# Reinforcement Learning for Improving Agent Design

David Ha
Google Brain
Tokyo, Japan
<EMAIL_ADDRESS>

###### Abstract

In many reinforcement learning tasks, the goal is to learn a policy to
manipulate an agent, whose design is fixed, to maximize some notion of
cumulative reward. The design of the agent’s physical structure is rarely
optimized for the task at hand. In this work, we explore the possibility of
learning a version of the agent’s design that is better suited for its task,
jointly with the policy. We propose an alteration to the popular OpenAI Gym
framework, where we parameterize parts of an environment, and allow an agent
to jointly learn to modify these environment parameters along with its policy.
We demonstrate that an agent can learn a better structure of its body that is
not only better suited for the task, but also facilitates policy learning.
Joint learning of policy and structure may even uncover design principles that
are useful for assisted-design applications. Videos of results at
https://designrl.github.io/.

##  1 Introduction

Embodied cognition [3, 40, 58] is the theory that an organism’s cognitive
abilities is shaped by its body. It is even argued that an agent’s cognition
extends beyond its brain, and is strongly influenced by aspects of its body
and also the experiences from its various sensorimotor functions [73, 25].
Evolution plays a vital role in shaping an organism’s body to adapt to its
environment; the brain and its ability to learn is only one of many body
components that is co-evolved together [47, 11]. We can recognize embodiment
in nature by observing that many organisms exhibit complex motor skills, such
as the ability to jump [12] or swim [10] even after brain death.

Figure 1: Learning to navigate over randomly generated terrain in
BipedalWalkerHardcore-v2 environment [13] (top). Agent learns a better body
design while jointly learning to navigate (bottom).

While evolution shapes the overall structure of the body of a particular
species, an organism can also change and adapt its body to its environment
during its life. For instance, professional athletes spend their lives body
training while also improving specific mental skills required to master a
particular sport [68]. In everyday life, regular exercise not only strengthens
the body but also improves mental conditions [49, 22]. We not only learn and
improve our skills and abilities during our lives, but also learn to shape our
bodies for the lives we want to live.

We are interested to investigate embodied cognition within the reinforcement
learning (RL) framework. Most baseline tasks [66, 36] in the RL literature
test an algorithm’s ability to learn a policy to control the actions of an
agent, with a predetermined body design, to accomplish a given task inside an
environment. The design of the agent’s body is rarely optimal for the task,
and sometimes even intentionally designed to make policy search challenging.
In this work, we explore enabling learning versions of an agent’s body that
are better suited for its task, jointly with its policy. We demonstrate that
an agent can learn a better structure of its body that is not only better for
its task, but also facilitates policy learning. We can even optimize our
agent’s body for certain desired characteristics, such as material usage. Our
approach may help uncover design principles useful for assisted-design.

Furthermore, we believe the ability to learn useful morphology is an important
area for the advancement of AI. Although morphology learning originally
initiated from the field of evolutionary computation, there has also been
great advances in RL in recent years, and we believe much of what happens in
ALife should be in principle be of interest to the RL community and vice
versa, since learning and evolution are just two sides of the same coin.

We believe that conducting experiments using standardized simulation
environments facilitate the communication of ideas across disciplines, and for
this reason we design our experiments based on applying ideas from ALife,
namely morphology learning, to standardized tasks in the OpenAI Gym
environment, a popular testbed for conducting experiments in the RL community.
We decide to use standardized Gym environments such as Ant (based on Bullet
physics engine) and Bipedal Walker (based on Box2D) not only for their
simplicity, but also because their difficulty is well-understood due to the
large number of RL publications that use them as benchmarks. As we shall see
later, the BipedalWalkerHardcore-v2 task, while simple looking, is especially
difficult to solve with modern Deep RL methods. By applying simple morphology
learning concepts from ALife, we are able to make a difficult task solvable
with much fewer compute resources. We also made the code for augmenting OpenAI
Gym for morphology learning, along with all pretrained models for reproducing
results in this paper available at https://github.com/hardmaru/astool.

We hope this paper can serve as a catalyst to precipitate a cultural shift in
both fields and encourage researchers to open up our minds to each other. By
drawing ideas from ALife and demonstrating them in the OpenAI Gym platform
used by RL, we hope this work can set an example to bring both the RL and
ALife communities closer together to find synergies and push the AI field
forward.

##  2 Related Work

There is a broad literature in evolutionary computation, artificial life and
robotics devoted to studying, and modelling embodied cognition [47]. In 1994,
Karl Sims demonstrated that artificial evolution can produce novel morphology
that resemble organisms observed in nature [60, 59]. Subsequent works further
investigated morphology evolution [11, 8, 9, 37, 64, 65, 44, 70, 4], modular
robotics [39, 45, 48, 75], and evolving soft robots [17, 20], using indirect
encoding [61, 23, 6, 7, 5].

Literature in the area of passive dynamics study robot designs that rely on
natural swings of motion of body components instead of deploying and
controlling motors at each joint [41, 19, 46, 18]. Notably, artist Theo Jansen
[33] also employed evolutionary computation to design physical Strandbeests
that can walk on their own consuming only wind energy to raise environmental
awareness.

Recent works in robotics investigate simultaneously optimizing body design and
control of a legged robot [29, 30] using constraint-based modelling, which is
related to our RL-based approach. Related to our work, [24, 1] employ CMA-ES
[31] to optimize over both the motion control and physical configuration of
agents. A related recent work [52, 53] employs RL to learn both the policy and
design parameters in an alternating fashion, where a single shared policy
controls a distribution of different designs, while in this work we simply
treat both policy and design parameters the same way.

##  3 Method

In this section, we describe the method used for learning a version of the
agent’s design better suited for its task jointly with its policy. In addition
to the weight parameters of our agent’s policy network, we will also
parameterize the agent’s environment, which includes the specification of the
agent’s body structure. This extra parameter vector, which may govern the
properties of items such as width, length, radius, mass, and orientation of an
agent’s body parts and their joints, will also be treated as a learnable
parameter. Hence the weights $w$ we need to learn will be the parameters of
the agent’s policy network combined with the environment’s parameterization
vector. During a rollout, an agent initialized with $w$ will be deployed in an
environment that is also parameterized with the same parameter vector $w$.

Figure 2: OpenAI Gym [13] framework for rolling out an agent in an environment
(left). We propose an alteration where we parameterize parts of an
environment, and allow an agent to modify its environment before a rollout,
and also augment its reward based on these parameters (right).

The goal is to learn $w$ to maximize the expected cumulative reward,
$E[R(w)]$, of an agent acting on a policy with parameters $w$ in an
environment governed by the same $w$. In our approach, we search for $w$ using
a population-based policy gradient method based on Section 6 of Williams’ 1992
REINFORCE [72]. The next section provides an overview of this algorithm.

Armed with the ability to change the design configuration of an agent’s own
body, we also wish to explore encouraging the agent to challenge itself by
rewarding it for trying more difficult designs. For instance, carrying the
same payload using smaller legs may result in a higher reward than using
larger legs. Hence the reward given to the agent may also be augmented
according to its parameterized environment vector. We will discuss reward
augmentation to optimize for desirable design properties later on in more
detail in Section 4.2.

###  3.1 Overview of Population-based Policy Gradient Method (REINFORCE)

In this section we provide an overview of the population-based policy gradient
method described in Section 6 of William’s REINFORCE [72] paper for learning a
parameter vector $w$ in a reinforcement learning environment. In this
approach, $w$ is sampled from a probability distribution $\pi(w,\theta)$
parameterized by $\theta$. We define the expected cumulative reward $R$ as:

$J(\theta)=E_{\theta}[R(w)]=\int R(w)\;\pi(w,\theta)\;dw.$ (1)

Using the log-likelihood trick allows us to write the gradient of $J(\theta)$
with respect to $\theta$:

$\nabla_{\theta}J(\theta)=E_{\theta}[\;R(w)\;\nabla_{\theta}\log\pi(w,\theta)\;].$
(2)

In a population size of $N$, where we have solutions $w^{1}$, $w^{2}$, …,
$w^{N}$, we can estimate this as:

$\nabla_{\theta}J(\theta)\approx\frac{1}{N}\sum_{i=1}^{N}\;R(w^{i})\;\nabla_{\theta}\log\pi(w^{i},\theta).$
(3)

With this approximated gradient $\nabla_{\theta}J(\theta)$, we then can
optimize $\theta$ using gradient ascent:

$\theta\rightarrow\theta+\alpha\nabla_{\theta}J(\theta)$ (4)

and sample a new set of candidate solutions $w$ from updating the pdf using
learning rate $\alpha$. We follow the approach in REINFORCE where $\pi$ is
modelled as a factored multi-variate normal distribution. Williams derived
closed-form formulas of the gradient $\nabla_{\theta}\log\pi(w^{i},\theta)$.
In this special case, $\theta$ will be the set of mean $\mu$ and standard
deviation $\sigma$ parameters. Therefore, each element of a solution can be
sampled from a univariate normal distribution $w_{j}\sim
N(\mu_{j},\sigma_{j})$. Williams derived the closed-form formulas for the
$\nabla_{\theta}\log N(z^{i},\theta)$ term in Equation 3, for each individual
$\mu$ and $\sigma$ element of vector $\theta$ on each solution $i$ in the
population:

$\nabla_{\mu_{j}}\log
N(w^{i},\theta)=\frac{w_{j}^{i}-\mu_{j}}{\sigma_{j}^{2}},\;\nabla_{\sigma_{j}}\log
N(w^{i},\theta)=\frac{(w_{j}^{i}-\mu_{j})^{2}-\sigma_{j}^{2}}{\sigma_{j}^{3}}.$
(5)

For clarity, we use subscript $j$, to count across parameter space in $w$, and
this is not to be confused with superscript $i$, used to count across each
sampled member of the population of size $N$. Combining Equations 5 with
Equation 4, we can update $\mu_{j}$ and $\sigma_{j}$ at each generation via a
gradient update.

We note that there is a connection between population-based REINFORCE, a
population-based policy gradient method, and particular formulations of
Evolution Strategies [50, 56], namely ones that are not elitist. For instance,
Natural Evolution Strategies (NES) [57, 71] and OpenAI-ES [51] are closely
based on Sec. 6 of REINFORCE. There is also a connection between natural
gradients (computed using NES) and CMA-ES [31]. We refer to Akimoto et al. [2]
for a detailed theoretical treatment and discussion about the connection
between CMA-ES and natural gradients.

##  4 Experiments

In this work, we experiment on the continuous control RoboschoolAnt-v1
environment from Roboschool [36], based on the open source Bullet [21] physics
engine, and also BipedalWalker-v2 from the Box2D [16] section of the OpenAI
Gym [13] set of environments. For simplicity, we first present results of
anecdotal examples obtained over a single representative experimental run to
convey qualitative results such as morphology and its relationship to
performance. A more comprehensive quantitative study based on multiple runs
using different random seeds will be presented in Section 4.3.

