# Seizure Detection using Least EEG Channels by
Deep Convolutional Neural Network

###### Abstract

This work aims to develop an end-to-end solution for seizure onset detection.
We design the SeizNet, a Convolutional Neural Network for seizure detection.
To compare SeizNet with traditional machine learning approach, a baseline
classifier is implemented using spectrum band power features with Support
Vector Machines (BPsvm). We explore the possibility to use the least number of
channels for accurate seizure detection by evaluating SeizNet and BPsvm
approaches using all channels and two channels settings respectively. EEG Data
is acquired from 29 pediatric patients admitted to KK Woman’s and Children’s
Hospital who were diagnosed as typical absence seizures. We conduct leave-one-
out cross validation for all subjects. Using full channel data, BPsvm yields a
sensitivity of 86.6% and 0.84 false alarm (per hour) while SeizNet yields
overall sensitivity of 95.8 % with 0.17 false alarm. More interestingly, two
channels seizNet outperforms full channel BPsvm with a sensitivity of 93.3%
and 0.58 false alarm. We further investigate interpretability of SeizNet by
decoding the filters learned along convolutional layers. Seizure-like
characteristics can be clearly observed in the filters from third and forth
convolutional layers.

††footnotetext: This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

##  1 INTRODUCTION

Monitoring brain activity through EEG is critical for epilepsy diagnosis. To
capture seizure events that may occur sparsely, neurologists have to visually
scan vast amount of EEG data. The process is extremely time consuming and may
be subjective due to inter observer variance. Computer aided seizure detection
approach would serve as valuable clinical tool for the scrutiny of EEG data in
an objective and much more efficient manner.

Traditional machine learning approaches for seizure detection usually
composite three stages: data pre-processing to eliminate artifacts, followed
by feature extraction and decision-making. A number of features have been
identified to describe the behavior of seizures, including those based on
time-domain, frequency-domain, time-frequency analysis, wavelet features and
chaotic features such as entropy etc. A pioneer work was presented in [1],
which created subject-specific seizure onset detection model using hand-
crafted features extracted from the raw EEG data followed by classification.
The subject-specific model reaches a sensitivity of 96% and false alarm rate
of 0.08 per hour on CHB-MIT dataset by using SVM over a combination of
spectral, spatial and temporal features.

The extracted features are believed to contain discriminative information for
systems to differentiate seizure from non-seizure states. Feature space are
highly compact as compare to raw EEG data space which is critical when
computing power is limited. Furthermore, the features may bring
interpretability for a machine learning system. However, there are limitations
for such seizure detection methods. Firstly, extracting features from raw data
may induce information loss. Secondly, the standard power band analysis which
splitting the spectrum bands (delta 0-4Hz, theta 4-8Hz etc.) could not take
into consideration of individual variance of spectrum distributions. Finally,
hand crafted feature extraction brings extra computational complexity for
real-time applications.

Deep learning (DL) solves such problems by representation learning which
enables computer to learn high level features from raw data without human
interference. Recently advance in DL such as batch normalization [2], dropout
[3] and various new network structures have largely prompt the applications
for DL in real world problems and some have achieved near human-like
performance. Various DL approaches have been proposed for seizure detection.
Hugle _et. al._ [4] proposed a Convolutional neural network (CNN) designed for
implantable microcontroller by using only 4 electrodes selected a priori by
expert. Ullah and colleagues [5] developed a pyramid CNN for Epilepsy
Detection. A typical 13-layer CNN model is created for seizure detection
problem in [6] using Bonn University database [7] which has only 3.27 hours of
EEG data. The deep CNN structure has about 100k parameters, however neither
dropout nor batch normalization is used to regularize the deep CNN. As a
result, they could not reach the state-of-art performance achieved by machine
learning approaches using hand crafted features.

In this work, we aim to develop an end-to-end solution for seizure onset
detection. A CNN structure called seizNet is carefully designed to enable an
efficient and effective representative learning for seizure onset detection,
equipped with dropout and batch normalization to prevent overfitting for a
more generalized solution. We explore the possibility to use least number of
channels for accurate detection across subjects. Finally, we attempt to
interpret the model by discovering signatures hidden in the filters from
different convolutional layers.

##  2 METHOD

###  2.1 Baseline method – BPsvm

We develop a SVM-based classifier using hand crafted spectrum band power
features, called BPsvm. Frequency spectrum components within the 0-25 Hz band
is considered, as suggested by [8]. In this study, we preserve the 5-second
epoch for analysis across different approaches. To obtain features in higher
resolution, we split the 5-second epoch into 5 1-second windows for spectrum
transformation and band power feature extraction. In Shoeb’s work [1], 8 bands
from 0.5-24 Hz are chosen. As our window size is 1 second, the least sampling
rate is 1 Hz, our sub-bands are defined as [1-3, 3-6, 6-9, 9-12, 12-15, 15-18,
18-21, 21-24] Hz. Spectrum band power feature in the sub-band signals on every
1-second are calculated and then are concatenated into one instance for every
5-second epoch. SVM classifier is trained using radial basis function (RBF)
kernel.

###  2.2 Deep learning method – SeizNet

We develop a deep CNN network named SeizNet for end-to-end seizure detection
solution. Comparing to [6], SeizNet contains additional dropout layers [3] and
batch normalization [2] after every convolution layer. Such layers are
designed to avoid model overfitting. Unlike the typical usage of dropout which
is only after fully connected layer, we used dropout in various parts of the
model as it is indeed suggested by the inventors of dropout [3]. The number of
filter at each convolution layer is multiplied by two every time like VGGNet
[9]. It enables SeizNet to have less number of filters at low levels in which
filters learn basic shapes, while having more filters at the higher levels
where filters are capable of grasping sophisticated patterns. As an activation
function ReLU is used and other hyper-parameters of the model such as number
of filters and filter sizes at each layer as well as number of unit in fully
connected layer are cross-validated over a broad range. Detailed architecture
of SeizNet can be found in table 1. The total number of parameters for
SeizNet-2chn and SeizNet-18chn are $200,592$ and $201,872$ respectively, both
include $240$ non-trainable parameters.

Table 1: SeizNet Architecture | Layer | Output
---|---|---
Input |  ($1000\times n^{*}$) |  ($1\times 1000\times n$)
Conv 1 | $8\times Conv2D(1\times 10)$ | $(1\times 991\times 8)$
|  MaxPool2D $(1\times 2)$ | $(1\times 495\times 8)$
| Dropout(0.2) | $(1\times 495\times 8)$
Conv 2 | $16\times Conv2D(1\times 10)$ | $(1\times 486\times 16)$
|  MaxPool2D $(1\times 2)$ | $(1\times 243\times 16)$
| Dropout (0.2) | $(1\times 243\times 16)$
Conv 3 | $32\times Conv2D(1\times 10)$ | $(1\times 224\times 32)$
|  MaxPool2D $(1\times 2)$ | $(1\times 112\times 32)$
| Dropout (0.2) | $(1\times 112\times 32)$
Conv 4 | $64\times Conv2D(1\times 10)$ | $(1\times 93\times 64)$
|  MaxPool2D ($1\times 2$) | $(1\times 46\times 64)$
| Dropout (0.2) | $(1\times 46\times 64)$
Flatten | Flatten | (2944)
| Dense (50) | 50
| Dropout (0.5) | (50)
Output | Dense (2) | (2)
*n=2 for 2-channel data, n=18 for 18-channel data

We conduct Leave-one-subject-out cross validation to evaluate our model. For
training, we use Adam [10] optimizer with a learning rate of $4.1e-3$, binary
cross entropy loss function and batch size of 128. We explored early-stopping
[11] approach by randomly selecting 20% of training data as validation split.
However, it is observed that SeizNet does not overfit due to effect of
aggressive usage of dropout and batch normalization applied after every
convolution layer. Therefore, 100 iterations have been decided to execute
without any validation split. It also enables us to take advantage of all the
data which possibly affects performance in a statistically important level
since deep learning approaches for BCI problems are often deprived of big
datasets.

