenging due to a lack of information
in the spectrum of the associated transfer operator, for example the strategy
of looking at the space of functions with high regularity does not work in
this situation. An important case for which the classic approach fails is that
of parabolic maps, i.e. maps with parabolic fixed points. The properties of
the dynamical zeta function have been studied in a symbolic dynamic context
equivalent to this case in [15], but the direct approach to differentiable
parabolic maps is to our knowledge still lacking.

A notable exception is the case of parabolic maps which have an analytic
extension to a complex domain. This case has been studied in [24] for maps of
the interval and in [21] for higher dimensional maps. In the analytic case the
transfer operator of the map, or of its induction, has strong properties which
allow for a complete understanding of the problem in the one-dimensional case.
This technique has been used also in [5] for the Farey map and its relations
with the Selberg zeta function on the modular surface. In the multidimensional
case the results in [21] give only a partial meromorphic extension of the
dynamical zeta function by using the powerful property of nuclearity for the
transfer operator of the induced map.

In this paper we deal with the case of differentiable parabolic maps of the
unit interval, under the simplifying assumption that the maps are piecewise
monotone with full branches (see Section 2 for all the assumptions). By using
induction and the thermodynamic formalism for uniformly hyperbolic systems, we
first show that results similar to those in [15] and [21] hold in this case.
Then we introduce a method to further extend the dynamical zeta function under
an additional assumption on the potential. Our conjecture is that this method
can be improved to obtain a complete understanding of the extension of the
dynamical zeta function also in the differentiable context.

The structure of the paper unfolds as follows: Section 2 introduces the maps
$T$ under examination and the key elements, namely, the transfer operator and
the dynamical zeta function associated with $T$. The primary outcome,
contained in Theorem 2.5, stands as the pivotal result of the paper. It
encompasses the meromorphic extension of the dynamical zeta function,
establishing a connection between its poles and the eigenvalues of the
weighted transfer operator of the induced map. Additionally, Theorem 2.6
delves into the potential extension of the dynamical zeta function’s domain
beyond the unit disc defined by the transfer operator associated to the
induced map.

Section 3 enumerates and proves the principal characteristics of the spectra
of the transfer operators studied. These features are subsequently employed in
Sections 4 and 5 in order to prove the main theorems, respectively.

##  2\. Settings and main results

We consider maps $T:[0,1]\to[0,1]$ satisfying the following assumptions:

  * (H1)

$T$ is _piecewise monotone_ with respect to a partition
$J=\left\\{J_{0},J_{1}\right\\}$ of the interval with $J_{0}=(0,a)$ and
$J_{1}=(a,1)$ for some $a\in(0,1)$;

  * (H2)

$T$ has _full branches_ , that is
$\overline{T(J_{0})}=\overline{T(J_{1})}=[0,1]$;

  * (H3)

For $r\in\mathbb{N}\cup\\{\infty\\}$, $r\geq 2$, the restriction
$T_{0}:=T|_{J_{0}}$ extends to a $C^{r}$ function on $(0,a]$ and
$T_{1}:=T|_{J_{1}}$ extends to a $C^{r}$ function on $[a,1]$. We denote by
$\psi_{i}:=T_{i}^{-1}:(0,1)\to J_{i}$ the local inverses of $T$;

  * (H4)

$0$ is an _indifferent fixed point_ , that is $T(0)=0$ and $T^{\prime}(0)=1$
with $T^{\prime}(x)-1\sim c\,x^{\alpha}$ as $x\to 0^{+}$ for some $c>0$ and
$\alpha>0$ (notice that $T_{0}$ is extendible to a $C^{r}$ function at $0$
only for $\alpha\geq r-1$);

  * (H5)

There exist $\varepsilon>0$ and $\rho>1$ such that $T^{\prime}(x)$ is
increasing in $(0,\varepsilon)$ and $|T^{\prime}(x)|\geq\rho$ for all
$x\in[\varepsilon,a)\cup(a,1)$.

######  Example 2.1.

A classical example of a family of maps satisfying our assumptions are the so-
called _Pomeau-Manneville maps_ $T_{PM}:[0,1]\to[0,1]$, introduced and studied
in [20, 22] to investigate the phenomenon of intermittency in physics and
defined as

$T_{PM}(x):=x+x^{1+\alpha}\,\text{(mod $1$),}\quad\text{for $\alpha>0$.}$

A somewhat simpler family of maps with the same dynamical properties have been
introduced in [18] and are now referred to as _Liverani-Saussol-Vaienti (LSV)
maps_. The maps have been defined as

$T_{LSV}(x):=\begin{cases}x(1+2^{\alpha}\,x^{\alpha}),&\text{for
$x\in[0,\frac{1}{2})$,}\\\\[2.84544pt] 2x-1,&\text{for
$x\in[\frac{1}{2},1]$},\end{cases}$

for $\alpha\in(0,1)$, and clearly can be considered for all $\alpha>0$. These
families of maps, together with generalizations, are studied in [6] in the
case $\alpha\geq 1$ as models of infinite measure-preserving interval maps.

The main objects we investigate in the paper are the transfer operator and the
dynamical zeta function associated to the map $T$. Let $v:[0,1]\to\mathbb{R}$
be a function satisfying:

  * (H6)

for some $k\in\mathbb{N}\cup\\{\infty\\}$, $k<r$, $v$ is in $C^{k}(J_{i})$ for
$i=0,1$ and extends to a $C^{k}$ function on $[0,a]$ and $[a,1]$.

Then we consider the _transfer operator $\mathcal{L}_{v}:L^{1}(0,1)\to
L^{1}(0,1)$ with potential $v$_ given by

$(\mathcal{L}_{v}f)(x):=e^{v(\psi_{0}(x))}\,f(\psi_{0}(x))+e^{v(\psi_{1}(x))}\,f(\psi_{1}(x)).$

In the following it is useful to write $\mathcal{L}_{v}f$ as the sum of two
terms, $\mathcal{L}_{0,v}f$ and $\mathcal{L}_{1,v}f$, which represent the
action of the _parabolic_ and _expanding_ part of the map respectively:

(2.1) $\mathcal{L}_{i,v}f:=(e^{v}\cdot f)\circ\psi_{i}\,,\qquad i=0,1\,.$

An important case is given by the choice $v=-\log|T^{\prime}|$, for which
$\mathcal{L}_{v}$ reduces to be the _Perron-Frobenius operator_ , which is
associated to the existence of an absolutely continuous invariant measure. It
is known that such invariant measure is finite if $\alpha\in(0,1)$ and
infinite for $\alpha\geq 1$. In the so-called thermodynamic formalism approach
it is also interesting to consider the more general choice
$v=-q\,\log|T^{\prime}|$ with $q\in[0,+\infty)$ (and sometimes
$q\in\mathbb{C}$, see [5]).

By now, a standard way to study parabolic maps of the interval is inducing on
the sets where the dynamics is expanding. Let

(2.2) $\tau(x):=\min\left\\{n\geq 0\,:\,T^{k}(x)\in J_{1}\right\\}$

be the _first passage function_ from $J_{1}$. It is finite on $[0,1]$ but for
a countable set. Let

$a_{0}:=1,\quad a_{1}:=a,\quad a_{\ell}:=\psi_{0}(a_{\ell-1})\quad\text{for
$\ell\geq 2$,}$

then $\tau(x)<\infty$ if and only if $x\not=a_{\ell}$ for all $\ell\geq 0$. In
addition we introduce the notation

$A_{\ell}:=(a_{\ell},\,a_{\ell-1})=\left\\{x\in(0,1)\,:\,\tau(x)=\ell-1\right\\}\quad\text{for
$\ell\geq 1$,}$

for the level sets of $\tau$. It is well known (see e.g. [13, Lemma 2.1]) that
there exists a positive constant only depending on $T$ such that222The
constants involved in the inequalities are written in the form $const(\cdot)$
to specify their dependence and they might be different from one line to the
next one even if written with the same notation.

(2.3) $a_{\ell}\sim\,const(T)\ \ell^{-1/\alpha}\,(1+O(\ell^{-1}))\quad\text{as
$\ell\to\infty$.}$

We are now ready to introduce the _jump transformation_ of $T$ on $J_{1}$
which is the map

$G:\bigcup_{\ell\geq 1}\,A_{\ell}\to(0,1),\quad G(x):=F^{1+\tau(x)}(x).$

It is immediate to verify that $G$ has the following properties.

######  Lemma 2.2.

Let (H1)-(H5) hold for $T$, then for any $\ell\geq 1$ the restriction
$G_{\ell}:=G|_{A_{\ell}}$ satisfies:

  1. (i)

$G_{\ell}=T_{1}\circ T_{0}^{\ell-1}$;

  2. (ii)

$G_{\ell}(A_{\ell})=(0,1)$;

  3. (iii)

It is monotone with inverse $\phi_{\ell}:(0,1)\to A_{\ell}$ given by
$\phi_{\ell}=\psi_{0}^{\ell-1}\circ\psi_{1}$;

  4. (iv)

It extends to a $C^{r}$ function on
$\overline{A_{\ell}}=[a_{\ell},\,a_{\ell-1}]$ and $\phi_{\ell}$ extends to a
$C^{r}$ function on $[0,1]$;

  5. (v)

$|G^{\prime}_{\ell}|\geq\rho$.

######  Example 2.3.

A classical example of a map which can be obtained as a jump transformation is
the _Gauss map_ $G(x)=1/x-\lfloor 1/x\rfloor$ for $x\in(0,1]$. Let

(2.4) $T_{F}(x):=\begin{cases}\frac{x}{1-x},&\text{for
$x\in[0,\frac{1}{2}]$,}\\\\[2.84544pt] \frac{1-x}{x},&\text{for
$x\in[\frac{1}{2},1]$},\end{cases}$

be the _Farey map_ , then the Gauss map is the jump transformation of $T_{F}$
on $(1/2,1)$. The Farey map satisfies (H1)-(H5) with $\alpha=1$ and
$r=\infty$, but can also be extended to an analytic function on a strip
containing the interval $(0,1)$. Hence the dynamical zeta function of $T_{F}$
has been studied by the method of Fredholm determinants for analytic maps (see
[24] and [5]).

The next step is the application of the inducing procedure to define the
transfer operator for $G$. Given a potential function $v:(0,1)\to\mathbb{R}$
satisfying (H6), we define the _induced potential_ to be the function

(2.5) $w:\bigcup_{\ell\geq 1}\,A_{\ell}\to\mathbb{R},\quad
w(x):=\sum_{j=0}^{\tau(x)}\,v(T^{j}(x)).$

For any $\ell\geq 1$, the function $w$ extends to a $C^{k}$ function on
$\overline{A_{\ell}}$. Then we introduce the _transfer operator-valued power
series $\mathcal{Q}_{w}(z)$ with potential $w$ and variable $z\in\mathbb{C}$_
as the formal series acting on a function $f\in L^{1}(0,1)$ as

(2.6)
$(\mathcal{Q}_{w}(z)f)(x):=\sum_{n=1}^{\infty}\,z^{n}\,e^{w(\phi_{n}(x))}\,f(\phi_{n}(x)).$

Notice that the choice $v=-q\log|T^{\prime}|$ leads to $w=-q\log|G^{\prime}|$,
and for $q=1$ the operator $\mathcal{Q}_{w}(1)$ is the Perron-Frobenius
operator of the map $G$.

In order to obtain good convergence properties for $\mathcal{Q}_{w}(z)$ we
assume that:

  * (H7)

there is a constant $c_{k}>0$ such that for all
$h\in\mathbb{N}\cup\\{\infty\\}$, $h\leq k$,

$\sum_{n=1}^{\infty}\,\|e^{w\circ\phi_{n}}\|_{h}\,\Big{(}1+\|\phi_{n}\|_{h}^{h}\Big{)}\leq
c_{k},$

where for $f\in C^{k}([0,1])$ we denote

$\|f\|_{h}:=\max_{0\leq j\leq h}\|f^{(j)}\|_{\infty},$

the max of the sup-norm of $f$ and its first $h$ derivatives.

For $w=-q\log|G^{\prime}|$ and $G$ the Gauss map of Example 2.3 we obtain by
(2.3) that (H7) holds for $\Re(q)>\alpha/(1+\alpha)$.