The RoboschoolAnt-v1111A compatible version of this environment is also
available in PyBullet [21] which was used for visualization. environment
features a four-legged agent called the Ant. The body is supported by 4 legs,
and each leg consists of 3 parts which are controlled by 2 motor joints. The
bottom left diagram of Figure 3 describes the initial orientation of the
agent. The length of each part of a leg is controlled by the $\Delta x$ and
$\Delta y$ distances from its joint connection. A size parameter also controls
the radius of each leg part.

Figure 3: Agent learning a policy to navigate forward in RoboschoolAnt-v1
environment [36] (left). Agent develops longer, thinner legs while supporting
the same body during training (right).

In our experiment, we keep the volumetric mass density of all materials, along
with the parameters of the motor joints identical to the original environment,
and allow the 36 parameters (3 parameters per leg part, 3 leg parts per leg, 4
legs in total) to be learned. In particular, we allow each part to be scaled
to a range of $\pm$ 75% of its original value. This allows us to keep the sign
and direction for each part to preserve the original intended structure of the
design.

| Top Left | Top Right | Bottom Left | Bottom Right
---|---|---|---|---
| Length | Radius | Length | Radius | Length | Radius | Length | Radius
Top | 141% | 33% | 141% | 25% | 169% | 35% | 84% | 51%
Middle | 169% | 26% | 164% | 26% | 171% | 31% | 140% | 29%
Bottom | 174% | 26% | 168% | 50% | 173% | 29% | 133% | 38%
Table 1: Learned agent body for RoboschoolAnt-v1 as % of original design
specification.

Figure 3 illustrates the learned agent design compared to the original design.
With the exception of one leg part, it learns to develop longer, thinner legs
while jointly learning to carry the body across the environment. While the
original design is symmetric, the learned design (Table 1) breaks symmetry,
and biases towards larger rear legs while jointly learning the navigation
policy using an asymmetric body. The original agent achieved an average
cumulative score of 3447 $\pm$ 251 over 100 trials, compared to 5789 $\pm$ 479
for an agent that learned a better body design.

The Bipedal Walker series of environments is based on the Box2D [16] physics
engine. Guided by LIDAR sensors, the agent is required to navigate across an
environment of randomly generated terrain within a time limit, without falling
over. The agent’s payload – its head, is supported by 2 legs. The top and
bottom parts of each leg is controlled by two motor joints. In the easier
BipedalWalker-v2 [34] environment, the agent needs to travel across small
random variations of a flat terrain. The task is considered solved if an agent
obtains an average score greater than 300 points over 100 rollouts.

Figure 4: Agent learning a policy to navigate forward in BipedalWalker-v2
environment [13] (top). Agent learns a body to allow it to bounce forward
efficiently (bottom).

Keeping the head payload constant, and also keeping the density of materials
and the configuration of the motor joints the same as the original
environment, we only allow the lengths and widths for each of the 4 leg parts
to be learnable, subject to the same range limit of $\pm$ 75% of the original
design. In the original environment in Figure 4 (top), the agent learns a
policy that is reminiscent of a joyful skip across the terrain, achieving an
average score of 347. In the learned version in Figure 4 (bottom), the agent’s
policy is to hop across the terrain using its legs as a pair of springs,
achieving a score of 359.

In our experiments, all agents were implemented using 3 layer fully-connected
networks with $\tanh$ activations. The agent in RoboschoolAnt-v1 has 28 inputs
and 8 outputs, all bounded between $-1$ and $+1$, with hidden layers of 64 and
32 units. The agents in BipedalWalker-v2 and BipedalWalkerHardcore-v2 has 24
inputs and 4 outputs all bounded between $-1$ and $+1$, with 2 hidden layers
of 40 units each.

Our population-based training experiments were conducted on 96-CPU core
machines. Following the approach described in [28], we used a population size
of 192, and had each agent perform the task 16 times with different initial
random seeds. The agent’s reward signal used by the policy gradient method is
the average reward of the 16 rollouts. The most challenging
BipedalWalkerHardcore agents were trained for 10000 generations, while the
easier BipedalWalker and Ant agents were trained for 5000 and 3000 generations
respectively. As done in [28], we save the parameters of the agent that
achieves the best average cumulative reward during its entire training
history.

###  4.1 Joint learning of body design facilitates policy learning:
BipedalWalkerHardcore-v2

Learning a better version of an agent’s body not only helps achieve better
performance, but also enables the agent to jointly learn policies more
efficiently. We demonstrate this in the much more challenging
BipedalWalkerHardcore-v2 [35] version of the task. Unlike the easier version,
the agent must also learn to walk over obstacles, travel up and down hilly
terrain, and even jump over pits. Figure 1 illustrates the original and
learnable versions of the environment.222As of writing, two methods are
reported to solve this task. Population-based training [28] (our baseline),
solves this task in 40 hours on a 96-CPU machine, using a small feed forward
policy network. A3C [43] adapted for continuous control [26] solves the task
in 48 hours on a 72-CPU machine, but requires an LSTM [32] policy.

Figure 5: Population-based training curves for both versions of
BipedalWalkerHardcore-v2 (left). Plot of performance of best agent in the
population over 100 random trials (right). Original version solved in under
4600 generations (40 hours); By allowing morphology to be learned, the task is
solved in under 1400 generations (12 hours).

In this environment, our agent generally learns to develop longer, thinner
legs, with the exception in the rear leg where it developed a thicker lower
limb to serve as useful stability function for navigation. Its front legs,
which are smaller and more manoeuvrable, also act as a sensor for dangerous
obstacles ahead that complement its LIDAR sensors. While learning to develop
this newer structure, it jointly learns a policy to solve the task in 30% of
the time it took the original, static version of the environment. The average
scores over 100 rollouts for the learnable version is 335 $\pm$ 37 compared to
the baseline score of 313 $\pm$ 53\. The full results are summarized in Table
2.

BipedalWalker-v2 | Top leg 1 | Bottom leg 1 | Top leg 2 | Bottom leg 2
---|---|---|---|---
| Avg. score | leg area | $w$ | $h$ | $w$ | $h$ | $w$ | $h$ | $w$ | $h$
Original | 347 $\pm$ 0.9 | 100% | 8.0 | 34.0 | 6.4 | 34.0 | 8.0 | 34.0 | 6.4 | 34.0
Learnable | 359 $\pm$ 0.2 | 33% | 2.0 | 57.3 | 1.6 | 46.0 | 2.0 | 48.8 | 1.6 | 18.9
Reward smaller leg | 323 $\pm$ 68 | 8% | 2.0 | 11.5 | 1.6 | 10.6 | 2.0 | 11.4 | 1.6 | 10.2
BipedalWalkerHardcore-v2 | Top leg 1 | Bottom leg 1 | Top leg 2 | Bottom leg 2
| Avg. score | leg area | $w$ | $h$ | $w$ | $h$ | $w$ | $h$ | $w$ | $h$
Original | 313 $\pm$ 53 | 100% | 8.0 | 34.0 | 6.4 | 34.0 | 8.0 | 34.0 | 6.4 | 34.0
Learnable | 335 $\pm$ 37 | 95% | 2.7 | 59.3 | 10.0 | 58.9 | 2.3 | 55.5 | 1.7 | 34.6
Reward smaller leg | 312 $\pm$ 69 | 27% | 2.0 | 35.3 | 1.6 | 47.1 | 2.0 | 36.2 | 1.6 | 26.7
Table 2: Summary of results for Bipedal Walker environments. Scaled Box2D
dimensions reported.

###  4.2 Optimize for both the task and desired design properties

Allowing an agent to learn a better version of its body obviously enables it
to achieve better performance. But what if we want to give back some of the
additional performance gains, and also optimize also for desirable design
properties that might not generally be beneficial for performance? For
instance, we may want our agent to learn a design that utilizes the least
amount of materials while still achieving satisfactory performance on the
task. Here, we reward an agent for developing legs that are smaller in area,
and augment its reward signal during training by scaling the rewards by a
utility factor of $1+\log(\frac{\text{orig leg area}}{\text{new leg area}})$.
Augmenting the reward encourages development of smaller legs.

Figure 6: Agent rewarded for smaller legs for the task in BipedalWalker-v2
environment [13] (top). Agent learns the smallest pair of legs that still can
solve BipedalWalkerHardcore-v2 (bottom).

This reward augmentation resulted in much a smaller agent that is still able
to support the same payload. In BipedalWalker, given the simplicity of the
task, the agent’s leg dimensions eventually shrink to near the lower bound of
$\sim$ 25% of the original dimensions, with the exception of the heights of
the top leg parts which settled at $\sim$ 35% of the initial design, while
still achieving an average (unaugmented) score of 323 $\pm$ 68\. For this
task, the leg area used is 8% of the original design.

However, the agent is unable to solve the more difficult BipedalWalkerHardcore
task using a similar small body structure, due to the various obstacles
presented. Instead, it learns to set the widths of each leg part close to the
lower bound, and instead learn the shortest heights of each leg part required
to navigate, achieving a score of 312 $\pm$ 69\. Here, the leg area used is
27% of the original.

###  4.3 Results over Multiple Experimental Runs

In the previous sections, for simplicity, we have presented results over a
single representative experimental run to convey qualitative results such as
morphology description corresponding to average score achieved. Running the
experiment from scratch with a different random seed may generate different
morphology designs and different policies that lead to different performance
scores. To demonstrate that morphology learning does indeed improve the
performance of the agent over multiple experimental runs, we run each
experiment 10 times and report the full range of average scores obtained in
Table 3 and Table 4. From multiple independent experimental runs, we see that
morphology learning consistently produces higher scores over the normal task.

Experiment | Statistics of average scores over 10 independent runs
---|---
(a) Ant | 3139 $\pm$ 189.3
(b) Ant + Morphology | 5267 $\pm$ 631.4
(c) Biped | 345 $\pm$ 1.3
(d) Biped + Morphology | 354 $\pm$ 2.2
(e) Biped + Morphology + Smaller Leg | 330 $\pm$ 3.9
(f) Biped Hardcore | 300 $\pm$ 11.9
(g) Biped Hardcore + Morphology | 326 $\pm$ 12.7
(h) Biped Hardcore + Morphology + Smaller Leg | 312 $\pm$ 11.9
Table 3: Summary of results for each experiment over 10 independent runs. Experiment | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | #10
---|---|---|---|---|---|---|---|---|---|---
(a) Ant | 3447 | 3180 | 3076 | 3255 | 3121 | 3223 | 3130 | 3096 | 3167 | 2693
(b) + Morphology | 5789 | 6035 | 5784 | 4457 | 5179 | 4788 | 4427 | 5253 | 6098 | 4858
(c) Biped | 347 | 343 | 347 | 346 | 345 | 345 | 345 | 346 | 346 | 344
(d) + Morphology | 359 | 354 | 353 | 354 | 353 | 352 | 353 | 352 | 353 | 356
(e) + Smaller Leg | 323 | 327 | 327 | 331 | 330 | 331 | 333 | 329 | 337 | 333
(f) Biped Hardcore | 313 | 306 | 300 | 283 | 311 | 295 | 307 | 309 | 292 | 279
(g) + Morphology | 335 | 331 | 330 | 330 | 332 | 292 | 327 | 331 | 316 | 330
(h) + Smaller Leg | 312 | 320 | 314 | 318 | 307 | 314 | 316 | 281 | 319 | 324
Table 4: Full results from each of the 10 experimental trials. Each number is
the average score of the trained agent over 100 rollouts in the environment.