###  2.3 Filter Decoding for SeizNet interpretation

CNNs extract spatial features hierarchically in a modular way throughout
convolution layers, the filters decompose the input space to set a mapping
between abstract features and labels. Decoding such filters help us discern
what decomposed components are, thereby how CNNs work. One technique to fathom
the characterization of hidden units is visualizing sample inputs that
maximize the selected units. A pioneer method, Activation Maximization (AM)
proposed by Erhan in 2009 [12] has turned this technique into an optimization
problem yielding artificial inputs that maximally activate any chosen hidden
unit/units by gradient ascent rather than selecting from the data set which is
shown to be problematic and inadequate in the sense of leading a conclusion.
Furthermore, its consistency has been shown with different initializations
producing mostly same salient features at the input[12].

To date, AM has been widely used to decode abstract spatial filters and to
make qualitative interpretations of CNNs. It was first applied by [13] to the
AlexNet [14], known as first modern CNN architecture as well as to the others
such as VGGNet [9], GoogleNet [15] etc.

##  3 Experiment and Result

###  3.1 Seizure EEG data and Pre-processing

Data used in this study is from KK Women’s and Children’s Hospital, Singapore.
IRB was acquired from the hospital review board. EEG data of 29 pediatric
patients diagnosed with typical absence seizures are included in this study.
The data are extracted from Nikon Kohden EEG-1200K and EEG-9100K recording
systems (reading setting: Cal Voltage=$50\mu V$, HFF=$70Hz$, LFF=$0.53Hz$,
Sensitivity=$7-10\mu V/mm$, sampling rate=$200\,\,or\,\,500Hz$). The length of
patients’ EEG recordings range from 25 to 66 minutes. In total the data
contains 1037.6 minutes of EEG recording with 24.95 minutes seizure data
distributed among 120 seizure onsets. EEG data is down-sampled to 200 Hz
across all subjects. Data from all channels for each subject is z-normalized.
Window size of 5 second is chosen and preserved for all the methods in order
to obtain a conclusive comparison based on performance metrics. A common
problem in CNN networks for seizure detection is that datasets are often
imbalanced meaning interictal phases outnumber the ictal phases by a wide
margin and it has been shown that imbalanced datasets lead to statistically
significant performance drop in CNN architectures [16]. To overcome this
issue, a data augmentation method during pre-preprocessing is preferred rather
than undersampling or oversampling. To increase the number ictal phases,
sliding is applied with different overlapping proportions according to
existence or absence of seizure. Namely, while shifting with 5 seconds (no
overlapping) is implemented to create interictal class, 0.075 second shifting
is used for ictal class to create balanced input for the SeizNet. For BPsvm,
however, no such technique is applied since SVM is shown to be robust against
imbalanced datasets [17].

###  3.2 Experiment Settings and Performance Metrics

We compared result of four experimental settings including 18-channel SVM,
18-channel CNN, 2-channel SVM and 2-channel CNN respectively. Performance of
the seizure detection algorithms are assessed with sensitivity and false alarm
rate by the community [18] and often extended with ‘latency’ in order to make
more comprehensive analysis across detector algorithms. The definitions are
described as follows:

Sensitivity(%): Proportion of seizures correctly detected

False alarm rate(fp/h): Number of false positive seizures per hour

Latency(second): Delay between electrographic onset and detection

While BPsvm yields determinate results, seizNet models produce different
result in every round, due to randomly parameter seeding. To evaluate the
result objectively, ten tests have been carried out for both seizNet-2-chn and
seizNet-18-chn models, and most frequent result which is statistically
corresponds the mode of sensitivity and false alarm for a subject is chosen as
a final result.

###  3.3 Results

Results of performance metrics obtained from different experimental settings
can be found in Table 2. In both BPsvm and seizNet, models trained using
18-chn reduce false alarms with a boost in sensitivity as compare to the 2-chn
models. But to our surprise, SeizNet-2-chn model though using much less number
of channels, outperforms BPsvm-18-chn model for all performance metrics
measured.

Table 2: Comparison of Performance Metrics  Model | BPsvm | SeizNet
---|---|---
Channel used | 2-chn | 18-chn | 2-chn | 18-chn
Seizure detected | 104/120 | 108/120 | 112/120 | 115/120
Sensitivity (%) | 86.6% | 90% | 93.3% | 95.8%
False alarms | 33 | 14 | 10 | 3
$FAR^{*}$ (fp/h) | 1.91 | 0.81 | 0.58 | 0.17
Mean Latency(sec) | 4.42 | 3.75 | 3.26 | 3.80
*FAR–false alarm rate

Detailed number of seizures identified correctly and false alarm for each
subject can be found in Fig 1.

Fig. 1: Mis-identified and false alarm seizures across subjects

###  3.4 Filters in SeizNet

To understand what representation features are learnt in seizNet, we use
Activation Maximization method that generates input patterns activating given
filters. It enables us to visualize the low level and high level features that
help to reveal feature hierarchy throughout convolution layers. In this work,
AM is implemented with keras-vis library [19] which provides methods to
generate inputs activating the given unit/units maximally. Unlike the baseline
method, activation maximization loss calculated based on given input is not
only dependent of model weights. In the library, there are two kind of
regularization terms namely LP norm and total variation added to the loss in
order to enforce natural image prior [19]. Default values are preserved for
the weights of these regularization terms. Only input range is changed
according to our pre-processing, and it is set to (-10,10). Finally, seed for
the optimization is initialized with random values.

Fig. 2: Filters from 4 convolutional layers

In image classification problems, filters at the first convolutional layer
usually encode the direction and color channels. Hierarchically moving along
the CNN, more and more complex features are found at the higher layers that
are indeed combination of features at the lower levels [12].

In SeizNet, the first layer filters present very basic shapes as shown in Fig
2.a. While interpreted as color channels in the perspective of image analysis,
for SeizNet the filters can be thought to encode EEG channel information, due
to the fact that there are filters that have high and constant value for the
channel 1 and low and constant value for the channel 2 and vice versa. Another
possible interpretation could come from EEG montage perspective, where bi-
polar signal represents the difference of two electrodes since the filters
indeed subtract the channels one another.

Fig. 3: Examples of seizure waveforms (3 seconds)

In the second convolution layer, various kind of filters have been found but
nothing meaningful to us has been observed. However, from the third
convolution layer onwards we observe substantial characteristics of seizures,
therefore our hypothesis for the second layer is that it serves as the middle
man to bring basic information in the lower layers into complex like-seizure
signals in the higher layers

Characteristics of the seizures are observed at the third convolution layer
and become clearer at the last convolution layer. It can be inferred that
SeizNet has learnt the fact that absence seizures create periodic and 3 Hz
signals. Nonetheless, clearly SeizNet focuses on spike-and-wave happening
three times in one second and try to capture it rather than capturing a whole
shape of a seizure. One possible reason behind is that seizure patterns often
vary from subject to subject, as can be seen from the Fig 3.

##  4 Discussion and Conclusion

It has been observed from the overall results that SeizNet is better than
BPsvm in terms of sensitivity and false alarm. It shows that CNN models are
more suitable for generalized models unlike the SVM which is indeed often
implemented as a subject-specific model. Nevertheless, it should be noted that
for subject 2 and subject 23, BPsvm-2chn could identify all the seizures while
not triggering more false alarm than seizNet-2chn which could not find all the
seizures for the mentioned subjects. It can be inferred that for some subjects
frequency domain features can be more discriminative than the features
extracted from time domain. It also consistent with the fact that even EEG
experts are to check time-frequency graphic in some cases in order to finalize
their decision. An interesting discovery in the study is that, developed using
CNN, seizNet model trained by only 2 channels’ data is able to outperform
traditional approach trained with full scalp EEG data.

End-to-end approach is more favorable for developing real time seizure
detection systems as it eliminates feature extraction, which can be a burden
for real time signal processing. A solution using data merely from 2 channels
makes the approach even more adoptable for light-weight, home based seizure
monitoring system.

