11institutetext: Department of Computer Science and Technology, Ocean
University of China, Qingdao, China
11email: wangzhenpeng<EMAIL_ADDRESS>22institutetext: School of Management Engineering, Qingdao University of
Technology, China
22email<EMAIL_ADDRESS>33institutetext: School of Information and
Electronics, Beijing Institute of Technology, Beijing 100080, China
33email<EMAIL_ADDRESS>
# RAUNE-Net: A Residual and Attention-Driven Underwater Image Enhancement
Method

Wangzhen Peng 11 0000-0002-2333-5415 Chenghao Zhou 22 Runze Hu 33 Jingchao Cao
11 Yutao Liu(✉) 11

###### Abstract

Underwater image enhancement (UIE) poses challenges due to distinctive
properties of the underwater environment, including low contrast, high
turbidity, visual blurriness, and color distortion. In recent years, the
application of deep learning has quietly revolutionized various areas of
scientific research, including UIE. However, existing deep learning-based UIE
methods generally suffer from issues of weak robustness and limited
adaptability. In this paper, inspired by residual and attention mechanisms, we
propose a more reliable and reasonable UIE network called RAUNE-Net by
employing residual learning of high-level features at the network’s bottle-
neck and two aspects of attention manipulations in the down-sampling
procedure. Furthermore, we collect and create two datasets specifically
designed for evaluating UIE methods, which contains different types of
underwater distortions and degradations. The experimental validation
demonstrates that our method obtains promising objective performance and
consistent visual results across various real-world underwater images compared
to other eight UIE methods. Our example code and datasets are publicly
available at https://github.com/fansuregrin/RAUNE-Net.

###### Keywords:

Underwater image enhancement Deep learning Image processing Deep Neural
Network Attention Residual.

##  1 Introduction

Underwater image enhancement (UIE), a challenging yet promising research
direction, has garnered growing interest among researchers in recent years.
Nowadays, a wide array of methods have been put forward for enhancing
underwater images [13]. These UIE techniques can be broadly categorized into
four types [8]: supplementary information-based, non-physical model-based,
physical model-based, and data-driven methods. With the rise of deep learning
and the utilization of large-scale datasets, data-driven methods have
revolutionized UIE. However, it is worth noting that data-driven UIE methods
still have several limitations. We have evaluated these previously proposed
deep learning-based methods on multiple real-world underwater image datasets.
As illustrated in Figure 1, these five methods tend to produce visually poor
results when dealing with images that they are not proficient at enhancing.
Therefore, further research and development are necessary to overcome these
challenges and create more reliable and adaptable UIE techniques.

Figure 1: Poor results of five UIE methods with weak robustness

Based on above analysis, we propose a Residual and Attention-driven UNderwater
image Enhancement Network called RAUNE-Net to process a wide variety of
underwater images. Our main contributions can be summarized as follows:

  * $\cdot$

We incorporate residual and attention mechanisms into UIE to construct an
effective and efficient network called RAUNE-Net, which is capable of handling
a wide range of underwater scenes comprehensively.

  * $\cdot$

Our method achieves remarkable objective performance and consistent subjective
results on both referenced and non-referenced real-world underwater image
testsets.

##  2 Related Work

###  2.1 Convolutional Neural Network-based UIE methods

In the early stages, Wang et al. [27] first presented an end-to-end framework
(i.e., UIE-Net) based on Convolutional Neural Network (CNN) for enhancing
visual quality of underwater images. In [7], an underwater image dataset was
synthesized firstly through an underwater image formulation algorithm based on
underwater scene priors, which includes two types of water environments and
five levels of distortions. Subsequently, they devised a lightweight CNN
network (i.e., UWCNN) to enhance images from different underwater scenes. In
contrast to aforementioned methods that used synthetic datasets for model
training, Li et al. [8] created a real-world underwater image dataset (i.e.,
UIEB). Then, they trained an enhancement network called Water-Net on this
dataset. In detail, they generated three input images by applying white
balance, histogram equalization, and gamma correction algorithms to the
original underwater image. These inputs were then transformed into refined
inputs using the Feature Transformation Units. Subsequently, Water-Net
utilized a gated fusion network to learn three confidence maps, which were
used to combine three refined inputs and produce an enhanced result. These
methods either rely on synthesized images for model training or require
preprocessing on raw underwater images, which still have much room for
improvement.

###  2.2 Generative Adversarial Network-based UIE methods

Li et al. [11] introduced an approach called WaterGAN, which adopted a GAN [4]
to learn realistic representations of water column properties. Then, they used
these generated underwater images along with their corresponding depth maps to
train a color restoration network capable of real-time compensation for water
column effects at a specific location. Unlike the previous method, where GAN
was used solely for generating training samples, in [3], a GAN-based network
(i.e., UGAN), which utilizes the Wasserstein GAN [1] with gradient penalty,
was employed to directly enhance distorted underwater images. Subsequently,
Islam et al. [6] proposed a network called FUnIE-GAN. In order to supervise
the adversarial training process, they designed several loss functions based
on global content, color, local texture, and style information. In [9], a
weakly supervised network based on CycleGAN was applied to the UIE task. It
maps color and style of underwater images to those of above-water natural
images, thereby correcting the color cast in underwater images.

###  2.3 Other deep learning-based UIE methods

Differing from CNN, which adopts local receptive kernels to extract features
from images, Vision Transformer (ViT) [2] treats an image as a sequence
composed of multiple patches and uses self-attention computation to capture
global information of the image. Inspired by this, Peng et al. [21] introduced
Transformer into UIE for the first time. They proposed a U-shape Transformer
network consisting of a channel-wise multi-scale feature fusion transformer
module and a spatial-wise global feature modeling transformer. In addition,
some UIE methods have reexamined physical models and combined deep neural
networks with them. SyreaNet [29] integrated both synthetic and real-world
data under the guidance of a revised underwater image formation model to
fulfil the enhancement aim. Furthermore, apart from using physical models to
guide the UIE models, there are also methods aiming to leverage high-level
information to guide learning process of the network. In [22], they managed to
use information from semantic segmentation to force network to learn region-
wise enhancement features from underwater images and their paired high-quality
reference images.

##  3 Methodology

###  3.1 RAUNE-Net

We have observed significant improvements in objective metrics for some deep
neural networks that utilized residual [5] and attention [30] mechanisms.
Consequently, we combine residual modules and attention blocks to design the
RAUNE-Net. As demonstrated in Figure 2, our network follows an end-to-end
manner. Given a raw underwater image, by feeding it through the network, we
can obtain an enhanced result. Mathematically, this enhancement process can be
written as:

$Y_{e}=\Phi(X_{i}),$ (1)

where $X_{i}$ are input underwater images, $\Phi$ is RAUNE-Net, and $Y_{e}$
are enhanced outputs.