We also visualize the variations of morphology designs over different runs in
Figure 7 to get a sense of the variations of morphology that can be discovered
during training. As these models may take up to several days to train for a
particular experiment on a powerful 96-core CPU machine, it may be costly for
the reader to fully reproduce the variation of results here, especially when
10 machines running the same experiment with different random seeds are
required. We also include all pretrained models from multiple independent runs
in the GitHub repository containing the code to reproduce this paper. The
interested reader can examine the variations in more detail using the
pretrained models.

Figure 7: Examples of learned morphology in Run #9. Biped + Morphology (top)
develops a thicker but short rear lower limb, unlike the agent in Figure 3.
Biped Hardcore + Morphology (bottom) develops a larger rear leg, but unlike
the agent in Figure 1, its thigh is larger than the lower limb.

##  5 Discussion and Future Work

We have shown that allowing a simple population-based policy gradient method
to learn not only the policy, but also a small set of parameters describing
the environment, such as its body, offer many benefits. By allowing the
agent’s body to adapt to its task within some constraints, it can learn
policies that are not only better for its task, but also learn them more
quickly.

The agent may discover design principles during this process of joint body and
policy learning. In both RoboschoolAnt and BipedalWalker experiments, the
agent has learned to break symmetry and learn relatively larger rear limbs to
facilitate their navigation policies. While also optimizing for material usage
for BipedalWalker’s limbs, the agent learns that it can still achieve the
desired task even by setting the size of its legs to the minimum allowable
size. Meanwhile, for the much more difficult BipedalWalkerHardcore-v2 task,
the agent learns the appropriate length of its limbs required for the task
while still minimizing the material usage.

This approach may lead to useful applications in machine learning-assisted
design, in the spirit of [15, 14]. While not directly related to agent design,
machine learning-assisted approaches have been used to procedurally generate
game environments that can also facilitate policy learning of game playing
agents [67, 42, 63, 69, 27]. Game designers can optimize the designs of game
character assets while at the same time being able to constrain the characters
to keep the essence of their original forms. Optimizing character design may
complement existing work on machine learning-assisted procedural content
generation for game design. By framing the approach within the popular OpenAI
Gym framework, design firms can create more realistic environments – for
instance, incorporate strength of materials, safety factors, malfunctioning of
components under stressed conditions, and plug existing algorithms into this
framework to optimize also for design aspects such as energy usage, easy-of-
manufacturing, or durability. The designer may even incorporate aesthetic
constraints such as symmetry and aspect ratios that suits her design sense.

In this work we have only explored using a simple population-based policy
gradient method [72] for learning. State-of-the-art model-free RL algorithms,
such as TRPO [54] and PPO [55] work well when our agent is presented with a
well-designed dense reward signal, while population-based methods offer
computational advantages for sparse-reward problems [51, 62]. In our setting,
as the body design is parameterized by a small set of learnable parameters and
is only set once at the beginning of a rollout, the problem of learning the
body along with the policy becomes more sparse. In principle, we could allow
an agent to augment its body during a rollout to obtain a dense reward signal,
but we find this unpractical for realistic problems. Future work may look at
separating the learning from dense-rewards and sparse-rewards into an inner
loop and outer loop, and also examine differences in performance and
behaviours in structures learned with various different RL algorithms.

Separation of policy learning and body design into inner loop and outer loop
will also enable the incorporation of evolution-based approaches to tackle the
vast search space of morphology design, while utilizing efficient RL-based
methods for policy learning. The limitations of the current approach is that
our RL algorithm can learn to optimize only existing design properties of an
agent’s body, rather than learn truly novel morphology in the spirit of Karl
Sims’ Evolving Virtual Creatures [60].

Nevertheless, our approach of optimizing the specifications of an existing
design might be practical for many applications. While a powerful evolutionary
algorithm that can also evolve novel morphology might come up with robot
morphology that easily outperforms the best bipedal walkers in this work, the
resulting designs may not be as useful to a game designer who is tasked to
work explicitly with bipedal walkers that fit within the game’s narrative
(although it is debatable whether the game can be more entertaining and
interesting if the designer is allowed to explore the space beyond given
specifications). Due to the vast search space of all possible morphology, a
search algorithm can easily come up with unrealistic or unusable designs that
exploits its simulation environment, as discussed in detail in [38], which may
be why subsequent morphology evolution approaches constrain the search space
of the agent’s morphology, such as constraining to the space of soft-body
voxels [17] or constraining to a set of possible pipe frame connection
settings [33]. We note that unrealistic designs may also result in our
approach, if we do not constrain the learned dimensions to be within $\pm$ 75%
of its original value. For some interesting examples of what REINFORCE
discovers without any constraints, we invite the reader to view the Bloopers
section of https://designrl.github.io/.

Just as REINFORCE [72] can also be applied to the discrete search problem of
neural network architecture designs [74], similar RL-based approaches could be
used for novel morphology design – not simply for improving an existing design
like in this work. We believe the ability to learn useful morphology is an
important area for the advancement of AI. Although morphology learning
originally initiated from the field of evolutionary computation, we hope this
work will engage the RL community to investigate the concept further and
encourage idea exchange across communities.

## Acknowledgments

We would like to thank the three reviewers from Artificial Life Journal, Luke
Metz, Douglas Eck, Janelle Shane, Julian Togelius, Jeff Clune, and Kenneth
Stanley for their thoughtful feedback and conversation. All experiments were
performed on CPU machines provided by Google Cloud.

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# Rotation bounds
for Hölder continuous homeomorphisms
with integrable distortion

A. Clop, L. Hitruhin, B. Sengupta

###### Abstract

We obtain sharp rotation bounds for the subclass of homeomorphisms
$f:\mathbb{C}\to\mathbb{C}$ of finite distortion which have distortion
function in $L^{p}_{loc}$, $p>1$, and for which a Hölder continuous inverse is
available. The interest in this class is partially motivated by examples
arising from fluid mechanics. Our rotation bounds hereby presented improve the
existing ones, for which the Hölder continuity is not assumed. We also present
examples proving sharpness.

##  1 Introduction

We say that $f:\mathbb{C}\to\mathbb{C}$ is a mapping of finite distortion if
it belongs to the Sobolev space $f\in W^{1,1}_{loc}(\mathbb{C};\mathbb{C})$,
its jacobian determinant $\det(Df)=J(\cdot,f)$ is locally integrable, and
there exists a measurable function
$\mathbb{K}(\cdot,f):\mathbb{C}\to[1,+\infty]$ such that

$|Df(z)|^{2}\leq\mathbb{K}(z,f)\cdot J(z,f)$

at almost every point $z\in\mathbb{C}$. Above, $|Df(z)|$ stands for the
operator norm of the differential matrix $Df(z)$ of $f$ at the point $z$. When
$\mathbb{K}(\cdot,f)\in L^{\infty}$ then $f$ is said to be $K$-quasiregular,
with $K=\|\mathbb{K}(\cdot,f)\|_{\infty}$ (or $K$-quasiconformal, if
bijective). In the same way quasiregular maps arose as a generalization of
holomorphic functions, mappings of finite distortion arose as a generalization
of quasiregular maps partially motivated by questions in nonlinear elasticity.
The authors address the interested reader to the monograph [11] for
quasiregular maps in the plane, and to [1] for a background on mappings of
finite distortion.

Recently there has been a growing interest in understanding the rotational
properties of planar homeomorphisms, see [2, 4, 7, 8, 9, 10]. Special
attention has been devoted to the spiraling rate of these maps. More
precisely, given a homeomorphism $f:\mathbb{C}\to\mathbb{C}$ normalized by
$f(0)=0$ and $f(1)=1$, one is interested in the growth of $|\arg(f(r))|$ as
$r\to 0$. This growth represents the number of times that the image $f([r,1])$
winds around the origin as $r\to 0$. This quantity has been proven to admit
several speeds of growth which depend on the class of maps under study. In
this way, it was proven in [2] that if $f$ is $K$-quasiconformal then

$|\arg(f(r))|\leq\frac{1}{2}\left(K-\frac{1}{K}\right)\,\log\left(\frac{1}{r}\right)+c_{K},\hskip
28.45274pt\text{for all }0<r<1.$ (1)

In contrast, if the maps under study are homeomorphisms of finite distortion,
the situation changes and the order of growth depends on the integrability of
the distortion function. Namely, the second named author discovered in [8]
that if $e^{\mathbb{K}(\cdot,f)}\in L^{p}_{loc}$ for some $p>0$ then

$|\arg(f(z))|\leq\frac{c}{p}\,\log^{2}\left(\frac{1}{|z|}\right),\hskip
28.45274pt\text{for small enough }|z|,$

and moreover this is sharp up to the value of the constant $c>0$. In other
words, the transition between boundedness and exponential integrability of
$\mathbb{K}(\cdot,f)$ results in a larger power of the logarithmic term.
Further optimal results were obtained later on in [9], in the case of
integrable distortion, that is, when $\mathbb{K}(\cdot,f)\in L^{p}_{loc}$ for
some $p>1$,

$|\arg(f(z))|\leq\frac{c}{|z|^{\frac{2}{p}}},\hskip 28.45274pt\text{for small
enough }|z|$ (2)

or even when $\mathbb{K}(\cdot,f)\in L^{1}_{loc}$,

$\lim_{|z|\to 0}|z|^{2}\,|\arg(f(z))|=0.$ (3)

The moral here is that more spiraling is allowed at the cost of relaxing the
integrability properties of $\mathbb{K}(\cdot,f)$. As explained in [2, 8, 9],
the local rotational properties go hand in hand with the local stretching
behavior. Especially important for the argument are the estimates for the
modulus of continuity of the inverse map.

It turns out mappings of finite distortion also have a role in fluid
mechanics. To be precise, let us think of the planar incompressible Euler
system of equations in vorticity form,

$\begin{cases}\frac{d}{dt}\omega+(\mathbf{v}\cdot\nabla)\omega=0\\\
\operatorname{div}(\mathbf{v})=0\\\ \omega(0,\cdot)=\omega_{0}.\end{cases}$
(4)

Here $\omega=\omega(t,z):[0,T]\times\mathbb{C}\to\mathbb{C}$ is the unknown,
$\omega_{0}\in L^{\infty}(\mathbb{C};\mathbb{C})$ is given, and $\mathbf{v}$
is the velocity field. The Biot-Savart law,

$\mathbf{v}=\frac{i}{2\pi\bar{z}}\ast\omega$

makes more precise the relation between $\mathbf{v}$ and $\omega$. As it is
well known, Yudovich [15] proved existence and uniqueness of a solution
$\omega\in L^{\infty}([0,T];L^{\infty}(\mathbb{C};\mathbb{C}))$ for any given
$\omega_{0}$. In particular, the corresponding velocity field $\mathbf{v}$
belongs to the Zygmund class, and therefore the classical Cauchy-Lipschitz
theory guarantees for the ODE

$\begin{cases}\frac{d}{dt}X(t,z)=\mathbf{v}(t,X(t,z))\\\ X(0,z)=z\end{cases}$

both existence and uniqueness of a flow map
$X:[0,T]\times\mathbb{C}\to\mathbb{C}$. It was proven in [6] that, for small
enough $t>0$, each of the flow homeomorphisms
$X_{t}=X(t,\cdot):\mathbb{C}\to\mathbb{C}$ is indeed a mapping of finite
distortion. Moreover, for each small value $t>0$ there is a number $p(t)>1$
such that the distortion function $\mathbb{K}(\cdot,X_{t})$ belongs to
$L^{p}_{loc}$ whenever $p<p(t)$.