## References

  * [1] Ali Hossam Shoeb,  Application of machine learning to epileptic seizure onset detection and treatment,  Ph.D. thesis, Massachusetts Institute of Technology, 2009.
  * [2] Sergey Ioffe and Christian Szegedy,  “Batch normalization: Accelerating deep network training by reducing internal covariate shift,”  arXiv preprint arXiv:1502.03167, 2015.
  * [3] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov,  “Dropout: a simple way to prevent neural networks from overfitting,”  The Journal of Machine Learning Research, vol. 15, no. 1, pp. 1929–1958, 2014.
  * [4] Maria Hügle, Simon Heller, Manuel Watter, Manuel Blum, Farrokh Manzouri, Matthias Dümpelmann, Andreas Schulze-Bonhage, Peter Woias, and Joschka Boedecker,  “Early seizure detection with an energy-efficient convolutional neural network on an implantable microcontroller,”  arXiv preprint arXiv:1806.04549, 2018.
  * [5] Ihsan Ullah, Muhammad Hussain, Hatim Aboalsamh, et al.,  “An automated system for epilepsy detection using eeg brain signals based on deep learning approach,”  Expert Systems with Applications, vol. 107, pp. 61–71, 2018.
  * [6] U Rajendra Acharya, Shu Lih Oh, Yuki Hagiwara, Jen Hong Tan, and Hojjat Adeli,  “Deep convolutional neural network for the automated detection and diagnosis of seizure using eeg signals,”  Computers in biology and medicine, vol. 100, pp. 270–278, 2018\.
  * [7] Ralph G Andrzejak, Klaus Lehnertz, Florian Mormann, Christoph Rieke, Peter David, and Christian E Elger,  “Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state,”  Physical Review E, vol. 64, no. 6, pp. 061907, 2001.
  * [8] J_ Gotman, JR Ives, and P Gloor,  “Frequency content of eeg and emg at seizure onset: possibility of removal of emg artefact by digital filtering,”  Clinical Neurophysiology, vol. 52, no. 6, pp. 626–639, 1981.
  * [9] Karen Simonyan and Andrew Zisserman,  “Very deep convolutional networks for large-scale image recognition,”  arXiv preprint arXiv:1409.1556, 2014.
  * [10] Diederik P Kingma and Jimmy Ba,  “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.6980, 2014.
  * [11] Yuan Yao, Lorenzo Rosasco, and Andrea Caponnetto,  “On early stopping in gradient descent learning,”  Constructive Approximation, vol. 26, no. 2, pp. 289–315, 2007.
  * [12] Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent,  “Visualizing higher-layer features of a deep network,”  University of Montreal, vol. 1341, no. 3, pp. 1, 2009.
  * [13] Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman,  “Deep inside convolutional networks: Visualising image classification models and saliency maps,”  arXiv preprint arXiv:1312.6034, 2013.
  * [14] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton,  “Imagenet classification with deep convolutional neural networks,”  in Advances in neural information processing systems, 2012, pp. 1097–1105.
  * [15] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich,  “Going deeper with convolutions,”  in Proceedings of the IEEE conference on computer vision and pattern recognition, 2015, pp. 1–9.
  * [16] Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski,  “A systematic study of the class imbalance problem in convolutional neural networks,”  Neural Networks, vol. 106, pp. 249–259, 2018.
  * [17] Rehan Akbani, Stephen Kwek, and Nathalie Japkowicz,  “Applying support vector machines to imbalanced datasets,”  in European conference on machine learning. Springer, 2004, pp. 39–50.
  * [18] Alexandros T Tzallas, Markos G Tsipouras, Dimitrios G Tsalikakis, Evaggelos C Karvounis, Loukas Astrakas, Spiros Konitsiotis, and Margaret Tzaphlidou,  “Automated epileptic seizure detection methods: a review study,”  in Epilepsy-histological, electroencephalographic and psychological aspects. InTech, 2012.
  * [19] Raghavendra Kotikalapudi and contributors,  “keras-vis,” https://github.com/raghakot/keras-vis, 2017.


condition with respect to the metric $d\varsigma$. By Proposition 7.1, we can
define inductively families of arcs $\alpha^{(n)}_{\theta}$, $n\geq 0$,
$\theta\in[0,1]$ by

$\displaystyle\alpha^{(0)}_{\theta}$ $\displaystyle=\alpha_{\theta},$
$\displaystyle\alpha^{(n+1)}_{\theta}$
$\displaystyle=f^{-1}_{n,\theta}\big{(}\alpha^{(n)}_{d\theta\bmod 1}\big{)},$

where $f^{-1}_{n,\theta}$ is the branch of $f^{-1}$ mapping
$\gamma_{n}(d\theta\bmod 1)$ to $\gamma_{n+1}(\theta)$. Then
$\alpha^{(n)}_{\theta}$ is a $C^{1}$-arc joining $\gamma_{n}(\theta)$ to
$\gamma_{n+1}(\theta)$ within $\overline{A_{n+1}}\setminus A_{n}$ and $f^{n}$
maps $\alpha^{(n)}_{\theta}$ injectively onto $\alpha_{d^{n}\theta\bmod 1}$.
Consequently,

$\operatorname{dist}_{\varsigma}(\gamma_{n}(\theta),\gamma_{n+1}(\theta))\leq\int_{\alpha^{(n)}_{\theta}}\varsigma(z)|dz|\leq\frac{\operatorname{length}_{\varsigma}\alpha_{d^{n}\theta\bmod
1}}{\inf_{\alpha^{(n)}_{\theta}}|(f^{n})^{\prime}|_{\varsigma}}<M^{\prime}b_{n}.$

It follows that for every $\theta\in[0,1]$ and $m>n>0$ we have

$\operatorname{dist}_{\varsigma}(\gamma_{n}(\theta),\gamma_{m}(\theta))\leq\operatorname{dist}_{\varsigma}(\gamma_{n+1}(\theta),\gamma_{n}(\theta))+\cdots+\operatorname{dist}_{\varsigma}(\gamma_{m}(\theta),\gamma_{m-1}(\theta))\leq
M^{\prime}\sum_{k=n}^{m}b_{k}.$

Since $\sum_{k=1}^{\infty}b_{k}$ is a convergent series, the sequence
$\gamma_{n}$ satisfies the uniform Cauchy condition with respect to the metric
$d\varsigma$. We show that this implies that $\gamma_{n}$ satisfies also the
uniform Cauchy condition with respect to the spherical metric. Suppose this
does not hold. Then there exists $\varepsilon>0$ and sequences
$n_{k},m_{k}\to\infty$, $\theta_{k}\in[0,1]$, such that

$\operatorname{dist}_{sph}(\gamma_{n_{k}}(\theta_{k}),\gamma_{m_{k}}(\theta_{k}))\geq\varepsilon.$

Passing to a subsequence, we can assume $\gamma_{n_{k}}(\theta_{k})\to z$,
$\gamma_{m_{k}}(\theta_{k})\to w$ for some distinct points
$z,w\in\widehat{\mathbb{C}}$, where at least one of them (say $z$) is not
equal to $\infty$. By Lemma 5.9, there exist $\delta,c>0$, such that
$w\notin\mathbb{D}(z,\delta)$ and $\varsigma>c$ on
$W_{N}\cap\mathbb{D}(z,\delta)$. Then for sufficiently large $k$ we have
$\gamma_{n_{k}}(\theta_{k})\in\mathbb{D}(z,\frac{\delta}{3})$ and
$\gamma_{m_{k}}(\theta_{k})\notin\mathbb{D}(z,\frac{2\delta}{3})$, so

$\operatorname{dist}_{\varsigma}(\gamma_{n_{k}}(\theta_{k}),\gamma_{m_{k}}(\theta_{k}))\geq\frac{c\delta}{3},$

which contradicts the uniform Cauchy condition with respect to the metric
$d\varsigma$. Therefore, the sequence $\gamma_{n}$ satisfies the uniform
Cauchy condition with respect to the spherical metric, which implies that it
has a limit $\gamma$, which is equal to the boundary of $U$ in
$\widehat{\mathbb{C}}$, providing its local connectivity.

