# Robust Change Detection Based on Neural Descriptor Fields

Jiahui Fu, Yilun Du, Kurran Singh, Joshua B. Tenenbaum, and John J. Leonard
All authors are with the MIT Computer Science and Artificial Intelligence
Laboratory, Cambridge, MA 02139, USA.
<EMAIL_ADDRESS>work was supported by ONR
MURI grant N00014-19-1-2571 and ONR grant N00014-18-1-2832.

###### Abstract

The ability to reason about changes in the environment is crucial for robots
operating over extended periods of time. Agents are expected to capture
changes _during operation_ so that actions can be followed to ensure a smooth
progression of the working session. However, varying viewing angles and
accumulated localization errors make it easy for robots to falsely detect
changes in the surrounding world due to low observation overlap and drifted
object associations. In this paper, based on the recently proposed category-
level Neural Descriptor Fields (NDFs), we develop an object-level online
change detection approach that is robust to partially overlapping observations
and noisy localization results. Utilizing the shape completion capability and
$\text{SE}(3)$-equivariance of NDFs, we represent objects with compact shape
codes encoding _full_ object shapes from partial observations. The objects are
then organized in a spatial tree structure based on object centers recovered
from NDFs for fast queries of object neighborhoods. By associating objects via
shape code similarity and comparing local object-neighbor spatial layout, our
proposed approach demonstrates robustness to low observation overlap and
localization noises. We conduct experiments on both synthetic and real-world
sequences and achieve improved change detection results compared to multiple
baseline methods. Project webpage: http://yilundu.github.io/ndf_change

##  I Introduction

The ability to perform robust long-term operations is critical for many
robotics applications such as room scanning and household cleaning. As these
tasks usually involve frequent visits to the same environment over extended
periods of time, during which the environment may experience changes, robots
are expected to understand these newly-emerged scene differences, e.g., the
introduction and removal of a coffee mug, as they may impact the proper
subsequent actions to be taken _during_ operation.

An intuitive way to conduct change detection is to perform scene differencing
between current observations and scenes reconstructed from previous sessions.
Using various scene representations such as point clouds and Truncated Signed
Distance Fields (TSDF), previous works detect changed areas in the environment
through global point-wise or voxel-wise differencing on two scenes
reconstructed and pre-aligned _post hoc_ from sequential data [1, 2, 3]. These
methods, which treat the environment as an unordered collection of points or
voxels, demand high inter-session viewpoint overlap and are susceptible to
noisy sensor data and localization errors. Considering the fact that changes
take place at the object level, recent works [4, 5] explore the use of
semantic consistency for local object-level verification on top of the common
global point- or voxel-wise scene comparison scheme. Having demonstrated
improved robustness to localization errors, they are nevertheless prone to
failure with noisy reconstruction input and little observation overlap, both
of which are frequently encountered during online change detection tasks.

Figure 1: Approach overview. Given a sequence of observations as the source
and the target, during streaming of the target sequence, the system takes in
partial object point clouds from depth sensors and outputs changed objects on-
the-fly in the target w.r.t the source. (a) Given each partial object point
cloud, Neural Descriptor Fields (NDFs) represent the object as a shape code
encoding the full object shape and recover the object center from full shape
reconstruction. (b) Based on recovered object centers, observed objects are
organized in a spatial object tree, which consists of two coordinate interval
trees $(T_{x},T_{y})$ and allows for fast query of neighboring objects.
(c)-(d): Corresponding neighborhoods of the current object are found from the
source and the target object tree using object locations, where objects of
similar shape codes are matched. For each matched object pair, object graphs
of the neighborhood are constructed and compared to determine if the local
object layouts are different, which implies a changed object.

It is therefore highly desirable to seek approaches that yield consistent
representations of the object surmounting viewing angle limitations, where we
then adopt the recently proposed Neural Descriptor Fields (NDFs) [6]. NDFs
have been developed as a category-level, $\text{SE}(3)$-equivariant object
representation for object manipulation tasks. By encoding object as a
continuous $\text{SE}(3)$-equivariant function, NDFs are able to reconstruct
the full object shape through a compact shape encoding and further recover
object centers (translation). Considering that NDFs are formulated to only
ensure identical shape codes given rotated point clouds that are otherwise
identical, we augment its shape completion capability by enforcing a new shape
similarity loss with partial object observations as inputs, making it encode
consistent full object shape and positions across different viewing angles.

Following the object-level interpretation of the world via NDFs-derived object
representations, in this paper, we propose an object change detection approach
(see Fig. 1) for mobile robots, targeting the most common online operating
scenarios with high viewing angle variation, potential localization errors,
and no pre-alignment tools available. Based on the $\text{SE}(3)$-equivariance
property and category-level shape completion ability of NDFs, we represent the
object with a compact _full_ shape code obtained from partial observations,
and organize the objects in a spatial tree structure in terms of object
centers recovered from NDFs for fast query of object neighborhoods. By
associating objects through shape code similarity and decomposing the global
differencing scheme into local object-neighbor layout comparison, our approach
shows improved robustness to little observation overlap and localization
errors. Our main contributions are as follows:

  1. 1.

First, we explore the use of NDFs for category-generalizable shape consistent
object representation across different viewing angles.

  2. 2.

Next, we propose an online change detection approach based on NDF-derived
object representations and local object layout comparison.

  3. 3.

Finally, we demonstrate the effectiveness of the proposed approach on both
synthetic and real-world testing sequences featuring novel same-category
object instances not seen during training time.

##  II Related Work

###  II-A Change Detection

Previous works conduct change detection based on inputs in various 2D and 3D
environment representations. Given 2D image inputs, Derner et al. [7] build a
visual database and compare classical 2D feature descriptors extracted from
greyscale images for query and reference feature matching. For works with 3D
representations, Finman et al. [1] discover new objects as changed parts of
multiple depth point clouds while Herbst et al. detect changes through
movement of surfaces [8]. Ambrus et al. [2] compute a meta-room static
reference map based on point clouds, and discover new objects from changes via
spatial clustering. Fehr et al. [3] propose a multi-layer TSDF grid structure
and perform volumeric differencing for object discovery and class recognition.
Langer et al. [4] combine semantic information and supporting plane
information to discover objects newly introduced to the scene. Schmid et al.
[5] propose a multi-TSDF panoptic mapping approach and conduct online change
detection based on TSDF value comparison.

Most previous works focus on point-wise or voxel-wise differencing on pre-
aligned reconstructions in an offline fashion, demanding high observation
overlap and decent localization results. They therefore do not always satisfy
the need for online change detection under various viewing angles, as commonly
encountered during mobile robot operation.

###  II-B Neural Implicit Representations

Neural implicit representations have been proposed as a continuous,
differentiable, and parameterized representation of 3D geometry [9, 10, 11],
appearance [12, 13], and tactile properties [14] of both objects and scenes.
Neural implicit representations represent shapes as continuous functions,
enabling the principled incorporation of symmetries, such as $\text{SE}(3)$
equivariance [15, 6], as well as the construction of latent spaces that encode
class information as well as 3D correspondences [16]. Owing to their
continuous parameterized nature, several works have applied [6, 17] them to
robotics tasks as intermediate object representations directly inferable from
raw perception. Simenov et al. [6] show that symmetries incorporated from such
a representation can generalize demonstrations of objects to novel poses,
while [17, 18, 19] demonstrate their ability to be integrated with online
robotic mapping tasks.

In this paper, we explore how the $\text{SE}(3)$-equivariance of NDFs [6]
enables them to represent objects from partial observations and thus allows
for robust online change detection with disparate viewing angles and
localization noises.

##  III Category-level Object Representation for Partial Observations

We base our object representation on the recently proposed Neural Descriptor
Fields (NDFs) [6], a function $f_{\theta}$ that encodes both object shape and
pose through a category-level $\text{SE}(3)$-equivariant latent
representation.

###  III-A Neural Descriptor Fields

NDFs consist of an encoder function $f_{\theta}({\bf{P})=z}$, which maps a
partial object point cloud $\bf P$ into a global latent code $\bf z$, and a
decoder function, $\Phi(\bf{x},f_{\theta}(\bf{P}))$, which maps an input point
$\bf{x}$ to its predicted occupancy value:

$\displaystyle f_{\theta}({\bf{P)=z}}:\mathbb{R}^{n\times
3}\rightarrow\mathbb{R}^{k\times 3}$ (1)
$\displaystyle\Phi({\bf{x},f_{\theta}({\bf{P}}))}=\Phi{\bf(\bf{x},{\bf{z}}}):\mathbb{R}^{3}\times\mathbb{R}^{k\times
3}\rightarrow[0,1].$

The encoder function $f_{\theta}(\cdot)$ is constructed such that by zero-
centering the input $\bf P$, the inferred global latent code $\bf z$ is
represented as a vector of points that is equivariant with respect to
$\text{SO}(3)$ rotations of the input point cloud $\bf P$. This means that if
a point cloud is rotated by $R$, the inferred latent will be equivalently
rotated by $R$, as ensured via the Vector Neuron encoder layers [15]. By
subtracting from $\bf x$ the point cloud center of $\bf P$ as the translation,
we then make $f_{\theta}(\cdot)$ an $\text{SE}(3)$-equivariant shape occupancy
predictor for different points $\bf x$ on and off $\bf P$.

NDFs are trained with partial object point clouds recovered from semantic-
segmented RGB-D images and corresponding 3D occupancy voxelgrids of objects’
complete geometry. During training, the full model $[f_{\theta},\Phi]$ is
trained to predict the complete 3D occupancy of an object using the standard
cross-entropy classification loss $L_{occ}$:

$L_{occ}=\mathcal{L}(\Phi({\bf{p}},f_{\theta}({\bf{P}}),v),$ (2)

where $\bf{p}$ is a point sampled from the object occupancy grid and $v$ is
the ground truth occupancy value at point $\bf{p}$.

By feeding $\Phi(\cdot,\cdot)$ with a query point cloud $\mathcal{X}$, which
is obtained via uniform sampling within a large bounding box centered around
$\bf P$, the full shape point cloud $\mathcal{S}$ of the object can be
reconstructed in terms of the predicted occupancy values:

$\mathcal{S}=\\{{\bf x}|\Phi({\bf x},f_{\theta}({\bf x}|{\bf{P}}))>v_{0},{\bf
x}\in\mathcal{X}\\},$ (3)

where $v_{0}$ is the occupancy threshold to mark a point location as occupied.