As mappings with $L^{p}$ distortion, the mappings $X_{t}$ are a bit special
because both $X_{t}$ and $X_{t}^{-1}$ are Hölder continuous, as shown in [14],
with a Hölder exponent that decays exponentially in time. This is not true in
general, and mappings of $L^{p}$ distortion need not have a Hölder continuous
inverse, as shown in [12]. Therefore, it is a question of interest to find out
if the Hölder nature of the inverse map results in better rotation bounds.
Indeed, even though the bounds obtained in [9] can be applied to $X_{t}$, the
Hölder continuous nature of $X_{t}^{-1}$ provides a significant improvement to
(2). We describe this improvement in our main Theorem.

######  Theorem 1.

Let $f:\mathbb{C}\to\mathbb{C}$ be a homeomorphism of finite distortion such
that $f(0)=0$ and $f(1)=1$, and assume that $\mathbb{K}(\cdot,f)\in
L^{p}_{loc}$ for some $p>1$. Suppose also that

$|f(x)-f(y)|\geq C\,|x-y|^{\alpha},\hskip 28.45274pt\text{if }|x-y|\text{ is
small,}$

for some $\alpha>1$. Then

$|\arg(f(z))|\leq
C\,\sqrt{\alpha}\,|z|^{-\frac{1}{p}}\,\log^{\frac{1}{2}}\left(\frac{1}{|z|}\right)$

whenever $|z|$ is small enough.

In contrast with (2) and (3), the existence of a Hölder continuous inverse
allows the power term exponent to be halved, although then the logarithmic
term needs to be included. As an application, we can estimate the spiraling
rate of $X_{t}$.

As an application, we can estimate the spiraling rate of $X_{t}$ for small
times. The rotational behavior of $X_{t}$ is nowadays studied a lot. For
instance, in the case of $\omega_{0}$ being _close_ to the characteristic
function of the unit disk, the article [5] provides bounds for the winding
number of most of the trajectories $\\{X_{t}(z)\\}_{t>0}$ as $t\to\infty$.
Here, instead, we do not evaluate the rotational behavior at large times, but
look instead at spiraling bounds _in the space variable_ for a fixed and small
enough time.

######  Corollary 2.

Given $\omega_{0}\in L^{\infty}(\mathbb{C};\mathbb{C})$, let $\mathbf{v}$ be
the velocity field of Yudovich’s solution to (4), and let $X_{t}$ be its flow.
Then there exists a constant $C>0$ such that

$\left|\arg\left(\frac{X_{t}(z)-X_{t}(0)}{X_{t}(1)-X_{t}(0)}\right)\right|\leq
C\,\log^{\frac{1}{2}}\left(\frac{1}{|z|}\right)\,|z|^{-t\|\omega_{0}\|_{\infty}}\,\exp\left(Ct\|\omega_{0}\|_{\infty}\right)\,$

if both $|z|$ and $t>0$ are small enough.

In particular, if $z=\frac{1}{n}$, $n=1,2,\dots$ and we fix a time $t_{0}$
small enough, then the curve $X_{t_{0}}([\frac{1}{n},1])$ cannot wind around
$X_{t_{0}}(0)$ more than a multiple of

$n^{t_{0}\|\omega_{0}\|_{\infty}}\,(\log
n)^{\frac{1}{2}}\,e^{Ct_{0}\|\omega_{0}\|_{\infty}}$

times. Towards the optimality of Theorem 1, we can show the following.

######  Theorem 3.

Given an increasing, onto homeomorphism $h:[0,+\infty)\to[0,+\infty)$, and a
real number $p>1$, there exists a homeomorphism
$\bar{f}:\mathbb{C}\to\mathbb{C}$ with the folllowing properties:

  * •

$\bar{f}$ is a mapping of finite distortion, with
$\mathbb{K}(\cdot,\bar{f})\in L^{p}_{loc}$.

  * •

$\bar{f}(0)=0$, $\bar{f}(1)=1$.

  * •

If $\alpha>\frac{3p}{p-1}$, then $|\bar{f}(x)-\bar{f}(y)|\geq C|x-y|^{\alpha}$
whenever $|x-y|<1$.

  * •

There exists a decreasing sequence $\\{r_{n}\\}$, with $r_{n}\to 0+$ as
$n\to\infty$, for which

$|\arg(\bar{f}(r_{n}))|\geq
r_{n}^{-\frac{1}{p}}\,\log^{\frac{1}{2}}\left(\frac{1}{r_{n}}\right)\,h(r_{n}).$

Since $h$ can be chosen to approach $0$ at any speed, Theorem 3 shows that the
order provided in Theorem 1 is sharp.

The paper is structured as follows. In Section 2 we give the basic
preliminaries. In Section 3 we prove Theorem 1 and Corollary 2. Finally, we
prove Theorem 3 in Section 4.

Acknowledgements. A.C. and B.S. are partially supported by projects
MTM2016-81703-ERC, MTM2016-75390 (Spanish Government) and 2017SGR395 (Catalan
Government). L.H. was partially supported by ICMAT Severo Ochoa project
SEV-2015-0554 grant MTM2017-85934-C3-2-P, the ERC grant 307179-GFTIPFD, the
ERC grant 834728 Quamap, by the Finnish Academy of Science project 13316965
and by a grant from The Emil Aaltonen Foundation.

##  2 Preliminaries

A mapping $f:\mathbb{C}\to\mathbb{C}$ is said to be Hölder continuous, or
simply Hölder from above, if there exist constants $C>0$, $d>0$ and $\alpha>0$
such that for any two points $x,y\in\mathbb{C}$ and
$\alpha\in\mathbb{R}^{+}\setminus\\{0\\}$ with $|x-y|<d$ one has

$|f(x)-f(y)|\leq C|x-y|^{\alpha}$

Similarly, we say $f$ is Hölder from below if there are constants $C,\beta>0$
such that for any two points $x,y\in\mathbb{C}$ and
$\beta\in\mathbb{R}^{+}\setminus\\{0\\}$ with $|x-y|<d$ one has

$|f(x)-f(y)|\geq\bar{C}|x-y|^{\beta}$

A mapping $f:\mathbb{C}\to\mathbb{C}$ is called bi-Hölder if it is both Hölder
from above and from below.

Let $f:\mathbb{C}\to\mathbb{C}$ be a mapping of finite distortion and fix a
point $z_{0}\in\mathbb{C}$. In order to study the pointwise rotation of $f$ at
the point $z_{0}$, one usually fixes an argument $\theta\in[0,2\pi)$, and then
looks at how the quantity

$\arg(f(z_{0}+te^{i\theta})-f(z_{0}))$

changes as the parameter $t$ goes from 1 to a small $r$. This can also be
understood as the winding of the path
$f\left([z_{0}+re^{i\theta},z_{0}+e^{i\theta}]\right)$ around the point
$f(z_{0})$. As we are interested in the maximal pointwise spiraling, we need
to normalize and then retain the maximum over all directions $\theta$,

$\sup_{\theta\in[0,2\pi)}|\arg(f(z_{0}+re^{i\theta})-f(z_{0}))-\arg(f(z_{0}+e^{i\theta})-f(z_{0}))|.$
(5)

Then, the maximal pointwise rotation is precisely the behavior of the above
quantity (5) when $r\to 0^{+}$. In this way, we say that the map $f$ _spirals
at the point $z_{0}$ with a rate $g$_, where $g:[0,\infty)\to[0,\infty)$ is a
decreasing continuous function, if

$\limsup_{r\to
0^{+}}\frac{\sup_{\theta\in[0,2\pi)}|\arg(f(z_{0}+re^{i\theta})-f(z_{0}))-\arg(f(z_{0}+e^{i\theta})-f(z_{0}))|}{g(r)}=C$
(6)

for some constant $0<C<\infty$. Finding maximal pointwise rotation for a given
class of mappings equals finding the maximal spiraling rate for this class.
Note that in (6) we must use limit superior as the limit itself might not
exist. Furthermore, for a given mapping $f$ there might be many different
sequences $r_{n}\to 0$ along which it has profoundly different rotational
behaviour.

Our proof of Theorem 1 relies heavily on the modulus of path families. We give
here the main definitions, and address the interested reader to [13] for a
closer look at the topic. The image of a line segment $I$ under a continuous
mapping is called a _path_ , and we denote by $\Gamma$ a family of paths.
Given a path family $\Gamma$, we say that a Borel measurable function $\rho$
is _admissible for $\Gamma$_ if any rectifiable $\gamma\in\Gamma$ satisfies

$\int_{\gamma}\rho(z)dz\geq 1.$

The _modulus of the path family $\Gamma$_ is defined by

$M(\Gamma)=\inf_{\rho\text{ admissible}}\int_{\mathbb{C}}\rho^{2}(z)\,dA(z),$

where $dA(z)$ denotes the Lebesgue measure on $\mathbb{C}$. As an intuitive
rule, the modulus is big if the family $\Gamma$ has _lots_ of short paths, and
it is small if the paths are long and there are _not many_ of them.

We will also need a weighted version of the modulus. Any measurable, locally
integrable function $\omega:\mathbb{C}\to[0,\infty)$ will be called _a weight
function_. In our case, $\omega$ will always be the distortion function
$\mathbb{K}(\cdot,f)$ of some map $f$. Then, we define the weighted modulus
$M_{\omega}(\Gamma)$ by

$M_{\omega}(\Gamma)=\inf_{\rho\text{
admissible}}\int_{\mathbb{C}}\rho^{2}(z)\,\omega(z)\,dA(z).$

Finally, we need the modulus inequality

$M(f(\Gamma))\leq M_{\mathbb{K}(\cdot,f)}(\Gamma)$ (7)

which holds for any mapping $f$ of finite distortion for which the distortion
$\mathbb{K}(\cdot,f)$ is locally integrable, proven by the second named author
in [9].

##  3 Spiraling bounds

We will write Theorem 1 in the following, clearly equivalent, form.