##  8\. Local connectivity of Julia sets: Proof of Theorem C

Throughout this section, we consider the transcendental meromorphic map

$f(z)=z-\tan z,$

which is the Newton map of the entire transcendental function $F(z)=\sin z$.
We will prove that the Julia set $J(f)$ is locally connected, establishing
Theorem C. See Figure 2. Throughout the section, we will understand the word
‘boundary’ as the boundary in $\widehat{\mathbb{C}}$.

A useful tool to prove local connectivity of a compact connected set in
$\widehat{\mathbb{C}}$ (like the Julia set of a rational or transcendental
map) is Whyburn’s Theorem, which reads as follows.

######  Theorem 8.1 ([Why63, Theorem 4.4, p. 113]).

A compact connected set $J\subset\widehat{\mathbb{C}}$ is locally connected if
and only if it satisfies the following properties.

  1. $($a$)$

The boundary of every connected component of $\widehat{\mathbb{C}}\setminus J$
is locally connected.

  2. $($b$)$

For every $\varepsilon>0$, only a finite number of connected components of
$\widehat{\mathbb{C}}\setminus J$ have spherical diameter greater than
$\varepsilon$.

To check the conditions of Whyburn’s Theorem for $J=J(f)$, we first recall
some properties of the map $f$. By [BFJK14], $J$ is connected as the Julia set
of Newton’s method for a transcendental entire map. Note that

(51) $f(z+\pi)=f(z)+\pi,\qquad z\in\mathbb{C},$

which implies a ‘translation invariance’ of the dynamical plane of $f$ (see
Figure 2). The properties listed below are proved in [BFJK17, Example 7.2] and
[BFJK20, Proposition 4.1].

######  Lemma 8.2.

The following statements hold.

  1. $($a$)$

$f$ has infinitely many simply connected immediate basins of attraction
$U_{k}$, $k\in\mathbb{Z}$, of superattracting fixed points

$c_{k}=k\pi,\quad k\in\mathbb{Z},$

such that $U_{k}=U_{0}+k\pi$ and $\deg_{c_{k}}f=\deg f|_{U_{k}}=3$. The points
$c_{k}$, $k\in\mathbb{Z}$, are the only critical points of $f$. They are
located in the vertical lines

$\ell_{k}(t)=k\pi+it,\qquad t\in\mathbb{R},\;k\in\mathbb{Z},$

which are invariant and contained in $U_{k}$, respectively. Moreover, $f$ has
no finite asymptotic values.

  2. $($b$)$

The poles of $f$,

$p_{k}=\frac{\pi}{2}+k\pi,\ k\in\mathbb{Z},$

are simple. They are located in the vertical lines

$r_{k}(t)=\frac{\pi}{2}+k\pi+it,\qquad t\in\mathbb{R},\;k\in\mathbb{Z},$

which are contained in $J(f)$.

  3. $($c$)$

The asymptotics of $f$ for $\textup{Im}(z)\to\pm\infty$ is given by

$f(z)=\begin{cases}z-i+\mathcal{O}(e^{-2\textup{Im}(z)})&\text{as}\;\;\textup{Im}(z)\to+\infty\\\
z+i+\mathcal{O}(e^{2\textup{Im}(z)})&\text{as}\;\;\textup{Im}(z)\to-\infty\end{cases}.$

  4. $($d$)$

We have

$J(f)\cap\mathbb{C}=\bigcup_{k\in\mathbb{Z}}J_{k},$

where $J_{k}$ is the connected component of $J(f)\cap\mathbb{C}$ containing
the line $r_{k}$. For every $k\in\mathbb{Z}$, the basin $U_{k}$ has exactly
two accesses to infinity, and $\partial U_{k}$ contains exactly two poles of
$f$, i.e. $p_{k-1},p_{k}$, which are accessible from $U_{k}$.

  5. $($e$)$

All Fatou components of $f$ are preperiodic and eventually mapped by iterates
of $f$ into $U_{k}$ for some $k\in\mathbb{Z}$. In particular, $f$ has no
wandering domains.

See Figure 8.

Figure 8. The dynamical plane of the map $f(z)=z-\tan z$.

The following proposition establishes the first condition in Whyburn’s
criterium (Theorem 8.1).

######  Proposition 8.3.

Every Fatou component of $f$ has locally connected boundary.

###### Proof.

First, we show that the boundaries of the attracting basins $U_{k}$ are
locally connected. To this end, we check that they fulfil the assumptions of
Theorem A. Recall that the basins $U_{k}$ are simply connected, since $J(f)$
is connected. By Lemma 8.2, we have $\deg f|_{U_{k}}=3$ and

(52)
$\operatorname{Crit}(f)=\mathcal{P}(f)=\\{c_{k}\\}_{k\in\mathbb{Z}}=\\{k\pi\\}_{k\in\mathbb{Z}}.$

This immediately implies the assumptions (a)–(b) of Theorem A. To check the
assumption (c), note that by Lemma 8.2c, for $M>0$ large enough, the map $f$
on the half planes

(53) $P_{+}=\\{z\in\mathbb{C}:\textup{Im}(z)>M\\},\qquad
P_{-}=\\{z\in\mathbb{C}:\textup{Im}(z)<-M\\}$

is arbitrarily close to $z\mapsto z\mp i$. In particular, this implies
$\overline{P_{\pm}}\subset f(P_{\pm})$ and $f(P_{+})\cap f(P_{-})=\emptyset$.
We show that $P_{\pm}$ are repelling petals of $f$ at infinity. In view of
Lemma 8.2c, the univalency of $f$ on $P_{\pm}$ follows easily from Rouché’s
Theorem, while the remaining conditions of Definition 4.1 with $g(t)\equiv 1$
are satisfied by Proposition 4.4, where we take $r=M$, $\delta=\frac{\pi}{2}$,
$d=1$, $a=\mp i$ and $j=1$. Hence, $P_{\pm}$ are repelling petals of $f$ at
infinity. Note that Lemma 8.2 asserts that the basins $U_{k}$ are located
between the vertical lines $r_{k-1}$ and $r_{k}$, both contained in the Julia
set of $f$. Hence, outside a compact set, $U_{k}$ is contained in the two
repelling petals $P_{\pm}$ of $f$ at infinity, which shows that $f$ and
$U_{k}$ satisfy the assumptions of Theorem A. Hence, the boundaries of $U_{k}$
are locally connected.

By Lemma 8.2(e), all Fatou components of $f$ are successive preimages of the
basins $U_{k}$ which have locally connected boundaries, so their boundaries
are also locally connected. ∎

It remains to show the second condition in Whyburn’s criterium. Note that in
contrast to the rational or even entire (hyperbolic) cases, the meromorphic
setting presents some additional difficulties, since for any given $n$,
components of preperiod $n$ can accumulate at poles and prepoles.

The idea of the proof is as follows. We show that the sum of the squares of
the spherical diameters of all Fatou components is finite, which immediately
implies that only a finite number of them can have diameter larger than any
given $\varepsilon$. To this end, we prove that the spherical distortion of
branches of $f^{-n}$ on Fatou components $U\subset f^{-1}(U_{k})\setminus
U_{k}$, $k\in\mathbb{Z}$, is uniformly bounded. From this, it follows that the
ratio between the square of the spherical diameter and the spherical area is
approximately the same for the component $U$ and its all inverse images by
branches of $f^{-n}$, $n>0$. Since the sum of spherical areas over all
components is finite (smaller than the spherical area of the whole sphere),
the same holds for the sum of the squares of spherical diameters.

Now we proceed to present the proof in detail. The following lemma is
straightforward to check.

######  Lemma 8.4.

For every $r_{0}>0$ there exist $c_{1},c_{2}>0$ such that

$\mathcal{D}_{sph}\Big{(}z,\frac{c_{1}r}{|z|^{2}+1}\Big{)}\subset\mathbb{D}(z,r)\subset\mathcal{D}_{sph}\Big{(}z,\frac{c_{2}r}{|z|^{2}+1}\Big{)}$

for every $z\in\mathbb{C}$ and every $0<r<r_{0}$.