Figure 2: The data-flow and architecture of RAUNE-Net

Overall, the RAUNE-Net consists of a wide-range perception module (WRPM), an
attention down-sampling module (ADM), a high-level features residual-learning
module (HFRLM), an up-sampling module (UM), and a feature map smoothing module
(FMSM) followed by a Tanh activation layer. The detail structure of our
network is illustrated in Figure 3.

In detail, the WRPM contains a reflection padding layer, a convolutional
layer, a normalization layer, and a non-linear activation layer. Specifically,
this convolutional layer has a large kernel size of $7\times 7$, which can
cover a larger receptive field, enabling the network to capture more extensive
contextual information, identify the overall structure and important features
in the image. We set the default number of convolutional kernels to 64, use
Instance Normalization [26] as the default normalization layer, and employ
ReLU as the activation function.

ADM consists of multiple blocks, each block containing a $4\times 4$ kernel
size convolutional layer with a stride of 2 and a padding size of 1, followed
by a normalization layer, a LeakyReLU [20] activation layer, an optional
dropout [25] layer, and an attention module. Specifically, the attention
module in ADM combines channel attention calculation and spatial attention
calculation in a sequential manner [30]. After down-sampling, HFRLM consists
of several residual blocks, where each residual block contains two
convolutional layers with padding layers placed before them and normalization
layers assigned after them. A ReLU activation layer and an optional dropout
layers are inserted between the two convolutional layers to prevent
overfitting and improve the network’s robustness.

Similar to down-sampling stage, the UM is composed of multiple up-sampling
blocks. Each up-sampling block includes a convolutional transpose layer with a
kernel size of $4\times 4$, stride of 2, and padding size of 1, followed by a
normalization layer, a non-linear activation layer, and an optional dropout
layer. After up-sampling, we do not directly output the result. Instead, we
add a module called FMSM, which consists of a reflection padding layer and a
convolutional layer with a kernel size of $7\times 7$. We adopt this module
because we have observed that the enhanced images often exhibit varying
degrees of checkerboard artifacts. By performing a convolution operation with
a large kernel, we can smooth the feature maps and reduce the occurrence of
jagged edges and noise. This makes the up-sampled images more natural, thereby
improving the visual quality [19, 14] of the output. Lastly, we incorporate a
Tanh non-linear activation layer to map the pixel values of the image to the
range of -1 to 1, enhancing its contrast and color variation range.

Figure 3: The detail structure of RAUNE-Net

###  3.2 Loss Function

To train our model, we use a weighted combination of three types of loss
functions as the overall loss, i.e., the pixel content loss
$\mathcal{L}_{pcont}$, the structural similarity loss $\mathcal{L}_{ssim}$,
and the semantic content loss $\mathcal{L}_{scont}$:

$\mathcal{L}=\lambda_{pcont}\cdot\mathcal{L}_{pcont}+\lambda_{ssim}\cdot\mathcal{L}_{ssim}+\lambda_{scont}\cdot\mathcal{L}_{scont}.$
(2)

####  3.2.1 Pixel Content Loss

The pixel content loss is designed to measure the discrepancy in pixel values
between the output image $Y_{e}$ generated by the model and the reference
image $Y_{ref}$ across the red, green, and blue color channels. Here, to
reduce computational cost, we utilize the absolute distance between pixel
values to represent the difference between the two images. The mathematical
formula is defined as:

$\mathcal{L}_{pcont}=\mathbbm{E}\left[{\left\|Y_{e}-Y_{ref}\right\|}_{1}\right].$
(3)

####  3.2.2 Structural Similarity Loss

Structural Similarity Index Measure (SSIM) [28, 18] is a metric that
quantifies the structural similarity between two images based on luminance,
contrast, and structural information. The main purpose of incorporating this
metric into the loss function is to encourage the network to generate enhanced
images that are closer to the ground truth in these three aspects.
Specifically, the value of SSIM ranges from 0 to 1, and a value closer to 1
indicates a higher similarity to the reference image. However, typically in
optimization, the value of objective function is minimized. Therefore, during
the training process, the expression for SSIM Loss is:

$\mathcal{L}_{ssim}=\mathbbm{E}\left[\frac{1-SSIM\left(Y_{e},Y_{ref}\right)}{2}\right],$
(4)

where $Y_{e}$ are enhanced images, $Y_{ref}$ are reference images, and the
$SSIM\left(*,*\right)$ [28] represents structural similarity index map between
two images. Additionally, the index map is calculated as:

$SSIM(x,y)=\frac{\left(2\mu_{x}\mu_{y}+c_{1}\right)\left(2\sigma_{xy}+c_{2}\right)}{\left(\mu_{x}^{2}+\mu_{y}^{2}+c_{1}\right)\left(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2}\right)},$
(5)

where the $\mu_{x}$ and $\mu_{y}$ represent the pixel sample mean of image $x$
and image $y$, the $\sigma_{y}^{2}$, $\sigma_{y}^{2}$ and $\sigma_{xy}$ stand
for variances and covariance of $x$ and $y$, and $c_{1}$ and $c_{2}$ are two
variables to stabilize the division with weak denominator, respectively.

####  3.2.3 Semantic Content Loss

In networks used for image classification, the features obtained from deeper
convolutional layers to some extent represent the semantic content of the
input image. Therefore, we consider extracting high-level features obtained
from last several layers of a classification network to evaluate the semantic
discrepancy between the enhanced image and the reference image. Specifically,
we utilize the pre-trained VGG19_BN [24] model trained on the ImageNet-1k_v1
[23] dataset. As shown in Figure 4, we extract feature maps obtained from the
last five convolutional layers for both the enhanced image and the reference
image. Finally, we calculate the weighted sum of the L1 distances between each
pair of feature maps, which serves as the semantic content loss.
Mathematically, this loss can be written as:

$\mathcal{L}_{scont}=\sum_{i=1}^{5}{k_{i}\cdot
MAE(\Omega_{i}(Y_{e}),\Omega_{i}(Y_{ref}))},$ (6)

where $k_{i}$ is the $i$-th weight for summing, $MAE(*,*)$ is the Mean
Absolute Error (i.e., L1 loss), and the $\Omega_{i}(*)$ is the last $i$-th
convolutional layer in VGG19_BN.