######  Theorem 4.

Let $f$ be a homeomorphism of finite distortion with distortion
$\mathbb{K}(\cdot,f)\in L^{p}(\mathbb{C})$, $p>1$, normalized by $f(0)=0$ and
$f(1)=1$. Assume that it satisfies the following condition,

$\displaystyle|f(x)-f(y)|\geq C|x-y|^{\alpha}$

whenever $|x-y|$ is small. Then the winding number $n(z_{0})$ of the image of
the line segment $\left[z_{0},\frac{z_{0}}{|z_{0}|}\right]$ around the image
of the origin is bounded from above by

$\displaystyle n(z_{0})$ $\displaystyle\leq
C\sqrt{\alpha}\,|z_{0}|^{-\frac{1}{p}}\,\log^{\frac{1}{2}}\left(\frac{1}{|z_{0}|}\right)$

###### Proof.

We would like to prove this theorem using the modulus inequality for
homeomorphisms of finite distortion (7) following the presentation in [9]. At
first, we would like to estimate the modulus term
$M_{\mathbb{K}(\cdot,f)}(\Gamma)$ from above. To this end, let us choose an
arbitrary point $z_{0}\in\mathbb{C}\setminus\\{0\\}$ such that $|z_{0}|<1$.
Without loss of generality, we may assume that $z_{0}$ lies on the positive
side of the real axis. Next, let us fix the line segments $E=[z_{0},1]$ and
$F=(-\infty,0]$, and $\Gamma$ be the family of paths connecting a point in $E$
to a point in $F$. Also, let us fix balls $B_{j}=B(2^{j}z_{0},2^{j}z_{0})$,
$j\in\\{0,1,...,n\\}$ and $n$ is the smallest integer such that
$2^{n}z_{0}\geq 1$. Define

$\displaystyle\rho_{0}(z)=\begin{cases}\frac{2}{r(B_{0})}\hskip
5.69046pt\mbox{if}\hskip 5.69046ptz\in B_{0}\\\ \frac{2}{r(B_{1})}\hskip
5.69046pt\mbox{if}\hskip 5.69046ptz\in B_{1}\setminus B_{0}\\\ \vdots\\\
\frac{2}{r(B_{n})}\hskip 5.69046pt\mbox{if}\hskip 5.69046ptz\in B_{n}\setminus
B_{n-1}\\\ 0\hskip 17.07182pt\mbox{otherwise}\end{cases}$

Note that any $z\in E$ belongs to some ball $\frac{1}{2}B_{j}$ and that
$\rho_{0}(z)\geq\frac{2}{r(B_{j})}$, whenever $z\in B_{j}$. This implies,
since $B_{j}\cap F=\emptyset$ for every $j$, that $\rho_{0}(z)$ is admissible
with respect to $\Gamma$. Hence we can estimate the modulus from above by

$\displaystyle M_{\mathbb{K}(\cdot,f)}(\Gamma)$
$\displaystyle=\inf_{\rho\text{
admissible}}\int_{\mathbb{C}}\mathbb{K}(\cdot,f)\rho^{2}(z)\,dA(z)$
$\displaystyle\leq\int_{\mathbb{C}}\mathbb{K}(\cdot,f)\rho_{0}^{2}(z)\,dA(z)$
$\displaystyle\leq\|\mathbb{K}(\cdot,f)\|_{L^{p}\left(B(0,4)\right)}\left(\int_{B(0,4)}\rho_{0}^{\frac{2p}{p-1}}(z)\,dA(z)\right)^{\frac{p-1}{p}}$
$\displaystyle\leq
c_{f,p}\left(\int_{B(0,4)}\rho_{0}^{\frac{2p}{p-1}}(z)\,dA(z)\right)^{\frac{p-1}{p}}$

Let us now estimate the integral term by using the definition of $\rho_{0}$.

$\displaystyle\int_{B(0,4)}\rho_{0}^{\frac{2p}{p-1}}(z)\,dA(z)$
$\displaystyle\leq\sum_{j=0}^{n}\int_{B_{j}}\left(\frac{2}{r(B_{j})}\right)^{\frac{2p}{p-1}}\,dA(z)$
$\displaystyle=\sum_{j=0}^{n}|B_{j}|\left(\frac{2}{r(B_{j})}\right)^{\frac{2p}{p-1}}$
$\displaystyle=c_{p}\sum_{j=0}^{n}\frac{\left(r(B_{j})\right)^{2}}{\left(r(B_{j})\right)^{\frac{2p}{p-1}}}$
$\displaystyle=c_{p}\sum_{j=0}^{n}\frac{1}{z_{0}^{\frac{2}{p-1}}}\frac{1}{2^{\frac{2j}{p-1}}}$
$\displaystyle=c_{p}z_{0}^{-\frac{2}{p-1}}\sum_{j=0}^{n}\frac{1}{2^{\frac{2j}{p-1}}}$

The series $\sum_{j=0}^{n}\frac{1}{2^{\frac{2j}{p-1}}}$ converges to a
constant depending on p for any fixed $p>1$. Therefore,

$\displaystyle M_{\mathbb{K}(\cdot,f)}(\Gamma)\leq
c_{f,p}z_{0}^{-\frac{2}{p}}$

Next, we would like to estimate the modulus term $M\left(f(\Gamma)\right)$
from below. Let us start with the definition of $M\left(f(\Gamma)\right)$ in
polar coordinates

$\displaystyle M\left(f(\Gamma)\right)$ $\displaystyle=\inf_{\rho\text{
admissible}}\int_{\mathbb{C}}\rho^{2}(z)\,dA(z)$
$\displaystyle=\inf_{\rho\text{
admissible}}\int_{0}^{2\pi}\int_{0}^{\infty}\rho^{2}(r,\theta)r\hskip
2.84544ptdrd\theta$

and provide a lower bound for

$\displaystyle\int_{0}^{\infty}\rho^{2}(r,\theta)rdr$

for an arbitrary direction $\theta\in[0,2\pi)$ and an arbitrary admissible
$\rho$. To this end, we fix a direction $\theta$ and consider the half line
$L_{\theta}$ starting from the origin in the direction $\theta$. Let us choose
points $z_{0}\leq t_{2}<t_{0}\leq 1$ so that the image set $f(E)$ winds once
around the origin when $z$ moves from a point $t_{0}$ to a point $t_{2}$ along
$E$ and $f(t_{0})\in L_{\theta}$. Since the mapping $f$ is a homeomorphism and
the path $f(F)$ contains both the origin and points with big modulus, as $F$
is unbounded, the path $f(F)$ must intersect the line segment
$(f(t_{2}),f(t_{0}))$ at least once, say at a point $f(t_{1})$, with $t_{1}\in
F$. We can choose $t_{1}$ and $t_{0}$ such that there are no points from the
paths $f(E)$ and $f(F)$ in the line segment $(f(t_{1}),f(t_{0}))$, which thus
belongs to the path family $f(\Gamma)$. Since the path $f(E)$ cycles around
the origin
$n(z_{0})=\left\lfloor{\frac{|\arg\left(f(z_{0})\right)-\arg\left(f(1)\right)|}{2\pi}}\right\rfloor$
times we can find at least

$\displaystyle
n(z_{0})=\left\lfloor{\frac{|\arg\left(f(z_{0})\right)-\arg\left(f(1)\right)|}{2\pi}}\right\rfloor-1$

such disjoint line segments belonging to the path family $f(\Gamma)$. Note
that $n(z_{0})$ does not depend on the direction $\theta$. Since we are
interested in extremal rotation, it can be assumed that $f(E)$ winds around
the origin at least once, which makes it clear that $n(z_{0})$ is non-
negative. Now, the $n(z_{0})$ disjoint line segments can be written in the
form $\left(x_{j}e^{i\theta},y_{j}e^{i\theta}\right)\subset L_{\theta}$, where
$j\in\\{1,2,...,n(z_{0})\\}$ and $x_{j}$,$y_{j}$ are positive real numbers
satisfying

$\displaystyle 0<r_{f}\leq x_{1}<y_{1}<...<x_{n(z_{0})}<y_{n(z_{0})}\leq
c_{f}$

where $c_{f}=\sup_{z\in E}|f(z)|$ and $r_{f}=\min_{z\in E}|f(z)|$. Here,
neither $c_{f}$ nor $r_{f}$ depends on $\theta$ or $z_{0}$. So, one could
write

$\displaystyle\int_{0}^{\infty}\rho^{2}(r,\theta)rdr\geq\sum_{j=1}^{n(z_{0})}\int_{x_{j}}^{y_{j}}\rho^{2}(r,\theta)rdr.$

Next, let us consider the Hölder inequality with the functions
$f(r)=\rho\sqrt{r}$ and $g(r)=\frac{1}{\sqrt{r}}$, which after squaring both
sides gives

$\displaystyle\int_{x_{j}}^{y_{j}}\rho^{2}(r,\theta)rdr\geq\left(\int_{x_{j}}^{y_{j}}\rho(r,\theta)dr\right)^{2}\left(\int_{x_{j}}^{y_{j}}\frac{1}{r}dr\right)^{-1}\geq\frac{1}{\log\left(\frac{y_{j}}{x_{j}}\right)}.$

The last inequality holds true as $\rho$ is admissible with respect to
$f(\Gamma)$ where the line segments $(x_{j}e^{i\theta},y_{j}e^{i\theta})$
belong to the path family $f(\Gamma)$. Therefore,

$\displaystyle\int_{0}^{\infty}\rho^{2}(r,\theta)rdr\geq\sum_{j=1}^{n(z_{0})}\frac{1}{\log\left(\frac{y_{j}}{x_{j}}\right)}.$

It is quite clear from the definition of $c_{f}$ that

$\displaystyle\sum_{j=1}^{n(z_{0})}\frac{1}{\log\left(\frac{y_{j}}{x_{j}}\right)}\geq\sum_{j=1}^{n(z_{0})-1}\frac{1}{\log\left(\frac{x_{j+1}}{x_{j}}\right)}+\frac{1}{\log\left(\frac{c_{f}}{x_{n(z_{0})}}\right)}.$

Next, let us consider the AM-HM inequality. For every positive real number
$a_{j}$,

$\displaystyle\sum_{j=1}^{n}a_{j}\geq\frac{n^{2}}{\sum_{j=1}^{n}\frac{1}{a_{j}}}.$

At this point, we would like to use AM-HM with the precise choices

$a_{j}=\frac{1}{\log\left(\frac{x_{j+1}}{x_{j}}\right)}\hskip 14.22636pt\text{
if }j\in\\{1,2,...,n(z_{0})-1\\},\text{ and }\hskip
28.45274pta_{n(z_{0})}=\frac{1}{\log\left(\frac{c_{f}}{x_{n(z_{0})}}\right)},$

which give

$\displaystyle\sum_{j=1}^{n(z_{0})}\frac{1}{\log\left(\frac{y_{j}}{x_{j}}\right)}\geq\frac{n^{2}(z_{0})}{\log\left(\frac{c_{f}}{x_{1}}\right)}\geq\frac{n^{2}(z_{0})}{\log\left(\frac{c_{f}}{r_{f}}\right)}.$

Therefore,

$\int_{0}^{\infty}\rho^{2}(r,\theta)rdr\geq\frac{n^{2}(z_{0})}{\log\left(\frac{c_{f}}{r_{f}}\right)}.$