###  III-B Shape Consistency

Thanks to the $\text{SO}(3)$-equivariance of ${\bf z}$, a shape code invariant
of view direction, ${\bf{s}}\in\mathbb{R}^{k}$, can then be extracted from its
rotation invariant portion:

${}s_{i}=||{\bf{z}}_{i}||_{2},i=1,2,...,k.$ (4)

Ideally, the shape code, serving as a compact representation of the full
object shape, should be consistently close among partial observations of the
same shape from various viewing perspectives, while discriminatively far apart
across observations of different shapes. While NDFs enable identical shape
codes given identical point clouds that are rotated, shape consistency across
partial inputs of the same shape is not inherently guaranteed.

To enforce $\bf{s}$ to be a discriminative yet consistent representation for
shapes seen from various viewing angles, as commonly seen during mobile robot
operation, we further formulate a shape similarity loss, $L_{shape}$,
fashioned after the idea of the triplet loss with [_anchor, positives,
negatives_] for person re-identification tasks [20]. Taking a partial point
cloud from object shape $i$ as the anchor ${\bf{A}}_{i}$, we assign it with a
positive sample ${\bf{P}}_{i}$ as the point cloud obtained from another
perspective of the same object, and any other partial observations of a
different object as the negative sample ${\bf{N}}_{i}$. With the distance
metric $D(\cdot,\cdot)$ chosen as the cosine similarity,
$L_{shape}({\bf{A}}_{i},{\bf{P}}_{i},{\bf{N}}_{i})$ then tends to pull the
shape codes of ${\bf{A}}_{i}$ and ${\bf{P}}_{i}$ closer, while pushing those
of ${\bf{A}}_{i}$ and ${\bf{N}}_{i}$ further apart:

$L_{shape}=-D({\bf{s}}_{{\bf{A}}_{i}},{\bf{s}}_{{\bf{P}}_{i}})+D({\bf{s}}_{{\bf{A}}_{i}},{\bf{s}}_{{\bf{N}}_{i}}).$
(5)

Moreover, to ensure that $L_{shape}$ always finds the more informative
$(\bf{A},\bf{P},\bf{N})$ triplet, we adopt the _batch hard_ way for triplet
forming [20], i.e., using the most dissimilar $(\bf{A},\bf{P})$ and the most
similar $(\bf{A},\bf{N})$ within each batch to guide training.

We hence populate each training batch $B$ with randomly generated observations
of $({o_{A}},o_{P})$ pairs of different (but likely repetitive) object shapes,
which ensures that at least two observations exist for each object within the
batch. Every sample within the batch is then treated as an anchor, and paired
batch-wise for the most dissimilar positive and the most similar negative
samples. Hence the final batch-hard shape similarity loss is formulated as:

$\displaystyle L_{b\\_shape}=\frac{1}{|B|}\sum_{i=1}^{N}\sum_{j=1}^{N_{i}}($
$\displaystyle-\min_{k\in[1,N_{i}]}D(o_{ij},o_{ik})$ (6)
$\displaystyle+\max_{m\neq i}D(o_{ij},o_{mn})),$

where $N$ is the number of objects whose observations are included in the
current batch, $N_{i}$ the number of samples of object $i$ and $o_{ij}$ the
$j$th observation of object $i$ within batch.

The final training objective is therefore formulated as the weighted
combination of the cross entropy loss and the shape similarity loss as:

$L=L_{occ}+\alpha L_{b\\_shape},$ (7)

where $\alpha$ is the weighting coefficient set as $\alpha=$ 0.01.

Figure 2: Training sample generation. Snapshots of the rendered scenes are
presented on the right. Along with a height uniformly varying within 0.1 m
from the tabletop, 2D camera locations (blue dots) are drawn uniformly from
the hollow rectangular regions centered around the table. The width of the
region is determined by the closest and furthest camera-to-table distance
$d_{c}$ and $d_{f}$. $2-4$ geometric object models are randomly selected and
placed on the table to simulate potential occlusions.

###  III-C Training in Simulation

To overcome the data availability issue, we train our category-level shape-
consistent NDFs fully in simulation using depth images rendered with Pybullet
[21]. To consider the commonly encountered variation in viewing angles,
observation distances, and occlusions during operation, for each training
sample, we place a randomly-posed object on the table, along with 2-4 randomly
drawn objects around it to create occlusion. The camera locations are sampled
in a hollow rectangular region with the table at the center, thus accounting
for both nearby and longer distance observations seen in real-world operations
(see Fig. 2).

###  III-D Object Representation

We hence base our object representation on the shape code in (4) and the
object center (translation) $\bf{t}$ recovered from NDFs as $(\bf{s},t)$,
where $\bf{t}$ is simply the predicted center of the reconstructed full point
cloud $\mathcal{S}$ in (3):

${\bf{t}}=\frac{1}{|\mathcal{S}|}\sum_{{\bf{x}}\in\mathcal{S}}{\bf{x}}.$ (8)

By converting $\bf{t}$ into the world frame with the given camera pose, the
object is characterized by $\bf{s}$ for its full shape and $\bf{t}$ for its
global location. With the shape completion ability of NDFs, the current object
representation is expected to provide a compact as well as robust way to
distinguish object instances, even under varying viewing angles during robot
motion.

##  IV NDF-based Object Change Detection

Inspired by the idea that local relative comparison is less sensitive to
global localization drift than its global counterpart [4], we organize the
observed object instances in a spatial object tree based on the predicted
object centers, allowing for the convenient query of object neighborhoods. By
comparing local neighboring object layout, we improve the method’s robustness
to localization drift by avoiding direct comparison on the absolute values of
localization results.

###  IV-A Spatial Object Tree Construction

To enable the online execution of our approach, we construct and maintain a
spatial object tree in the world coordinate to organize the incoming data
stream.

Built on top of the coordinate interval tree proposed in [22], our object tree
comprises two translation interval trees, $T=(T_{x},T_{y})$, in the $x$ and
$y$ direction, respectively, which are initialized and updated simultaneously
in the same fashion each time a new object measurement $(\bf s,t)$ arrives.
Considering that adjacent objects are located mostly on the same plane, for
our case, it suffices to maintain trees only in the $x$ and $y$ direction for
neighborhood query (while an extension to the $z$ direction should be
straightforward).

Tree Construction.  Consider $T_{x}$ for instance. $T_{x}$ divides the $x-y$
plane into several equi-distant interval slices in the $x$ direction (Fig.
1(b)), where each instantiated interval $[x_{min},x_{max}]$ of fixed length
$l=x_{max}-x_{min}$ is represented by a node $n$. Each node stores a set of
object instances whose translation component ${\bf t}$ follows
${t}_{x}\in[x_{min},x_{max}]$. The $x_{min}$ and $x_{max}$ are determined by
the first ${t}_{x}$ that initializes this new node as
$[{t}_{x}-l/2,{t}_{x}+l/2]$. The new node is then inserted in the tree such
that the upper interval limit of a left child $n_{l}$ is always smaller than
the lower interval limit of its parent node $n_{p}$, i.e.,
$x_{l,max}<x_{p,min}$ and similarly for the right child $n_{r}$, we have
$x_{r,min}>x_{p,max}$.

Intra-tree Update.  We associate the incoming object measurement $\bf m=(\bf
s,t)$ with existing object instances in the tree through spatial proximity and
the shape cosine similarity used in (5). We first traverse through $T_{x}$ and
$T_{y}$ to locate the corresponding coordinate interval nodes $n_{x}$ and
$n_{y}$ that ${t}_{x}$ and ${t}_{y}$ land in, respectively. By finding the
intersection of the objects within $n_{x}$ and $n_{y}$, we obtain the
neighborhood $\mathcal{N}$ that $\bf m$ should be adjacent to as
$\mathcal{N}=\\{o|o\in n_{x}\wedge o\in n_{y}\\}$. The measurement is then
successfully associated with an object instance ${o}_{0}=({\bf s}_{0},{\bf
t}_{0})$ in the neighborhood when:

$\displaystyle||{\bf t}_{0}-{\bf t}||<\delta_{d}\wedge D({\bf s}_{0},{\bf
s})>\delta_{s},$ (9)

where $\delta_{d}$ and $\delta_{s}$ are the maximum distance threshold and
minimum shape similarity threshold for valid association. We then replace the
old shape code and the recovered object center of the object with $\bf m$ as
an update if the average occupancy value of the reconstructed full shape point
cloud for $\bf m$ (calculated in (3)) is higher than that of $o_{0}$.

###  IV-B NDF-based Change Detection

Given two sequences as the source and the target, we construct the source
object tree $T_{s}$ in advance and perform object change detection during
construction of the target tree $T_{t}$.

Sequence Registration.  We wish to apply our approach in an online mobile
operating scenario where post-alignment tools or motion capture systems are
not always readily available. Therefore, we first conduct a rough alignment
between the target and the source sequence at the beginning of the target
traverse, in the hope that target objects can correctly locate the
corresponding neighborhood patch in the source so as to enable _valid_
neighboring object comparison.

With all the shape codes of the source objects easily obtainable from $T_{s}$,
after intra-tree update for $T_{t}$, we determine each corresponding source
object ${o}^{\prime}$ for the current target object ${o}$ as the one sharing
the highest shape code similarity among all the objects (if exist) whose shape
similarity with $o$ is above $\delta_{s}$ (similar to (9) amid intra-tree
measurement-object association).

After $N$ pairs of object correspondences ${(o^{\prime},o)}$ have been
accumulated, we apply Single Value Decomposition (SVD) with RANSAC onto the
$N$ object center pairs $({\bf t^{\prime}}_{i},{\bf t}_{i})$ and obtain the
relative transform $T_{rel}$ between the two sequences. This considers the
potential inclusion of changed object instances, while assuming that they only
take up a small portion of the environment to allow for alignment when the
target sequence starts. Here, we set $N=6$ to account for transform accuracy
versus timing to start change detection.

Figure 3: Neighborhood comparison as object graph matching. (1) Shape $a$ in
the target neighborhood $\mathcal{N}$ forms two shape-similar pairs $(a,b)$
and $(a,c)$ from the corresponding source neighborhood $\mathcal{N}^{\prime}$
(dots colored in orange). (2) Object graphs centered at $a$, $b$, and $c$ are
constructed respectively, demonstrating local inter-object layout. Graphs of
each pair are compared as shown in (2.1) for $(a,b)$ and (2.2) for $(a,c)$.
Each object graph consists of vertices as all the objects in the neighborhood
(colored dots) and directed edges pointing from the object to the neighbor
with length as the object center difference (colored arrows, e.g., ${\bf
e}_{cb}={\bf t}_{b}-{\bf t}_{c}$). (3): Graph (neighborhood layout) comparison
is conducted edge-wise, where edge similarity between vertex-matched edges
(shown in arrows of the same color) is measured by the Euclidean distance
between the two edge vectors.