######  Remark 2.4.

By definition, $\phi_{n}(x)\in A_{n}$ for all $n\geq 1$ and
$T^{j}(A_{n})=A_{n-j}$ for all $0\leq j\leq n-1$. Hence

$w(\phi_{n}(x))=\sum_{j=0}^{n-1}\,v(T^{j}(\phi_{n}(x)))$

satisfies

$\sum_{i=1}^{n}\,\min_{x\in A_{i}}\,v(x)\leq
w(\phi_{n}(x))\leq\sum_{i=1}^{n}\,\max_{x\in A_{i}}\,v(x).$

Therefore, it follows $w(\phi_{n}(x))\sim n\cdot v(0)$ as $n\to\infty$. Hence
(H7) implies $v(0)\leq 0$.

In the following we need to consider iterates of $\mathcal{Q}_{w}(z)$ which
involve compositions of the local inverse of $G$ and of the potential $w$.
Given the multi-indices
$\beta=(\beta_{1},\,\beta_{2},\dots,\beta_{m})\in\mathbb{N}^{m}$ for which we
set $|\beta|:=\beta_{1}+\beta_{2}+\dots+\beta_{m}$, we introduce the following
notations:

(2.7)
$\displaystyle\phi_{\beta}:=\phi_{\beta_{1}}\circ\phi_{\beta_{2}}\circ\dots\circ\phi_{\beta_{m}},$
$\displaystyle\phi_{{}_{\beta_{i}\dots\beta_{h}}}:=\phi_{\beta_{i}}\circ\phi_{\beta_{i+1}}\circ\dots\circ\phi_{\beta_{h}},\quad\forall\,1\leq
i\leq h\leq m,$ $\displaystyle
W_{\beta}(x):=\exp\Big{(}\sum_{j=1}^{m}\,w(\phi_{{}_{\beta_{j}\dots\beta_{m}}}(x))\Big{)}.$

The main results use the following quantities:

(2.8)
$\Lambda_{m}(z):=\sup_{x\in(0,1)}\,\sum_{\beta\in\mathbb{N}^{m}}\,|z|^{|\beta|}\,W_{\beta}(x),$

and the limit

(2.9) $\Lambda(z):=\limsup_{m\to\infty}\,[\Lambda_{m}(z)]^{\frac{1}{m}}.$

Their existence is studied in Theorem 3.1 below.

Finally, we recall the definition of the _dynamical zeta function_ of the map
$T$ with potential $v$, which is defined for $z\in\mathbb{C}$ as

(2.10) $\zeta_{{}_{T,v}}(z):=\exp\Big{(}\sum_{n\geq
1}\,\frac{z^{n}}{n}\,\sum_{T^{n}(x)=x}\,\exp\Big{(}\sum_{j=0}^{n-1}\,v(T^{j}(x))\Big{)}\Big{)}.$

It is a straightforward computation that the series in (2.10) converges for

$|z|<\exp(-P_{T}(v)),\quad
P_{T}(v):=\limsup_{n\to\infty}\,\frac{1}{n}\,\log\Big{(}\sum_{T^{n}(x)=x}\,\exp\Big{(}\sum_{j=0}^{n-1}\,v(T^{j}(x))\Big{)}\Big{)},$

where $P_{T}(v)$ is the so-called _topological pressure_ of the map $T$ for
the potential $v$ and it is finite by (H7). In the case $v=-\log|T^{\prime}|$,
the topological pressure of $T$ vanishes. The behaviour of the function
$P_{T}(-q\,\log|T^{\prime}|)$ for $q\in[0,\infty)$ is studied in the
thermodynamic formalism approach to the map $T$. For example, $P_{T}(0)$ is
the topological entropy of $T$ and the lack of regularity of this function at
$q=1$ is an interesting phenomenon called phase transition.

Our main results are a contributions to the important open problem of the
extension of $\zeta_{{}_{T,v}}(z)$ outside its $z$-disc of convergence. For
the proofs see Section 4 and 5.

######  Theorem 2.5.

Under assumptions (H1)-(H7), the dynamical zeta function $\zeta_{{}_{T,v}}(z)$
is defined on the disc $\\{z<\exp(-P_{T}(v))\\}$ and has a meromorphic
extension to the set

$\\{z\in\mathbb{C}\,:\,\text{$\mathcal{Q}_{w}(z)$ is
defined},\,\Lambda(z)<\rho^{r-2}\\},$

with singularities at $z_{0}=e^{-v(0)}$ and at the values $\bar{z}$ for which
1 is an isolated eigenvalue of $\mathcal{Q}_{w}(\bar{z})$, and with zeroes at
the values $\tilde{z}$ for which 1 is an isolated eigenvalue of
$\mathcal{Q}_{w-\log G^{\prime}}(\tilde{z})$ 333See Proposition 4.3 for the
definition of $\mathcal{Q}_{w-\log G^{\prime}}(\tilde{z})$..

Applying Theorem 2.5, the meromorphic extension of $\zeta_{{}_{T,v}}(z)$
strongly depends on the set of $z$ for which $\mathcal{Q}_{w}(z)$ is defined.
As we shall see later in the proofs, we obtain that $\mathcal{Q}_{w}(z)$ is
defined in the domain $D=\\{|z|\leq 1\\}$, therefore Theorem 2.5 guarantees
the extension to the subset of $D$ on which $\\{\Lambda(z)<\rho^{r-2}\\}$. Our
second result tells us that in some situations the domain of meromorphic
extension becomes bigger and is unbounded for $r=\infty$.

######  Theorem 2.6.

Under assumptions (H1)-(H7), if $v(0)<0$ the dynamical zeta function
$\zeta_{{}_{T,v}}(z)$ is defined on the disc $\\{z<\exp(-P_{T}(v))\\}$ and has
a meromorphic extension to the set

$\\{z\in D\cup(\mathbb{C}\setminus\bar{\Delta})\,:\,\Lambda(z)<\rho^{r-2}\\},$

where

$\bar{\Delta}:=\\{z\in\mathbb{C}\,:\,|\arg z|\leq\max\\{0,\pi+v(0)\\}\\}.$

In addition $\zeta_{{}_{T,v}}(z)$ has singularities at $z_{0}=e^{-v(0)}$ and
at the values $\bar{z}$ for which 1 is an isolated eigenvalue of
$\mathcal{Q}_{w}(\bar{z})$, and has zeroes at the values $\tilde{z}$ for which
1 is an isolated eigenvalue of $\mathcal{Q}_{w-\log G^{\prime}}(\tilde{z})$.

When $\pi+v(0)\leq 0$ the sector $\bar{\Delta}$ reduces to the line
$(0,+\infty)$, so that we obtain the extension to the cut domain
$(\mathbb{C}\setminus(1,+\infty))\cap\\{\Lambda(z)<\rho^{r-2}\\}$. For
$r=\infty$, the extension is then proved to the cut plane
$\mathbb{C}\setminus(1,+\infty)$.

######  Remark 2.7.

Unfortunately, this method proves ineffective for the case where $v(0)=0$, as
seen in the Perron-Frobenius scenario where $v=-\log|T^{\prime}|$. The problem
is to find a suitable function $h$ that satisfies Lemma 5.2 with
$e_{{}_{\Pi}}<\pi$ in this instance, we believe this is an open interesting
question. In general, we conjecture that in all cases the dynamical zeta
function $\zeta_{{}_{T,v}}(z)$ has a meromorphic extension to the set
$\\{z\in\mathbb{C}\setminus(1,+\infty):\Lambda(z)<\rho^{r-2}\\}$.

##  3\. The spectrum of the transfer operators

In this section we study the spectral properties of the transfer operators of
$T$ and of the corresponding induced map $G$ on the space of differentiable
functions. The results for $G$ are expected by the previous literature on the
subject. We give some proofs for completeness.

######  Theorem 3.1.

Let (H1)-(H7) hold. Then for all $z\in D:=\\{|z|\leq 1\\}$, the power series
$\mathcal{Q}_{w}(z)$ converges to a bounded operator from $C^{k}([0,1])$ to
$C^{k}([0,1])$. In addition, recalling (2.8) and (2.9), we have for all $z\in
D$:

  1. (i)

The limit

$\Lambda(z)=\lim_{m\to\infty}\,[\Lambda_{m}(z)]^{\frac{1}{m}}$

exists and $\Lambda(z)\leq c_{k}$;

  2. (ii)

The essential spectral radius of $\mathcal{Q}_{w}(z)$ on $C^{k}([0,1])$
satisfies $r_{ess}(\mathcal{Q}_{w}(z)|_{C^{k}})\leq\rho^{-k}\Lambda(z)$;

  3. (iii)

If $z\in(0,1]$ then $\Lambda(z)$ is an eigenvalue of $\mathcal{Q}_{w}(z)$ on
$C^{k}([0,1])$.

###### Proof.

The proof of the first assertion goes as follows. Let us write
$\mathcal{Q}_{w}(z)$ as

$(\mathcal{Q}_{w}(z)f)(x)=\sum_{n=1}^{\infty}\,z^{n}\,(Q_{n,w}f)(x),$

where

(3.1) $Q_{n,w}f:=(e^{w}\cdot f)\circ\phi_{n},\quad n\geq 1.$

Then we estimate $\|Q_{n,w}f\|_{k}$ for all $n\geq 1$. We use two formulas to
compute the derivatives of the product and composition of functions. One is
the well known Leibniz’s formula

$(f\cdot g)^{(j)}=\sum_{i=0}^{j}\,{j\choose i}\,f^{(i)}\,g^{(j-i)}.$

The other is the Faà di Bruno’s formula (see e.g. [12]). Given a multi-index
$a=(a_{1},a_{2},\dots,a_{j})\in\mathbb{N}_{0}^{j}$ the formula writes

$(f\circ\phi)^{(j)}=\sum_{s=1}^{j}\,(f^{(s)}\circ\phi)\,\sum_{a\in
B_{j,s}}\,\frac{j!}{a_{1}!\dots
a_{j}!}\,\prod_{\ell=1}^{j}\,\Big{(}\frac{1}{\ell!}\,\phi^{(\ell)}\Big{)}^{a_{\ell}},$

where
$B_{j,s}=\\{a\in\mathbb{N}_{0}^{j}\,:\,a_{1}+\dots+a_{j}=s,\,a_{1}+2a_{2}+\dots+ja_{j}=j\\}$.
Thus for all $j=1,\dots,k$,

$\|(Q_{n,w}f)^{(j)}\|_{\infty}\leq\sum_{i=0}^{j}\,{j\choose
i}\,\|(e^{w\circ\phi_{n}})^{(j-i)}\|_{\infty}\,\|(f\circ\phi_{n})^{(i)}\|_{\infty}$

and

$\displaystyle\|(f\circ\phi_{n})^{(j)}\|_{\infty}\leq$
$\displaystyle\sum_{s=1}^{j}\,\|f\|_{s}\,\sum_{a\in
B_{j,s}}\,const(j,s)\,\prod_{\ell=1}^{j}\,\|\phi_{n}^{(\ell)}\|_{\infty}^{a_{\ell}}$
$\displaystyle\leq$ $\displaystyle
const(j)\,\sum_{s=1}^{j}\,\|f\|_{s}\,\|\phi_{n}\|_{j}^{j}\leq\,const(j)\,\|f\|_{j}\,\|\phi_{n}\|_{j}^{j}.$

Therefore for all $n\geq 1$

$\|Q_{n,w}f\|_{k}\leq
const(k)\,\|e^{w\circ\phi_{n}}\|_{k}\,\|\phi_{n}\|_{k}^{k}\,\|f\|_{k}.$

It follows by (H7) that for all $z\in D$

$\|\mathcal{Q}_{w}(z)f\|_{k}\leq
const(k)\,\sum_{n=1}^{\infty}\,|z|^{n}\,\|e^{w\circ\phi_{n}}\|_{k}\,\|\phi_{n}\|_{k}^{k}\,\|f\|_{k}\leq
const(k)\,\|f\|_{k}.$

Hence $\mathcal{Q}_{w}(z)$ converges absolutely on $D$ to a bounded operator
from $C^{k}([0,1])$ to $C^{k}([0,1])$.