The constant $c_{f}$ can be defined as $\max_{z\in\mathbb{D}}|f(z)|$, which is
finite and does not depend on either $\theta$ or $z_{0}$, and thus it is
irrelevant at the limit $z_{0}\to 0$. On the other hand, the constant $r_{f}$
must be estimated using the Hölder modulus of continuity assumption on our
mapping $f$, that is

$|f(z_{0})|\geq C|z_{0}|^{\alpha}$

for sufficiently small $z_{0}$. This combined with the estimate above gives
that

$M\left(f(\Gamma)\right)\geq\frac{n^{2}(z_{0})}{C\alpha\log\left(\frac{1}{|z_{0}|}\right)}$

Now, using the modulus inequality we get

$\frac{n^{2}(z_{0})}{C\alpha\log\left(\frac{1}{|z_{0}|}\right)}\leq
c_{f,p}\left(\frac{1}{|z_{0}|}\right)^{\frac{2}{p}}$

which implies the desired estimate. ∎

###### Proof of Corollary 2.

Corollary 2 follows immediately after noting that one can take $f=X_{t}$ in
Theorem 1. Indeed, from [6, Corollary 3] we know that $X_{t}$ belongs to
$W^{1,p}$ for any $p<\frac{2}{t\|\omega_{0}\|_{\infty}}$, provided that
$0<t<\frac{2}{\|\omega_{0}\|_{\infty}}$. Since $J(\cdot,X_{t})=1$ due to the
incompressibility, it then follows $X_{t}$ is a homeomorphism with finite
distortion, and moreover $\mathbb{K}(\cdot,X_{t})\in L^{p}_{\text{loc}}$ for
$p<\frac{1}{t\|\omega_{0}\|_{\infty}}$. Especially, if $t$ is so small that
$0<t<\frac{1}{\|\omega_{0}\|_{\infty}}$ then one may take $p>1$. Also, we
recall from [14] (see also [3]) that $X_{t}^{-1}$ is $\alpha$-Hölder
continuous with some exponent $\alpha\geq e^{-ct\,\|\omega_{0}\|_{\infty}}$
for some $c>0$. Hence, Theorem 1 applies to $f=X_{t}$ and the claim follows. ∎

##  4 Optimality of spiraling

We will get Theorem 3 in two steps. In the first step, we will construct a map
which _only rotates_. This map will already give us the optimal result (in the
power scale). In the second step, we will strengthen this up with a second
map, that _both rotates and stretches_. This second map is going to be the
optimal one.

Given an arbitrary annulus $A=B(0,R)\setminus B(0,r)$ we define the
corresponding rotation map as

$\displaystyle\phi_{A}(z)=\begin{cases}z&|z|>R\\\
z\,e^{i\alpha\log\left|\frac{z}{R}\right|}&r\leq|z|\leq R\\\
z\,e^{i\alpha\,\log\frac{r}{R}}&|z|<r\end{cases}$

Here $0<r<R$, and $\alpha\in\mathbb{R}$. One must note that
$\phi_{A}:\mathbb{C}\to\mathbb{C}$ is bilipschitz (i.e. both $\phi_{A}$ and
its inverse are Lipschitz), hence quasiconformal (its quasiconformality
constant depends only on $\alpha$), and moreover it is conformal outside the
annulus $A$. Note also that $\phi_{A}$ leaves fixed all circles centered at
$0$, since $|\phi_{A}(te^{i\theta})|=t$ for each $t>0$ and
$\theta\in\mathbb{R}$. Finally, a direct calculation shows for the jacobian
determinant that $J(z,\phi_{A})=1$ for each $z$.

Next, we fix a sequence $\\{r_{n}\\}$ such that $0<r_{n+1}<\frac{r_{n}}{2e}$
and $r_{1}<\frac{1}{e}$. Also, let $R_{n}=er_{n}$. These assumptions make sure
that $2r_{n+1}<R_{n+1}<\frac{r_{n}}{2}$. Let us now construct disjoint annuli
$A_{n}=B(0,R_{n})\setminus B(0,r_{n})$, and set $\\{f_{n}\\}_{n}$ to be a
sequence of maps, constructed in an iterative way as follows. For $n=1$, we
set

$\displaystyle f_{1}(z)=\phi_{A_{1}}(z)=\begin{cases}z&|z|>R_{1}\\\
z\,e^{i\alpha_{1}\,\log\frac{|z|}{R_{1}}}&r_{1}\leq|z|\leq R_{1}\\\
z\,e^{-i\alpha_{1}}&|z|<r_{1}\\\ \end{cases}$

where $\alpha_{1}\in\mathbb{R}$, $\alpha_{1}\geq 1$, is to be determined
later. We then define $f_{n}$ for $n\geq 2$ as

$\displaystyle f_{n}(z)=\phi_{f_{n-1}(A_{n})}\circ f_{n-1}(z)$

again for some values $\alpha_{n}\in\mathbb{R}$, $\alpha_{n}\geq 1$, to be
determined later. Clearly, each $f_{n}:\mathbb{C}\to\mathbb{C}$ is
quasiconformal, and conformal outside the annuli $A_{i}$, $i=1,\dots,n$. It is
also clear that $f_{n}(z)=f_{n-1}(z)$ on the unbounded component of
$\mathbb{C}\setminus f_{n-1}(A_{n})$ (i.e. outside of $B(0,R_{n})$). This
proves that the sequence $f_{n}$ is uniformly Cauchy and hence it converges to
a map $f$, that is,

$\displaystyle f=\lim_{n\to\infty}f_{n}$

which is again a homeomorphism by construction. Now, since $f_{n}$ is
quasiconformal for every $n$ and $f_{n}(z)=f_{n-1}(z)$ everywhere except
inside the ball $B(0,R_{n})$, where $R_{n}\to 0$ as $n\to\infty$, the limit
map $f$ is absolutely continuous on almost every line parallel to the
coordinate axes and differentiable almost everywhere.

It is helpful to note that each $f_{n}$ leaves fixed all circles centered at
the origin, so in particular we have $f_{n}(A_{j})=A_{j}$ for each $j$, and
therefore $\phi_{f_{n-1}(A_{n})}=\phi_{A_{n}}$. Direct calculation shows that

$|D\phi_{A_{n}}(z)|=|\partial\phi_{A_{n}}(z)|+|\overline{\partial}\phi_{A_{n}}(z)|=\begin{cases}1&|z|>R_{n}\\\
\frac{|2+i\alpha_{n}|+|\alpha_{n}|}{2}&r_{n}\leq|z|\leq R_{n}\\\
1&|z|<r_{n}\end{cases}$

which allows us to estimate that

$\displaystyle|\partial f(z)|+|\bar{\partial}f(z)|\leq 2\alpha_{n}\hskip
28.45274pt\text{whenever }z\in A_{n},$

and $|Df(z)|\leq 1$ otherwise. Therefore, in order to have $Df(z)\in
L^{1}_{loc}(\mathbb{C})$ it suffices that

$\sum_{n}\alpha_{n}\,r_{n}^{2}<+\infty.$ (8)

This, together with the absolute continuity, guarantees $f\in
W^{1,1}_{loc}(\mathbb{C})$. Also, since $f$ is a homeomorphism, we have that
$J_{f}(z)\in L^{1}_{loc}(\mathbb{C})$, and in fact $J(z,f)=1$ at almost every
$z\in\mathbb{C}$. Therefore, $f$ is a homeomorphism of finite distortion, with
distortion function

$\displaystyle\mathbb{K}(z,f)=\frac{|Df(z)|^{2}}{J(z,f)}\leq\begin{cases}4\alpha_{n}^{2}&z\in
A_{n},\\\ 1&\text{otherwise.}\end{cases}$

Especially, in order to have $\mathbb{K}(\cdot,f)\in L^{p}_{loc}$, it suffices
to ensure the convergence of the series

$\sum_{n=1}^{\infty}|A_{n}|(4\alpha_{n}^{2})^{p}\simeq\sum_{n=1}^{\infty}\alpha_{n}^{2p}\,r_{n}^{2}$
(9)

which can be done by choosing $\alpha_{n}$ properly. Note that if (9) holds,
then also (8) holds, because our choice of $\alpha_{n}$ will guarantee
$\alpha_{n}\geq 1$. The last restriction to choose our $\alpha_{n}$ comes from
rotational behavior of $f$. It is clear from the above construction that
$f(0)=0$, $f(1)=1$ and

$\left|\arg\left(f(r_{n})\right)\right|\geq\left|\arg\left(\left(\frac{1}{e}\right)^{1+i\alpha_{n}}\right)\right|=\alpha_{n}$

for every $r_{n}$. Since we want our map to be optimal for Theorem 1, we may
be tempted to choose $\alpha_{n}=r_{n}^{-1/p}\,\log^{1/2}(1/r_{n})$.
Unfortunately such a choice does not meet the requirement (9). The same
problem occurs if we simply choose $\alpha_{n}=r_{n}^{-1/p}$. So we choose

$\alpha_{n}=h(r_{n})\,r_{n}^{-1/p}.$

Here $h:[0,\infty)\to[0,\infty)$ is any monotonically decreasing gauge
function such that $\lim_{r\to 0^{+}}h(r)=0$. With this choice, (9) is
fulfilled if the series

$\sum_{n=1}^{\infty}h(r_{n})^{2p}<+\infty.$

But this can always be done by simply reducing the already chosen values of
$r_{n}$, for instance if $h(r_{n})<\frac{1}{n^{1/2}}$. Note that this does not
provide full optimality for Theorem 1, but it already gives the right order
(in the power scale).

We now show that $f$ is Hölder continuous with exponent $1-\frac{1}{p}$. For
this, let us recall that our map $f$ is a limit of iterates of logarithmic
spiral maps inside the annuli $A_{n}=B(0,R_{n})\setminus B(0,r_{n})$. In
particular, as shown in [2], if $\gamma\in\mathbb{R}$ then the basic
logarithmic spiral map $g(z)=z|z|^{i\alpha}=ze^{i\gamma\log|z|}$ is
$L$-bilipschitz, for a constant $L$ such that $|\gamma|=L-\frac{1}{L}$. When
$|\gamma|$ is large, $L$ is large as well and so one roughly has
$|\gamma|\simeq L$. Since our $f_{n}$ behaves on the annulus $A_{n}$ as a
spiral map with $|\gamma|=\alpha_{n}$, we deduce the bilipschitz constant of
$f_{n}$ on $A_{n}$ is

$L\simeq|\gamma|=\alpha_{n}=h(r_{n})\,r_{n}^{-1/p}.$

Let us now start the proof. To this end, let us consider two arbitrary points
$x$ and $y$ in $\mathbb{D}\setminus\\{0\\}$. We first consider the case where
$x,y\in A_{n}$. In this case, $f(x)=f_{n}(x)$ and $f(y)=f_{n}(y)$. Since
$r_{n}>C|x-y|$, we have

$\displaystyle|f(x)-f(y)|=|f_{n}(x)-f_{n}(y)|$ $\displaystyle\lesssim
h(r_{n})\,r_{n}^{-1/p}|x-y|$ $\displaystyle\leq
h(r_{n})\left(\frac{C}{|x-y|}\right)^{\frac{1}{p}}|x-y|$ $\displaystyle\leq
C|x-y|^{1-\frac{1}{p}}$

where we have used the bilipschitz nature of $f_{n}$ on $A_{n}$.