For $n\geq 0$ define

$\displaystyle\mathcal{F}_{n}=\\{U:\ $ $\displaystyle U\text{ is a Fatou
component of $f$,}$ $\displaystyle\text{and $n$ is minimal such that
}f^{n}(U)\subset U_{k}\text{ for some $k\in\mathbb{Z}$}\\}.$

In particular, $\mathcal{F}_{0}=\\{U_{k}\\}_{k\in\mathbb{Z}}$. In the
following lemma we describe the Fatou components from $\mathcal{F}_{1}$. Let

$R_{k}=R_{k}(\delta,R)=\\{z\in\mathbb{C}:\textup{Re}(z)\in[k\pi+\delta,(k+1)\pi-\delta],\;\textup{Im}(z)\in[-R,R]\\},\quad
k\in\mathbb{Z}$

for $\delta,R>0$. See Figure 8.

######  Lemma 8.5.

We have

$\mathcal{F}_{1}=\\{U_{k,l}:k\in\mathbb{Z},\>l\in\mathbb{Z}\setminus\\{k,k+1\\}\\},$

where

$U_{k,l}\subset R_{k},\qquad U_{k,l}=U_{0,l-k}+k\pi,\qquad f(U_{k,l})=U_{l}$

and $R_{k}=R_{k}(\delta,R)$ for some $\delta,R>0$. Moreover, $f$ is univalent
on $U_{k,l}$.

###### Proof.

Fix a large $\tilde{R}>0$ such that
$\\{z\in\mathbb{C}:|\textup{Im}(z)|>\tilde{R}\\}\subset P_{+}\cup P_{-}$ for
the repelling petals $P_{\pm}$ from (53). Take an arbitrary point $z_{0}\in
U_{0}$ with $|\textup{Im}(z_{0})|>\tilde{R}$ (note that such points exist
since $\ell_{0}\subset U_{0}$) and let $z_{l}=z_{0}+l\pi$ for
$l\in\mathbb{Z}$. By (51) and Lemma 8.2(a), we have $z_{l}\in U_{l}$ and
$f^{-1}(z_{l})\subset P_{+}\cup
P_{-}\cup\bigcup_{k\in\mathbb{Z}}\mathbb{D}(p_{k},\varepsilon)$ for a small
fixed $\varepsilon>0$, provided $\tilde{R}$ was chosen sufficiently large.
Moreover, since $f$ is univalent on $P_{\pm}$ with $f(z)\sim z\mp i$, the set
$f^{-1}(z_{l})\cap(P_{+}\cup P_{-})$ consists of exactly one point
$z^{\prime}_{l}\in P_{\pm}$ close to $z_{l}\pm i$. Similarly, as the pole
$p_{0}$ is simple, the map $f$ is univalent on $\mathbb{D}(p_{0},\varepsilon)$
and $f^{-1}(z_{l})\cap\mathbb{D}(p_{0},\varepsilon)$ consists of exactly one
point $z_{0,l}$. Consequently, by (51), $f$ is univalent on
$\mathbb{D}(p_{k},\varepsilon)$ and
$f^{-1}(z_{l})\cap\mathbb{D}(p_{k},\varepsilon)$ consists of exactly one point
$z_{k,l}$, where

$z_{k,l}\in\mathbb{D}(p_{k},\varepsilon),\qquad z_{k,l}=z_{0,l-k}+k\pi,\qquad
k,l\in\mathbb{Z}.$

Hence,

(54) $f^{-1}(z_{l})=\\{z^{\prime}_{l}\\}\cup\\{z_{k,l}:k\in\mathbb{Z}\\},$

where all the points $z_{k,l}$, $k\in\mathbb{Z}\setminus\\{l-1,l\\}$ are
outside $P_{+}\cup P_{-}\cup S_{l}$ for

$S_{l}=\Big{\\{}z\in\mathbb{C}:\textup{Re}(z)\in\Big{[}\frac{\pi}{2}+(l-1)\pi,\frac{\pi}{2}+l\pi\Big{]}\Big{\\}}.$

This implies $f^{-1}(z_{l})\cap(P_{+}\cup P_{-}\cup
S_{l})\subset\\{z^{\prime}_{l},z_{l-1,l},z_{l,l}\\}$. On the other hand, since
$\deg f|_{U_{l}}=3$, the point $z_{l}$ has exactly three preimages under $f$
in $U_{l}$. As $U_{l}\subset S_{l}$, we conclude

(55) $f^{-1}(z_{l})\cap(P_{+}\cup P_{-}\cup S_{l})=f^{-1}(z_{l})\cap
U_{l}=\\{z^{\prime}_{l},z_{l-1,l},z_{l,l}\\}.$

Take $U\in\mathcal{F}_{1}$. Then
$U\cap\bigcup_{k\in\mathbb{Z}}U_{k}=\emptyset$ and $f(U)\subset U_{l}$ for
some $l\in\mathbb{Z}$. Suppose there exists a point $w\in U$ with
$|\textup{Im}(w)|>\tilde{R}+2$. Then $w\in P_{+}\cup P_{-}\setminus U_{l}$ and
$f(w)\in U_{l}$ with $|\textup{Im}(f(w))|>\tilde{R}$, contradicting (55) for
$z_{0}=f(w)-l\pi$. Hence, $U\subset\\{z\in\mathbb{C}:|\textup{Im}(z)|\leq
R\\}$ for $R=\tilde{R}+2$. Moreover, since $\ell_{k}\subset U_{k}$ and
$U_{k}=U_{0}+k\pi$ for $k\in\mathbb{Z}$ by Lemma 8.2(a), the set
$\\{z\in\mathbb{C}:\textup{Re}(z)\in(k\pi-\delta,k\pi+\delta),\;|\textup{Im}(z)|\leq
R\\}$ is contained in $U_{k}$ for a sufficiently small $\delta>0$ independent
of $k\in\mathbb{Z}$. Consequently, $U\subset\bigcup_{k\in\mathbb{Z}}R_{k}$ for
$R_{k}=R_{k}(\delta,R)$. Since the sets $R_{k}$ are disjoint and compact, and
$U$ is connected, in fact $U\subset R_{k}$ for some $k\in\mathbb{Z}$. As
$U_{l}$ is simply connected and $R_{k}\cap\operatorname{Crit}(f)=\emptyset$ by
(52), the component $U$ is the image of $U_{l}$ under a well-defined branch
$g$ of $f^{-1}$. Note that by (54) and (55), we have
$g(z_{l})=z_{k^{\prime},l}\in U$ for some
$k^{\prime}\in\mathbb{Z}\setminus\\{l-1,l\\}$. As $z_{k^{\prime},l}\in
R_{k^{\prime}}$ and the sets $R_{k^{\prime}}$ are disjoint, in fact
$k^{\prime}=k$, so $z_{k,l}\in U$ and $k\neq l-1,l$.

We have proved that for every component $U\in\mathcal{F}_{1}$ there exist
$k,l\in\mathbb{Z}$ with $k\neq l-1,l$, such that $z_{k,l}\in U\subset R_{k}$
and $f$ maps $U$ univalently onto $U_{l}$. On the other hand, (54) and (55)
imply that for every $k,l\in\mathbb{Z}$ with $k\neq l-1,l$, there exists a
component from $\mathcal{F}_{1}$ containing $z_{k,l}$. This together with (51)
and Lemma 8.2(a) ends the proof. ∎

Now we can show the second condition in Whyburn’s criterium.

######  Proposition 8.6.

For every $\varepsilon>0$ there are only a finite number of Fatou components
$U$ of $f$ with $\operatorname{diam}_{sph}U\geq\varepsilon$.

###### Proof.

Recall that by Lemma 8.2(e), all Fatou components of $f$ are elements of
$\bigcup_{n=0}^{\infty}\mathcal{F}_{n}$. Since $U_{k}\subset
S_{k}\subset\mathbb{C}\setminus\mathbb{D}(0,\frac{\pi}{2}+(|k|-1)\pi)$ for
$k\in\mathbb{Z}\setminus\\{0\\}$, the spherical diameter of $U_{k}$ tends to
zero as $|k|\to\infty$, so we only need to consider components in
$\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$.