(i) By definition, for all $z\in D$ and $m\geq 1$,

$\displaystyle\Lambda_{m}(z)\leq$
$\displaystyle\Big{\|}\sum_{\beta_{1},..,\beta_{m-1}=1}^{\infty}\,|z|^{\beta_{1}+\beta_{2}+..+\beta_{m-1}}\,\exp\Big{(}\sum_{\ell=1}^{m-1}\,w\circ\phi_{\ell}\Big{)}\Big{\|}_{\infty}\,\sum_{\beta_{m}=1}^{\infty}\,|z|^{\beta_{m}}\,\|e^{w\circ\beta_{m}}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle
c_{k}\,\Big{\|}\sum_{\beta_{1},..,\beta_{m-1}=1}^{\infty}\,|z|^{\beta_{1}+..+\beta_{m-1}}\,\exp\Big{(}\sum_{\ell=1}^{m-1}\,w\circ\phi_{\ell}\Big{)}\Big{\|}_{\infty}=c_{k}\,\Lambda_{m-1}(z),$

hence $\Lambda_{m}(z)\leq c_{k}^{m}$. In addition, it is immediate to show
that
$\Lambda_{m+m^{\prime}}(z)\leq\Lambda_{m}(z)\cdot\Lambda_{m^{\prime}}(z)$,
hence $\Lambda(z)$ exists and satisfies $\Lambda(z)\leq c_{k}$.

(ii) follows as in [16, Thm. 3.1] (see also [9, Thm. 6] and [2, Thm. 2.5]).
The main idea is to notice that for all $j=1,\dots,k$ one can write as above

$(\mathcal{Q}_{w}(z)f)^{(j)}=\sum_{n=1}^{\infty}\,z^{n}\,e^{w\circ\phi_{n}}\,(f^{(j)}\circ\phi_{n})\,(\phi_{n}^{\prime})^{j}+\mathcal{R}_{z,j,w}f\,,$

with $\mathcal{R}_{z,j,w}$ compact as an operator from $C^{j}([0,1])$ to
$C^{0}([0,1])$. Applying this idea to $(\mathcal{Q}_{w}(z))^{m}$ for all
$m\geq 1$ implies that, denoting with $\mathcal{K}(C^{k})$ the set of compact
operators from $C^{k}([0,1])$ to $C^{k}([0,1])$,

$\inf_{\mathcal{R}\in\mathcal{K}(C^{k})}\,\|(\mathcal{Q}_{w}(z))^{m}-\mathcal{R}\|\leq\sup_{x\in(0,1)}\,\sum_{\beta\in\mathbb{N}^{m}}\,|z|^{|\beta|}\,W_{\beta}(x)\,|\phi^{\prime}_{\beta}(x)|^{k}\,.$

The result then follows from (2.8)-(2.9) and the classical Nussbaum formula
for the essential spectral radius.

(iii) follows as in [15, Thm. 3.1]. ∎

Properties 3.1-(ii) and (iii) tell us that $\mathcal{Q}_{w}(z)$ has a spectral
gap on $C^{k}$ for $z\in(0,1]$, which implies exponential decay of
correlations for $C^{k}$ observables for the map $G$ and has implications for
the analytic extension of the dynamical determinants we study in the next
section.

It is well known that we should not expect the operators $\mathcal{L}_{v}$ to
have a spectral gap on $C^{k}$ for all $k\in\mathbb{N}\cup\\{\infty\\}$ or
even on the space of holomorphic functions for maps $T$ with an analytic
extension. Nevertheless, in the analytic case, it has been proved in [24] that
the spectrum of the operator $\mathcal{L}_{0,v}$ on a space of holomorphic
functions is the unit interval $[0,1]$. This property has important
implications in the definition of a dynamical determinant for the map $T$ in
the analytic case. This approach, which we discuss in the next section, has
been used also in [14, 5] for the Farey map (2.4). We show that this approach
may fail in the differentiable case as the next result holds.

######  Proposition 3.2.

Let $T:[0,1]\to[0,1]$ be a map satisfying (H1)-(H5) with $\alpha\geq 1$, and
let $\mathcal{L}_{0}$ be the part of the Perron-Frobenius operator of $T$
associated to the parabolic branch, that is
$\mathcal{L}_{0}f=\psi^{\prime}_{0}\,(f\circ\psi_{0})$. The essential spectral
radius of $\mathcal{L}_{0}$ on $C^{1}$ is greater than or equal to 1.

###### Proof.

We show that every $\lambda\in\mathbb{C}$ with $|\lambda|<1$ is an eigenvalue
for $\mathcal{L}_{0}$ on $C^{1}$. Let $f\in C^{1}([0,1])$ and let $F\in C^{2}$
satisfy $F^{\prime}=f$. Notice that $T$ is $C^{2}$ at 0 and the same is true
for $\psi_{0}$. Then

$\mathcal{L}_{0}f=\lambda\,f\quad\Leftrightarrow\quad\psi^{\prime}_{0}\,(f\circ\psi_{0})=\lambda\,f\quad\Leftrightarrow\quad(F\circ\psi_{0})^{\prime}=\lambda\,F^{\prime}.$

Therefore $\lambda\in\mathbb{C}$ is an eigenvalue for $\mathcal{L}_{0}$ on
$C^{1}$ if and only if there exists $F\in C^{2}$ such that

(3.2) $(F\circ\psi_{0})(x)=\lambda\,F(x)+const(F),\qquad\forall\,x\in[0,1].$

We show the existence of a solution to (3.2) for all $\lambda\in\mathbb{C}$
with $|\lambda|<1$.

First of all, since $\psi_{0}(0)=0$ and $\psi_{0}^{\prime}(0)=1$, then a
solution $F$ to (3.2) with $\lambda\not=1$ satisfies $F^{\prime}(0)=0$, and
the same must be true for the second derivative, that is
$F^{\prime\prime}(0)=0$.

Let now $h:\mathbb{R}\to\mathbb{R}$ be a $C^{2}$ function with support in
$\overline{A_{0}}=[a,1]$ and such that $h^{(j)}(a)=h^{(j)}(1)=0$ for $j=1,2$.
Then for all $\lambda\in\mathbb{C}$ with $|\lambda|<1$

(3.3)
$F_{\lambda}(x):=h(x)+\sum_{\ell=1}^{+\infty}\,\lambda^{\ell}\,\chi_{{}_{\overline{A_{\ell}}}}(x)\,h(T^{\ell}(x)),\qquad\text{for
$x\in(0,1]$}$

with $F_{\lambda}(0)=0$ is a solution to (3.2) with $const(F)=0$, where
$\chi_{A}$ is the indicator function of the set $A$. Since $|\lambda|<1$ we
show that the series is totally convergent in $C^{2}$. The function
$\chi_{{}_{\overline{A_{\ell}}}}(x)\,h(T^{\ell}(x))$ is in $C^{2}([0,1])$ for
all $\ell\geq 1$, and we can compute

$\|\chi_{{}_{\overline{A_{\ell}}}}(x)\,h(T^{\ell}(x))\|_{\infty}\leq\|h\|_{2},\quad\forall\,\ell\geq
1,$
$\|\Big{(}\chi_{{}_{\overline{A_{\ell}}}}(x)\,h(T^{\ell}(x))\Big{)}^{\prime}\|_{\infty}\leq\|h\|_{2}\,\prod_{i=1}^{\ell}\,\|T^{\prime}\|_{\infty,A_{i}},$

where $\|\cdot\|_{\infty,A_{i}}$ denotes the sup-norm of a function restricted
to $A_{i}$, and

$\|\Big{(}\chi_{{}_{\overline{A_{\ell}}}}(x)\,h(T^{\ell}(x))\Big{)}^{\prime\prime}\|_{\infty}\leq\|h\|_{2}\,\prod_{i=1}^{\ell}\,\|T^{\prime}\|_{\infty,A_{i}}^{2}\,\Big{(}1+\ell\|T\|_{2}\Big{)}.$

By (H5), for $\ell$ big enough so that $a_{\ell-1}<\varepsilon$, we have

$\|T^{\prime}\|_{\infty,A_{\ell}}=T^{\prime}(a_{\ell-1})\sim
1+ca_{\ell-1}^{\alpha}\sim 1+const(T)\,\ell^{-1}\quad\text{as
$\ell\to\infty$,}$

using (H4) and (2.3), where $const(T)>0$. Hence

$\prod_{i=1}^{\ell}\,\|T^{\prime}\|_{\infty,A_{i}}\sim\ell^{const(T)}\quad\text{as
$\ell\to\infty$.}$

This implies that $F_{\lambda}\in C^{2}([0,1])$. In addition, using that
$\psi_{0}([0,1])=[0,a]$, $\psi_{0}(\overline{A_{\ell-1}})=\overline{A_{\ell}}$
and $T^{\ell}\circ\psi_{0}(x)=T^{\ell-1}(x)$ for $x\in\overline{A_{\ell-1}}$,
we can compute

$\displaystyle F_{\lambda}(\psi_{0}(x))=$ $\displaystyle
h(\psi_{0}(x))+\sum_{\ell=1}^{+\infty}\,\lambda^{\ell}\,\chi_{{}_{\overline{A_{\ell}}}}(\psi_{0}(x))\,h(T^{\ell}(\psi_{0}(x)))$
$\displaystyle=$
$\displaystyle\sum_{\ell=1}^{+\infty}\,\lambda^{\ell}\,\chi_{{}_{\overline{A_{\ell-1}}}}(x)\,h(T^{\ell-1}(x))$
$\displaystyle=$
$\displaystyle\lambda\,h(x)+\lambda\,\sum_{\ell=1}^{+\infty}\,\lambda^{\ell}\,\chi_{{}_{\overline{A_{\ell}}}}(\psi_{0}(x))=\lambda\,F_{\lambda}(x)$

for all $x\in(0,1]$. This completes the proof. ∎

######  Remark 3.3.

In the proof we have used that $0$ is an indifferent fixed point to obtain
that the term $\prod_{i=1}^{\ell}\,\|T^{\prime}\|_{\infty,A_{i}}$ diverges
with polynomial speed in $\ell$. This is false for uniformly expanding maps
for which $T^{\prime}(0)\geq\rho>1$, and in this case the construction of the
eigenfunction $F_{\lambda}$ works only for $|\lambda|<1/\rho$, which is the
classical bound on the essential spectral radius of the Perron-Frobenius
operator of $T$ on $C^{1}$.

######  Remark 3.4.

We expect this proposition to hold on $C^{k}$ for $k\geq 2$ under suitable
assumptions on the higher derivatives of $T$ which guarantee the total
convergence of the series (3.3) in $C^{k}$ for $|\lambda|<1$. However, this
construction does not work in the analytic case since the solution to (3.2)
has vanishing derivatives of all order at 0. Hence, we are describing a
property which is typical of the differentiable case.

##  4\. The dynamical zeta function of $T$ and $G$

We now discuss the properties of the dynamical zeta function $\zeta_{T,v}$
defined in (2.10). It is known that it is a good strategy to use the induced
map. Hence we consider the dynamical zeta function of the jump transformation
$G$ with the induced potential $w$ (given in (2.5)) for $u\in\mathbb{C}$ given
by

$\zeta_{{}_{G,w}}(u):=\exp\Big{(}\sum_{m\geq
1}\,\frac{u^{m}}{m}\,\sum_{G^{m}(x)=x}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}\Big{)},$

which converges for $|u|<\exp(-P_{G}(w))$ where

$P_{G}(w):=\limsup_{m\to\infty}\,\frac{1}{m}\,\log\Big{(}\sum_{G^{m}(x)=x}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}\Big{)}$

is the topological pressure of $G$ for the potential $w$.