We now assume that $x,y\in D_{n}=B(0,r_{n})\setminus B(0,R_{n+1})$. On that
set $f$ is of the form $ze^{i\beta}$, where
$\beta\in\mathbb{R}\setminus\\{0\\}$, which is clearly an isometry.

Next, we take $x\in A_{n}$ and $y\in D_{n}$. In particular, $|x|\geq|y|$. Then
let $w$ be any point on the outer boundary of $D_{n}$ joining $x$ and $y$. We
have

$\displaystyle|f(x)-f(y)|$ $\displaystyle\leq|f(x)-f(w)|+|f(w)-f(y)|$
$\displaystyle\leq C|x-w|^{1-\frac{1}{p}}+|w-y|$ $\displaystyle\leq
2C|x-y|^{1-\frac{1}{p}}$

The same happens if $x\in D_{n-1}$ and $y\in A_{n}$.

So it just remains to see what happens when $x\in A_{n}=B(0,R_{n})\setminus
B(0,r_{n})$ and $y\in B(0,R_{n+1})$. Let $L$ be the line joining $x$ and $y$.
We divide it into three parts, viz., $L_{1}$, $L_{2}$ and $L_{3}$. $L_{1}$
connects $x$ to a point $a$ on the inner boundary of $A_{n}$, so that

$|f(x)-f(a)|=|f_{n}(x)-f_{n}(a)|\leq C|x-a|^{1-\frac{1}{p}}$

Next, $L_{2}$ connects $a$ to $b$, which is the closest point to $y$ where the
line $L$ crosses the inner boundary of $D_{n}$. From
$2R_{n+1}<r_{n}<\frac{R_{n}}{2}$ we get that $|f(a)|>2|f(b)|$. Also, since
$a,b\in D_{n}$ and $f$ is an isometry there, we get

$|f(b)-f(y)|\leq 2|f(b)|\leq 2|f(a)-f(b)|=2|a-b|$

Summarizing

$\displaystyle|f(x)-f(y)|$
$\displaystyle\leq|f(x)-f(a)|+|f(a)-f(b)|+|f(b)-f(y)|$ $\displaystyle\leq
C|x-a|^{1-\frac{1}{p}}+|a-b|+2|a-b|$ $\displaystyle\leq
C|x-y|^{1-\frac{1}{p}}$

The case $x\in D_{n}$ and $y\in B(0,r_{n+1})$ can be proved in a similar
manner. Therefore, we have covered all the possible cases. Since the set
$\mathbb{D}\setminus\\{0\\}$ is partitioned by separated annuli $A_{n}$ and
$D_{n}$, it is clear that we have proved that $f$ is Hölder continuous with
exponent $1-\frac{1}{p}$. At this point, it is worth noting that this
regularity could also be proven by means of the Sobolev embedding. Indeed, we
proved above that $\mathbb{K}(\cdot,f)\in L^{p}_{loc}$, and also that the
Jacobian determinant is constantly $1$. This together implies that $Df\in
L^{2p}_{loc}$.

Now we show that also $f^{-1}$ is Hölder continuous. Indeed, let us recall
that $f$ is the limit of iterates of logarithmic spiral maps inside the annuli
and conformal outside. Now, $f^{-1}$ can be constructed using the same
building blocks as $f$ itself, just changing the sign of $\alpha_{n}$ at each
step. This is possible because the inverse of a logarithmic spiral map is the
same spiral map, just the direction of rotation is opposite of the original
map. Since it is clear that the direction of rotation does not play any role
in the proof of Hölder continuity of $f$, this implies that $f^{-1}$ is also
Hölder from above. Thus $f$ is Hölder from below as well.

As we said before, the above example approaches the borderline stated in
Theorem 1, but it does not attain full optimality yet. To this end, we have to
modify it by adding to our building blocks a stretching factor. This is done
by replacing, at each iterate, the logarithmic spiral map
$z|z|^{i\alpha}=ze^{i\alpha\log|z|}$ by a complex power
$z|z|^{q+i\alpha}=z|z|^{q}\,e^{i\alpha\log|z|}$. We now proceed with the
details.

So, similarly as in the previous construction, we fix a rapidly decreasing
sequence $\\{r_{n}\\}$ such that $r_{n+1}<\frac{r_{n}}{2e}$ and
$r_{1}<\frac{1}{e}$. Also, let $R_{n}=er_{n}$. Given an arbitrary annulus
$A=B(0,R)\setminus B(0,r)$ we define the corresponding radial stretching
combined with rotation map as follows:

$\displaystyle\phi_{A}(z)=\begin{cases}z&|z|>R\\\
z\,\left|\frac{z}{R}\right|^{q-1}\,e^{i\alpha\log\frac{|z|}{R}}&r\leq|z|\leq
R\\\
z\left(\frac{r}{R}\right)^{q-1}\,e^{i\alpha\log\frac{r}{R}}&|z|<r\end{cases}$
(10)

Note that this time we will have $q\geq 1$. Direct calculation shows that

$|\partial\phi_{A}(z)|+|\overline{\partial}\phi_{A}(z)|=\begin{cases}1&|z|>R\\\
R^{1-q}|z|^{q-1}\frac{|q+1+i\alpha|+|q-1+i\alpha|}{2}&r\leq|z|\leq R\\\
R^{1-q}r^{q-1}&|z|<r\end{cases}$

and also that

$J(z,\phi_{A})=\begin{cases}1&|z|>R\\\
q\,\left(\frac{|z|}{R}\right)^{2(q-1)}&r\leq|z|\leq R\\\
\left(\frac{r}{R}\right)^{2(q-1)}&|z|<r\end{cases}$

whence

$\mathbb{K}(z,\phi_{A})=\begin{cases}1&|z|>R\\\
\frac{(|q+1+i\alpha|+|q-1+i\alpha|)^{2}}{4q}&r\leq|z|\leq R\\\
1&|z|<r\end{cases}$

In particular, if $2\leq q+1<\alpha$ then one may estimate
$\|\mathbb{K}(\cdot,\phi_{A})\|_{\infty}\leq\frac{4\alpha^{2}}{q}$. Next, let
us construct the sequence of maps $f_{n}$ in an iterative way as follows. For
$n=1$, we set

$\displaystyle f_{1}(z)=\phi_{A_{1}}(z)=\begin{cases}z&|z|<R_{1}\\\
z\,\left|\frac{z}{R_{1}}\right|^{q_{1}-1}\,e^{i\alpha_{1}\log\frac{|z|}{R_{1}}}&r_{1}\leq|z|\leq
R_{1}\\\ z\left(\frac{1}{e}\right)^{q_{1}-1}\,e^{-i\alpha_{1}}&|z|<r_{1}\\\
\end{cases}$

where $q_{1}$ and $\alpha_{1}$ are to be determined later. Next, assuming we
have $f_{1},\dots,f_{n-1}$, we define $f_{n}$ for $n\geq 2$ as:

$\displaystyle f_{n}(z)=\phi_{f_{n-1}(A_{n})}\circ f_{n-1}(z)$

Note that $\phi_{f_{n-1}(A_{n})}$ is determined by the inner and outer radii
of $\phi_{f_{n-1}(A_{n})}$ (which are already available since
$f_{1},\dots,f_{n-1}$ are known) as well as for the parameters $q_{n}$ and
$\alpha_{n}$, which will be determined later. Clearly, each
$f_{n}:\mathbb{C}\to\mathbb{C}$ is quasiconformal, and conformal outside the
annuli $A_{i}$, $i\in\\{1,...,n\\}$. Moreover, one can easily show that

$\mathbb{K}(\cdot,f_{n})=\prod_{j=1}^{n}\mathbb{K}(\cdot,f_{n-j}\circ\phi_{f_{n-j}(A_{n-j+1})})\\\
=\prod_{j=1}^{n}\mathbb{K}(\cdot,\phi_{A_{n-j+1}})$

so that $\mathbb{K}(z,f_{n})\leq C\frac{\alpha_{j}^{2}}{q_{j}}$ whenever $z\in
A_{j}$, $j=1\dots n$ while $\mathbb{K}(\cdot,f_{n})=1$ otherwise. In a similar
way, we can use that $|D\phi_{A}(z)|\leq C\alpha$ when $z\in A$ (and
$|D\phi_{A}(z)|\leq 1$ at all other points) to obtain that $|Df_{n}|\leq
C\alpha_{j}$ on $A_{j}$, $j=1\dots n$, and $|Df_{n}|\leq 1$ otherwise.

By construction, we have $f_{n}(z)=f_{n-1}(z)$ whenever $z\notin B(0,R_{n})$.
Thus $\\{f_{n}\\}_{n}$ converges uniformly to a map $\bar{f}(z)$, that is,

$\displaystyle\bar{f}=\lim_{n\to\infty}f_{n}$

which is again a homeomorphism by construction. A similar argument to the one
before shows that $\bar{f}$ is absolutely continuous on almost every line
parallel to the coordinate axis. For almost every fixed $z_{0}\neq 0$ there is
a neighbourhood of $z_{0}$ such that the sequence $\\{f_{n}(z)\\}_{n}$ remains
constant for $n$ very large and $z$ in that neighbourhood. Therefore the same
happens to the sequences $Df_{n}(z)$, $J(z,f_{n})$ and $\mathbb{K}(z,f_{n})$,
and so their limits are precisely $D\bar{f}(z)$, $J(z,\bar{f})$ and
$\mathbb{K}(z,\bar{f})$. Especially, in order to have $D\bar{f}\in
L^{1}_{loc}$ it suffices that

$\sum_{n=1}^{\infty}|A_{n}|\,\alpha_{n}<+\infty$ (11)

In case this holds true, then $\bar{f}$ is a homeomorphism in $W^{1,1}_{loc}$,
and as a consequence its jacobian determinant $J(\cdot,\bar{f})\in
L^{1}_{loc}$. Moreover, in order to have $\mathbb{K}(\cdot,\bar{f})\in
L^{p}_{loc}$ one needs to require that

$\sum_{n=1}^{\infty}|A_{n}|\,\frac{\alpha_{n}^{2p}}{q_{n}^{p}}<\infty$ (12)

Again, as it was the case for $f$, (12) implies (11) when
$q_{n}^{\frac{p}{2p-1}}<\alpha_{n}$ and so our parameters $\alpha_{n}$ and
$q_{n}$ need to be chosen according to (12) as well as the purpose of
$\bar{f}$ to be optimal for Theorem 1. For this, again as before, we have
$\bar{f}(0)=0$, $\bar{f}(1)=1$ and

$\left|\arg\left(\bar{f}(r_{n})\right)\right|\geq\left|\arg\left(\left(\frac{1}{e}\right)^{q_{n}+i\alpha_{n}}\right)\right|=|\alpha_{n}|$

which motivates us to choose

$\alpha_{n}=h(r_{n})\,\left(\log\frac{1}{r_{n}}\right)^{1/2}\,r_{n}^{-\frac{1}{p}}\hskip
56.9055ptq_{n}=\log\frac{1}{r_{n}},$

where $h$ is any gauge function such that $h(r)\to 0$ as $r\to 0$ and the
condition $q_{n}^{\frac{p}{2p-1}}<\alpha_{n}$ is satisfied. Indeed, with these
choices (12) becomes

$\sum_{n}h(r_{n})^{2p}<\infty$

which, as before, may always be granted by choosing smaller $r_{n}$, if
needed. Having (12) fulfilled, our map $\bar{f}$ is a mapping of finite
distortion with $\mathbb{K}(\cdot,\bar{f})\in L^{p}_{loc}$. Also, the
resulting map $\bar{f}$ attains the optimal rotational behavior stated at
Theorem 1 modulo the gauge function $h$ which can be chosen to converge to $0$
as slowly as desired.