Figure 4: Diagram of the semantic content loss

##  4 Experiments

###  4.1 Datasets

Firstly, we select the LSUI3879 dataset for training, which consists of 3879
real-world underwater images and their corresponding reference images randomly
partitioned from the LSUI [21] dataset. For testing, we use four different
datasets, namely LSUI400 (the remaining 400 pairs of images from LSUI),
UIEB100 (100 pairs of images randomly selected from UIEB [7]), OceanEx, and
EUVP_Test515 (515 pairs of test samples from EUVP [6]). In particular, OceanEx
is a testset collected and constructed by ourselves. We gather 40 high-quality
underwater images from NOAA Ocean Exploration. Then, we apply CycleGAN [31] to
add underwater distortions and degradation styles to these images, making them
the samples to be enhanced, while keeping the original high-quality images as
reference images. These four datasets are used for objective evaluation of
networks’ performance, with the evaluation metrics being peak signal-to-noise
ratio (PSNR) and structural similarity (SSIM). In addition to that, we also
employ three datasets for subjective evaluation. They are U45 [10],
RUIE_Color90 (90 images randomly selected from the UCCS subset of RUIE [12]
dataset), and UPoor200. Among them, UPoor200 is a dataset we collected and
curated from the internet, which consists of 200 real-world underwater images
with poor visual quality. It includes distortions [17] such as blue-green
color cast, low lighting, blurriness [16], noise, and haze.

###  4.2 Implementation Details

Before training, the input images are first resized to a resolution of
$256\times 256$ and then normalized. The normalization is performed by setting
the mean and standard deviation to 0.5. During training, the optimizer we used
is Adam with a learning rate of 0.0001. The two coefficients (i.e, betas) of
this optimizer are set to 0.9 and 0.999 respectively. The weights assigned to
pixel content loss, SSIM loss, and semantic content loss are 1.0, 1.0, and 1.0
respectively. During each training process, the network is trained for 100
epochs, and the trainable parameters are saved every five epochs. Every 500
iterations, the enhanced results of four sampled images from the testset are
displayed. We conducted a series of experiments on the number of attention
modules and residual modules in RAUNE-Net. Additionally, we compared RAUNE-Net
with eight other UIE methods (i.e., UT-UIE [21], SyreaNet [29], WaterNet [8],
UGAN [3], FUnIE-GAN [6], UWCNN [7], SGUIE-Net [22], Cycle-GAN [9]). To ensure
a fair comparison, we retrained some networks mentationed in these methods
(i.e., WaterNet, UGAN, and FUnIE-GAN) using the LSUI3879 dataset.

###  4.3 Objective Evaluations

The results of objective evaluation [15] of nine UIE methods, including RAUNE-
Net and other eight UIE methods compared to it, are demonstrated in Tabel 1.
From the table, our RAUNE-Net achieves highest PSNR values and SSIM values,
which are marked as red color, on LSUI400, EUVP_Test515 and UIEB100 testsets.
On OceanEx, RAUNE-Net achieves the highest PSNR value and the second highest
SSIM value, but the SSIM value is already very close to the highest value
obtained by WaterNet. It is worth noting that WaterNet achieves slightly lower
performance values (except for SSIM value on OceanEx), which are indicated
with blue color, than RAUNE-Net on the four test sets. The reason for the
improved performance after retraining is the use of SSIM loss and the larger
training dataset (i.e., LSUI), which contains more diverse underwater scenes.
The UWCNN consistently achieves the lowest objective values on all four
datasets. This is mainly due to its training on synthesized images using a
physical model, which results in weak generalization and poor adaptability.

Table 1: Objective evaluation results of different UIE methods Methods | LSUI400 | EUVP_Test515 | UIEB100 | OceanEx
---|---|---|---|---
PSNR | SSIM | PSNR | SSIM | PSNR | SSIM | PSNR | SSIM
UT-UIE [21] | 24.351 | 0.829 | 25.214 | 0.813 | 20.916 | 0.764 | 21.270 | 0.822
SyreaNet [29] | 18.050 | 0.766 | 17.721 | 0.743 | 16.501 | 0.836 | 20.243 | 0.865
WaterNet [8] | 26.688 | 0.874 | 25.285 | 0.833 | 22.279 | 0.868 | 22.132 | 0.887
UGAN [3] | 25.117 | 0.846 | 23.636 | 0.805 | 21.368 | 0.825 | 22.436 | 0.822
FUnIE-GAN [6] | 23.272 | 0.818 | 24.077 | 0.794 | 19.614 | 0.813 | 20.448 | 0.855
UWCNN [7] | 17.366 | 0.725 | 17.725 | 0.704 | 14.155 | 0.686 | 15.960 | 0.724
SGUIE-Net [22] | 19.910 | 0.819 | 19.187 | 0.760 | 21.178 | 0.872 | 18.677 | 0.834
Cycle-GAN [9] | 18.320 | 0.749 | 17.963 | 0.709 | 17.714 | 0.758 | 21.007 | 0.828
RAUNE-Net | 26.812 | 0.876 | 26.331 | 0.845 | 22.751 | 0.879 | 22.728 | 0.876

###  4.4 Subjective Evaluations

As can be observed from Figure 5, RAUNE-Net is capable of enhancing bluish,
greenish and hazy underwater images, while the other methods have exhibited
improper or insufficient handling of these images. Specifically, UT-UIE and
FUnIE-GAN incompletely remove the green color tone when processing the second
and third images, resulting in noticeable green color residue in the results.
Although SyreaNet effectively eliminates the blue-greenish color tone and
foggy effect, it reduces the clarity and brightness of the source images. From
the first four images, we can clearly see that WaterNet only partially removes
the blue-green bias, as there are still blue or green artifacts preserved at
the edges of objects in the images. UGAN, while not showing unpleasant
artifacts, fails to effectively remove the blue and green color bias. For
example, in the second result image, a large area of green color is retained,
and in the fourth result image, the blue color style is still prominent.

Unfortunately, UWCNN completely fails to enhance the original image and
instead makes the input image appear more turbid and with reduced visibility.
This is particularly evident in the last two result images. Similarly, SGUIE-
Net’s ability to handle color shifts is not as impressive as that of RAUNE-
Net. CycleGAN also failed to properly handle the green color shift and
introduced strange textures that altered the original image content.
Furthermore, as shown in Figure 6 and 7, we can observe consistent results
from subjective evaluations on other two testsets. All in all, compared to
other eight enhancement methods, RAUNE-Net coherently achieves remarkable
visual effects.