The relation between the map $T$ and its jump transformation $G$ reflects into
well-known relations between the associated transfer operators and dynamical
zeta functions. From (3.1), using (2.5) with $\tau(x)=n-1$ and Lemma 2.2-(iii)
we write

$Q_{n,w}f=\mathcal{L}_{1,v}\,\mathcal{L}_{0,v}^{n-1}f,$

see (2.1), hence the power series $\mathcal{Q}_{w}(z)$ in (2.6) can be
formally written as

$\mathcal{Q}_{w}(z)f=\sum_{n=1}^{\infty}\,z^{n}\,\mathcal{L}_{1,v}\,\mathcal{L}_{0,v}^{n-1}f=z\,\mathcal{L}_{1,v}\Big{(}\sum_{n=1}^{\infty}\,(z\,\mathcal{L}_{0,v})^{n-1}f\Big{)}.$

A straighforward manipulation implies that

(4.1)
$\Big{(}1-\mathcal{Q}_{w}(z)\Big{)}\,\Big{(}1-z\,\mathcal{L}_{0,v}\Big{)}=1-z\,\mathcal{L}_{v}.$

Proving that (4.1) holds on a suitable space of functions gives the relation
between the spectral properties of the two operators, in particular relating
functions fixed by $\mathcal{Q}_{w}(z)$ with eigenfunctions of
$\mathcal{L}_{v}$. This approach has been used in [14, 5] for the Farey and
the Gauss map. It is precisely because of (4.1) that it is interesting to
study the spectrum of $\mathcal{L}_{0,v}$: If the spectrum of
$\mathcal{L}_{0,v}$ is the unit interval $[0,1]$ we can write
$(1-\mathcal{Q}_{w}(z))$ as
$(1-z\,\mathcal{L}_{v})(1-z\,\mathcal{L}_{0,v})^{-1}$ for all $z$ in the cut
complex plane $\mathbb{C}\setminus[1,\infty)$, obtaining the analytic
extension of the power series $\mathcal{Q}_{w}(z)$ to
$\mathbb{C}\setminus[1,\infty)$. However, by Proposition 3.2 this approach
does not work for the maps $T$ we are studying in this paper.

From (4.1), it is possible to get the intuition that a relation should hold
also between $\zeta_{{}_{T,v}}$ and $\zeta_{{}_{G,w}}$. The immediate idea is
to write the zeta function $\zeta_{{}_{T,v}}$ as the determinant, in some
sense, of $(1-z\,\mathcal{L}_{v})$ and to do the same for the jump
transformation. Building on this idea, one obtains a relation between the two
dynamical zeta functions by using the relation between the periodic points of
$T$ and $G$. If $G^{m}(x)=x$ then $T^{n}(x)=x$ where
$n=\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))$. Then, following [21, 14] we introduce
the _two-variable dynamical zeta function_ $Z_{w}(z,u)$ defined for
$z,u\in\mathbb{C}$ as

(4.2) $Z_{w}(z,u):=\exp\Big{(}\sum_{m\geq
1}\,\frac{u^{m}}{m}\,\sum_{G^{m}(x)=x}\,z^{\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}\Big{)}.$

The function $Z_{w}(z,u)$ is well-defined for $z\in D=\\{|z|\leq 1\\}$ and
$|u|<\exp(-P_{G}(u))$. It is immediate to get for $z\not=0$

(4.3) $\zeta_{{}_{G,w}}(z)=Z_{w}(1,u)\quad\text{and}\quad
Z_{w}(z,u)=\zeta_{{}_{G,w+(\log z)(1+\tau)}}(u),$

and the relation with the dynamical zeta function of $T$ is given by the next
result.

######  Proposition 4.1 ([21, 14]).

We can write

(4.4) $\zeta_{{}_{T,v}}(z)=(1-z\,e^{v(0)})^{-1}\,Z_{w}(z,1)$

whenever the three terms make sense.

In particular, $(1-z\,e^{v(0)})^{-1}$ is a meromorphic function on
$\mathbb{C}$ and it remains to study the meromorphic extensions of
$Z_{w}(z,1)$. To this aim we recall the notions of _flat trace_ and _flat
determinant_ of a linear operator. See [3, Section 3.2.2] for more details and
the proofs of the properties we recall. Let $\gamma:[0,1]\to[0,\infty)$ be a
$C^{\infty}$ compactly supported function with $\int_{0}^{1}\gamma(x)dx=1$.
Setting $\gamma_{\varepsilon}(x):=\varepsilon^{-1}\gamma(x/\varepsilon)$ for
small $\varepsilon>0$, one introduces the _mollifier operator_
$I_{\varepsilon}$ associated to $\gamma$ on $L^{1}(0,1)$ as

$\Big{(}I_{\varepsilon}(f)\Big{)}(x):=\int_{0}^{1}\,\gamma_{\varepsilon}(x-y)\,f(y)\,dy.$

Basic properties of $I_{\varepsilon}$ include that $I_{\varepsilon}(f)\in
C^{\infty}$ for all $f\in L^{1}$ and there exists a kernel
$K_{\varepsilon}(x,y)\in C^{\infty}([0,1]\times[0,1])$ with support converging
to the diagonal $\\{(x,x)\,:\,x\in[0,1]\\}$ such that

$\Big{(}I_{\varepsilon}(f)\Big{)}(x)=\int_{0}^{1}\,K_{\varepsilon}(x,y)\,f(y)\,dy.$

######  Definition 4.2.

Given a bounded operator $\mathcal{P}$ on $C^{k}([0,1])$ set

$\mathcal{P}_{\varepsilon}(x,y):=\Big{(}\mathcal{P}(K_{\varepsilon}(\cdot,y))\Big{)}(x)\quad\text{and}\quad\text{tr}^{b}_{\varepsilon}(\mathcal{P}):=\int_{0}^{1}\,\mathcal{P}_{\varepsilon}(x,x)\,dx.$

Then $\mathcal{P}$ admits a _flat trace_ if the following limit exists:

$\text{tr}^{b}(\mathcal{P}):=\lim_{\varepsilon\to
0}\,\text{tr}^{b}_{\varepsilon}(\mathcal{P}).$

If $\mathcal{P}^{k}$ admits a flat trace for all $k\geq 1$, then we call _flat
determinant_ of $\mathcal{P}$ the formal power series defined for
$u\in\mathbb{C}$ as

$\text{det}^{b}(1-u\,\mathcal{P}):=\exp\Big{(}-\sum_{m\geq
1}\,\frac{u^{m}}{m}\,\text{tr}^{b}(\mathcal{P}^{k})\Big{)}.$

We now show that the transfer operators $\mathcal{Q}_{w}(z)$ admit a flat
determinant and give a relation with $Z_{w}(z,u)$.

######  Proposition 4.3.

Under the assumptions (H1)-(H7), for all $z\in D$ the transfer operator-valued
power series $(\mathcal{Q}_{w}(z))^{m}$ admits a flat trace for all $m\geq 1$
and

$\text{tr}^{b}((\mathcal{Q}_{w}(z))^{m})=\sum_{G^{m}(x)=x}\,\frac{z^{\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}}{1-((G^{m})^{\prime}(x))^{-1}}.$

It follows that

(4.5) $Z_{w}(z,u)=\frac{\text{det}^{b}(1-u\,\mathcal{Q}_{w-\log
G^{\prime}}(z))}{\text{det}^{b}(1-u\,\mathcal{Q}_{w}(z))}$

whenever the three terms make sense, where we are using the notation444By
definition of $G$, since $T^{\prime}_{0}(x)>0$, the sign of $G^{\prime}$ is
always positive or always negative where it exists. When $G^{\prime}>0$ the
notation is justified by the remark that $\mathcal{Q}_{w-\log G^{\prime}}$ is
the operator associated to the potential $\tilde{w}=w-\log G^{\prime}$.

$(\mathcal{Q}_{w-\log
G^{\prime}}(z)f)(x)=\sum_{n=1}^{\infty}\,z^{n}\,\frac{e^{w(\phi_{n}(x))}}{G^{\prime}(\phi_{n}(x))}\,f(\phi_{n}(x)).$

###### Proof.

For $z\in D$ and all $m\geq 1$ we can write

$((\mathcal{Q}_{w}(z))^{m}f)(x)=\sum_{\beta\in\mathbb{N}^{m}}\,z^{|\beta|}\,W_{\beta}(x)\,f(\phi_{\beta}(x))$

where $\beta=(\beta_{1},\,\beta_{2},\dots,\beta_{m})$ and
$\phi_{\beta}:(0,1)\to A_{\beta_{1}}$ is a strict contraction (see (2.7)) with
a unique fixed point $x_{\beta}\in\phi_{\beta}(0,1)$. Setting

$(Q_{\beta}f)(x):=W_{\beta}(x)\,f(\phi_{\beta}(x)),$

let us show that $Q_{\beta}$ admits a flat trace. For all small
$\varepsilon>0$ we have

$\text{tr}^{b}_{\varepsilon}(Q_{\beta})=\int_{0}^{1}\,(Q_{\beta})_{\varepsilon}(x,x)\,dx=\int_{0}^{1}\,\gamma_{\varepsilon}(\phi_{\beta}(x)-x)\,W_{\beta}(x)\,dx,$

and using the notation $\Phi_{\beta}(x)$ for the injective map $(0,1)\ni
x\mapsto\phi_{\beta}(x)-x$, we can write

$\text{tr}^{b}_{\varepsilon}(Q_{\beta})=\int_{\Phi_{\beta}(0,1)}\,\gamma_{\varepsilon}(z)\,\frac{W_{\beta}(\Phi_{\beta}^{-1}(z))}{|1-\phi^{\prime}_{\beta}(\Phi_{\beta}^{-1}(z))|}\,dz\,.$

By the properties of $\gamma_{\varepsilon}$ and using the fixed point
$x_{\beta}=\Phi_{\beta}^{-1}(0)$, it follows that

$\lim_{\varepsilon\to
0}\,\text{tr}^{b}_{\varepsilon}(Q_{\beta})=\frac{W_{\beta}(x_{\beta})}{|1-\phi^{\prime}_{\beta}(x_{\beta})|}=\text{tr}^{b}(Q_{\beta}).$

In addition, for all $\beta\in\mathbb{N}^{m}$ we have
$|1-\phi^{\prime}_{\beta}(x_{\beta})|\geq 1-\rho^{-m}$ and

$\displaystyle\sum_{\beta\in\mathbb{N}^{m}}\,W_{\beta}(x_{\beta})=$
$\displaystyle\,\sum_{\beta_{2},..,\beta_{m}\in\mathbb{N}}\Big{(}\sum_{\beta_{1}\in\mathbb{N}}\,\exp(w((\phi_{\beta_{1}}\circ\phi_{{}_{\beta_{2}..\beta_{m}}})(x)))\Big{)}\exp\Big{(}\sum_{j=2}^{m}\,w(\phi_{{}_{\beta_{j}..\beta_{m}}}(x))\Big{)}$
$\displaystyle\leq$
$\displaystyle\,c_{k}\,\sum_{\beta_{2},..,\beta_{m}\in\mathbb{N}}\,\exp\Big{(}\sum_{j=2}^{m}\,w(\phi_{{}_{\beta_{j}..\beta_{m}}}(x))\Big{)}\leq
c_{k}^{m}$

by (H7). Hence we can apply [3, Lemma 3.21] and write for all $m\geq 1$ and
all $z\in D$

$\text{tr}^{b}((\mathcal{Q}_{w}(z))^{m})=\sum_{\beta\in\mathbb{N}^{m}}\,z^{|\beta|}\,\text{tr}^{b}(Q_{\beta}).$

Rearranging the terms in the sum using the first passage function $\tau(x)$
defined in (2.2) proves the first part of the statement.

Let us now consider the two-variable zeta function $Z_{w}(z,u)$ where it makes
sense. Let us first assume that $G^{\prime}(x)$ is positive where it is
defined. Since

$\sum_{G^{m}(x)=x}\,z^{\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}=$
$=\sum_{G^{m}(x)=x}\,\frac{z^{\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}}{1-((G^{m})^{\prime}(x))^{-1}}\,(1-((G^{m})^{\prime}(x))^{-1})=$
$=\text{tr}^{b}((\mathcal{Q}_{w}(z))^{m})-\sum_{G^{m}(x)=x}\,\frac{z^{\sum_{j=0}^{m-1}(1+\tau(G^{j}(x)))}\,\exp\Big{(}\sum_{j=0}^{m-1}\,w(G^{j}(x))\Big{)}\,((G^{m})^{\prime}(x))^{-1}}{1-((G^{m})^{\prime}(x))^{-1}}\,,$

and

$((G^{m})^{\prime}(x))^{-1}=\exp\Big{(}-\sum_{j=0}^{m-1}\,\log
G^{\prime}(G^{j}(x))\Big{)},$

we obtain by (4.2)

$Z_{w}(z,u)=\exp\Big{(}\sum_{m\geq
1}\,\frac{u^{m}}{m}\,\Big{(}\text{tr}^{b}((\mathcal{Q}_{w}(z))^{m})-\text{tr}^{b}((\mathcal{Q}_{w-\log
G^{\prime}}(z))^{m})\Big{)}\,\Big{)},$

and (4.5) follows by the definition of the flat determinant. In the case
$G^{\prime}(x)<0$, the proof follows in the same way. ∎

At this point, using (4.5) and (4.4), in order to obtain the analytic
properties of $\zeta_{{}_{T,v}}(z)$ we are reduced to study the function
$z\mapsto\text{det}^{b}(1-\mathcal{Q}_{w}(z))$.