Therefore, Theorem 3 will be proven if we are able to show that $\bar{f}$ is
Hölder from below. Furthermore, we also show that $\bar{f}$ is Hölder from
above, highlighting regularity of our mappings.

To do this, we first observe that the composition of $z\mapsto
ze^{i\alpha\log|z|}$ followed by $z\mapsto z|z|^{q-1}$ is precisely $z\mapsto
z|z|^{q-1}e^{i\alpha\log|z|}$. This observation suggests us to decompose
$\bar{f}=g\circ f$, where $f$ is essentially the first example in this section
(with different choice of $\alpha_{n}$) and $g$ is constructed by building
blocks (10) with $\alpha=0$ at each step. Morally, $f$ leaves fixed all
circles centered at $0$, and only rotates the annuli $A_{n}$, while $g$
conveniently stretches each $A_{n}$.

For any $p>1$, the bi-Hölder nature of $f$ has already been proven when
$\alpha_{n}=h(r_{n})\,r_{n}^{-1/p}$. Hence we can directly use the same proof
there after we estimate

$h(r_{n})\,\left(\log\frac{1}{r_{n}}\right)^{1/2}\,r_{n}^{-\frac{1}{p}}\leq
h(r_{n})\,r_{n}^{-1/(p-\epsilon)}$

for all small $r_{n}$ and $\epsilon=(p-1)/2$. Therefore, it only remains to
show that $g$ is bi-Hölder as well. To this end, we first show that $g$ is
Hölder from above using the fundamental theorem of calculus.

Let $x,y\in B(0,1)$ be given. Without loss of generality let us assume that
$|y|\geq|x|$ and let $w$ be the point for which $|w|=|x|$ and
$\arg(w)=\arg(y)$. Now

$|g(x)-g(y)|\leq|g(w)-g(x)|+|g(y)-g(w)|,$ (13)

and we will show that both of these are Hölder. First, since $g$ maps circles
centered at the origin radially to similar circles with equal or smaller
radius (as $q_{n}\geq 1$) it is clear that

$|g(x)-g(w)|\leq|x-w|\leq|x-y|.$

Let us then concentrate of the second part. First we note, that we can without
loss of generality assume that $y$ and $w$ are real numbers as $g$ is a radial
mapping. From our construction we see that the line segments $[r_{n},R_{n}]$,
$(R_{n+1},r_{n})$ and $(R_{1},1]$ partition the line segment $(0,1]$.
Furthermore, from (10) it is clear that the differential is bounded from above
by $1$ in the segments $(R_{n+1},r_{n})$ and $(R_{1},1]$. On the other hand,
in segments $[r_{n},R_{n}]$ we can estimate

$|g^{\prime}(t)|\leq\log\left(\frac{1}{r_{n}}\right)\leq\frac{C}{\sqrt{t}}$

for any $t\in[r_{n},R_{n}]$ with fixed $C$ that does not depend on $n$ or $t$.
This is so because of our choice of $q_{n}$. Combining these two estimates we
have

$|g^{\prime}(t)|\leq\frac{C}{\sqrt{t}}$

for any $t\in(0,1)$. Thus we can use fundamental theorem of calculus to
estimate

$\displaystyle|g(y)-g(w)|$ $\displaystyle=\int_{w}^{y}|g^{\prime}(t)|dt$
$\displaystyle\leq\int_{w}^{y}\frac{C}{\sqrt{t}}dt$
$\displaystyle=2C\left(\sqrt{y}-\sqrt{w}\right)$ $\displaystyle\leq
2C\sqrt{y-w}.$

This proves that also the second part in (13) is Hölder, and thus we obtain

$|g(y)-g(x)|\leq|g(y)-g(w)|+|g(w)-g(x)|\leq C\sqrt{|y-w|}+\sqrt{|x-w|}\leq
2C\sqrt{|x-y|},$

which shows $g$ is Hölder from above.

Let us next prove that $g$ is Hölder from below. To this end, given any two
points $x,y\in B(0,1)$ we again without loss of generality assume that
$|y|\geq|x|$ and let $w$ be the point for which $|w|=|x|$ and
$\arg(w)=\arg(y)$. Now, as $g$ is a radial homeomorphism, it follows that

$\displaystyle|g(x)-g(y)|\geq\max\\{|g(x)-g(w)|,|g(y)-g(w)|\\}$

Moreover,

$\displaystyle\max\\{|x-w|,|y-w|\\}\geq\frac{1}{2}|x-y|$

Therefore, it is enough to show that both $|g(x)-g(w)|$ and $|g(y)-g(w)|$
satisfy Hölder bounds from below. Note that if $x=0$ then clearly $w=0$ and we
have only the radial part $|g(y)-g(w)|$.

Let us first check the term $|g(x)-g(w)|$. Since $g$ maps radially circles
centered at the origin to similar circles we see that $|g(x)-g(w)|$ gets
contracted the same amount as the modulus $|g(w)|$ is contracted under $g$.
Now we must consider two possibilities, either $x,w\in A_{n}$ or $x,w\in
D_{n}$ for some $n$. Let us first assume $x,w\in A_{n}=B(0,R_{n})\setminus
B(0,r_{n})$ for some $n$. Here we impose an additional assumption that

$r_{n}<\left(\frac{1}{e}\right)^{q_{n-1}+q_{n-2}+...+q_{1}-(n-1)},$ (14)

which we can do as the radii $r_{n}$ can be assumed to decrease as fast as we
want. Then we can estimate

$\displaystyle|g(x)|$
$\displaystyle=\left(\frac{1}{e}\right)^{q_{n-1}+q_{n-2}+...+q_{1}-(n-1)}\cdot|x|\,\left(\frac{|x|}{R_{n}}\right)^{q_{n}-1}$
$\displaystyle\geq r_{n}\cdot|x|\,\left(\frac{|x|}{R_{n}}\right)^{q_{n}-1}$
$\displaystyle\geq r_{n}\cdot|x|\,\left(\frac{1}{e}\right)^{q_{n}-1}=e\cdot
r_{n}^{2}\cdot|x|$

for any $x\in A_{n}$. Therefore,

$\displaystyle|g(x)-g(w)|\geq e\cdot r_{n}^{2}\cdot|x-w|$ $\displaystyle\geq
C\cdot|x-w|^{3}$

since $|x-w|<C\cdot r_{n}$ for some fixed constant $C>0$ when $x,w\in A_{n}$.

Next, let $x,w\in D_{n}=B(0,r_{n})\setminus B(0,R_{n+1})$ for some $n$. Using
(14) we get

$\displaystyle|g(x)|$ $\displaystyle\geq
c\left(\frac{1}{e}\right)^{q_{n-1}+q_{n-2}+...+q_{1}-(n-1)}\cdot
r_{n}\cdot|x|$ $\displaystyle\geq c\cdot r_{n}^{2}\cdot|x|.$

Thus we can use a similar argument as in the previous case to estimate

$\displaystyle|g(x)-g(w)|$ $\displaystyle\geq c\cdot r_{n}^{2}\cdot|x-w|$
$\displaystyle\geq c\cdot|x-w|^{3}$

since $|x-w|<c\cdot r_{n}$ for some fixed constant $c>0$ when $x,w\in D_{n}$.

Since the set $\mathbb{D}\setminus\\{0\\}$ is partitioned by separated annuli
$A_{n}$ and $D_{n}$ we have thus proven that $|g(x)-g(w)|$ satisfies Hölder
estimates from below.

Finally, let us prove the Hölder estimates from below for the term
$|g(y)-g(w)|$. As the mapping $g$ is radial, we can again assume that $y$ and
$w$ are real. We aim to use again the Fundamental Theorem of Calculus, and
thus have to estimate the differential from below. Using (14), as well as the
fact that $q_{n}>1$, we can estimate for any real number $t\in[r_{n},R_{n}]$
that

$\displaystyle g^{\prime}(t)$
$\displaystyle=\left(\frac{1}{e}\right)^{q_{n-1}+q_{n-2}+...+q_{1}-(n-1)}\cdot
q_{n}\cdot\left(\frac{t}{R_{n}}\right)^{q_{n}-1}$ $\displaystyle\geq
r_{n}\,q_{n}\cdot\left(\frac{r_{n}}{R_{n}}\right)^{q_{n}-1}$
$\displaystyle=e\,q_{n}\,r_{n}^{2}$ $\displaystyle\geq c\cdot
t^{2}\,\log\frac{1}{t}.$

Next, if $t\in[R_{n+1},r_{n}]$, we have

$\displaystyle g^{\prime}(t)$
$\displaystyle=\left(\frac{1}{e}\right)^{q_{n-1}+q_{n-2}+...+q_{1}-(n-1)}\cdot\left(\frac{1}{e}\right)^{q_{n}-1}$
$\displaystyle\geq e\cdot r_{n}^{2}$ $\displaystyle\geq c\cdot t^{2}$

Thus, as before, since $(0,1)$ is partitioned by the intervals
$[r_{n},R_{n}]$, $[R_{n+1},r_{n}]$ and $[R_{1},1)$, we end up getting that

$\displaystyle g^{\prime}(t)\geq c\cdot t^{2}$

for every $t\in(0,1)$. Now, we use the fundamental theorem of calculus to get

$\displaystyle|g(y)-g(w)|$ $\displaystyle=\int_{w}^{y}g^{\prime}(t)dt$
$\displaystyle\geq\int_{w}^{y}c\cdot t^{2}dt$
$\displaystyle=C\left(y^{3}-w^{3}\right)$ $\displaystyle\geq C|y-w|^{3}$

This proves that the second term is Hölder from below as well, which in turn
proves that $g$ is Hölder from below. This finishes the proof of Theorem 3.

## References

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Albert Clop
Department of Mathematics and Computer Science
Universitat de Barcelona
08007-Barcelona
CATALONIA
<EMAIL_ADDRESS>

Lauri Hitruhin
Department of Mathematics and Systems Analysis
Aalto University
P.O. Box 11100 FI-00076 Aalto
Helsinki, Finland
<EMAIL_ADDRESS>

Banhirup Sengupta
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193-Bellaterra
CATALONIA
<EMAIL_ADDRESS>