Change Detection. Rather than examining object-wise correspondences through
spatial proximity and shape similarity (as in (9)), we draw support from
neighboring objects and detect changes through the relative spatial layout
consistency of the object within its target neighborhood and that of its
corresponding neighborhood (if exists) in the source.

When an object measurement $\bf m=(s,t)$ arrives, we update the $T_{t}$ by
finding/instantiating the associated object instance $O$. Considering the
online operation setting, change detection is then performed on $O=({\bf
s}_{0},{\bf t}_{0})$ that has _not_ been marked as changed before.

Given $O=({\bf s}_{0},{\bf t}_{0})$ along with its neighborhood
$\mathcal{N}_{0}$ in the target, we find the projected location in $T_{s}$
with $T_{rel}$ and query the corresponding source neighborhood
$\mathcal{N}^{\prime}=\\{o^{\prime}_{i}=({\bf s}_{i},{\bf t}_{i})\\}$.
Potential object matches are found based on shape code similarity between
${\bf s}_{0}$ and each ${\bf s}_{i}$ in $\mathcal{N}^{\prime}$.

The most straightforward change cases are that either $\mathcal{N}^{\prime}$
is empty or no matched $o_{i}$ is found in the source neighborhood, as both
indicate newly added objects to new locations or else new shapes.

Then with object matches found within $\mathcal{N}^{\prime}$, we conduct local
neighborhood comparison for each matched pair $(O,o^{\prime}_{i})$ (see Fig. 3
for a concrete example). This also serves as a validation step for rejecting
false positives in dealing with noise-corrupted localization. When object
position changes at a scale similar to cross-session localization errors, it
can be hard to determine if the absolute object center difference between
$(O,o^{\prime}_{i})$ is brought by actual changes (e.g., object moves within
the plane) or merely localization/projection drift, while the latter can be
validated through the relatively more stable inter-object spatial layout.

For each of the matched pairs, neighborhood comparison is accomplished via
object graph matching.

Figure 4: Synthetic scene layout and camera trajectories. Mug instance are
shown in colored dots around eight tables as numbered squares. Changes of
different types are shown with different icons. A snapshot of the rendered
scene is shown in the upper right corner.

A local object graph $G_{o}=(V,E)$ for object $o$ in neighborhood
$\mathcal{N}_{0}$ is constructed as a directed graph with vertices as all the
objects in $\mathcal{N}_{0}$, $V=\\{o_{i}|o_{i}\in\mathcal{N}_{0}\\}$, and
directed edges as the object center difference pointing from $o$ to its
neighbors, $E=\\{{\bf e}_{oi}|{\bf e}_{oi}={\bf t}_{i}-{\bf t}_{o},\forall
i\in\mathcal{N}_{0},i\neq o\\}$. Considering the limited number of objects
within a neighborhood (size set similar to a tabletop), the relatively small
graph size makes it possible for edge-wise matching. We find corresponding
edges ${\bf e}_{oj}$ and ${\bf e}_{o^{\prime}j^{\prime}}$ through vertex shape
matching. Similar edges, indicating unchanged inter-object layout, is
determined by the Euclidean distance between them:

$\displaystyle\sum\limits_{j}\mathds{1}(||{\bf e}_{oj}-{\bf
e}_{o^{\prime}j^{\prime}}||_{2}\leq\delta_{e})=\begin{cases}0,&\text{changed}\\\
\geq 1,&\text{unchanged}.\end{cases}$ (10)

If at least one pair of edges is found to be closer than $\delta_{e}$, i.e.,
at least one out of the few neighboring objects remains the same, the local
object layout is marked as consistent. This implies an unchanged object, and
vice versa. Note that in the rarer case when either $O$ or $o^{\prime}_{i}$ is
the only object in the neighborhood, we revert back to the object-wise
comparison in terms of spatial proximity and shape similarity.

Figure 5: Real-world scene layout and camera trajectories. Changes take place
on table 1 and 3, and are indicted by the colored letter block highlighted in
the top view pictures of the three tables.

Lastly, for detecting objects removed from the source, during target streaming
we label each source object as _observed_ if it ever participates in shape
matching with any target objects, then _removed_ objects can be found after
the target sequence finishes as those _observed_ but never matched with a
target object.

##  V Experiments and Results

Targeting robust online change detection in the face of little observation
overlap and localization errors, we would like to answer two questions: (1)
Can we use the shape-consistent NDF-based representation as a valid category-
level object representation that robustly generalizes to unseen object
instances under partial observations? (2) Can our change detection approach
perform well on sequences with little observation overlap and demonstrate
robustness to localization errors? We evaluate our approach on both synthetic
and real-world sequences consisting of various mug instances, where mugs are
added, removed, and switched places between sequences.

###  V-A Datasets

To better examine the effectiveness of our _category-level_ object
representation and the change detection approach built on top of it, we choose
to have our testing sequences composed of objects from the same category. We
hence design two pairs of testing sequences, in simulation and in real world,
respectively, featuring coffee mugs of diverse shapes observed from distinct
viewing angles. Considering the extensive multi-category results reported in
relevant works [6, 23], we argue that the effectiveness of our approach should
be extendable to the multi-category case by incorporating more object
categories into the training data.

Figure 6: 3-view shape reconstruction of five unseen mugs (each column) from
the synthetic and real-world testing sequences. For each view, Left: Partial
observation. Right: Partial point clouds (red) and full shape reconstruction
(green). For the first two columns of synthetic mugs, the two target mugs for
reconstruction is highlighted in green. The point clouds are rendered in an
orientation that showcases the boundary between partial observation and the
predicted shape completion, whose shape fitness between the two colored point
clouds demonstrates NDFs’ effectiveness in completing partial shapes.

Synthetic Sequences. We set up an environment in Pybullet with 35-40 instances
of 20 ShapeNet [24] mugs models scattered around eight tables and obtain RGB-D
images along with segmentation masks from two preset camera trajectories. The
mug models are unseen by NDFs during training. Each table supports 3-5 mugs
with mugs newly added, removed, or switched locations, creating in total 12
changes between the two sessions. The two camera trajectories are designed
such that the camera always faces towards the nearest table, prompting less
observation overlap between sequences around table 1,2, and 5 (see Fig. 4).

Real-world Sequences. We collect two sequences featuring 14 mugs of diverse
shapes randomly placed on three rectangular tables. We mount an Intel
RealSense L515 camera onto a Jackal robot close to mugs’ height and collect
RGB-D data with the trajectories shown in Fig. 5. The camera trajectories are
recovered using ORB-SLAM3 [25]. Since the camera is mounted facing sideways,
while the camera circles around table 1, 2, 3 in the source, it only weaves
through table 1 and 3 in the target, thereby creating observation disparity on
mugs along the inner side of table 1 and 3.

###  V-B Metrics

We adopt the commonly used precision and recall rate at the _object_ level,
i.e., the number of objects, for evaluation. We report the number of correctly
detected changes (true positives, TP), falsely detected changes (false
positives, FP), and undetected changes (false negatives, FN). Precision and
recall rate are computed as $\frac{TP}{TP+FP}$ and $\frac{TP}{TP+FN}$.

###  V-C Implementation Details

To train NDFs using an occupancy network, we set the camera-table distances as
$d_{c}=$ 0.2 m and $d_{f}=$ 5 m and generate 50,000 RGB-D partial observations
with 94 ShapeNet [24] mug models following the sample generation strategy in
III-C. Partial object point clouds are obtained by extracting corresponding
depth points in the camera frame indicated by the segmentation masks on RGB
images. We build our model upon the network implementation provided by [6] and
train it on two NVIDIA RTX 3090 GPUs using a learning rate of $6\times
10^{-4}$ with the Adam optimizer. The length of the interval tree $l$ is set
to be 1.2 m and 1.6 m for the synthetic and real-world sequence, respectively,
intending to group most objects on the same table plane. For parameters in IV,
we set $\delta_{s}=$ 0.9, $\delta_{d}=$0.02 m, and $\delta_{e}=$ 0.03 m.

###  V-D Generalization to Unseen Instances

To demonstrate the robust generalization of the shape-consistent NDFs to
unseen object instances under various viewing angles, in Fig. 6, we render the
reconstructed 3D shapes of five mugs drawn from the synthetic and real-world
testing sequences. The three views are selected in the hope of representing
some of the most typical perspectives when observing a mug, e.g., handles
visible in different directions and handle obscured. We can conclude that
despite the variety of the selected mug shapes, the shape-consistent NDFs
still demonstrate satisfactory performance in encoding and completing the full
shape under varying viewing angles, justifying the adoption of NDF-derived
object representation in our change detection task.

This can be attributed to the fact that by learning purely from geometric
structures embedded in the partial point clouds, NDFs are able to transfer
this structural knowledge to unseen instances regardless of object color and
texture. We also include cases with occlusion and shape ambiguity. For
occluded observations, as shown in the first two columns of “View 2” and “View
3” , we can still see decent reconstruction results by virtue of the occlusion
scenes incorporated in the training data. For the case of shape ambiguity,
i.e., the handle is obscured in the second and last column of “View 2”, the
major body parts of the mugs are still reconstructed reasonably well. While
the handle location was incorrectly predicted, such mistakes only result in
negligible errors to object center recovery.

###  V-E Results on Change Detection

We further present the change detection results on the two sets of testing
sequences based on our proposed approach and compare them to two baselines.

The first baseline is the typical nearest neighboring point search (NN)
commonly used for offline scene differencing [4, 2]. Given two spatially
aligned observations $S$ and $T$, the change point set $C$ of $T$ w.r.t $S$ is
found as $C=\\{p|p\in T,\forall s\in S,||p-s||_{2}>d\\}$. We adapt it to the
object-level and define that an object is _changed_ if a certain proportion
$r$ of its target point cloud finds no neighbors in the source. To make the
results interpretable to online object-level change detection evaluation, we
mark an object as _changed_ the first time during data streaming it is
detected as _changed_ , as during real-world operation, corresponding actions
will be taken right after changes are detected and usually no correction can
be made for false positives.

The second baseline is the recently proposed panoptic multi-TSDFs (PMT)
mapping method by Schimd et al. [5], which represents panoptic entities as
TSDF submaps and captures online object-level scene changes based on voxel-
wise weight counting informed by TSDF value differences. After tuning the
default parameters for better performance, we count the total number of
changed objects based on the number of conflicting submaps determined by the
baseline.