The jump transformation $G$ is piecewise smooth on $[0,1]$ with full branches
and expanding, and the flat determinant for maps with these properties has
been studied since the 1990s. The classical results can be found in [23, 17]
and we refer to the bibliography in [3, Chapter 3] for more recent results and
an account of the situation. In our context these results imply that the
function $u\mapsto\text{det}^{b}(1-u\,\mathcal{Q}_{w}(1))$ admits a
holomorphic extension to the disc555Recall that $w\in C^{k}$ with $1\leq k\leq
r-1$. $\\{|u|\leq\rho^{-k}\,\exp(-P_{G}(w))\\}$, and the function vanishes at
some $u_{0}$ in this disc if and only if $1/u_{0}$ is an eigenvalue of
$\mathcal{Q}_{w}(1)$.

Hence, in order to study the function
$z\mapsto\text{det}^{b}(1-\mathcal{Q}_{w}(z))$, using the relations in (4.3)
we might look at the transfer operator of $G$ with potential $w+(\log
z)(1+\tau)$ and consider the set of $z$ for which
$\rho^{-k}\,\exp(-P_{G}(w+(\log z)(1+\tau))\geq 1$. A more straightforward
approach is to use the method of [19].

Let us recall Theorem 3.1 and the definitions of $\Lambda_{m}(z)$ in (2.8) and
$\Lambda(z)$ in (2.9) for $|z|\leq 1$.

######  Proposition 4.4.

Let $z\in D$ and let $\sigma(z)=(1/\rho)^{k-1}\Lambda(z)$ for $k\in\mathbb{N}$
and $k\leq r-1$. Then $\text{det}^{b}(1-u\,\mathcal{Q}_{w}(z))$ is holomorphic
on the disc $\\{|u|<(\sigma(z))^{-\frac{1}{2}}\\}$ on which

$\text{det}^{b}(1-u\,\mathcal{Q}_{w}(z))=\det(1-u\,R(z))\,\exp\Big{(}-p(u)-\sum_{n=1}^{\infty}\,a_{n}\,u^{n}\Big{)},$

where $R(z)$ is a finite range operator, $p_{0}(u)$ is a polynomial of finite
order, and $|a_{n}|=O((\sigma(z))^{\frac{n}{2}})$.

###### Proof.

For a fixed $z\in D$ for which Theorem 3.1 holds for $\mathcal{Q}_{w}(z)$ on
$C^{k}([0,1])$, following [19] we first consider the adjoint operator
$\mathcal{Q}_{w}^{*}(z)$ given by
$(\mathcal{Q}_{w}(z)f,g)_{L^{2}}=(f,\mathcal{Q}_{w}^{*}(z)g)_{L^{2}}$ which
reads

$(\mathcal{Q}_{w}^{*}(z)g)(y)=z^{1+\tau(y)}\,e^{w(y)}\,|G^{\prime}(y)|\,g(G(y)).$

Then we define the operator $\mathcal{M}_{w}(z)$ on $C^{k}([0,1]\times[0,1])$
as the extension of the operator
$\mathcal{Q}_{w}(z)\otimes\mathcal{Q}_{w}^{*}(z)$ defined on
$C^{k}([0,1])\otimes C^{k}([0,1])$. We obtain

$(\mathcal{M}_{w}(z)f)(x,y)=\sum_{n=1}^{\infty}\,z^{n+1+\tau(y)}\,e^{w(\phi_{n}(x))+w(y)}\,|G^{\prime}(y)|\,f(\phi_{n}(x),G(y)).$

For $p,q\in\mathbb{N}_{0}$ such that $p,q\leq k-1$, let $\mathcal{B}^{p,q}$
denote the completion of $C^{k}([0,1]\times[0,1])$ with respect to the norm

$\|f(x,y)\|_{p,q}:=\sup_{x\in(0,1)}\,\sup_{\alpha\leq
p}\,\sup_{\stackrel{{\scriptstyle\xi\in C^{q}_{0}(0,1)}}{{\|\xi\|_{q}\leq
1}}}\,\int_{0}^{1}\,\partial^{\alpha}_{x}f(x,y)\,\xi(y)\,dy.$

We obtain an example of the Banach spaces introduced and studied in [11] (see
also [10, Section 3.4]). We prove that for all $f\in C^{k}([0,1]\times[0,1])$
and all $m\geq 1$

(4.6) $\displaystyle\|(\mathcal{M}_{w}(z))^{m}f\|_{p,q}\leq
const\cdot\,\Lambda_{m}(z)\,\|f\|_{p,q}$ (4.7)
$\displaystyle\|(\mathcal{M}_{w}(z))^{m}f\|_{p,q}\leq
const\cdot\,\Lambda_{m}(z)\,\Big{(}\rho^{-\min\\{p,q\\}\,m}\,\|f\|_{p,q}+\|f\|_{p-1,q+1}\Big{)},$

where the constants do not depend on $m$ and $f$. From (4.6) and (4.7) it
follows that $\mathcal{M}_{w}(z)$ on $\mathcal{B}^{p,q}$ has spectral radius
$\Lambda(z)$ and its essential spectral radius is bounded by
$\rho^{-\min\\{p,q\\}}\,\Lambda(z)$ (see [11, Theorem 2.3]).

Let’s begin with the case $m=1$. For $p=0$ we have

$\begin{split}&\|\mathcal{M}_{w}(z)f\|_{0,q}\\\
&=\sup_{x\in(0,1)}\,\sup_{\stackrel{{\scriptstyle\xi\in
C^{q}_{0}(0,1)}}{{\|\xi\|_{q}\leq 1}}}\,\sum_{n\geq
1}\,z^{n}\,e^{w(\phi_{n}(x))}\,\int_{0}^{1}\,z^{1+\tau(y)}\,e^{w(y)}\,|G^{\prime}(y)|\,f(\phi_{n}(x),G(y))\,\xi(y)\,dy\end{split}$

and since

$\begin{split}&\int_{0}^{1}\,z^{1+\tau(y)}\,e^{w(y)}\,|G^{\prime}(y)|\,f(\phi_{n}(x),G(y))\,\xi(y)\,dy=\\\
&=\sum_{\ell\geq
1}\,z^{\ell}\,\int_{A_{\ell}}\,e^{w(y)}\,|G^{\prime}(y)|\,f(\phi_{n}(x),G(y))\,\xi(y)\,dy=\\\
&=\sum_{\ell\geq
1}\,z^{\ell}\,\int_{0}^{1}\,e^{w(\phi_{\ell}(s))}\,f(\phi_{n}(x),s)\,\xi(\phi_{\ell}(s))\,ds=\\\
&=\int_{0}^{1}\,f(\phi_{n}(x),s)\,(\mathcal{Q}_{w}(z)\xi)(s)\,ds,\end{split}$

and by (2.8)

$\sum_{n\geq 1}\,z^{n}\,e^{w(\phi_{n}(x))}=\Lambda_{1}(z),$

we use that $\|\mathcal{Q}_{w}(z)\xi\|_{q}\leq const(q)\,\|\xi\|_{q}$ to
obtain (4.6) for $m=1$ and $p=0$.

For $p=1$ we need to control the term involving
$\partial_{x}\mathcal{M}_{w}(z)f$. Since

$\displaystyle\partial_{x}(\mathcal{M}_{w}(z)f)(x,y)=\sum_{n\geq 1}$
$\displaystyle
z^{n+1+\tau(y)}\,e^{w(\phi_{n}(x))+w(y)}\,|G^{\prime}(y)|\,\phi^{\prime}_{n}(x)\,\cdot$
$\displaystyle\cdot\,\Big{(}w^{\prime}(\phi_{n}(x))\,f(\phi_{n}(x),G(y))+\partial_{x}f(\phi_{n}(x),G(y))\Big{)},$

we have

(4.8)
$\Big{|}\int_{0}^{1}\,\partial_{x}(\mathcal{M}_{w}(z)f)(x,y)\,\xi(y)\,dy\Big{|}\leq
const(w)\,\|M_{w}f\|_{0,q}+\rho^{-1}\,\|M_{w}(z)(\partial_{x}f)\|_{0,q}.$

Then we use (4.6) and the definition of $\|\cdot\|_{1,q}$ to write

$\|M_{w}(z)(\partial_{x}f)\|_{0,q}\leq
const\cdot\,\Lambda_{1}(z)\,\|f\|_{1,q},$

and we show that

(4.9) $\|M_{w}f\|_{0,q}\leq
const\cdot\Lambda_{1}(z)\,\Big{(}\rho^{-q}\,\|f\|_{1,q}+\rho^{q}\,\|f\|_{0,q+1}\Big{)},$

to obtain in (4.8)

$\Big{|}\int_{0}^{1}\,\partial_{x}(\mathcal{M}_{w}(z)f)(x,y)\,\xi(y)\,dy\Big{|}\leq
const\cdot\Lambda_{1}(z)\,\Big{(}\rho^{-1}\,\|f\|_{1,q}+\rho^{-q}\,\|f\|_{1,q}+\rho^{q}\,\|f\|_{0,q+1}\Big{)}$

which gives (4.7) for $m=1$ and $p=1$. It remains to prove (4.9). Consider a
fixed $\gamma:(0,1)\to[0,\infty)$ of class $C_{0}^{q+1}$ with
$\int_{0}^{1}\,\gamma(s)\,ds=1$. For $\varepsilon>0$ small let
$\gamma_{\varepsilon}(s):=\varepsilon^{-1}\gamma(s/\varepsilon)$, then we
denote $\xi_{\varepsilon}(y):=(\xi\star\gamma_{\varepsilon})(y)\in C^{q+1}$
for which we have666The claim is straightforward since
$\int\gamma_{\varepsilon}(x-y)\xi(y)dy=\int\xi(x-y)\gamma_{\varepsilon}(y)dy.$

(4.10) $\|\xi_{\varepsilon}\|_{q+1}\leq
const\,\varepsilon^{-1},\quad\|\xi-\xi_{\varepsilon}\|_{q-1}\leq\varepsilon\,\|\xi\|_{q},\quad\|\xi-\xi_{\varepsilon}\|_{q}\leq
const,$

where the constants do not depend on $\varepsilon$. We write

$\begin{split}&\int_{0}^{1}\,(\mathcal{M}_{w}(z)f)(x,y)\,\xi(y)\,dy=\\\
&=\int_{0}^{1}\,(\mathcal{M}_{w}(z)f)(x,y)\,\xi_{\varepsilon}(y)\,dy+\int_{0}^{1}\,(\mathcal{M}_{w}(z)f)(x,y)\,(\xi(y)-\xi_{\varepsilon}(y))\,dy\end{split}$

and as above

$\begin{split}&\int_{0}^{1}\,(\mathcal{M}_{w}(z)f)(x,y)\,(\xi-\xi_{\varepsilon})(y)\,dy=\\\
&=\sum_{n,\ell\geq
1}\,z^{n+\ell}\,e^{w(\phi_{n}(x))}\,\int_{0}^{1}\,e^{w(\phi_{\ell}(s))}\,f(\phi_{n}(x),s)\,(\xi-\xi_{\varepsilon})(\phi_{\ell}(s))\,ds.\end{split}$

Using (4.10), we have for all $\ell\geq 1$

$\begin{split}\|(\xi-\xi_{\varepsilon})(\phi_{\ell}(s))\|_{q}\leq&\,\|(\xi-\xi_{\varepsilon})^{(q)}(\phi_{\ell}(s))\cdot(\phi_{\ell}(s))^{q}\|_{\infty}+const\,\|(\xi-\xi_{\varepsilon})(\phi_{\ell}(s))\|_{q-1}\leq\\\\[5.69046pt]
\leq&\,const\,(\rho^{-q}+\varepsilon).\end{split}$

And finally

$\|M_{w}f\|_{0,q}\leq
const\,\Big{(}\varepsilon^{-1}\,\|M_{w}f\|_{0,q+1}+(\rho^{-q}+\varepsilon)\,\|M_{w}f\|_{0,q}\Big{)}.$

Choosing $\varepsilon=\rho^{-q}$, using (4.6) and the definition of
$\|\cdot\|_{1,q}$ we obtain (4.9).