Figure 5: Subjective evaluation results of different methods on U45 Figure 6:
Subjective evaluation results of different methods on RUIE_Color90 Figure 7:
Subjective evaluation results of different methods on UPoor200

##  5 Conclusion

In this paper, we devised a novel underwater image enhancement structure named
RAUNE-Net driven by residual learning, channel attention and spatial
attention. We found that the residual learning on high-level features can help
the network to learn more detail information to enhance input images and the
channel attention and spatial attention blocks connected sequentially can
assist the enhancing system to notice important areas to be enhanced and how
to enhance them. By employing objective and subjective evaluations on seven
testsets, we discovered that our network obtain an outstanding and reliable
enhancing capability on various underwater attenuation types, such as bluish
or greenish color shift, hazy or foggy distortion, and low-light condition. To
contrive a fair and generalized assessment for different UIE methods’
performances, we also constructed two datasets (a set with reference images
called OceanEX and the other set with no ground-truth images named UPoor200),
which contains diverse kinds of real-world underwater images. Furthermore, we
found using PSNR and SSIM as indicators of the objective performance of
enhanced images is still inaccurate, a non-reference quality assessment of
these enhanced images is a focal point of our future research.

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Updated determinations of $\left|V_{us}\right|$ with tau decays using new
recent estimates of radiative corrections for light-meson leptonic decay rates

A. Lusiani1*,

1 Scuola Normale Superiore and INFN sezione di Pisa, Italy

*<EMAIL_ADDRESS>
16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021 10.21468/SciPostPhysProc.?

## Abstract

We update the $\left|V_{us}\right|$ determinations using the HFLAV 2018 report
tau branching fraction results with recent new estimates of the $\pi\ell 2$
and $K\ell 2$ radiative corrections. There are minor changes of the central
values and uncertainties.

###### Contents

  1. 1 Introduction
  2. 2 $\left|V_{us}\right|$ from $B(\tau\to K\nu)/B(\tau\to\pi\nu)$ and $B(\tau\to K\nu)$
  3. 3 Conclusions

##  1 Introduction

Recent measurements of $\left|V_{ud}\right|$, $\left|V_{us}\right|$ and
$\left|V_{ub}\right|$ are not consistent with the unitarity condition on the
first row of the CKM matrix [1, 2]. Tau decay measurements are used to
determine $\left|V_{us}\right|$ [3, 4], supplementing the more precise
determinations that are obtained using kaon decays. All these estimates also
rely on lattice QCD estimates of form factors and decay constants[5, 6], with
the exception of the $\left|V_{us}\right|$ determinations based on the total
branching fraction of the tau lepton into strange final states. Two of the
$\left|V_{us}\right|$ determinations using tau measurements rely on estimates
of the radiative corrections for the branching fractions $B(\tau\to\pi/K\nu)$,
which are computed using also the radiative corrections for $B(\pi\to\ell\nu)$
and $B(K\to\ell\nu)$ [7]. New estimates of these radiative corrections have
been computed with a novel approach using lattice QCD+QED [8]. We evaluate in
the following the impact of these new estimates on the $\left|V_{us}\right|$
determinations using tau decay measurements.

##  2 $\left|V_{us}\right|$ from $B(\tau\to K\nu)/B(\tau\to\pi\nu)$ and
$B(\tau\to K\nu)$

We use the tau branching fractions of the HFLAV 2018 report fit [3] and we
compute $\left|V_{us}\right|$ using the updated external inputs provided by
the Review of Particle Physics [9], by the FLAG review of lattice QCD
calculations [5, 6], by CODATA 2018 [10]. In updating the CODATA constants
from the values used in the HFLAV 2018 report, a numerical trascription error
that slightly affected this $\left|V_{us}\right|$ determination in the HFLAV
2018 report has been corrected. Finally, we obtain $\left|V_{us}\right|$ using
both the original [7] and the recently published [8] radiative corrections.

We compute $\left|V_{us}\right|$ as in the HFLAV 2018 report from the ratio of
branching fractions $B(\tau^{-}\to
K^{-}\nu_{\tau})/B(\tau^{-}\to\pi^{-}\nu_{\tau})$ and from the branching
fraction $B(\tau^{-}\to K^{-}\nu_{\tau})$ using the equations

$\displaystyle\frac{B(\tau^{-}\to
K^{-}\nu_{\tau})}{B(\tau^{-}\to\pi^{-}\nu_{\tau})}$
$\displaystyle=\frac{f_{K\pm}^{2}\left|V_{us}\right|^{2}}{f_{\pi\pm}^{2}\left|V_{ud}\right|^{2}}\frac{\left(m_{\tau}^{2}-m_{K}^{2}\right)^{2}}{\left(m_{\tau}^{2}-m_{\pi}^{2}\right)^{2}}\frac{1+\delta
R_{\tau/K}}{1+\delta R_{\tau/\pi}}(1+\delta R_{K\mu 2/\pi\mu 2})~{},$
$\displaystyle B(\tau^{-}\to K^{-}\nu_{\tau})$
$\displaystyle=\frac{G^{2}_{F}}{16\pi{\vphantom{h}\ooalign{\kern
0.50003pt\smash{\raisebox{3.87495pt}{\scalebox{0.7}{\rotatebox[origin={c}]{18.0}{$-$}}}}\cr$h$\cr}\kern
0.50003pt}}f^{2}_{K\pm}\left|V_{us}\right|^{2}\tau_{\tau}m_{\tau}^{3}\left(1-\frac{m_{K}^{2}}{m_{\tau}^{2}}\right)^{2}(1+\delta
R_{\tau/K})(1+\delta R_{K\mu 2})~{},$

respectively. We use Refs. [11, 12, 13, 14] to compute the radiative-
correction factor

$\displaystyle\frac{1+\delta R_{\tau/K}}{1+\delta
R_{\tau/\pi}}=\frac{1+(0.90\pm 0.22)\%}{1+(0.16\pm 0.14)\%}~{}.$

$\left|V_{ud}\right|=0.97373\pm 0.00031$ is taken from a 2020 updated
determination [1].

The radiation correction terms $\delta R_{K\mu 2/\pi\mu 2}=-0.69\pm 0.17$ and
$\delta R_{K\mu 2}=1.07\pm 0.21$ are provided without the isospin-breaking
corrections by Refs. [15, 7, 16, 17, 18]. The same sources report also $\delta
R_{\pi\mu 2}=1.76\pm 0.21$. The three estimates are consistent with a
correlation of $0.67$ between $\delta R_{\pi\mu 2}$ and $\delta R_{K\mu 2}$,
which is used when computing $\left|V_{us}\right|$. These radiative correction
terms are used with the lattice QCD decay constants that include isospin-
breaking corrections from the FLAG 2019 lattice QCD averages with
$N_{f}=2+1+1$: $f_{K\pm}/f_{\pi\pm}=1.1932\pm 0.0021$ [5, 19, 20, 21, 22] and
$f_{K\pm}=155.7\pm 0.3\,\,\text{Me\kern-1.00006ptV}$ [5, 20, 23, 21].