Figure 7:  The source and target sequences share little observation overlap
for the green and red mugs. Despite the distinct viewing angles and occlusion,
(blue: from source, yellow: from target), our approach can still recover the
shape and accurate object center from the NDFs representation, thereby
successfully associating the target point clouds to the correct source mugs.

Since both baselines require pre-aligned sequences, we feed the baselines with
ground truth camera poses from Pybullet and those aligned by ORB-SLAM3,
respectively. For NN, the reference partial point cloud of each source object
is fused from the whole sequence of the source depth images. Data associations
between the target partial mug observations and the source point clouds are
determined by the instance ID in Pybullet for synthetic sequences and manual
labeling based on the panoptic masks produced by Detectron2 [26] (used in PMT)
for real-world sequences.

All results are obtained on a laptop with an Intel Core i7-9750H CPU and an
Nvidia GeForce RTX 2070 GPU. The network inference speed for NDF is around
0.019 s per frame, and the change detection operation with an expanding
spatial tree executes at an average speed of 0.023 s per frame given the
magnitude of the object number in our experiment setting (up to 40 mugs).
These result in an overall speed of 24 FPS for the complete pipeline, thus
showing no obvious latency for online operation.

Results on Synthetic Sequences.  We present the change detection results of
the three methods in Table. I. Here for NN, we set $d=$ 0.002 m and $r=$ 0.3.
We apply a 6 DoF random transform to the target camera poses when running our
approach so as to simulate the unavailability of motion capture systems during
common mobile robot operations.

We can see that our method accurately detects all the changes without any
false positives. On the one hand, NN and PMT respectively ignore four and one
changes for the two pairs of mugs switching positions at table 5 and 7, as the
similar viewing angles of these mugs during the source and target traverse
induce high overlap between the target and source point clouds, thereby
confounding the baselines by the existence of neighboring points and similar
local geometry.

On the other, both baselines falsely mark the four mugs on table 1 and 2 as
_changed_ , which is due to the remarkable viewing angle difference on these
four mugs between the source and target trajectory. As explained in Fig. 7,
the two mugs highlighted in red and green are observed in almost opposite
directions, which then leads to little/no overlap between the two partial
point clouds (blue and yellow). Therefore, few neighboring points can be found
between the source and target observations, while at the same time leads to
drastic differences in the resulted TSDF values. In contrast, our method is
capable of encoding the full mug shape with a compact shape code from the
partial observations and hence produces no false positives.

Results on Real-world Sequences. We report change detection results of the
three methods on the real sequence in Table. II and present the
reconstructions obtained from PMT and our method (for mugs) in Fig. 8. For
fair comparison, we obtain the mug point clouds for all the three methods
using the panoptic mask adopted in PMT, which are generated by Detectron2 [26]
and preprocessed by DBSCAN [27]. Here, the parameters of NN are set as $d=$
0.002 m and $r=$ 0.4 considering the noises in the real-world point clouds.
Due to localization drift between two sessions, for PMT, the final number of
_changed_ objects are counted after manual merging of the few submaps
instantiated for the same object. In order to mimic the real operating
scenarios, the camera poses to our approach are obtained through running ORB-
SLAM3 separately on the two traverses.

Figure 8: Top to bottom: real scenes, result from PMT, and results from NDFs. (a) Full reconstruction of the source real-world sequence. Front views of the three tables (first row) are included for mug layout illustration. Fully trained in simulation, NDFs are able to reconstruct 12 out of the 14 unseen mugs given partial observations, and the two missing mugs are highlighted in green in the front view pictures. (b) PMT and NDFs reconstruction results for the two mugs falsely detected as changes. As shown in the first row, two mugs (highlighted in green) are observed in almost opposite views in the source and target sequence. As shown in the second row, the missing part in the back of the PMT reconstruction misleads PMT to mark these two mugs as changed, while our NDF-based approach correctly recognize the mugs by virtue of the robust recovery of the full mug shape. TABLE I: Change Detection Results on the Synthetic Sequences. Best results are marked in bold. Approach | TP | FP | FN | Precision | Recall
---|---|---|---|---|---
NN | 8 | 4 | 4 | 66.7% | 66.7%
PMT | 11 | 4 | 1 | 73.3% | 91.7%
Ours | 12 | 0 | 0 | 100% | 100%
TABLE II: Change Detection Results on the Real-world Sequences. Best results are marked in bold. Approach | TP |  FP | FN | Precision | Recall
---|---|---|---|---|---
NN | 3 | 3 | 2 | 50% | 60%
PMT | 5 | 3 | 0 | 62.5% | 100%
Ours | 5 | 1 | 0 | 83.3% | 100%

When confronted with noisy point cloud inputs and localization results,
despite the reconstruction failure of the beige mug on table 1 and the blue
mug on table 2 (as shown Fig. 8(a)) due to poor point cloud and panoptic mask
quality, our method still maintains better detection efficacy in terms of both
precision and recall rate. From Fig. 8(b), we see that the confusion brought
by disparate viewing angles persists, as both NN and PMT wrongly mark the
black and blue mug on table 3 (highlighted in green) as _changed_. This
matches the PMT reconstruction results shown in the second column, as the two
mugs have just one side partially reconstructed during each session.
Furthermore, another false detection emerges as the dark blue mug on table 3
(highlighted in yellow in Fig. 8(a)), which results from the larger
localization error at the starting point of the target trajectory. Our method
reduces its reliance on the absolute accuracy of camera pose estimation by
conducting local neighborhood layout matching, i.e., the relative spatial
relationship among neighboring mugs, thereby successfully recognizing the mug
through its unchanged relative positions with its neighbors.

Limitation and Extension. The proposed approach works well under the
assumption that the changed objects do not take up a predominant portion of
all the objects in the scene, which is a common case for doing frequent visits
of the same environment whose changes occur incrementally. For scenes with
drastic changes between the two scans, our approach may not work well with the
absence of a consistent local layout to refer to. Our approach can be further
extended to detect objects that only rotate but do not move by utilizing the
$\text{SO}(3)$ equivariance of NDFs, i.e., computing an extra $\text{SO}(3)$
transform between the two $\bf{z}$’s after the shape-similarity-based
registration step. This situation was not included in the experiment as it is
rare for changed daily objects to have pure rotation but no translation.

##  VI Conclusion and Future Work

In this paper, we propose an online object-level change detection approach
based on an NDF-derived object representation, demonstrating improved
robustness to viewing angle disparity and localization drift. For future work,
we would like to test the approach’s scalability to larger scenes if
incorporated as part of an object-level SLAM system targeting long-term
operation, and further explore the potential of using NDFs for providing
object pose constraints to help improve camera localization.

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# Spectra of Weighted Composition Operators with Quadratic Symbols

Jessica Doctor  Timothy Hodges  Scott Kaschner  Alexander McFarland  Derek
Thompson  Department of Mathematics, Butler University, 4600 Sunset Ave,
Indianapolis, IN 46208  Department of Mathematics, Taylor University, 236 W.
Reade Ave, Upland, IN 46989<EMAIL_ADDRESS>
###### Abstract

Previously, spectra of certain weighted composition operators
$W_{\psi,\varphi}$ on $H^{2}$ were discovered under one of two hypotheses:
either $\varphi$ converges under iteration to the Denjoy-Wolff point on all of
$\mathbb{D}$ rather than compact subsets, or $\varphi$ is “essentially linear
fractional”. We show that if $\varphi$ is a quadratic self-map of $\mathbb{D}$
of parabolic type, then the spectrum of $W_{\psi,\varphi}$ can be found when
these maps exhibit both of the aforementioned properties, as most of them do.

###### keywords:

uniform convergence , essentially linear fractional , composition operator ,
weighted composition operator

###### MSC:

[2010] 40A30 ,

###### MSC:

[2010] 47B33 ,

###### MSC:

[2010] 47B35 ,

###### MSC:

[2010] 47A10

††journal: Journal of Mathematical Analysis and Applications t1t1footnotetext:
The authors were funded by an NSF-CURM grant.

##  1 Introduction

Let $H^{2}$ denote the classical Hardy space, the Hilbert space of analytic
functions $\displaystyle f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ on the open unit
disk $\mathbb{D}$ such that

$\|f\|^{2}=\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty.$

A composition operator $C_{\varphi}$ on $H^{2}$ is given by
$\mbox{$C_{\varphi}$}f=f\circ\mbox{$\varphi$}$. We call $\varphi$ the symbol
of the associated composition operator. When $\varphi$ is an analytic self-map
of $\mathbb{D}$, the operator $C_{\varphi}$ is bounded. Composition operators
on $H^{2}$ have been extensively studied for several decades; see [1] and [2]
for seminal books on the subject. One reason for their study is the deep
connection to Toeplitz operators: $T_{\psi}$ on $H^{2}$ is given by
$T_{\psi}f=P(\psi f)$ where $P$ is the projection back to $H^{2}$. When
$\psi\in\mbox{$H^{\infty}$}$, the space of bounded analytic functions on
$\mathbb{D}$, we simply have $T_{\psi}f=\psi f$, since $\psi f$ is guaranteed
to be in $H^{2}$, and all such Toeplitz operators are bounded. Throughout this
paper, we will assume $\psi\in\mbox{$H^{\infty}$}$. We write
$\mbox{$W_{\psi,\varphi}$}:=T_{\psi}\mbox{$C_{\varphi}$}$ and call such an
operator a weighted composition operator.

Another reason for the study of composition operators is the interesting
interplay between the operator-theoretic behavior of $C_{\varphi}$ and the
behavior of $\varphi$ on $\mathbb{D}$. The Denjoy-Wolff theorem guarantees
that any analytic self-map of $\mathbb{D}$ (apart from elliptic automorphisms)
will have a unique attracting fixed point $w$ in
$\overline{\mbox{$\mathbb{D}$}}$, and $\varphi$ converges, uniformly on
compact subsets of $\mathbb{D}$, to that point under iteration. This behavior
generally splits into three categories, each with different results for the
behavior of $C_{\varphi}$: either $\varphi$ has a fixed point properly in
$\mathbb{D}$ with $|\mbox{$\varphi$}^{\prime}(w)|<1$ (elliptic), or $w$ is on
the boundary (in a radial limit sense; $\varphi$ need not be analytic on
$\partial\mbox{$\mathbb{D}$}$), and $\mbox{$\varphi$}^{\prime}(w)<1$
(hyperbolic) or $\mbox{$\varphi$}^{\prime}(1)=1$ (parabolic). When the fixed
point is on $\partial\mbox{$\mathbb{D}$}$, we define
$\mbox{$\varphi$}^{\prime}(w)$ as an angular derivative, which coincides with
the typical derivative when $\varphi$ is analytic on
$\partial\mbox{$\mathbb{D}$}$. We will only be considering maps for which
$\varphi$ is analytic on all of $\overline{D}$. Generally, the parabolic case
is considered the most difficult of the three (see, e.g., [1, Chapter 7].)