By Lemma 8.5, for every $U\in\mathcal{F}_{1}$ we have $U=U_{k,l}$ for some
$k\in\mathbb{Z}$, $l\in\mathbb{Z}\setminus\\{k,k+1\\}$, such that

(56) $U_{k,l}=U_{0,l-k}+k\pi\subset R_{k}$

and $f$ maps $U_{k,l}$ univalently onto $U_{l}$. Since
$U_{l}=U_{0}+l\pi\subset S_{l}$, $\mathbb{D}(l\pi,r_{0})\subset U_{l}$ for
some $r_{0}>0$ independent of $l$, and $p_{0}$ is a simple pole of $f$, there
exists $c_{1}>0$ such that

$\operatorname{diam}U_{0,l}\leq\frac{c_{1}}{|l|+1},\qquad\operatorname{area}U_{0,l}\geq\frac{c_{1}}{l^{4}+1}$

for $l\in\mathbb{Z}\setminus\\{0,1\\}$. Consequently, by (56) and since
$\sigma_{sph}|_{R_{k}}\asymp\frac{1}{k^{2}+1}$, we have

(57)
$\operatorname{diam}U_{k,l}\leq\frac{c_{1}}{|l-k|+1},\quad\operatorname{area}_{sph}U_{k,l}\geq\frac{c_{2}}{((l-k)^{4}+1)(k^{4}+1)}$

for $k\in\mathbb{Z}$, $l\in\mathbb{Z}\setminus\\{k,k+1\\}$ and some constant
$c_{2}>0$.

By (52), all branches of $f^{-n}$, $n\geq 1$, are well-defined on
$\\{z\in\mathbb{C}:\textup{Re}(z)\in(k\pi,(k+1)\pi)\\}\supset R_{k}$ for
$k\in\mathbb{Z}$. Hence, for $n\geq 1$,

$\mathcal{F}_{n}=\\{g_{k,n}(U_{k,l}):k\in\mathbb{Z},\>l\in\mathbb{Z}\setminus\\{k,k+1\\},\>g_{k,n}\in\mathcal{G}_{k,n}\\},$

where $\mathcal{G}_{k,n}$ is the family of all branches of $f^{-(n-1)}$ on
$R_{k}$ (note that we include the case $n=1$, for which
$g_{k,1}=\operatorname{Id}$). We claim that all the branches
$g_{k,n}\in\mathcal{G}_{k,n}$ have spherical distortion bounded uniformly with
respect to $k,n$. Indeed, for given $r>0$, any two points $z,w\in R_{k}$ can
be joined by $N=\lceil 2(2R+\pi)/r\rceil$ Euclidean disks
$\mathbb{D}(z_{1},r),\ldots,\mathbb{D}(z_{N},r)$, such that
$z_{1},\ldots,z_{N}\in R_{k}$, $z_{1}=z$, $z_{N}=w$ and

(58) $\mathbb{D}(z_{j},r)\cap\mathbb{D}(z_{j+1},r)\neq\emptyset\quad\text{for
}\;j=1,\dots,N-1.$

Define

$r=\frac{c_{1}\delta}{2c_{2}},\qquad r_{j}=\frac{c_{1}\delta}{|z_{j}|^{2}+1}$

for the constants $c_{1},c_{2}$ from Lemma 8.4, and the constant $\delta$ from
Lemma 8.5. Then,

$\mathbb{D}(z_{j},r)\subset\mathcal{D}_{sph}\Big{(}z_{j},\frac{r_{j}}{2}\Big{)},\qquad\mathcal{D}_{sph}(z_{j},r_{j})\subset\mathbb{D}(z_{j},\delta)$

by Lemma 8.4, so by (58) and (52),

$\mathcal{D}_{sph}\Big{(}z_{j},\frac{r_{j}}{2}\Big{)}\cap\mathcal{D}_{sph}\Big{(}z_{j+1},\frac{r_{j}}{2}\Big{)}\neq\emptyset,\qquad\mathcal{D}_{sph}(z_{j},r_{j})\cap\mathcal{P}(f)=\emptyset,$

and hence using repeatedly Theorem 2.1 for the chain of spherical disks with
$\lambda=1/2$, we conclude that

(59) $\frac{|g^{\prime}_{k,n}(z)|_{sph}}{|g^{\prime}_{k,n}(w)|_{sph}}\leq
c_{3}$

for some $c_{3}>0$ independent of $k,n,g_{k,n},z,w$. Let

$M_{g_{k,n}}=\max_{R_{k}}|g^{\prime}_{k,n}|_{sph}.$

By (57) and since $\sigma_{sph}|_{R_{k}}\asymp\frac{1}{k^{2}+1}$,

$\operatorname{diam}_{sph}g_{k,n}(U_{k,l})\leq\frac{c_{4}M_{g_{k,n}}\operatorname{diam}U_{k,l}}{k^{2}+1}\leq\frac{c_{1}c_{4}M_{g_{k,n}}}{(|l-k|+1)(k^{2}+1)}$

and

$\displaystyle\operatorname{area}_{sph}g_{k,n}(U_{k,l})$
$\displaystyle\geq\big{(}\min_{R_{k}}|g^{\prime}_{k,n}|_{sph}^{2}\big{)}\operatorname{area}_{sph}U_{k,l}$
$\displaystyle\geq\frac{M_{g_{k,n}}^{2}}{c_{3}^{2}}\operatorname{area}_{sph}U_{k,l}\geq\frac{c_{2}}{c_{3}^{2}}\frac{M_{g_{k,n}}^{2}}{((l-k)^{4}+1)(k^{4}+1)},$

for a constant $c_{4}>0$. Hence, for some constants $c_{5},c_{6},c_{7}>0$,

$\displaystyle\sum_{U\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}}(\operatorname{diam}_{sph}U)^{2}$
$\displaystyle=\sum_{k\in\mathbb{Z}}\sum_{l\in\mathbb{Z}\setminus\\{k,k+1\\}}\sum_{\begin{subarray}{c}n\geq
1,\\\
g_{k,n}\in\mathcal{G}_{k,n}\end{subarray}}(\operatorname{diam}_{sph}g_{k,n}(U_{k,l}))^{2}$
$\displaystyle\leq
c_{5}\sum_{k\in\mathbb{Z}}\sum_{l\in\mathbb{Z}\setminus\\{k,k+1\\}}\sum_{n,g_{k,n}}\frac{M_{g_{k,n}}^{2}}{((l-k)^{2}+1)(k^{4}+1)}$
$\displaystyle=c_{5}\sum_{l\in\mathbb{Z}\setminus\\{0,1\\}}\frac{1}{l^{2}+1}\sum_{k\in\mathbb{Z}}\sum_{n,g_{k,n}}\frac{M_{g_{k,n}}^{2}}{k^{4}+1}\leq
c_{6}\sum_{l\in\mathbb{Z}\setminus\\{0,1\\}}\frac{1}{l^{4}+1}\sum_{k\in\mathbb{Z}}\sum_{n,g_{k,n}}\frac{M_{g_{k,n}}^{2}}{k^{4}+1}$
$\displaystyle=c_{6}\sum_{k\in\mathbb{Z}}\sum_{l\in\mathbb{Z}\setminus\\{k,k+1\\}}\sum_{n,g_{k,n}}\frac{M_{g_{k,n}}^{2}}{((l-k)^{4}+1)(k^{4}+1)}$
$\displaystyle\leq
c_{7}\sum_{k\in\mathbb{Z}}\sum_{l\in\mathbb{Z}\setminus\\{k,k+1\\}}\sum_{n,g_{k,n}}\operatorname{area}_{sph}g_{k,n}(U_{k,l})\leq
c_{6}\sum_{U\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}}\operatorname{area}_{sph}U$
$\displaystyle\leq c_{7}\operatorname{area}_{sph}\widehat{\mathbb{C}}<\infty,$

as $U\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$ are pairwise disjoint.
Concluding, we have showed that the series

$\sum_{U\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}}(\operatorname{diam}_{sph}U)^{2}$

is convergent, which immediately implies that for every $\varepsilon>0$ there
can be only a finite number of $U\in\bigcup_{n=1}^{\infty}\mathcal{F}_{n}$
with $\operatorname{diam}_{sph}U\geq\varepsilon$, proving the proposition. ∎

Propositions 8.3 and 8.6 complete the proof of Theorem C.