New estimates of the radiation correction terms inclusive of isospin-breaking
corrections $\delta R^{\prime}_{K\mu 2/\pi\mu 2}=-1.26\pm 0.14$ and $\delta
R^{\prime}_{K\mu 2}=0.24\pm 0.10$ are provided by Ref. [8]. The same source
reports also $\delta R^{\prime}_{\pi\mu 2}=1.53\pm 0.19$. The three estimates
are consistent with a correlation of $0.63$ between $\delta R^{\prime}_{\pi\mu
2}$ and $\delta R^{\prime}_{K\mu 2}$, which originates from a correlation of
0.794 between $\delta R^{\prime}_{\pi\mu 2}$ the non-isospin-breaking
component of $\delta R^{\prime}_{K\mu 2}$ [24]. These radiative correction
terms are used with the isospin-symmetric lattice QCD decay constants with
$N_{f}=2+1+1$ provided by Ref. [8]: $f_{K}/f_{\pi}=1.1966\pm 0.0018$ and
$f_{K}=156.1\pm 0.2\,\,\text{Me\kern-1.00006ptV}$.

Table 1 reports the values of $\left|V_{us}\right|_{\tau K/\pi}$ from
$B(\tau\to K\nu)/B(\tau\to\pi\nu)$ and $\left|V_{us}\right|_{\tau K}$ from
$B(\tau\to K\nu)$ using both the original and the new $\pi\ell 2$ and $K\ell
2$ radiative corrections. The improvements in the precision of the radiative
corrections results in minor improvements on the $\left|V_{us}\right|$
determinations because of other larger contributing uncertainties. The changes
of the central values are also minor compared with the total uncertainties.
Figure 1 reports the $\left|V_{us}\right|$ determinations in this document
compared with $\left|V_{us}\right|$ from the CKM matrix unitarity and
$\left|V_{us}\right|$ from kaon decay measurements.

Table 1: $\left|V_{us}\right|$ determinations. $\left|V_{us}\right|_{\tau s}$
denotes $\left|V_{us}\right|$ obtained with the $\tau\to X_{s}\nu$ inclusive
method using the HFLAV 2018 report [3], complemented with the updated external
inputs described in the text. $\left|V_{us}\right|_{\text{uni}}$ denotes the
value of $\left|V_{us}\right|$ assuming the CKM matrix unitarity,
$\left|V_{ud}\right|$ from Ref. [1] and $\left|V_{us}\right|$ from Ref. [9].
The determinations of $\left|V_{us}\right|_{\tau K/\pi}$ and
$\left|V_{us}\right|_{\tau K}$ using the exclusive tau branching fractions is
reported using both the original and the new estimates of the $\pi\ell 2$ and
$K\ell 2$ radiative corrections. $\left|V_{us}\right|_{\tau}$ denotes the
combination of all three $\left|V_{us}\right|$ determinations with tau decay
measurements. The signed difference with respect to
$\left|V_{us}\right|_{\text{uni}}$ in standard deviations is reported for all
$\left|V_{us}\right|$ determinations.

.

| $\left|V_{us}\right|$
---|---
$\left|V_{us}\right|_{\text{uni}}$ |  $0.2277\pm 0.0013$ $\hphantom{-}0.0\,\sigma$
$\left|V_{us}\right|_{\tau s}$ |  $0.2192\pm 0.0019$ $-3.6\,\sigma$
| Cirigliano & Neufeld 2011 | Di Carlo et al.2019
$\left|V_{us}\right|_{\tau K/\pi}$ | $0.2234\pm 0.0015$ | $-2.0\,\sigma$ | $0.2235\pm 0.0015$ | $-2.0\,\sigma$
$\left|V_{us}\right|_{\tau K}$ | $0.2226\pm 0.0015$ | $-2.6\,\sigma$ | $0.2229\pm 0.0014$ | $-2.6\,\sigma$
$\left|V_{us}\right|_{\tau}$ | $0.2217\pm 0.0013$ | $-3.2\,\sigma$ | $0.2219\pm 0.0013$ | $-3.1\,\sigma$

\begin{overpic}[trim=0.0pt 0.0pt 0.0pt 0.0pt,width=346.89731pt,clip]{hflav-
tau-work-plots/vus-tau-hflav-2018-upd2021.pdf} \end{overpic}

Figure 1: $\left|V_{us}\right|$ determinations using tau decay measurements
with the new radiative corrections in Ref. [8], compared with
$\left|V_{us}\right|$ from the CKM matrix unitarity and $\left|V_{us}\right|$
from kaon measurements in the Review of Particle Physics [9].

##  3 Conclusions

Recent new slightly more precise estimates of the $\pi\ell 2$ and $K\ell 2$
radiative corrections have been used to update the $\left|V_{us}\right|$
determination using the tau branching fractions of the HFLAV 2018 report,
resulting in minor changes of the central values and uncertainties.

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# Transcendental Semigroup that has Simply Connected Fatou Components

Bishnu Hari Subedi Central Department of Mathematics
Institute of Science and Technology
Tribhuvan University
Kirtipur, Kathmandu
Nepal<EMAIL_ADDRESS>/<EMAIL_ADDRESS>and  Ajaya Singh
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal<EMAIL_ADDRESS>/
<EMAIL_ADDRESS>
###### Abstract.

We prove that there exists a non-trivial transcendental semigroup $S$ such
that the wandering (or pre-periodic or periodic) components of Fatou set
$F(S)$ has at least a simply connected domain $D$.

###### Key words and phrases:

Fatou set, pre-periodic component, periodic component, wandering component,
transcendental semigroup, Carleman set

###### 2010 Mathematics Subject Classification:

37F10, 30D05

This research work of first author is supported by PhD faculty fellowship from
University Grants Commission, Nepal.