The spectrum of an operator $T$ on a Hilbert space $H$, denoted $\sigma(T)$,
is given by $\\{\lambda\in\mathbb{C}:T-\lambda I\textrm{ is not
invertible}\\}$. The essential spectrum $\sigma_{e}(T)$ is the spectrum of $T$
in the Calkin algebra $\mathcal{B}(H)/\mathcal{K}(H)$, the bounded operators
on a Hilbert space $H$ modulo the compact operators. The essential spectrum is
always a subset of the spectrum. Spectra of weighted composition operators on
$H^{2}$ were studied in [3] and [4]; both papers depended on the behavior of
$\varphi$ on $\mathbb{D}$. In [3] it was assumed that $\mathbb{D}$ converges
uniformly to the Denjoy-Wolff point on all of $\mathbb{D}$ rather than compact
subsets (denoted UCI for uniformly convergent iteration). In [4], $\varphi$
was assumed to be essentially linear fractional, a generalization of linear
fractional maps, which play nicely with the various standard tools used to
study composition operators on $H^{2}$ (e.g. the reproducing kernels on
$H^{2}$ are linear fractional.) Both papers were able to completely
characterize $\sigma(\mbox{$W_{\psi,\varphi}$})$ if
$\psi\in\mbox{$H^{\infty}$}$ and $\varphi$ is hyperbolic or elliptic, and both
give partial results if $\varphi$ is parabolic.

Therefore, we are interested in maps for which $\varphi$ is parabolic, so that
we may combine the results of these prior publications. In this paper, we
study the spectrum of $W_{\psi,\varphi}$ when $\varphi$ is a quadratic self-
map of $\mathbb{D}$ of parabolic type. For most of these maps, we give a
complete characterization of $\sigma(\mbox{$W_{\psi,\varphi}$})$.

In the next section, we establish preliminary facts about $\varphi$ in this
setting and divide it into two cases. In Section 3, we establish that
$\varphi$ exhibits UCI in both cases. In Section 4, we characterize when
$\varphi$ is essentially linear fractional and then give our main spectral
results in Section 5. We end with some commentary and questions about further
work in this direction in Section 6.

##  2 $\varphi$ is of parabolic type

Throughout this paper we assume that $\varphi$ is analytic in $\overline{D}$
with a single fixed point $w$ on the boundary, with
$\mbox{$\varphi$}^{\prime}(w)=1$. Since $C_{e^{i\theta}z}$ is a unitary
operator, we can conjugate $C_{\varphi}$ by $C_{e^{i\theta}z}$ to rotate the
fixed point without affecting the spectrum; therefore we may assume that
$w=1$. We can then simplify the generic quadratic
$\mbox{$\varphi$}(z)=a_{2}z^{2}+a_{1}z+a$ by noting

$\displaystyle a_{2}+a_{1}+a=1$ (2.1) $\displaystyle 2a_{2}+a_{1}=1$ (2.2)

which gives $a_{1}=1-2a_{2}$, and $a_{2}=a$, so we have
$\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$.

We now need to confirm exactly when $\varphi$ maps $\mathbb{D}$ into
$\mathbb{D}$.

######  Proposition 2.1.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ so that
$\mbox{$\varphi$}(1)=1,\mbox{$\varphi$}^{\prime}(1)=1$. Then $\varphi$ is a
self-map of $\mathbb{D}$ if and only if $|a-\frac{1}{4}|\leq\frac{1}{4}$.
Furthermore, the only elements of $\partial\mbox{$\mathbb{D}$}$ that are not
mapped into $\mathbb{D}$ are $\mbox{$\varphi$}(1)=1$, and when
$|a-\frac{1}{4}|=\frac{1}{4}$, $\mbox{$\varphi$}(-1)=4a-1$.

###### Proof.

Note that $\mbox{$\varphi$}(-1)=4a-1$, so we require that $|4a-1|\leq 1$.
Equivalently, we have $|a-\frac{1}{4}|\leq\frac{1}{4}$, so that $a$ is
contained in the circle of radius $\frac{1}{4}$ centered at $\frac{1}{4}$.

Now suppose instead that $|a-\frac{1}{4}|\leq\frac{1}{4}$ is given. Then, we
will find the image of the unit circle under $\varphi$. Thinking of $\varphi$
as a function of $a$, we know by the Maximum Modulus Principle that $|f(z)|$
will be maximized when $|a-\frac{1}{4}|=\frac{1}{4}$. Therefore, writing
$a=A+Bi$, we have $A^{2}+B^{2}=\frac{1}{2}A$ and if $z=X+Yi$, $X^{2}+Y^{2}=1$.

If
$\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a=(A+Bi)(X+Yi)^{2}+(1-2A-2Bi)(X+Yi)+(A+Bi)$,
then $|\mbox{$\varphi$}(z)|^{2}$ can be directly computed, resulting in a
28-term algebraic expression we will not suffer the reader to witness.
However, using the aforementioned equations, we are able simplify the
expression to $2AX^{2}-2A+1$. This is a real-valued quadratic in $X$ defined
on $[-1,1]$, with vertex $(0,-2A+1)$ (which has a nonnegative $y-$value for
$0\leq A\leq\frac{1}{2}$, as needed). Since the graph is a parabola whose
defining equation has a positive leading coefficient, the maximum values in
the domain are found at the endpoints $x=1$ and $x=-1$, which both result in
$|\mbox{$\varphi$}(z)|^{2}=1$. Note, however, equality is always attained at
$\mbox{$\varphi$}(1)=1$, but equality at $x=-1$ only happens for
$\mbox{$\varphi$}(-1)=4a-1$ when $|a-\frac{1}{4}|=\frac{1}{4}$; otherwise
$\mbox{$\varphi$}(-1)$ is properly in $\mathbb{D}$. In any case, for any other
value of $X$, we have $|\mbox{$\varphi$}(z)|^{2}<1$, so $\mathbb{D}$ is mapped
by $\varphi$ into $\mathbb{D}$ as desired. ∎

While the primary purpose of Proposition 2.1 is discovering exactly when
$\varphi$ is a self-map of $\mathbb{D}$, we should also note a few key facts
from this result that we will use later. In particular, we see that
$\mbox{Re\,}\textrm{ }a>0$ and $|a|\leq\frac{1}{2}$. Furthermore, we will take
advantage of knowing exactly which points on the unit circle are mapped by
$\varphi$ back onto the unit circle. As we have seen in this proof,
$|a-\frac{1}{4}|<\frac{1}{4}$ is a separate case from
$|a-\frac{1}{4}|=\frac{1}{4}$. This will remain true throughout the paper.

##  3 $\varphi$ converges uniformly under iteration on all of $\mathbb{D}$
(UCI)

Rather than iterating our quadratic maps directly, we will circumvent this
issue by taking alternate approaches to showing that
$\mbox{$\varphi$}_{n}\rightarrow 1$ uniformly on all of $\mathbb{D}$. For
$|a-\frac{1}{4}|<\frac{1}{4}$, we rely on arguments from complex dynamics.
When $|a-\frac{1}{4}|=\frac{1}{4}$, we work directly with the following
alternate characterization of uniform convergence.

######  Proposition 3.1.

A sequence of functions $f_{n}$ converges uniformly to its pointwise limit $f$
on all of a domain $D$ if and only if

$\lim_{n\rightarrow\infty}\sup_{z\in\mbox{$\mathbb{D}$}}|f_{n}(z)-f(z)|\rightarrow
0.$

######  Theorem 3.2.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ and
$|a-\frac{1}{4}|=\frac{1}{4}$, $a\neq 0$. Then the iterates of $\varphi$
converge uniformly to $1$ on the entire open disk $\mathbb{D}$.

###### Proof.

We will find a recursive pattern for $\sup_{z\in
D}|\mbox{$\varphi$}_{n}(z)-1|$ that approaches $0$.

Suppose $\sup_{z\in\mbox{$\mathbb{D}$}}|\mbox{$\varphi$}_{n}(z)-1|=r$ and
consider $|\mbox{$\varphi$}_{n+1}(z)-1|$. This factors as

$|a||\mbox{$\varphi$}_{n}(z)-1||\mbox{$\varphi$}_{n}(z)-(1-\frac{1}{a})|.$

We know that $\mbox{$\varphi$}_{n}(z)$ lies on the circle with center $1$ and
radius $r$ (call this circle $C$), and wish to find an upper bound on the
distance of $\mbox{$\varphi$}_{n}(z)$ from $z_{0}=1-\frac{1}{a}$. To do this,
we also need to find the image of the circle $|a-\frac{1}{4}|=\frac{1}{4}$
(ignoring $a=0$) under this transformation. Again, if $a=A+Bi$, then
$A^{2}+B^{2}=\frac{1}{2}A$. Then we have

$\frac{1}{a}=\frac{1}{A+Bi}=\frac{A-Bi}{A^{2}+B^{2}}=\frac{2(A-Bi)}{A}=2-\frac{B}{A}i,$

so we have $z_{0}=1-\frac{1}{a}=-1+\frac{B}{A}i$, and we know
$0<A\leq\frac{1}{2}$. Without loss of generality, assume $B$ is positive. Then
the furthest point from on $C$, still within $\overline{\mbox{$\mathbb{D}$}}$,
from $z_{0}$ is the point where $C$ intersects the unit circle in the fourth
quadrant, say $z_{1}$ (see Figure 1). Writing $x^{2}+y^{2}=1$ and
$(x-1)^{2}+y^{2}=r^{2}$, we can write $z_{1}$ in terms of $r$:
$\left(1-r^{2}/2\right)-\left(r\sqrt{1-\frac{1}{4}r^{2}}\right)i.$ We now wish
to find $|z_{1}-z_{0}|:$

$\left|\mbox{$\varphi$}_{n}(z)-(1-\frac{1}{a})\right|\leq|z_{1}-z_{0}|=\sqrt{1+2r^{2}+\frac{B^{2}}{A^{2}}-\frac{2B}{A}r\sqrt{1-\frac{1}{4}r^{2}}}.$

To incorporate the rest of our original expression for
$|\mbox{$\varphi$}_{n+1}(z)-1|$, we recall that
$|a|=\sqrt{A^{2}+B^{2}}=\sqrt{\frac{1}{2}A}$ and
$|\mbox{$\varphi$}_{n}(z)-1|=r$. We can also substitute
$B=\sqrt{\frac{1}{2}A-A^{2}}$ to get

$\sup_{z\in\mbox{$\mathbb{D}$}}|\mbox{$\varphi$}_{n+1}(z)-1|\leq
r\sqrt{r^{2}A+\frac{1}{4}-r\sqrt{\left(\frac{1}{2}A-A^{2}\right)\left(1-\frac{1}{4}r^{2}\right)}}.$

Figure 1: An example scenario for $z_{1}$ and $z_{0}$.