The general case $m\geq 1$ is proved in the same way using
$(\mathcal{Q}_{w}(z))^{m}$. For $p>1$ we can argue as we did in the proof of
Theorem 3.1.

At this point the proof is finished by choosing $p=q=k-1$ and applying the
arguments of [19, Section 4]. ∎

By Propositions 4.1, 4.3, and 4.4, we obtain the proof of Theorem 2.5 on
meromorphic extension for the zeta function $\zeta_{{}_{T,v}}(z)$.

Proof of Theorem 2.5. Using Propositions 4.1 and 4.3, it is enough to apply
Proposition 4.4 to $\mathcal{Q}_{w}(z)$ and $\mathcal{Q}_{w-\log
G^{\prime}}(z)$ and notice that $\sigma(z)=\rho\cdot
r_{ess}(\mathcal{Q}_{w}(z)|_{C^{k}})$ and that $r_{ess}(\mathcal{Q}_{w-\log
G^{\prime}}(z)|_{C^{k}})\leq r_{ess}(\mathcal{Q}_{w}(z)|_{C^{k}})$. ∎

##  5\. The meromorphic extension outside the unit disc

Applying Theorem 2.5, the meromorphic extension of $\zeta_{{}_{T,v}}(z)$
strongly depends on the set of $z$ for which $\mathcal{Q}_{w}(z)$ is defined.
For the moment we can define $\mathcal{Q}_{w}(z)$ on $D=\\{|z|\leq 1\\}$ and
all the results are therefore restricted to this domain. We now show a
strategy to extend the definition of $\mathcal{Q}_{w}(z)$ to larger domains.

We recall the following result by LeRoy and Lindelöf as stated in [1]. Let us
set

$\displaystyle\Delta_{c}:=\left\\{z\in\mathbb{C}\,:\,|\arg
z|<\frac{c}{2}\right\\},\quad\text{for $c\in[0,2\pi)$;}$
$\displaystyle\Pi:=\left\\{z\in\mathbb{C}\,:\,\text{Re($z$)}\geq 0\right\\}.$

Let us denote by $H(\Pi)$ the set of holomorphic functions in some
neighborhood of $\Pi$. Then we say that $h\in H(\Pi)$ is _of exponential type
$e_{{}_{\Pi}}\in[0,+\infty)$ on $\Pi$_ if

$e_{{}_{\Pi}}=\limsup_{z\to\infty,\,z\in\Pi}\,|z|^{-1}\,\log^{+}|h(z)|.$

Then the following result holds for the analytic continuation of power series.

######  Theorem 5.1.

For a function $h\in H(\Pi)$, assume that the power series

$q(z)=\sum_{n=0}^{\infty}\,h(n)\,z^{n}$

converges in the unit disc $D$. If $h$ is of exponential type
$e_{{}_{\Pi}}<\pi$ on $\Pi$ then $q(z)$ admits an analytic continuation to the
sector $\mathbb{C}\setminus\Delta_{2e_{{}_{\Pi}}}$.

In the case $e_{{}_{\Pi}}=0$ the theorem yields a sufficient condition for the
analytic continuation of $q(z)$ to the cut complex plane
$\mathbb{C}\setminus(1,\infty)$.

Theorem 5.1 has been improved in [1] by stating a necessary and sufficient
condition on a function $h\in H(\Pi)$ for the analytic continuation of the
power series $q(z)$ to a sector $\mathbb{C}\setminus\Delta_{c}$. However, for
our aims it is enough to consider the sufficient condition considered by LeRoy
and Lindelöf. It is shown in [1] that the proof of Theorem 5.1 follows by
choosing for each compact set
$K\subset\mathbb{C}\setminus\Delta_{2e_{{}_{\Pi}}}$ an unbounded contour
$\Gamma$ on which the integral

$\int_{\Gamma}\,\frac{h(s)\,z^{s}}{e^{2\pi is}-1}\,ds$

converges uniformly to $q(z)$ for all $z\in K$.

Let $f\in C^{k}([0,1])$ and consider the series

$(\mathcal{Q}_{w}(z)f)(x)=\sum_{n=1}^{\infty}\,z^{n}\,e^{w(\phi_{n}(x))}\,f(\phi_{n}(x))$

which, by assumption (H7), converges for $z\in D$ uniformly in $x$. Fixed
$x\in[0,1]$, we study the existence of a function $h\in H(\Pi)$ such that
$h(n)=e^{w(\phi_{n}(x))}\,f(\phi_{n}(x))$ for all $n\geq 1$.

######  Lemma 5.2.

Let $a_{n}=e^{w(\phi_{n}(x))}\,f(\phi_{n}(x))$ and assume that $v(0)<0$. Then
for all $\varepsilon>0$, the function

$h(s):=\frac{\sin(\pi
s)}{\pi}\,\sum_{i=1}^{\infty}\,(-1)^{i}\,\frac{a_{i}}{s-i}\,e^{(1-\varepsilon)\,(s-i)\,v(0)}$

converges in $\mathbb{C}$ and satisfies $h(n)=a_{n}$ for all $n\geq 1$.
Moreover it is of exponential type
$e_{{}_{\Pi}}=\max\\{0,\pi+(1-\varepsilon)v(0)\\}<\pi$.

###### Proof.

Given the sequence $\\{a_{n}\\}$, an application of the Pringsheim
interpolation formula gives the function $h$ we are looking for. In details,
the function $g(s)=\sin(\pi s)$ has zeroes of first order at all
$n\in\mathbb{Z}$, therefore the formal power series

$\tilde{h}(s):=\sum_{i=1}^{\infty}\,\frac{g(s)}{g^{\prime}(i)\,(s-i)}$

satisfies $\tilde{h}(n)=1$ for all $n\in\mathbb{N}$. Since
$g^{\prime}(i)=(-1)^{i}\,\pi$ and $a_{n}\sim e^{n\,v(0)}\,f(0)$ as showed in
Remark 2.4, the series defining $h(s)$ is convergent in $\mathbb{C}$ and
satisfies $h(n)=a_{n}$ for all $n\geq 1$.

In addition, for all $s\in\Pi$ we have

$|h(s)|=|\sin(\pi
s)|\,e^{(1-\varepsilon)\,v(0)\,\Re(s)}\,\sum_{i=1}^{\infty}\,\frac{|a_{i}|}{\pi\,|s-i|}\,e^{-(1-\varepsilon)\,v(0)\,i}.$

Since the series converges if $\varepsilon>0$ and $|\sin(\pi s)|\leq
2\,e^{\pi\,|s|}$, we obtain that
$e_{{}_{\Pi}}=\max\\{0,\pi+(1-\varepsilon)v(0)\\}$. ∎

Proof of Theorem 2.6. Put together Theorem 2.5 and Theorem 5.1 for all
$\varepsilon>0$. ∎

### Acknowledgements

This work was started when RC was working at University of Pisa, Italy.
CB acknowledges the MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Pisa, CUP I57G22000700001.
The authors are partially supported by the research project PRIN 2022NTKXCX
“Stochastic properties of dynamical systems” funded by the Ministry of
University and Scientific Research of Italy.
This research is part of the authors’ activity within the the UMI Group
“DinAmicI” www.dinamici.org and the INdAM (Istituto Nazionale di Alta
Matematica) group GNFM.
It is a pleasure to thank Paolo Giulietti for many useful discussions at the
beginning of the project.

### Declarations

The authors have no relevant financial or non-financial interests to disclose.
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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# Finding active galactic nuclei through Fink

Etienne Russeil
Université Clermont Auvergne
CNRS/IN2P3, LPC
F-63000 Clermont-Ferrand, France
<EMAIL_ADDRESS>Emille E. O. Ishida
Université Clermont Auvergne
CNRS/IN2P3, LPC
F-63000 Clermont-Ferrand, France
<EMAIL_ADDRESS>Roman Le Montagner
Université Paris-Saclay
CNRS/IN2P3, IJCLab
91405 Orsay, France
<EMAIL_ADDRESS>Julien Peloton
Université Paris-Saclay
CNRS/IN2P3, IJCLab
91405 Orsay, France
<EMAIL_ADDRESS>Anais Möller
Centre for Astrophysics and Supercomputing
Swinburne University of Technology
Mail Number H29, PO Box 218, 31122, Hawthorn, VIC, Australia
<EMAIL_ADDRESS>
###### Abstract

We present the Active Galactic Nuclei (AGN) classifier as currently
implemented within the Fink broker. Features were built upon summary
statistics of available photometric points, as well as color estimation
enabled by symbolic regression. The learning stage includes an active learning
loop, used to build an optimized training sample from labels reported in
astronomical catalogs. Using this method to classify real alerts from the
Zwicky Transient Facility (ZTF), we achieved 98.0% accuracy, 93.8% precision
and 88.5% recall. We also describe the modifications necessary to enable
processing data from the upcoming Vera C. Rubin Observatory Large Survey of
Space and Time (LSST), and apply them to the training sample of the Extended
LSST Astronomical Time-series Classification Challenge (ELAsTiCC). Results
show that our designed feature space enables high performances of traditional
machine learning algorithms in this binary classification task.

##  1 Introduction

Active Galactic Nuclei (AGN) are bright, variable astrophysical sources
associated with the inflow of circumstellar matter into central galactic black
holes [1]. From the observer perspective, they comprise a large set of light
curve behaviors, including instances where observational patterns evolve with
time [2, 3]. Beyond being paramount for the study of accretion and
photoionization physics [4], they can trace star formation regions [5] and
have the potential to enrich cosmological studies [6]. Thus, reliable and
cheap identification of large populations of AGNs are crucial to enable a
better understanding of their mechanisms and their impact in the galactic
environment.

The Vera C. Rubin Observatory Legacy Survey of Space and
Time111https://www.lsst.org/ (LSST), expected to start operations in 2024,
will produce a large volume of photometric data, including a diverse AGN
population. Each time a brightness variability beyond 5-$\sigma$ from the
background is detected, an alert will be generated. We expect 10 million of
such alerts per night, which will be streamed to chosen community brokers.
Fink222https://fink-broker.org/, is one of the official LSST brokers, whose
task is to receive this data, extract meaningful information from it and re-
distribute it to scientific communities. The broker contains a series of
machine learning based modules, which enables fast processing of the data
stream. In preparation for the start of LSST, Fink is currently operating on
data from the Zwicky Transient Facility (ZTF) ([7]), which produces around 300
000 alerts per night.

This work presents details about the AGN classifier within the Fink broker. It
includes a tailored feature extraction procedure followed by the construction
of an optimized training sample using uncertainty sampling active learning and
a random forest classifier [8].

##  2 Data

Figure 1: Example of Fink-data light curve. The object belongs to the BLLac
class, with a ZTF objectID = "ZTF18abzwaiw" and an alert ID =
1424109966315015005. Bold symbols show observed points. The continuous lines
show flux estimation using Equation 1 and open symbols denote flux value used
for color estimation. ZTF filters $g$ and $r$ are shown in dark blue and
orange, respectively.

Two databases were used in this work. Fink-data corresponds to all ZTF data
collected333https://zenodo.org/record/5645609 by Fink from Nov/2019 to
Mar/2021, for which an associated label was found in
SIMBAD444https://simbad.unistra.fr/simbad/ or in Transient Name Server
(TNS555https://www.wis-tns.org/). For each filter, the maximum flux was
normalized to 1 and the time of first observation was shifted to zero. We then
randomly sampled 100k AGN and 1 million non-AGNs alerts, and only considered
objects with a minimum of 4 observed points in each filter. The resulting
database contained 607772 alerts. Following [9], we regrouped objects into 2
larger categories, "AGN" and "non-AGN". The AGN category encloses [AGN, LINER,
Blazar, BLLac, QSO]666SIMBAD tags were used before the June 2022 taxonomy
modification.. Among Fink-data, 536621 alerts belong to the non-AGNs, while
71151 are AGNs. The resulting set was then equally divided into two samples:
training and testing. Since each object can produce multiple alerts, we
ensured that alerts from a given object were only present in one of the two
samples. The Fink-data represents the state of the art of what can be done
with real data.