##  1\. Introduction

We denote the complex plane by $\mathbb{C}$, extended complex plane by
$\mathbb{C_{\infty}}$ and set of integers greater than zero by $\mathbb{N}$.
We assume the function $f:\mathbb{C}\rightarrow\mathbb{C}$ is transcendental
entire function (TEF) unless otherwise stated. For any
$n\in\mathbb{N},\;\;f^{n}$ always denotes the nth iterates of $f$. If
$f^{n}(z)=z$ for some smallest $n\in\mathbb{N}$, then we say that $z$ is
periodic point of period n. In particular, if $f(z)=z$, then $z$ is a fixed
point of $f$. If $|(f^{n}(z))^{\prime}|<1$, where $\prime$ represents complex
differentiation of $f^{n}$ with respect to $z$, then $z$ is called attracting
periodic point. A family $\mathscr{F}=\\{f:f\;\text{is meromorphic on some
domain}\;X\;of\;\mathbb{C}_{\infty}\\}$ forms normal family if every sequence
$(f)_{i\in\mathbb{N}}$ of functions contains a subsequence which converges
uniformly to a finite limit or converges to $\infty$ on every compact subset
$D$ of $X$. The Fatou set of $f$ denoted by $F(f)$ is the set of points
$z\in\mathbb{C}$ such that sequence $(f^{n})_{n\in\mathbb{N}}$ forms a normal
family in some neighborhood of $z$. That is, $z\in F(f)$ if $z$ has a
neighborhood $U$ on which the family $\mathscr{F}$ is normal. By definition,
Fatou set is open and may or may not be empty. Fatou set is non-empty for
every entire function with attracting periodic points. If $U\subset F(f)$
(Fatou component), then $f(U)$ lies in some component $V$ of $F(f)$ and
$V-f(U)$ is a set which contains at most one point (see for instance [3]). Let
$U\subset F(f)$ (a Fatou component) such that $f^{n}(U)$ for some
$n\in\mathbb{N}$, is contained in some component of $F(f)$, which is usually
denoted by $U_{n}$. A Fatou component $U$ is called pre-periodic if there
exist integers $n>m\geqslant 0$ such that $U_{n}=U_{m}$. In particular, if
$U_{n}=U_{0}=U$ ( that is, $f^{n}(U)\subset U$) for some smallest positive
integer $n\geqslant 1$, then $U$ is called periodic Fatou component of period
n and $\\{U_{0},U_{1}\ldots,U_{n-1}\\}$ is called the periodic cycle of $U$.
In the case, if $U_{1}=f(U)\subset U$, then $U$ is called invariant domain. A
component of Fatou set $F(f)$ which is not pre-periodic is called wandering
domain.

For the complex plane $\mathbb{C}$, let us denote the set of all holomorphic
functions of $\mathbb{C}$ by $\text{Hol}(\mathbb{C})$. If
$f\in\text{Hol}(\mathbb{C})$, then $f$ is a polynomial or transcendental
entire function. The set $\text{Hol}(\mathbb{C})$ forms a semigroup with
semigroup operation being the functional composition.

######  Definition 1.1 (Transcendental semigroup).

Let $A=\\{f_{i}:i=1,2,\ldots\\}\subset\text{Hol}(\mathbb{C})$ be a set of
transcendental entire functions $f_{i}:\mathbb{C}\rightarrow\mathbb{C}$. A
transcendental semigroup $S$ is a semigroup generated by the set $A$ with
semigroup operation being the functional composition. We denote this semigroup
by $S=\langle A\rangle=\langle f_{i}:i=1,2,3,\ldots\rangle$ or simply by
$S=\langle f_{1},f_{2},f_{3},\cdots,f_{n},\cdots\rangle$.

Here, each $f\in S$ is the transcendental entire function and $S$ is closed
under functional composition. Thus $f\in S$ is constructed through the
composition of finite number of functions $f_{i_{k}},\;(k=1,2,3,\ldots,m)$.
That is, $f=f_{i_{1}}\circ f_{i_{2}}\circ f_{i_{3}}\circ\cdots\circ
f_{i_{m}}$.

A semigroup generated by finitely many functions $f_{i},(i=1,2,3,\ldots,n)$ is
called finitely generated transcendental semigroup. We write $S=\langle
f_{1},f_{2},\ldots,f_{n}\rangle$. If $S$ is generated by only one
transcendental entire function $f$, then $S$ is cyclic or trivial
transcendental semigroup. We write $S=\langle f\rangle$. In this case each
$g\in S$ can be written as $g=f^{n}$, where $f^{n}$ is the nth iterates of $f$
with itself. Note that in our study of semigroup dynamics, we say $S=\langle
f\rangle$ a trivial semigroup.

Based on the Fatou-Julia theory of a complex analytic function, The Fatou set
and Julia set in the settings of semigroup are defined as follows.

######  Definition 1.2 (Fatou set, Julia set).

Fatou set of the transcendental semigroup $S$ is defined by

$F(S)=\\{z\in\mathbb{C}:S\;\ \textrm{is normal in a neighborhood of}\;\ z\\}$

The connected component of Fatou set $F(S)$ is called Fatou component. The
Julia set of $S$ is defined by $J(S)=\mathbb{C}-F(S)$.

From the definition 1.2, it is clear that $F(S)$ is the open set and
therefore, it complement $J(S)$ is closed set. Indeed, these definitions
generalize the definitions of Julia set and Fatou set of the iteration of
single holomorphic map. If $S=\langle f\rangle$, then $F(S)$ and $J(S)$ are
respectively the Fatou set and Julia set in classical iteration theory of
complex dynamics. In this situation we simply write: $F(f)$ and $J(f)$.

The fundamental contrast between classical complex dynamics and semigroup
dynamics appears by different algebraic structure of corresponding semigroups.
In fact, non-trivial semigroup (rational or transcendental) need not be, and
most often will not be abelian. However, trivial semigroup is cyclic and
therefore abelian. As we discussed before, classical complex dynamics is a
dynamical study of trivial (cyclic) semigroup whereas semigroup dynamics is a
dynamical study of non-trivial semigroup.

Note that for any semigroup $S$, we have

  1. (1)

$F(S)\subset F(h)$ all $h\in S$ and hence $F(S)\subset\cap_{h\in S}F(h)$.

  2. (2)

$J(h)\subset J(S)$.

Since, in classical complex dynamics, Fatou set $F(f)$ may be empty. So from
the above first relation, we can say that Fatou set $F(S)$ of semigroup $S$
may also be empty. In this paper we are interested to find a non-trivial
semigroup $S$ that has non-empty Fatou set $F(S)$. Basically, we prove that
there is a non-trivial transcendental semigroup that has simply connected
Fatou component.

######  Proposition 1.1.

There is a non trivial transcendental semigroup $S$ such that the Fatou set
$F(S)$ has at least a simply connected component.

Note that if the semigroup $S$ is trivial, that is, semigroup $S=\langle
f\rangle$ generated by a single transcendental entire function $f$, then
Bergweiler[1] proved that the Fatou set $F(S)$ has both a simply and a
multiply connected wandering domains. However, in the case of non-trivial
transcendental semigroup, the proof is not so easy. The reason behind is that
the dynamics of individual transcendental entire functions differ largely from
the dynamics of their composites.

##  2\. Some Essential Lemmas

To workout a proof of the proposition 1.1, first of all we need a notion of
approximation theory of entire functions. In our case, we can use the notion
of Carleman set from which we obtain approximation of any holomorphic map by
entire functions.

######  Definition 2.1 (Carleman Set).

Let $F$ be a closed subset of $\mathbb{C}$ and
$C(F)=\\{f:F\to\mathbb{C}:f\;\text{is continuous on }\;S\;\text{and analytic
in the interior of}\;F^{\circ}\;\text{of}\;F\\}$. Then $F$ is called a
Carleman set (for $\mathbb{C}$) if for any $g\in C(F)$ and any positive
continuous function $\epsilon$ on $F$, there exists entire function $h$ such
that $|g(z)-h(z)|<\epsilon$ for all $z\in F$.