The value of $r$ actually depends on which iterate of $\varphi$ we are
considering, so we can rename it $r_{n}$. For our base case, consider
$r_{1}=|\mbox{$\varphi$}(z)-1|$. Then for
$z=X+Yi\in\partial\mbox{$\mathbb{D}$}$, if $a=A+Bi$, by direct computation we
have

$|\mbox{$\varphi$}(z)-1|^{2}=2-2AX^{2}+4AX-2A+4(X-1)\sqrt{A/2-A^{2}}\sqrt{1-X^{2}}-2X.$

For smaller values of $A$, this value is maximized at $X=-1$. However, when
$|(4a-1)-1|^{2}=|4A-2+4Bi|^{2}=1$, we have $A=\frac{3}{8}$, and the situation
changes. For $\frac{3}{8}\leq A\leq\frac{1}{2}$, we have
$|\mbox{$\varphi$}(z)-1|\leq 1$ for all $z\in\overline{\mbox{$\mathbb{D}$}}$.
For $0<A<\frac{3}{8}$, the maximum value of $|\mbox{$\varphi$}(z)-1|$ is
$2\sqrt{1-2A}$, so $r_{1}=\max\\{2\sqrt{1-2A},1\\}$.

For $0<A\leq\frac{1}{2}$, define
$f(r)=\sqrt{r^{2}A+\frac{1}{4}-r\sqrt{\left(\frac{1}{2}A-A^{2}\right)\left(1-\frac{1}{4}r^{2}\right)}}$,
so $f$ satisfies $0<f(r)<1$ for $0\leq r\leq\max\\{2\sqrt{1-2A},1\\}$. Then
$r_{n+1}=r_{n}\sqrt{r_{n}^{2}A+\frac{1}{4}-r_{n}\sqrt{\left(\frac{1}{2}A-A^{2}\right)\left(1-\frac{1}{4}r_{n}^{2}\right)}}<r_{n}$,
and thus, our sequence is decreasing. Since $r$ and $f(r)$ are both positive,
we know that $r_{n}$ is bounded below by $0$. Since our sequence is
monotonically decreasing and bounded below, it converges. Then $c=\lim
r_{n+1}=\lim{r_{n}}$, which means that $c=cf(c)$ where $f$ is defined as
above. We have already established that $f$ is positive, so the only solution
is $c=0$. Therefore, $\lim_{n\rightarrow\infty}\sup_{z\in
D}|\mbox{$\varphi$}_{n}(z)-1|=0$, so $\mbox{$\varphi$}_{n}$ converges under
iteration to $1$ uniformly on all of $\mathbb{D}$.

∎

For $|a-\frac{1}{4}|<\frac{1}{4}$, we now turn to traditional results in
complex dynamics, via Beardon [5].

######  Theorem 3.3.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ and
$|a-\frac{1}{4}|<\frac{1}{4}$. Then the iterates of $\varphi$ converge
uniformly to $1$ on the entire open disk $\mathbb{D}$.

###### Proof.

Consider the family of degree two polynomial maps
$\varphi\colon\mathbb{D}\rightarrow\mathbb{D}$ given by

$\varphi(z)=az^{2}+(1-2a)z+a,$

where $|a-1/4|<1/4$. These maps all have a parabolic fixed point at $z=1$ and
as maps of $\mathbb{C}$ are conjugate to the map $z\mapsto z-z^{2}$.
Specifically, for the maps $g,\sigma\colon\mathbb{C}\rightarrow\mathbb{C}$
given by

$\displaystyle g(z)$ $\displaystyle=$ $\displaystyle z-z^{2}\quad\mbox{and}$
$\displaystyle\sigma(z)$ $\displaystyle=$ $\displaystyle-\frac{1}{a}z+1,$

we have $g=\sigma^{-1}\circ\varphi\circ\sigma$ on $\mathbb{C}$. Since
$\varphi$ is forward invariant on the disk, $g$ is forward invariant on
$\mathbb{D}_{a}:=\sigma^{-1}(\mathbb{D})$, and we have the following commuting
diagram

$\textstyle{\mathbb{D}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{D}_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\mathbb{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathbb{D}}$
(3.1)

In particular, $\varphi_{n}$ converges uniformly to 1 on $\mathbb{D}$ if
$g_{n}$ converges uniformly to 0 on $\mathbb{D}_{a}$.

For specific details regarding the dynamics of one complex variable, we refer
the reader to ([5, 6, 7]). The domain of $g$ is partitioned into two totally
invariant sets, the Julia set, denoted $J(g)$, and Fatou set, denoted
$\mathcal{F}(g)$. The Fatou set is the set of points for which the sequence of
iterates forms a normal family, and the Julia set is the complement of the
Fatou set. In this case, $\mathcal{F}(g)$ is just the disjoint union of the
two open sets:

$\displaystyle B(g,0)$ $\displaystyle=$ $\displaystyle\\{z\in\mathbb{C}\colon
g_{n}(z)\rightarrow 0\\}\quad\mbox{and}$ $\displaystyle B(g,\infty)$
$\displaystyle=$ $\displaystyle\\{z\in\mathbb{C}\colon
g_{n}(z)\rightarrow\infty\\},$

called the basin of zero and the basin of infinity, respectively. $J(g)$ has
no interior, so it must be that $\mathbb{D}_{a}$ is a subset of $B(g,0)$ or
$B(g,\infty)$. Note that $\sigma^{-1}(0)=a$, and we have assumed that
$|a-1/4|<1/4$. Define

$A=\\{a\in\mathbb{C}\colon|a-1/4|<1/4\\}.$

Since

$\left|g(a)-\frac{1}{4}\right|=\left|a-a^{2}-\frac{1}{4}\right|=\left|a-\frac{1}{2}\right|^{2}\leq\left(\left|a-\frac{1}{4}\right|+\frac{1}{4}\right)^{2}<\frac{1}{4},$

we have that $A$, an open set with nonempty interior, is forward invariant by
$g$, so $A$ is a subset of either $B(g,0)$ or $B(g,\infty)$. It is easily
verified that for real $a\in A$, $g_{n}(a)\rightarrow 0$. Thus, we have
$A\subset B(g,0)$ (See Figure 2), so $\sigma^{-1}(0)\in B(g,0)$ as well. It
follows that $\mathbb{D}_{a}\subset B(g,0)$, and $g_{n}$ converges uniformly
on compact subsets of $B(g,0)$.

$B(g,0)$$B(g,\infty)$$A$$J(g)$

Figure 2: $B(g,0)$ contains $A$.

We complete the proof by adapting the proof of the Petal Theorem from [5], for
which we need one more conjugacy. Observe that $g$ is conjugate by
$\sigma_{0}(z)=1/z$ to $h\colon\mathbb{C}\rightarrow\mathbb{C}$ given by

$h(z)=z+1+\frac{1}{z-1}.$

The fixed point $z=0$ for $g$ corresponds to the fixed point at $\infty$ for
$h$. It is easy to show that the half plane
$\\{z\in\mathbb{C}\colon\mbox{Re\,}z>3\\}$ is forward invariant by $h$, so it
is contained in $B(h,\infty)$. The image of this set by $\sigma_{0}$ is a disk
of radius $1/6$ centered at $1/6$, contained in $B(g,0)$ (since the original
set was contained in $B(h,\infty)$). However, based on the picture of this
basin in Figure 2, it appears we can construct a much larger forward invariant
set.

Instead of starting with a half plane for $h$, we’ll use the following
parabolic region. For each $t\geq 0$, let

$\Pi+t:=\\{z=x+iy\colon y^{2}>12(3+t-x)\\}.$

It can be shown that $\Pi:=\Pi+0$ is forward invariant by $h$. In particular,
if $z\in\Pi$, we will show that $h(z)\in\Pi+1/2$. Let $z=x+iy$, $h(z)=X+iY$,
and $1/(z-1)=u+iv$, so $X=x+1+u$ and $Y=y+v$; then

$\displaystyle Y^{2}-12(3+1/2-X)$ $\displaystyle=$
$\displaystyle(y+v)^{2}-12(3+1/2-(x+1+u))$ $\displaystyle=$ $\displaystyle
y^{2}-12(3+1/2-x)+v^{2}+2yv+12(1+u)$ $\displaystyle>$ $\displaystyle
v^{2}+2yv+12(1+u)$ $\displaystyle\geq$ $\displaystyle 2yv+12(1+u)$
$\displaystyle\geq$ $\displaystyle 12-2|yv|-12|u|$ $\displaystyle>$
$\displaystyle 0,$

where the last inequality can be derived from the fact that $z\in\Pi$ implies
$|z|>3$. Since $Y^{2}>12(3+1/2-X)$, we have $h(z)\in\Pi+1/2$. We also have
inductively that if $z\in\Pi$, then for all positive integers $n$,

$h_{n}(z)\in\Pi+\frac{n}{2}.$

Thus, for $h_{n}(x)+X_{n}+iY_{n}$,

$|h_{n}(z)|^{2}=X_{n}^{2}+Y_{n}^{2}>X_{n}^{2}+12(2+n/2-X_{n})=(X_{n}-6)^{2}+6n>n,$

so $|h_{n}(z)|>\sqrt{n}$ and $h_{n}(z)\rightarrow\infty$ uniformly on $\Pi$.
The image of $\Pi$ by $\sigma_{0}$ is the cardioid

$P:=\sigma_{0}(\Pi)=\\{z=re^{i\theta}\colon 6r<1+\cos\theta\\},$

and $g_{n}(z)\rightarrow 0$ uniformly on $P$. The set $P$ is the “petal”
referred to in the Petal Theorem.