Aiming to estimate the performance of our classifier in a LSST-like data
environment, we put together a second database, hereafter, ELAsTiCC_data using
data from the Extended LSST Astronomical Time-series Classification
Challenge777https://portal.nersc.gov/cfs/lsst/DESC_TD_PUBLIC/ELASTICC/
(ELAsTiCC). It represents a more complex version of the Photometric LSST
Astronomical Time-series Classification
Challenge888https://www.kaggle.com/c/PLAsTiCC-2018 (PLAsTiCC), held in 2018.
Focusing in anticipating an LSST-like data and software environment, ELAsTiCC
aims to test not only the classification power of broker systems, but also the
resilience of their infrastructure and link with LSST facilities. It uses 32
different transient template models to generate simulated light curves using
the Supernova Analysis (SNANA) [10] code. These are cut into alerts (each new
photometric observation above detection threshold generate an alert) and daily
streamed to broker teams. The brokers are expected to process the stream and
send back their probability scores. Results will compare the performance of
all teams who provided scores after an initial period of 3 months. A training
sample composed of full light curves, and with a different cadence from the
one used to create the test sample, was made available so broker teams could
prepare/train their models for the challenge. This training sample is our
starting point for the construction of the ELAsTiCC_data database. From it, we
selected 50k AGNs and 50k non-AGNs objects with at least 4 points in 2
consecutive filters for training and another 50k AGN/50k non-AGN for testing.

##  3 Methodology

Figure 2: Broad categories of SIMBAD and TNS labels within the sample of
identified AGN candidates. It follows SIMBAD taxonomy and the + marks classes
which may include AGN sub-types.

For each filter, we used the photometric points to compute the following
features: maximum flux before normalization; standard deviation of the flux;
number of points and mean signal to noise ratio. Moreover, we also added
metadata to the features table, meaning: right ascension; declination;
standard deviation and maximum absolute value of each color. Additionally,
only for ELASTiCC_data, we added metadata information regarding: host galaxy
redshift, host galaxy redshift error and host galaxy distance to the object.

The color calculation for this type of data is specially delicate. Since the
sampling of the points is irregular, no pair of points exist at the exact same
time in different filters, therefore we must interpolate each light curve so a
proper colour estimation can be performed. In searching for a smooth function
which would correctly capture the overall behavior of AGNs, we opted to use:

$f(t)=\frac{1}{1+e^{-At-e^{Bt}+C}}+D$ (1)

This form was obtained by applying a traditional symbolic regression algorithm
to noiseless simulated transient light curves via
gplearn999https://github.com/trevorstephens/gplearn. Once the code had
converged and an analytical form propose, we manually substitute the float
values by parameters, resulting in the reported functional form (Equation 1).
Therefore parameters A, B and C do not have any direct physical
interpretation. However the parameter D was added latter on and represents the
flux baseline. The minimization is performed using the least squared function
from the python module iMinuit ([11]). Figure 1 shows an example of a real
alert from Fink-data, alongside the estimated fluxes. Color was computed by
subtracting flux values between two consecutive filters, [g-r] for Fink-data
and [u-g, g-r, r-i, i-z, z-Y] for ELAsTiCC_data, at the condition that both
concerned filters contains at least 4 observed points (considering forced
photometry but not upper limits).

Once the feature matrix was constructed, classification proceeded using the
random forest algorithm from the python module scikit-learn ([12]). This
method uses an ensemble of decision trees, each built from a different sub-
sample of the data. When new data is inputted, the answer from each tree is
counted as a vote, and the probability outputted by the forest comes from the
proportion of each vote. In what follows, all results were obtained using 100
trees. Conducted attempts using a gradient boosted tree algorithm, as well as
attempts to increase the number of trees, lead to no further improvements in
the classification score.

For Fink-data, an optimized training sample was built by using an active
learning strategy based on uncertainty sampling [13]. We started the learning
cycle with 5 AGN and 5 non-AGNs randomly selected from the sample of available
labels, and trained a random forest using these 10 objects while the rest of
the data was used for testing. The alert in the testing sample for which the
random forest is the most uncertain about (probability closest to 50% percent)
is moved from testing sample to training sample in each cycle. We performed
2000 loops, resulting in a final training sample of 2010 objects. The entire
process was performed 10 times in order to briefly access the impact of
different initial conditions. In the case of ELAsTiCC_data, representativeness
was ensured by construction, thus no active learning loop was required.

##  4 Results and conclusion

Figure 3: Confusion matrix results from the ELAsTiCC_data sample.

The active learning process will, by construction, build a training data set
approximately made of 50% AGNs and 50% non-AGNs. In the case of unbalanced
data set such as Fink-data, this property is crucial. A balanced and
informative training set ensures that the random forest is learning to
separate types of alert, and not just learning the statistical distribution of
this particular dataset. Furthermore a balanced training dataset guarantees
the classifier optimal threshold to be around 0.5.

When applied to all the non queried objects, the classifiers built with active
learning achieved on average 98.6 % accuracy, 95.9 % precision and 91.9 %
recall on finding AGNs, with a standard deviation of 0.1%, 0.7% and 0.8%
respectively.

We conclude that the uncertainty sampling was efficient at building a training
sample. It also ensures that the classifier will perform well when used on
another dataset with different statistical properties. Among the 10 runs, we
kept the classifier with highest product recall $\times$ score, which in our
case also satisfies the F1 criteria. Applying this classifier to the testing
sample (not involved in training), we achieve a score of 98.0 % accuracy with
93.8 % precision and 88.5 % recall on AGNs. Finally the classifier has been
tested directly on the Fink alert stream. All alerts between Jan/2019 to
Jan/2020 were processed and we evaluate the results for every labelled alerts
(excluding objects used in the training sample). Figure 2 shows the
distribution of labels within the photometrically predicted AGNs. Results from
the ELAsTiCC_data sample are shown in Figure 3. Feature importance analysis on
results obtained from the Fink-data shows that all features play a similar
role in enabling classification. For the ELAsTiCC_data, the number of points
acts like a first layer of classification. This was expected, since this data
set contain a high percentage of transient classes. These are objects whose
brightness are only significant during a limited time window, which makes them
more easily distinguishable from AGNs, that are persistent sources.

Both models described in this work are now integrated to the Fink broker,
lively processing alerts from ZTF and ELAsTiCC. The filtered ZTF stream will
be directed to the AGN community interested in spectroscopic follow-up of
individual events. We foresee the development of similar models focused on
sub-types of AGNs chosen by Fink users. Finally, we intend to use the AGN
scores to further filter out the data given to the supernova classifiers in
Fink, thus allowing for an even higher precision in the estimates delivered by
the broker.

All the work presented in this paper is publicly available on GitHub 101010
https://github.com/astrolabsoftware/fink-science/tree/master/fink_science/agn.

## Acknowledgements

We thank David O. Jones for making simulations available for this project.
This work was developed within the Fink community and made use of the Fink
community broker resources. Fink is supported by LSST-France and CNRS/IN2P3.

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11institutetext: Athens University of Economics and Business, Athens, Greece
11email: {ckarats<EMAIL_ADDRESS>
# Attention-based method for categorizing different types of online harassment
language

Christos Karatsalos  Yannis Panagiotakis

###### Abstract

In the era of social media and networking platforms, Twitter has been doomed
for abuse and harassment toward users specifically women. Monitoring the
contents including sexism and sexual harassment in traditional media is easier
than monitoring on the online social media platforms like Twitter, because of
the large amount of user generated content in these media. So, the research
about the automated detection of content containing sexual or racist
harassment is an important issue and could be the basis for removing that
content or flagging it for human evaluation. Previous studies have been
focused on collecting data about sexism and racism in very broad terms.
However, there is no much study focusing on different types of online
harassment attracting natural language processing techniques. In this work, we
present an multi-attention based approach for the detection of different types
of harassment in tweets. Our approach is based on the Recurrent Neural
Networks and particularly we are using a deep, classification specific multi-
attention mechanism. Moreover, we tackle the problem of imbalanced data, using
a back-translation method. Finally, we present a comparison between different
approaches based on the Recurrent Neural Networks.

###### Keywords:

Text classification Twitter Hate Speech Deep Learning Attention Mechanism.

##  1 Introduction

In the era of social media and networking platforms, Twitter has been doomed
for abuse and harassment toward users specifically women. In fact, online
harassment becomes very common in Twitter and there have been a lot of critics
that Twitter has become the platform for many racists, misogynists and hate
groups which can express themselves openly. Online harassment is usually in
the form of verbal or graphical formats and is considered harassment, because
it is neither invited nor has the consent of the receipt. Monitoring the
contents including sexism and sexual harassment in traditional media is easier
than monitoring on the online social media platforms like Twitter. The main
reason is because of the large amount of user generated content in these
media. So, the research about the automated detection of content containing
sexual harassment is an important issue and could be the basis for removing
that content or flagging it for human evaluation. The basic goal of this
automatic classification is that it will significantly improve the process of
detecting these types of hate speech on social media by reducing the time and
effort required by human beings.

Previous studies have been focused on collecting data about sexism and racism
in very broad terms or have proposed two categories of sexism as benevolent or
hostile sexism [1], which undermines other types of online harassment.
However, there is no much study focusing on different types online harassment
alone attracting natural language processing techniques.

In this paper we present our work, which is a part of the SociaL Media And
Harassment Competition of the ECML PKDD 2019 Conference. The topic of the
competition is the classification of different types of harassment and it is
divided in two tasks. The first one is the classification of the tweets in
harassment and non-harassment categories, while the second one is the
classification in specific harassment categories like indirect harassment,
physical and sexual harassment as well. We are using the dataset of the
competition, which includes text from tweets having the aforementioned
categories. Our approach is based on the Recurrent Neural Networks and
particularly we are using a deep, classification specific attention mechanism.
Moreover, we present a comparison between different variations of this
attention-based approach like multi-attention and single attention models. The
next Section includes a short description of the related work, while the third
Section includes a description of the dataset. After that, we describe our
methodology. Finally, we describe the experiments and we present the results
and our conclusion.

##  2 Related Work

Waseem et al. [2] were the first who collected hateful tweets and categorized
them into being sexist, racist or neither. However, they did not provide
specific definitions for each category. Jha and Mamidi [1] focused on just
sexist tweets and proposed two categories of hostile and benevolent sexism.
However, these categories were general as they ignored other types of sexism
happening in social media. Sharifirad S. and Matwin S. [3] proposed
complimentary categories of sexist language inspired from social science work.
They categorized the sexist tweets into the categories of indirect harassment,
information threat, sexual harassment and physical harassment. In the next
year the same authors proposed [4] a more comprehensive categorization of
online harassment in social media e.g. twitter into the following categories,
indirect harassment, information threat, sexual harassment, physical
harassment and not sexist.

For the detection of hate speech in social media like twitter, many approaches
have been proposed. Jha and Mamidi [1] tested support vector machine, bi-
directional RNN encoder-decoder and FastText on hostile and benevolent sexist
tweets. They also used SentiWordNet and subjectivity lexicon on the extracted
phrases to show the polarity of the tweets. Sharifirad et al. [5] trained,
tested and evaluated different classification methods on the SemEval2018
dataset and chose the classifier with the highest accuracy for testing on each
category of sexist tweets to know the mental state and the affectual state of
the user who tweets in each category. To overcome the limitations of small
data sets on sexist speech detection, Sharifirad S. et al. [6] have applied
text augmentation and text generation with certain success. They have
generated new tweets by replacing words in order to increase the size of our
training set. Moreover, in the presented text augmentation approach, the
number of tweets in each class remains the same, but their words are augmented
with words extracted from their ConceptNet relations and their description
extracted from Wikidata. Zhang et al. [7] combined convolutional and gated
recurrent networks to detect hate speech in tweets. Others have proposed
different methods, which are not based on deep learning. Burnap and Williams
[8] used Support Vector Machines, Random Forests and a meta-classifier to
distinguish between hateful and non-hateful messages. A survey of recent
research in the field is presented in [9]. For the problem of the hate speech
detection a few approaches have been proposed that are based on the Attention
mechanism. Pavlopoulos et al. [10] have proposed a novel, classification-
specific attention mechanism that improves the performance of the RNN further
for the detection of abusive content in the web. Xie et al. [11] for emotion
intensity prediction, which is a similar problem to ours, have proposed a
novel attention mechanism for CNN model that associates attention-based
weights for every convolution window. Park and Fung [14] transformed the
classification into a 2-step problem, where abusive text ﬁrst is distinguished
from the non-abusive, and then the class of abuse (Sexism or Racism) is
determined. However, while the first part of the two step classification
performs quite well, it falls short in detecting the particular class the
abusive text belongs to. Pitsilis et al. [15] have proposed a detection scheme
that is an ensemble of RNN classiﬁers, which incorporates various features
associated with user related information, such as the users’ tendency towards
racism or sexism

Table 1: Class distribution of the dataset. Dataset | Tweets | Harassment | Harassment(%) | Indirect (%) | Sexual (%) | Physical (%)
---|---|---|---|---|---|---
train | 6374 | 2713 | 42.56 | 0.86 | 40.50 | 1.19
validation | 2125 | 632 | 29.74 | 3.34 | 24.76 | 1.69
test | 2123 | 611 | 28.78 | 9.28 | 14.69 | 4.71

##  3 Dataset description

The dataset from Twitter that we are using in our work, consists of a train
set, a validation set and a test set. It was published for the ”First workshop
on categorizing different types of online harassment languages in social
media”. The whole dataset is divided into two categories, which are harassment
and non-harassment tweets. Moreover, considering the type of the harassment,
the tweets are divided into three sub-categories which are indirect
harassment, sexual and physical harassment. We can see in Table 1 the class
distribution of our dataset. One important issue here is that the categories
of indirect and physical harassment seem to be more rare in the train set than
in the validation and test sets. To tackle this issue, as we describe in the
next section, we are performing data augmentation techniques. However, the
dataset is imbalanced and this has a significant impact in our results.