The following important characterization of Carleman set has been proved by A.
Nersesjan in 1971 but we have been taken this result from [2].

######  Theorem 2.1 ([2, Theorem 4, page 157]).

Let $F$ be proper subset of $\mathbb{C}$. Then $F$ is a Carleman set for
$\mathbb{C}$ if and only if $F$ satisfies:

  1. (1)

$\mathbb{C}_{\infty}-F$ is connected;

  2. (2)

$\mathbb{C}_{\infty}-F$ is locally connected at $\infty$;

  3. (3)

for every compact subset $K$ of $\mathbb{C}$, there is a neighborhood $V$ of
$\infty$ in $\mathbb{C}_{\infty}$ such that no component of $F^{\circ}$
intersects both $K$ and $V$.

It is well known in classical complex analysis that the space
$\mathbb{C}_{\infty}-F$ is connected if and only if each component $Z$ of open
set $\mathbb{C}-F$ is unbounded. This fact together with above theorem 2.1
will be a nice tool whether a set is a Carleman set for $\mathbb{C}$. The sets
given in the following examples are Carlemen sets for $\mathbb{C}$.

######  Example 2.1 ([2, Example-page 133]).

The set
$E=\\{z\in\mathbb{C}:|z|=1,\text{Re}z>0\\}\cup\\{z=x:x>1\\}\cup\big{(}\bigcup_{n=3}^{\infty}\\{z=re^{i\theta}:r>1,\theta=\pi/n\\}\big{)}$
is a Carleman set for $\mathbb{C}$.

######  Example 2.2 ([5, Set S, page-131]).

The set $E=G_{0}\cup\big{(}\bigcup_{k=1}^{\infty}(G_{k}\cup B_{K}\cup
L_{k}\cup M_{k})\big{)}$, where $G_{0}=\\{z\in\mathbb{C}:|z-2|\leqslant 1\\}$;

$\displaystyle G_{k}$ $\displaystyle=$
$\displaystyle\\{z\in\mathbb{C}:|z-(4k+2)|\leqslant
1\\}\cup\\{z\in\mathbb{C}:\text{Re}z=4k+2,\;\text{Im}z\geqslant 1\\}$
$\displaystyle\cup\\{z\in\mathbb{C}:\text{Re}z=4k+2,\;\text{Im}z\leqslant-1\\},\;\;(k=1,2,3,\ldots);$
$\displaystyle B_{k}$ $\displaystyle=$
$\displaystyle\\{z\in\mathbb{C}:|z+(4k+2)|\leqslant
1\\}\cup\\{z\in\mathbb{C}:\text{Re}z=-(4k+2),\;\text{Im}z\geqslant 1\\}\cup$
$\displaystyle\\{z\in\mathbb{C}:\text{Re}z=-(4k+2),\;\text{Im}z\leqslant-1\\},\;\;(k=1,2,3,\ldots);$
$L_{k}=\\{z\in\mathbb{C}:\text{Re}z=4k\\},\;\;(k=1,2,3,\ldots);$

and

$M_{k}=\\{z\in\mathbb{C}:\text{Re}z=-4k\\},\;\;(k=1,2,3,\ldots)$

is a Carleman set for $\mathbb{C}$ by the theorem 2.1 .

From the help of the Carleman set of the example 2.2, A.P. Singh [5, Theorem
2] proved the following result.

######  Lemma 2.1.

There are transcendental entire functions $f$ and $g$ such that there exists a
domain which lies in the wandering component of the $F(f),F(g),F(f\circ g)$
and $F(g\circ f)$.

In fact, A. P. Singh [5] also had proved other results regarding the dynamics
of two individual functions and their composites (see for instance [5, Theorem
1, Theorem 3 and Theorem 4]) which are also stricly based on the Carleman set
of example 2.2. Dinesh Kumar, Gopal Datt and Sanjay Kumar extended these
result of A.P. Singh in [4, Theorem 2.1 to Theorem 2.15]. For our purpose, we
cite the following two results from [4].

######  Lemma 2.2 (Theorem 2.2).

There are TEFs $f$ and $g$ such that there exists infinite number of domains
which lie in the wandering component of the $F(f),F(g)$, $F(f\circ g)$ and
$F(g\circ f)$.

######  Lemma 2.3 (Theorem 2.13).

There are TEFs $f$ and $g$ such that there exist infinite number of domains
which lie in the pre-periodic component of the $F(f),F(g)$, $F(f\circ g)$ and
$F(g\circ f)$.

We extended above lemmas 2.1. 2.3 and 2.2 in [6, Theorem 1.1][7, Theorem 1.1]
and [8, Theorem 1.1] to the following results:

######  Lemma 2.4.

There are transcendental entire functions $f$, $g$ and $h$ such that there
exist infinite number of domains which lie in the wandering component of the
$F(f),F(g),F(h),F(f\circ g),F(g\circ f),F(f\circ h),F(g\circ h),F(h\circ
f),F(h\circ g),F(f\circ g\circ h),F(f\circ h\circ g),F(g\circ f\circ
h),F(g\circ h\circ f),F(h\circ f\circ g)$ and $F(h\circ g\circ f)$.

######  Lemma 2.5.

There are transcendental entire functions $f$, $g$ and $h$ such that there
exist infinite number of domains which lie in the pre-periodic component of
the $F(f),F(g),F(h),F(f\circ g),F(g\circ f),F(f\circ h),F(g\circ h),F(h\circ
f),F(h\circ g),F(f\circ g\circ h),F(f\circ h\circ g),F(g\circ f\circ
h),F(g\circ h\circ f),F(h\circ f\circ g)$ and $F(h\circ g\circ f)$.

######  Lemma 2.6.

There are transcendental entire functions $f$, $g$ and $h$ such that there
exist infinite number of domains which lie in the periodic component of the
$F(f),\;F(g),\;F(h),\;F(f\circ g),\;F(g\circ f),\;F(f\circ h),\;F(g\circ
h),\;F(h\circ f),\;F(h\circ g),\;F(f\circ g\circ h),\;F(f\circ h\circ
g),\;F(g\circ f\circ h),\;F(g\circ h\circ f),\;F(h\circ f\circ g)$ and
$F(h\circ g\circ f)$.