The preimage of $\mathbb{D}_{a}$ by $\sigma_{0}$ is the half plane

$H_{a}:=\sigma_{0}(\mathbb{D}_{a})=\\{z=x+iy\colon
2\mbox{Re\,}a+2\mbox{Im\,}a>1\\}.$

Note that $\partial H_{a}$, the boundary of $H_{a}$, intersects the $x$-axis
at $x=1/(2\mbox{Re\,}a)$, and since $\mbox{Re\,}a>0$, it is also never a
horizontal line. Moreover, $\partial\Pi$ is a horizontally oriented parabola,
intersecting the $x$-axis at $x=3$. See Figure 3. Thus, if
$1/6<\mbox{Re\,}a<1/4$, then $\partial H_{a}$ intersects the $x$-axis at
$x<3$, so $\partial H_{a}$ must intersect $\partial\Pi$ at exactly two finite
points. If $0<\mbox{Re\,}a\leq 1/6$, then $\partial H_{a}$ intersects
$\partial\Pi$ at exactly two finite points, one point (at which $\partial
H_{a}$ is tangent to $\partial\Pi$), or zero points. In the last two cases, we
have $H_{a}\subset\Pi$.

Returning to the coordinates centered at zero, this implies that either
$\mathbb{D}_{a}\subset P$ or $\partial\mathbb{D}_{a}$ intersects $\partial P$
at exactly two nonzero points. Recall that the set $\mathbb{D}_{a}$ is a disk
of radius $|a|$ centered at $\bar{a}$, so both $\partial P$ and
$\partial\mathbb{D}_{a}$ always contains the origin.

$\sigma_{0}$$P$$\mathbb{D}_{a}\backslash
P$$\mathbb{D}_{a}$$\Pi$$H_{a}$$H_{a}\backslash\Pi$

Figure 3: On the left are $\Pi$, $H$, and $H_{a}\backslash\Pi$. On the right
are the images of these sets by $\sigma_{0}$: the cardioid, $P$, and
$\mathbb{D}_{a}$ with boundaries intersecting at two nonzero points.

If $\partial\mathbb{D}_{a}$ intersects $\partial P$ at exactly two nonzero
points, $\overline{\mathbb{D}}_{a}\backslash P$ is nonempty with a boundary
comprised of the curve segment of $\partial P$ between the two nonzero
intersections and the curve segment of $\partial\overline{\mathbb{D}}_{a}$
outside $P$ and between the two nonzero intersections. Thus,
$\overline{\mathbb{D}}_{a}\backslash P$ is closed and bounded, so it is
compact. It follows that $\overline{\mathbb{D}}_{a}\backslash P$ is always a
strict, compact (sometimes trivially) subset of $B(g,0)$.

Since $g_{n}$ converges uniformly on $P$ and
$\overline{\mathbb{D}}_{a}\backslash P\subset B(g,0)$, we have that $g_{n}$
converges uniformly on $\mathbb{D}_{a}$. Therefore, $\mbox{$\varphi$}_{n}$
converges uniformly on $\mathbb{D}$.

∎

Now we have the full picture:

######  Corollary 3.4.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ and
$|a-\frac{1}{4}|\leq\frac{1}{4}$. Then the iterates of $\varphi$ converge
uniformly to $1$ on the entire open disk $\mathbb{D}$.

##  4 $\varphi$ is essentially linear fractional

We first provide the definition of an essentially linear fractional map as
given in [4].

######  Definition 4.1 ([4]).

An analytic self-map $\varphi$ of $\mathbb{D}$ is essentially linear
fractional if

  1. 1.

$\varphi$($\mathbb{D}$) is contained in a proper subsdisk of $\mathbb{D}$
internally tangent to the unit circle at some point
$\eta\in\partial\mbox{$\mathbb{D}$}$;

  2. 2.

$\mbox{$\varphi$}^{-1}(\\{\eta\\}):=\\{\gamma\in\partial\mbox{$\mathbb{D}$}:\eta\textrm{
belongs to the cluster set of }\eta\textrm{ of }\mbox{$\varphi$}\ \textrm{ at
}\gamma\\}$ consists of one element, say
$\zeta\in\partial\mbox{$\mathbb{D}$}$; and

  3. 3.

$\mbox{$\varphi$}^{\prime\prime\prime}$ extends continuously to
$\mbox{$\mathbb{D}$}\cup\\{\zeta\\}$.

We will quickly verify that most of the the maps
$\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ satisfy these conditions. (Again, we
assume $a\neq 0$.) To do so, we need another result from [4].

######  Proposition 4.2 ([4], Proposition 1.3).

Let $\varphi$ be an analytic self-map of $\mathbb{D}$ that extends to be
continuous on $\partial\mbox{$\mathbb{D}$}$. Suppose that $\mbox{$\varphi$}\in
C^{2}(1)$, that $\mbox{$\varphi$}(1)=1$, and that
$|\mbox{$\varphi$}(\zeta)|<1$ for
$\zeta\in\partial\mbox{$\mathbb{D}$}\backslash\\{1\\}$. If

$\mbox{Re\,}\left(\frac{1}{\mbox{$\varphi$}^{\prime}(1)}-1+\frac{\mbox{$\varphi$}^{\prime\prime}(1)}{\mbox{$\varphi$}^{\prime}(1)^{2}}\right)>0$

then $\mbox{$\varphi$}(\mbox{$\mathbb{D}$})$ is contained in a proper subdisk
of $\mathbb{D}$ internally tangent to $\partial\mbox{$\mathbb{D}$}$ at $1$.

Now to verify which our family of symbols $\varphi$ satisfy Definition 4.1:

  1. 1.

For our symbols, the equation in Proposition 4.2 simplifies to $2a$, and we
know $\mbox{Re\,}a>0$. However, Proposition 4.2 also requires that the rest of
the unit circle is mapped into the $\mathbb{D}$. If
$|a-\frac{1}{4}|<\frac{1}{4}$, Proposition 4.2 applies and
$\mbox{$\varphi$}(\mbox{$\mathbb{D}$})$ is contained in a subdisk of
$\mathbb{D}$ internally tangent at $1$. However, we cannot use this
proposition if $|a-\frac{1}{4}|=\frac{1}{4}$, since $-1$ maps to $4a-1$. If
$a\neq\frac{1}{2}$, however, it is worth noting that
$\mbox{$\varphi$}\circ\mbox{$\varphi$}$ satisfies the definition.

  2. 2.

Since our function is analytic on the boundary, we are asking that
$\mbox{$\varphi$}^{-1}(\\{1\\})$ contains only a single point from
$\partial\mbox{$\mathbb{D}$}$. The points that map to $1$ are the zeroes of
$\mbox{$\varphi$}(z)-1$, $1$ and $1-\frac{1}{a}$, and the latter is outside of
$\overline{\mbox{$\mathbb{D}$}}$ if $a\neq\frac{1}{2}$. If $a=\frac{1}{2}$,
then $\varphi$ is not essentially linear fractional since $-1$ also maps to
$1$, and neither is any iterate of $\varphi$, so we will handle that case
separately.

  3. 3.

Since $\mbox{$\varphi$}^{\prime\prime\prime}(z)\equiv 0$, this is trivial.

Unsurprisingly, just as with our work on UCI, we see that the defition of
essentially linear fractional splits our work into the cases when
$|a-\frac{1}{4}|<\frac{1}{4}$ and $|a-\frac{1}{4}|=\frac{1}{4}$. This
continues in the following spectral results.

##  5 Spectrum of $W_{\psi,\varphi}$

The uniform convergence of the iterates $\varphi$ on all of $\mathbb{D}$ give
the following spectral containment, found in [3].

######  Theorem 5.1.

[3, Corollary 10]. Suppose
$\mbox{$\varphi$}:\mbox{$\mathbb{D}$}\rightarrow\mbox{$\mathbb{D}$}$ is
analytic with Denjoy-Wolff point $a$, $\mbox{$\varphi$}_{n}\rightarrow a$
uniformly in $\mathbb{D}$, and $\psi\in\mbox{$H^{\infty}$}$ is continuous at
$z=a$ with $\psi(a)\neq 0$. Then we have

$\overline{\sigma_{p}(\psi(a)\mbox{$C_{\varphi}$})}\subseteq\overline{\sigma_{ap}(T_{\psi}\mbox{$C_{\varphi}$})}\subseteq\sigma(T_{\psi}\mbox{$C_{\varphi}$})\subseteq\sigma(\psi(a)\mbox{$C_{\varphi}$})$

In particular, if
$\overline{\sigma_{p}(\mbox{$C_{\varphi}$})}=\sigma(\mbox{$C_{\varphi}$})$,
then
$\sigma(T_{\psi}\mbox{$C_{\varphi}$})=\sigma(\psi(a)\mbox{$C_{\varphi}$})$.

Composition operators with parabolic symbols are notoriously more difficult
when it comes to spectral problems, and this is no different. We have little
information about $\sigma_{p}(\mbox{$C_{\varphi}$})$; instead our goal is to
use the fact that Theorem 5.1 gives us
$\sigma(\mbox{$W_{\psi,\varphi}$})\subseteq\sigma(\psi(a)\mbox{$C_{\varphi}$})$.
We now turn to two results from [4] regarding essentially linear fractional
maps.

######  Theorem 5.2.

[4, Theorem 3.3] Suppose that $\varphi$ is an essentially linear fractional
self-map of $\mathbb{D}$ fixing $1$. Suppose also that for
$s=\mbox{$\varphi$}^{\prime\prime}(1)$, Re  $s>0$. Then

$\sigma(\mbox{$C_{\varphi}$})=\sigma_{e}(\mbox{$C_{\varphi}$})=\\{e^{-st}:t\geq
0\\}\cup\\{0\\}.$

######  Theorem 5.3.

[4, Theorem 4.3] Suppose $\varphi$ is essentially linear fractional with
$\varphi$(1) = 1, and $\psi\in\mbox{$H^{\infty}$}$ is continuous at $1$. Then
$\mbox{$W_{\psi,\varphi}$}\equiv\psi(1)\mbox{$C_{\varphi}$}$ modulo the
compact operators.

Putting these facts together, we arrive at our main theorem.