### Acknowledgements

The authors thank the Centro di Ricerca Matematica Ennio De Giorgi (Pisa) for
its hospitality.

## References

  * [ARS22] Mashael Alhamed, Lasse Rempe, and Dave Sixsmith, _Geometrically finite transcendental entire functions_ , J. Lond. Math. Soc. (2) 106 (2022), no. 2, 485–527. MR 4477198
  * [BD00] Irvine N. Baker and Patricia Domínguez, _Some connectedness properties of Julia sets_ , Complex Variables Theory Appl. 41 (2000), no. 4, 371–389. MR 1785150
  * [BFJK14] Krzysztof Barański, Núria Fagella, Xavier Jarque, and Bogusława Karpińska, _On the connectivity of the Julia sets of meromorphic functions_ , Invent. Math. 198 (2014), no. 3, 591–636. MR 3279533
  * [BFJK15] by same author, _Absorbing sets and Baker domains for holomorphic maps_ , J. Lond. Math. Soc. (2) 92 (2015), no. 1, 144–162. MR 3384509
  * [BFJK17] by same author, _Accesses to infinity from Fatou components_ , Trans. Amer. Math. Soc. 369 (2017), no. 3, 1835–1867. MR 3581221
  * [BFJK20] by same author, _Fatou components and singularities of meromorphic functions_ , Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), no. 2, 633–654. MR 4080454
  * [BFRG15] Walter Bergweiler, Núria Fagella, and Lasse Rempe-Gillen, _Hyperbolic entire functions with bounded Fatou components_ , Comment. Math. Helv. 90 (2015), no. 4, 799–829. MR 3433280
  * [BH88] Bodil Branner and John H. Hubbard, _The iteration of cubic polynomials. I. The global topology of parameter space_ , Acta Math. 160 (1988), no. 3-4, 143–206. MR 945011
  * [BKZ12] Krzysztof Barański, Bogusława Karpińska, and Anna Zdunik, _Bowen’s formula for meromorphic functions_ , Ergodic Theory Dynam. Systems 32 (2012), no. 4, 1165–1189. MR 2955309
  * [BM02] Walter Bergweiler and Shunsuke Morosawa, _Semihyperbolic entire functions_ , Nonlinearity 15 (2002), no. 5, 1673–1684. MR 1925433
  * [BW91] Irvine N. Baker and J. Weinreich, _Boundaries which arise in the dynamics of entire functions_ , Rev. Roumaine Math. Pures Appl. 36 (1991), no. 7-8, 413–420, Analyse complexe (Bucharest, 1989). MR 1144573 (93e:30056)
  * [CG93] Lennart Carleson and Theodore W. Gamelin, _Complex dynamics_ , Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383 (94h:30033)
  * [Cow81] Carl C. Cowen, _Iteration and the solution of functional equations for functions analytic in the unit disk_ , Trans. Amer. Math. Soc. 265 (1981), no. 1, 69–95. MR 607108 (82i:30036)
  * [DG87] Robert L. Devaney and Lisa R. Goldberg, _Uniformization of attracting basins for exponential maps_ , Duke Math. J. 55 (1987), no. 2, 253–266. MR 894579 (88h:30035)
  * [DH84] Adrien Douady and John Hamal Hubbard, _Étude dynamique des polynômes complexes. Partie I_ , Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984. MR 762431
  * [DH85] by same author, _Étude dynamique des polynômes complexes. Partie II_ , Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985, With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac. MR 812271
  * [DS22] Kostiantyn Drach and Dierk Schleicher, _Rigidity of Newton dynamics_ , Adv. Math. 408 (2022), Paper No. 108591, 64. MR 4458154
  * [Fat26] Pierre Fatou, _Sur l’itération des fonctions transcendantes Entières_ , Acta Math. 47 (1926), no. 4, 337–370. MR 1555220
  * [FJ23] Núria Fagella and Anna Jové, _A model for boundary dynamics of Baker domains_ , Math. Z. 303 (2023), no. 4, Paper No. 95, 36. MR 4562228
  * [Her98] M. E. Herring, _Mapping properties of Fatou components_ , Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 2, 263–274. MR 1642181
  * [Hub93] J. H. Hubbard, _Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz_ , Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 467–511. MR 1215974
  * [Kis97] Masashi Kisaka, _On the local connectivity of the boundary of unbounded periodic Fatou components of transcendental functions_ , no. 988, 1997, Complex dynamical systems and related areas (Japanese) (Kyoto, 1996), pp. 113–119. MR 1605767
  * [KL09] Jeremy Kahn and Mikhail Lyubich, _Local connectivity of Julia sets for unicritical polynomials_ , Ann. of Math. (2) 170 (2009), no. 1, 413–426. MR 2521120
  * [KvS09] Oleg Kozlovski and Sebastian van Strien, _Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials_ , Proc. Lond. Math. Soc. (3) 99 (2009), no. 2, 275–296. MR 2533666
  * [LvS98] Genadi Levin and Sebastian van Strien, _Local connectivity of the Julia set of real polynomials_ , Ann. of Math. (2) 147 (1998), no. 3, 471–541. MR 1637647
  * [MB10] Helena Mihaljević-Brandt, _A landing theorem for dynamic rays of geometrically finite entire functions_ , J. Lond. Math. Soc. (2) 81 (2010), no. 3, 696–714. MR 2650792
  * [MB12] by same author, _Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds_ , Trans. Amer. Math. Soc. 364 (2012), no. 8, 4053–4083. MR 2912445
  * [McM94] Curtis T. McMullen, _Complex dynamics and renormalization_ , Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994\. MR 1312365
  * [Mil00] John Milnor, _Local connectivity of Julia sets: expository lectures_ , The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 67–116. MR 1765085
  * [Mil06] by same author, _Dynamics in one complex variable_ , third ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006\. MR 2193309 (2006g:37070)
  * [Mor99] Shunsuke Morosawa, _Local connectedness of Julia sets for transcendental entire functions_ , Nonlinear analysis and convex analysis (Niigata, 1998), World Sci. Publ., River Edge, NJ, 1999, pp. 266–273. MR 1742079
  * [Osb13] John W. Osborne, _Spiders’ webs and locally connected Julia sets of transcendental entire functions_ , Ergodic Theory Dynam. Systems 33 (2013), no. 4, 1146–1161. MR 3082543
  * [PS22] Leticia Pardo-Simón, _Orbifold expansion and entire functions with bounded Fatou components_ , Ergodic Theory Dynam. Systems 42 (2022), no. 5, 1807–1846. MR 4403147
  * [PZ04] Carsten Lunde Petersen and Saeed Zakeri, _On the Julia set of a typical quadratic polynomial with a Siegel disk_ , Ann. of Math. (2) 159 (2004), no. 1, 1–52. MR 2051390
  * [Roe08] P. Roesch, _On local connectivity for the Julia set of rational maps: Newton’s famous example_ , Ann. of Math. (2) 168 (2008), no. 1, 127–174. MR 2415400
  * [RY08] Pascale Roesch and Yongcheng Yin, _The boundary of bounded polynomial Fatou components_ , C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 877–880. MR 2441925
  * [RY22] by same author, _Bounded critical Fatou components are Jordan domains for polynomials_ , Sci. China Math. 65 (2022), no. 2, 331–358. MR 4372925
  * [Sul83] Dennis Sullivan, _Conformal dynamical systems_ , Geometric dynamics (Rio de Janeiro, 1981), Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR 730296
  * [TY96] Lei Tan and Yongcheng Yin, _Local connectivity of the Julia set for geometrically finite rational maps_ , Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233
  * [Why63] Gordon Thomas Whyburn, _Analytic topology_ , American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR 0182943 (32 #425)
  * [WYZ23] Xiaoguang Wang, Yongcheng Yin, and Jinsong Zeng, _Dynamics of Newton maps_ , Ergodic Theory Dynam. Systems 43 (2023), no. 3, 1035–1080. MR 4544153
  * [Yoc99] Jean-Christophe Yoccoz, _Dynamique des polynômes quadratiques_ , Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, Notes prepared by Marguerite Flexor, pp. x, xii, 187–222. MR 1760845