##  4 Proposed methodology

Figure 1: Projection Layer

###  4.1 Data augmentation

As described before one crucial issue that we are trying to tackle in this
work is that the given dataset is imbalanced. Particularly, there are only a
few instances from indirect and physical harassment categories respectively in
the train set, while there are much more in the validation and test sets for
these categories. To tackle this issue we applying a back-translation method
[16], where we translate indirect and physical harassment tweets of the train
set from english to german, french and greek. After that, we translate them
back to english in order to achieve data augmentation. These ”noisy” data that
have been translated back, increase the number of indirect and physical
harassment tweets and boost significantly the performance of our models.

Another way to enrich our models is the use of pre-trained word embeddings
from 2B Twitter data [17] having 27B tokens, for the initialization of the
embedding layer.

###  4.2 Text processing

Before training our models we are processing the given tweets using a tweet
pre-processor111https://pypi.org/project/tweet-preprocessor/. The scope here
is the cleaning and tokenization of the dataset.

###  4.3 RNN Model and Attention Mechanism

Figure 2: Attention mechanism, MLP with l Layers

We are presenting an attention-based approach for the problem of the
harassment detection in tweets. In this section, we describe the basic
approach of our work. We are using RNN models because of their ability to deal
with sequence information. The RNN model is a chain of GRU cells [18] that
transforms the tokens $w_{1},w_{2},...,w_{k}$ of each tweet to the hidden
states $h_{1},h_{2},...,h_{k}$, followed by an LR Layer that uses $h_{k}$ to
classify the tweet as harassment or non-harassment (similarly for the other
categories). Given the vocabulary V and a matrix E $\in$ $R^{d\times|V|}$
containing d-dimensional word embeddings, an initial $h_{0}$ and a tweet
$w=<w_{1},..,w_{k}>$, the RNN computes $h_{1},h_{2},...,h_{k}$, with $h_{t}\in
R^{m}$, as follows:

$h^{{}^{\prime}}_{t}=\tanh(W_{h}x_{t}+U_{h}(r_{t}\odot h_{t-1})+b_{h})$
$h_{t}=(1-z_{t})\odot h_{t-1}+z_{t}\odot h^{{}^{\prime}}_{t}$
$z_{t}=\sigma(W_{z}x_{t}+U_{z}h_{t-1}+b_{z})$
$r_{t}=\sigma(W_{r}x_{t}+U_{r}h_{t-1}+b_{r})$

where $h^{{}^{\prime}}_{t}\in R^{m}$ is the proposed hidden state at position
t, obtained using the word embedding $x_{t}$ of token $w_{t}$ and the previous
hidden state $h_{t-1}$, $\odot$ represents the element-wise multiplication,
$r_{t}\in R^{m}$ is the reset gate, $z_{t}\in R^{m}$ is the update gate,
$\sigma$ is the sigmoid function. Also $W_{h},W_{z},W_{r}\in R^{m\times d}$
and $U_{h},U_{z},U_{r}\in R^{m\times m}$, $b_{h},b_{z},b_{r}\in R^{m}$. After
the computation of state $h_{k}$ the LR Layer estimates the probability that
tweet w should be considered as harassment, with $W_{p}\in R^{1\times
m},b_{p}\in R$:

$P_{RNN}(harassment|w)=\sigma(W_{p}h_{k}+b_{p}).$

We would like to add an attention mechanism similar to the one presented in
[10], so that the LR Layer will consider the weighted sum $h_{sum}$ of all the
hidden states instead of $h_{k}$:

$h_{sum}=\sum_{t=1}^{k}\alpha_{t}h_{t}$

$P_{attentionRNN}=\sigma(W_{p}h_{sum}+b_{p})$

Alternatively, we could pass $h_{sum}$ through an MLP with k layers and then
the LR layer will estimate the corresponding probability. More formally,

$P_{attentionRNN}=\sigma(W_{p}h_{*}+b_{p})$

where $h_{*}$ is the state that comes out from the MLP. The weights
$\alpha_{t}$ are produced by an attention mechanism presented in [10] (see
Fig. 2), which is an MLP with l layers. This attention mechanism differs from
most previous ones [19, 20], because it is used in a classification setting,
where there is no previously generated output sub-sequence to drive the
attention. It assigns larger weights $\alpha_{t}$ to hidden states $h_{t}$
corresponding to positions, where there is more evidence that the tweet should
be harassment (or any other specific type of harassment) or not. In our work
we are using four attention mechanisms instead of one that is presented in
[10]. Particularly, we are using one attention mechanism per category. Another
element that differentiates our approach from Pavlopoulos et al. [10] is that
we are using a projection layer for the word embeddings (see Fig. 1). In the
next subsection we describe the Model Architecture of our approach.

###  4.4 Model Architecture

The Embedding Layer is initialized using pre-trained word embeddings of
dimension 200 from Twitter data that have been described in a previous sub-
section. After the Embedding Layer, we are applying a Spatial Dropout Layer,
which drops a certain percentage of dimensions from each word vector in the
training sample. The role of Dropout is to improve generalization performance
by preventing activations from becoming strongly correlated [13]. Spatial
Dropout, which has been proposed in [12], is an alternative way to use dropout
with convolutional neural networks as it is able to dropout entire feature
maps from the convolutional layer which are then not used during pooling.
After that, the word embeddings are passing through a one-layer MLP, which has
tanh as activation function and 128 hidden units, in order to project them in
the vector space of our problem considering that they have been pre-trained
using text that has a different subject. In the next step the embeddings are
fed in a unidirectional GRU having 1 Stacked Layer and size 128. We prefer GRU
than LSTM, because it is more efficient computationally. Also the basic
advantage of LSTM which is the ability to keep in memory large text documents,
does not hold here, because tweets supposed to be not too large text
documents. The output states of the GRU are passing through four self-
attentions like the one described above [10], because we are using one
attention per category (see Fig. 2). Finally, a one-layer MLP having 128 nodes
and ReLU as activation function computes the final score for each category. At
this final stage we have avoided using a softmax function to decide the
harassment type considering that the tweet is a harassment, otherwise we had
to train our models taking into account only the harassment tweets and this
might have been a problem as the dataset is not large enough.

##  5 Experiments

###  5.1 Training Models

In this subsection we are giving the details of the training process of our
models. Moreover, we are describing the different models that we compare in
our experiments.

Batch size which pertains to the amount of training samples to consider at a
time for updating our network weights, is set to 32, because our dataset is
not large and small batches might help to generalize better. Also, we set
other hyperparameters as: epochs = 20, patience = 10. As early stopping
criterion we choose the average AUC, because our dataset is imbalanced.

The training process is based on the optimization of the loss function
mentioned below and it is carried out with the Adam optimizer [21], which is
known for yielding quicker convergence. We set the learning rate equal to
0.001:

$L=\frac{1}{2}BCE(harassment)+\frac{1}{2}(\frac{1}{5}BCE(sexualH)+\frac{2}{5}BCE(indirectH)+\frac{2}{5}BCE(physicalH))$

where BCE is the binary cross-entropy loss function,

$BCE=-\frac{1}{n}\sum_{i=1}^{n}[y_{i}log(y^{{}^{\prime}}_{i})+(1-y_{i})log(1-y^{{}^{\prime}}_{i}))]$

$i$ denotes the $i$th training sample, $y$ is the binary representation of
true harassment label, and $y^{{}^{\prime}}$ is the predicted probability. In
the loss function we have applied equal weight to both tasks. However, in the
second task (type of harassment classification) we have applied higher weight
in the categories that it is harder to predict due to the problem of the class
imbalance between the training, validation and test sets respectively.

###  5.2 Evaluation and Results

Table 2: The results considering F1 Score.

Model | sexual_f1 | indirect_f1 | physical_f1 | harassment_f1 | f1_macro
---|---|---|---|---|---
attentionRNN | 0.674975 | 0.296320 | 0.087764 | 0.709539 | 0.442150
MultiAttentionRNN | 0.693460 | 0.325338 | 0.145369 | 0.700354 | 0.466130
MultiProjectedAttentionRNN | 0.714094 | 0.355600 | 0.126848 | 0.686694 | 0.470809
ProjectedAttentionRNN | 0.692316 | 0.315336 | 0.019372 | 0.694082 | 0.430276
AvgRNN | 0.637822 | 0.175182 | 0.125596 | 0.688122 | 0.40668
LastStateRNN | 0.699117 | 0.258402 | 0.117258 | 0.710071 | 0.446212
ProjectedAvgRNN | 0.655676 | 0.270162 | 0.155946 | 0.675745 | 0.439382
ProjectedLastStateRNN | 0.696184 | 0.334655 | 0.072691 | 0.707994 | 0.452881

Each model produces four scores and each score is the probability that a tweet
includes harassment language, indirect, physical and sexual harassment
language respectively. For any tweet, we first check the score of the
harassment language and if it is less than a specified threshold, then the
harassment label is zero, so the other three labels are zero as well. If it is
greater than or equal to that threshold, then the harassment label is one and
the type of harassment is the one among these three having that has the
greatest score (highest probability). We set this threshold equal to 0.33.

We compare eight different models in our experiments. Four of them have a
Projected Layer (see Fig. 1), while the others do not have, and this is the
only difference between these two groups of our models. So, we actually
include four models in our experiments (having a projected layer or not).
Firstly, LastStateRNN is the classic RNN model, where the last state passes
through an MLP and then the LR Layer estimates the corresponding probability.
In contrast, in the AvgRNN model we consider the average vector of all states
that come out of the cells. The AttentionRNN model is the one that it has been
presented in [10]. Moreover, we introduce the MultiAttentionRNN model for the
harassment language detection, which instead of one attention, it includes
four attentions, one for each category.

We have evaluated our models considering the F1 Score, which is the harmonic
mean of precision and recall. We have run ten times the experiment for each
model and considered the average F1 Score. The results are mentioned in Table
2. Considering F1 Macro the models that include the multi-attention mechanism
outperform the others and particularly the one with the Projected Layer has
the highest performance. In three out of four pairs of models, the ones with
the Projected Layer achieved better performance, so in most cases the addition
of the Projected Layer had a significant enhancement.

##  6 Conclusion - Future work

We present an attention-based approach for the detection of harassment
language in tweets and the detection of different types of harassment as well.
Our approach is based on the Recurrent Neural Networks and particularly we are
using a deep, classification specific attention mechanism. Moreover, we
present a comparison between different variations of this attention-based
approach and a few baseline methods. According to the results of our
experiments and considering the F1 Score, the multi-attention method having a
projected layer, achieved the highest performance. Also, we tackled the
problem of the imbalance between the training, validation and test sets
performing the technique of back-translation.

In the future, we would like to perform more experiments with this dataset
applying different models using BERT [22]. Also, we would like to apply the
models presented in this work, in other datasets about hate speech in social
media.

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