##  3\. Proof of the Proposition 1.1

From all of above lemmas (Lemmas 2.1, 2.3, 2.2, 2.4, 2.5 and 2.6), we can say
that whatever domains that lie in the wandering, pre-periodic or periodic
components of $F(f),F(g)$, $F(h),F(f\circ g),F(g\circ f),F(f\circ h),F(g\circ
h),F(h\circ f),F(h\circ g),F(f\circ g\circ h),F(f\circ h\circ g),F(g\circ
f\circ h),F(g\circ h\circ f),F(h\circ f\circ g)$ and $F(h\circ g\circ f)$,
they also lie respectively in the wandering, pre-periodic or periodic
components of their successive composites. In this context, we can also prove
the following two results:

######  Lemma 3.1.

If $D$ is a set which lies in the wandering (or pre-periodic or periodic)
component of $F(f),F(g),F(f\circ g)$ and $F(g\circ f)$, then it also lies in
wandering (or pre-periodic or periodic) component of $F(f^{n_{k}}\circ
g^{n_{k-1}}\circ\ldots\circ g^{n_{1}})$ and $F(g^{n_{k}}\circ
f^{n_{k-1}}\circ\ldots\circ f^{n_{1}})$, where $n_{k},\ldots
n_{1}\in\mathbb{N}$.

###### Proof.

By the lemmas 2.1, 2.2 and 2.3, such a set $D$ exists. Since $F(f)=F(f^{n})$
and $F(g)=F(g^{n})$ for any $n\in\mathbb{N}$. So $D$ lies in the wandering (or
pre-periodic or periodic) component of $F(f^{n})$ and $F(g^{n})$ for all
$n\in\mathbb{N}$. As $D$ lies in the wandering (or pre-periodic or periodic)
component of $F(f\circ g)$, it also lies in the wandering (or pre-periodic or
periodic) component of $F(f^{n}\circ g^{n})$ for all $n\in\mathbb{N}$. By the
same argument we are using here, $D$ also lies in the wandering (or pre-
periodic or periodic) component of $F(f\circ g)^{n}$ for all $n\in\mathbb{N}$.
Since $F(f\circ g)^{n}=F(f\circ g\circ\ldots\circ f\circ g)$ (n -times $f\circ
g$), $D$ lies in the wandering (or pre-periodic or periodic) component of
$F(f^{n}\circ g^{n}\circ\ldots\circ f^{n}\circ g^{n})$(n-times $f^{n}\circ
g^{n}$) for all $n\in\mathbb{N}$. Since $n\in\mathbb{N}$ is arbitrary, so we
conclude that $D$ lies in the wandering (or pre-periodic or periodic)
component of $F(f^{n_{k}}\circ g^{n_{k-1}}\circ\ldots\circ g^{n_{1}})$ for all
$n_{k},\ldots n_{1}\in\mathbb{N}$. Similarly, we can show that $D$ lies in the
wandering (or pre-periodic or periodic) component of $F(g^{n_{k}}\circ
f^{n_{k-1}}\circ\ldots\circ f^{n_{1}})$ for all $n_{k},\ldots
n_{1}\in\mathbb{N}$. ∎

######  Lemma 3.2.

If $D$ is a set which lies in the wandering (or pre-periodic or periodic)
component of $F(f),F(g),F(h)$, $F(f\circ g)$, $F(g\circ f),\;F(f\circ
h),\;F(g\circ h),\;F(h\circ f),\;F(h\circ g),\;F(f\circ g\circ h),\;F(f\circ
h\circ g),\;F(g\circ f\circ h),\;F(g\circ h\circ f),\;F(h\circ f\circ g)$ and
$F(h\circ g\circ f)$, then it also lies in the wandering (pre-periodic or
periodic) component of $F(f^{n_{k}}\circ g^{n_{k-1}}\circ
h^{n_{k-2}}\ldots\circ f^{n_{1}})$, $F(g^{n_{k}}\circ f^{n_{k-1}}\circ
h^{n_{k-2}}\circ\ldots\circ g^{n_{1}})$ and $F(h^{n_{k}}\circ f^{n_{k-1}}\circ
g^{n_{k-2}}\circ\ldots\circ h^{n_{1}})$ etc.

###### Proof.

By lemmas 2.4, 2.5, 2.6, such a set $D$ exists. By the similar argument of
above lemma 3.1, the proof of this lemma follows. ∎

We prove the proposition 1.1 for a semigroup generated by two or three
transcendental entire functions as defined in above lemmas 2.1, 2.2, 2.3, 2.4,
2.5 and 2.6.

###### Proof of the Proposition 1.1.

Let $S$ be a transcendental semigroup generated by two or three transcendental
entire functions. If $S$ is generated by two transcendental entire functions
$f$ and $g$ as defined in the lemmas 2.1, 2.2 and 2.3, then by the lemma 3.1,
there is at least a domain which lies in the wandering (or pre-periodic or
periodic) component of the $F(f^{n_{k}}\circ g^{n_{k-1}}\circ\ldots\circ
g^{n_{1}})$ and $F(g^{n_{k}}\circ f^{n_{k-1}}\circ\ldots\circ f^{n_{1}})$ for
all $n_{k},\ldots n_{1}\in\mathbb{N}$. By the definition of transcendental
semigroup, any $h\in S=\langle f,g\rangle$ can be written in either of the
form $h=f^{n_{k}}\circ g^{n_{k-1}}\circ\ldots\circ g^{n_{1}}$ or
$h=g^{n_{k}}\circ f^{n_{k-1}}\circ\ldots\circ f^{n_{1}}$ for all $n_{k},\ldots
n_{1}\in\mathbb{N}$. Therefore, there is a domain $D$ which lies in the
wandering (or pre-periodic or periodic) component of the Fatou set $F(h)$ for
every function $h$ of transcendental semigroup $S$. This shows that this
domain lies in the wandering (or pre-periodic or periodic) component of the
Fatou set $F(S)$. Since for transcendental entire function, pre-peridic (or
periodic) domains are simply connected and so a domain within simply connected
domains is also simply connected. In the construction of functions in the
lemmas 2.1, 2.2 and 2.3, the domain which lies in the wandering domains is
simply connected. If $S$ is generated by three transcendental entire functions
$f$, $g$ and $h$ as defined in the lemmas 2.4, 2.5 and 2.6, then by lemma 3.2
and similar argument as above, Fatou set $F(S)$ contains simply connected
domain. ∎

We restricted our proof of the proposition 1.1 to the transcendental semigroup
generated by two or three transcendental entire functions . Rigorously, it is
not known that the essence of this proposition holds if a semigroup is
generated by more than three transcendental entire functions. We can only say
intuitively, the essence of this proposition may hold if semigroup $S$ is
generated by n- transcendental entire functions. There is another strong
aspect of this proposition which is- if a transcendental semigroup $S$
generated by such type of two or three transcendental entire functions, then
the Fatou set $F(S)$ is non-empty.

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