Let $\varphi:M\rightarrow N$ be a twisted differential operator of order at
most $n$ and $\psi:L\rightarrow M$ be a twisted differential operator of order
at most $m$. We have an exact sequence:

$\textstyle{P_{A}\otimes_{A}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{A}\otimes_{A}P_{A}\otimes_{A}L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{Id}\otimes\tilde{\psi}}$$\textstyle{P_{A}\otimes_{A}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\varphi}}$$\textstyle{N.}$

Moreover, $\tilde{\varphi}$ factorizes throught
$P_{A,(n)_{\underline{\sigma}}}\otimes_{A}^{\prime}M$ and
$\text{Id}\otimes_{A}\tilde{\psi}$ factorizes throught
$P_{A}\otimes_{A}P_{A,(m)_{\underline{\sigma}}}\otimes_{A}^{\prime}L$. Hence,
the composition factorizes throught
$P_{A,(n)_{\underline{\sigma}}}\otimes_{A}P_{A,(m)_{\underline{\sigma}}}\otimes_{A}^{\prime}L$.
We conclude using proposition 7.11 that it factorizes throught
$P_{A,(n+m)_{\underline{\sigma}}}\otimes_{A}^{\prime}L$. ∎

######  Proposition 8.4.

We have the following equalities

$\emph{Diff}_{0,\underline{\sigma}}(A)=A\text{ and
}\emph{Diff}_{1,\underline{\sigma}}(A)=A\oplus\text{T}_{A,\underline{\sigma}}.$

###### Proof.

The first one is direct by construction. For the second one, we can use
proposition 6.2 and the fact that $\text{T}_{A,\underline{\sigma}}$ is the
dual of $\Omega^{1}_{A,\underline{\sigma}}$. ∎

In the case where $x_{1},\ldots,x_{d}$ are classical coordinates, we denote by
$\overline{\text{D}}_{A,\underline{\sigma}}$ the smallest subring of
$\text{End}_{R}(A)$ containing $A$ and $\text{T}_{A,\underline{\sigma}}$ and
we denote by $\text{D}_{A,\underline{\sigma}}$ the non-commutative polynomial
ring $\partial_{\underline{\sigma},1},\ldots,\partial_{\underline{\sigma},d}$
on A with the following rules

(4) $\forall i\neq
j,\partial_{\underline{\sigma},i}\circ\partial_{\underline{\sigma},j}=\partial_{\underline{\sigma},j}\circ\partial_{\underline{\sigma},i}$
(5) $\text{and }\forall a\in
A,~{}\partial_{\underline{\sigma},i}a=\partial_{\underline{\sigma},i}(a)+\sigma_{i}(a)\partial_{\underline{\sigma},i}.$

######  Theorem 8.5.

If there exists some elements $\underline{q}=\\{q_{1},\ldots,q_{d}\\}$ of $R$
such that $x_{1},\ldots,x_{d}$ are classical $\underline{q}$-coordinates and

$\forall i=1,\ldots,d~{}\forall n\in\mathbb{N},~{}(n)_{q_{i}}\in R^{\times},$

then there exists an equivalence of categories

$\nabla_{\underline{\sigma}}^{\emph{Int}}\text{-}\emph{Mod}(A)\simeq\emph{D}_{A,\underline{\sigma}}^{(\infty)}\text{-}\emph{Mod}.$

###### Proof.

This can be deduced from proposition 7.14 and formula 3. ∎

######  Proposition 8.6.

If $x_{1},\ldots,x_{d}$ are classical coordinates, there exists a
factorization

$\emph{D}_{A,\underline{\sigma}}\twoheadrightarrow\overline{\emph{D}}_{A,\underline{\sigma}}\hookrightarrow\emph{D}^{(\infty)}_{A,\underline{\sigma}}.$

###### Proof.

The injection comes from proposition 8.4. For the surjection we can consider
for $i=1,\ldots,d,$ the map that sends $\partial_{\underline{\sigma},i}$ on
the corresponding endomorphism of $A$. The coordinates being symmetrical and
classical, the conditions 4 and 5 are satisfied. It ensures that the maps is
well defined and surjective by definition of
$\text{T}_{A,\underline{\sigma}}$. ∎

## References

  * [Abb10] Ahmed Abbes.  Éléments de Géométrie Rigide : Volume I. Construction et étude géométrique des espaces rigides.  Progress in Mathematics, vol. 286. Birkhäuser, 2010.
  * [ADV04] Yves André and Lucia Di Vizio.  $q$-difference equations and $p$-adic local monodromy.  Astérisque, (296):55–111, 2004.  Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. I.
  * [Aut23] The Stacks Project Authors.  Stacks Project.  2023\.
  * [BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze.  Integral p $p$ -adic Hodge theory.  Publications Mathématiques de L’IHÉS, 128(1):219–397, November 2018.
  * [Bou98] Nicolas Bourbaki.  Commutative algebra. Chapters 1–7.  Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998.
  * [BS22] Bhargav Bhatt and Peter Scholze.  Prisms and prismatic cohomology.  Annals of Mathematics, 196(3):1135 – 1275, 2022.
  * [DV02] Lucia Di Vizio.  Arithmetic theory of q-difference equations.  Inventiones mathematicae, 150(3):517–578, Dec 2002.
  * [GLQ19] Michel Gros, Bernard Le Stum, and Adolfo Quirós.  Twisted divided powers and applications.  Journal of Number Theory, pages 285–331, 2019.
  * [GLS14] Michel Gros and Bernard Le Stum.  Une neutralisation explicite de l’algèbre de Weyl quantique complétée.  Comm. Algebra, 42(5):2163–2170, 2014.
  * [Gro67] Alexander Grothendieck.  éléments de géométrie algébrique : IV. étude locale des schémas et des morphismes de schémas, Quatrième partie.  Publications Mathématiques de l’IHÉS, 32:5–361, 1967.
  * [Hub93] R. Huber.  Continuous valuations.  Math. Z., 212(3):455–477, 1993.
  * [LSQ15] Bernard Le Stum and Adolfo Quirós.  On quantum integers and rationals.  In Trends in number theory, volume 649 of Contemp. Math., pages 107–130. 2015.
  * [LSQ18a] Bernard Le Stum and Adolfo Quirós.  Formal confluence of quantum differential operators.  Pacific Journal of Mathematics, 292(2):427–478, 2018.
  * [LSQ18b] Bernard Le Stum and Adolfo Quirós.  Twisted calculus.  Communications in Algebra, 46(12):5290–5319, 2018.
  * [LSQ20] Bernard Le Stum and Adolfo Quirós.  Twisted calculus on affinoid algebras.  Pacific J. Math., 304(2):523–560, 2020.
  * [Mor19] Sophie Morel.  Adic Spaces.  2019\.
  * [MT21] Matthew Morrow and Takeshi Tsuji.  Generalised representations as q-connections in integral $p$-adic hodge theory, 2021.
  * [Pul08] Andrea Pulita.  p-adic confluence of q-difference equations.  Compositio Mathematica, 144(4):867–919, 2008.
  * [Sab93] Claude Sabbah.  Systèmes holonomes d’équations aux $q$-différences.  In $D$-modules and microlocal geometry (Lisbon, 1990), pages 125–147. de Gruyter, Berlin, 1993.
  * [Wed19] Torsten Wedhorn.  Adic spaces.  2019\.