######  Theorem 5.4.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ maps $\mathbb{D}$ into
$\mathbb{D}$ and $|a-\frac{1}{4}|<\frac{1}{4}$. Then for any
$\psi\in\mbox{$H^{\infty}$}$ continuous at $1$, we have

$\sigma(\mbox{$W_{\psi,\varphi}$})=\sigma(\psi(1)\mbox{$C_{\varphi}$})=\\{\psi(1)e^{-2at}:t\geq
0\\}\cup\\{0\\}.$

###### Proof.

By Theorem 5.1, we have
$\sigma(\mbox{$W_{\psi,\varphi}$})\subseteq\sigma(\psi(1)\mbox{$C_{\varphi}$})$.
By Theorem 5.2, we have
$\sigma(\mbox{$C_{\varphi}$})=\sigma_{e}(\mbox{$C_{\varphi}$})$. By Theorem
5.3, we have
$\sigma_{e}(\mbox{$W_{\psi,\varphi}$})=\sigma_{e}(\psi(1)\mbox{$C_{\varphi}$})$.
Putting these together, we have

$\sigma(\psi(1)\mbox{$C_{\varphi}$})=\sigma_{e}(\psi(1)\mbox{$C_{\varphi}$})=\sigma_{e}(\mbox{$W_{\psi,\varphi}$})\subseteq\sigma(\mbox{$W_{\psi,\varphi}$})\subseteq\sigma(\psi(1)\mbox{$C_{\varphi}$}),$

and since the first and last sets in the containment are equal, we have
$\sigma(\psi(1)\mbox{$C_{\varphi}$})=\sigma(\mbox{$W_{\psi,\varphi}$})$. By
Theorem 5.2, noting $\mbox{$\varphi$}^{\prime\prime}(1)=2a$, we have

$\sigma(\mbox{$W_{\psi,\varphi}$})=\\{\psi(1)e^{-2at}:t\geq 0\\}\cup\\{0\\}.$

∎

We now investigate the situation when $|a-\frac{1}{4}|=\frac{1}{4}$. While we
have shown that $\varphi$ still converges under iteration to $1$ uniformly on
all of $\mathbb{D}$, it is not essentially linear fractional since
$|\mbox{$\varphi$}(-1)|=4a-1|=1$. Here, we must actually divide our special
case yet again: if $a\neq\frac{1}{2}$, then
$\mbox{$\varphi$}_{2}=\mbox{$\varphi$}\circ\mbox{$\varphi$}$ is essentially
linear fractional (since $4a-1$ is then mapped into $\mathbb{D}$) and of
course $\mbox{$\varphi$}_{2}$ is uniformly convergent under iteration on all
of $\mathbb{D}$. Therefore, we as a corollary to Theorem 5.4, we get the
following:

######  Corollary 5.5.

Suppose $\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ maps $\mathbb{D}$ into
$\mathbb{D}$ and $|a-\frac{1}{4}|=\frac{1}{4}$, $a\neq 0,\frac{1}{2}$. Then
for any $\psi\in\mbox{$H^{\infty}$}$ continuous at $1$, we have

$\sigma(W_{\psi,\mbox{$\varphi$}_{2}})=\sigma(\psi(1)C_{\mbox{$\varphi$}_{2}})=\\{\psi(1)e^{-4at}:t\geq
0\\}\cup\\{0\\}$

where $\mbox{$\varphi$}_{2}=\mbox{$\varphi$}\circ\mbox{$\varphi$}$.

###### Proof.

The proof follows exactly as before, except that
$\mbox{$\varphi$}_{2}^{\prime\prime}(1)=4a$. ∎

The result of Corollary 5.5 suggests that it is most likely true that
$\sigma(\mbox{$W_{\psi,\varphi}$})$ is the same as shown in 5.4 when
$|a-\frac{1}{4}|=\frac{1}{4}$, but we do not have a proof.

However, even then, $a=\frac{1}{2}$ proves itself to be an entirely distinct
case. Here, instead of functional behavior, we now rely on a linear algebra
trick also used in [3].

######  Lemma 5.6.

[3, Lemma 3] If $A$ and $B$ are bounded linear operators on a Hilbet space
$\mathcal{H}$, then $\sigma(AB)\cup\\{0\\}=\sigma(BA)\cup\\{0\\}$.

Using this, we can now finish the story with $a=\frac{1}{2}$, which actually
varies just slightly from the result in 5.4.

######  Theorem 5.7.

Suppose $\mbox{$\varphi$}(z)=\frac{1}{2}z^{2}+\frac{1}{2}$, an analytic self-
map of $\mathbb{D}$. Then for any $\psi\in\mbox{$H^{\infty}$}$ continuous at
$1$, we have

$\sigma(\mbox{$W_{\psi,\varphi}$})=\\{\psi(1)e^{-t/2}:t\geq 0\\}\cup\\{0\\}.$

###### Proof.

If $f(z)=\frac{1}{2}z+\frac{1}{2}$, then we have
$\mbox{$C_{\varphi}$}=C_{z^{2}}C_{f}$. By Lemma 5.6, we have
$\sigma(\mbox{$C_{\varphi}$})\cup\\{0\\}=\sigma(C_{z^{2}}C_{f})\cup\\{0\\}=\sigma(C_{f}C_{z^{2}})\cup\\{0\\}=\sigma(C_{f^{2}})\cup\\{0\\}$.
Note that $f^{2}(z)=\frac{1}{4}z^{2}+\frac{1}{2}z+\frac{1}{4}$, which falls
under Theorem 5.2 with $s=\frac{1}{2}$. Since we also know $C_{\varphi}$ is
not invertible, we have $\sigma(\mbox{$C_{\varphi}$})=\\{\psi(1)e^{-t/2}:t\geq
0\\}\cup\\{0\\}.$

Likewise, let $\psi\in\mbox{$H^{\infty}$}$ be continuous at $1$ and consider
$\mbox{$T_{\psi}$}\mbox{$C_{\varphi}$}=\mbox{$T_{\psi}$}C_{z^{2}}C_{f}$. Again
by Lemma 5.6, we have
$\sigma(\mbox{$T_{\psi}$}C_{z^{2}}C_{f})\cup\\{0\\}=\sigma(C_{f}\mbox{$T_{\psi}$}C_{z^{2}})\cup\\{0\\}=\sigma(T_{\psi\circ
f}C_{f^{2}})\cup\\{0\\}$. Since $f$ maps $\mathbb{D}$ into $\mathbb{D}$
analytically and $f$ is continuous at $1$, we still have that $\psi\circ
f\in\mbox{$H^{\infty}$}$ and and $\psi\circ f$ is continuous at $1$ (and
$\psi\circ f(1)=\psi(1)$). Again, we also know that $T_{\psi}$$C_{\varphi}$ is
not invertible. Then, by Theorem 5.4 we have

$\sigma(\mbox{$W_{\psi,\varphi}^{\textstyle\ast}$})=\sigma(\psi(1)\mbox{$C_{\varphi}$})=\\{\psi(1)e^{-t/2}:t\geq
0\\}\cup\\{0\\}.$

∎

While our guess is that Theorem 5.4 holds for $|a-\frac{1}{4}|=\frac{1}{4}$
when $a$ is complex, the exponent in our result for $a=\frac{1}{2}$ does not
match up with Theorem 5.4, presumably because it is more distinct in its
failure to be essentially linear fractional. However, the final result is the
same in practicality: for $0<a\leq\frac{1}{2}$,
$\sigma(\mbox{$W_{\psi,\varphi}$})$ is the closed line segment connecting
$\psi(1)$ to the origin.

##  6 Implications and Further Questions

There are two important concepts in this paper that could be pursued further.
The first is extending our methods for showing quadratics exhibit uniformly
convergent iteration to higher-degree polynomials. Certainly Corollary 3.4
implies more than it says; e.g. $\mbox{$\varphi$}\circ\mbox{$\varphi$}$ is a
quartic self-map of $\mathbb{D}$ that also converges uniformly on all of
$\mathbb{D}$. For any polynomial map fixing $1$, $\mbox{$\varphi$}_{n}(z)-1$
will be a factor of $\mbox{$\varphi$}_{n+1}(z)-1$, suggesting that our
recursive approach used in Theorem 3.2 could be generalized, but it will
require deeper geometric intuition than we use here for quadratic maps.

The second important concept is the intersection of self-maps of $\mathbb{D}$
that exhibit uniformly convergent iteration on all of $\mathbb{D}$, and
essentially linear fractional maps. Certainly they do not perfectly align; we
have already seen that $\frac{1}{2}z^{2}+\frac{1}{2}$ converges under
iteration to $1$ uniformly on all of $\mathbb{D}$, but is not essentially
linear fractional. Likewise, a linear fractional map with both an interior and
a boundary fixed point (e.g. $\mbox{$\varphi$}(z)=\frac{z}{2-z}$) cannot
converge uniformly on all of $\mathbb{D}$; it must have only one fixed point
in $\overline{\mbox{$\mathbb{D}$}}$ [3, Theorem 3]. However, the intersection
of the two concepts is non-trivial, and leads to the following conjecture.

######  Conjecture 6.1.

Suppose $\varphi$ is an essentially linear fractional self-map of $\mathbb{D}$
with exactly one fixed point $w$ in $\overline{\mbox{$\mathbb{D}$}}$. Then
$\varphi$ converges under iteration to $w$ uniformly on all of $\mathbb{D}$.

Were this conjecture true, it would immediately provide a full description of
$\sigma(\mbox{$W_{\psi,\varphi}$})$ for a rather broad class of symbols, by
the same arguments made in this paper. Thus we end with the following list of
questions:

  1. 1.

What is $\sigma(\mbox{$W_{\psi,\varphi}$})$ if
$\mbox{$\varphi$}(z)=az^{2}+(1-2a)z+a$ and $|a-\frac{1}{4}|=\frac{1}{4},a\neq
0,\frac{1}{2}$ ?

  2. 2.

When do essentially linear fractional maps converge uniformly to their Denjoy-
Wolff point on all of $\mathbb{D}$?

  3. 3.

Which polynomial self-maps of $\mathbb{D}$ converge uniformly on all of
$\mathbb{D}$ to the Denjoy-Wolff point?

  4. 4.

If $\psi\in\mbox{$H^{\infty}$}$ is continuous at the Denjoy-Wolff point $w$ of
$\varphi$, and $\varphi$ is not an automorphism, then when, if ever, is
$\sigma(\mbox{$W_{\psi,\varphi}$})\neq\sigma(\psi(w)\mbox{$C_{\varphi}$})?$

## Acknowledgements

The authors would like to thank NSF for funding this research, and the
directors of CURM (Kathryn Leonard, Maria Mercedes Franco) for their counsel.
We would also like to thank Michal Misiurewicz, Paul Bourdon, and Carl Cowen
for their help with certain difficulties in the proofs.

## References

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  * [2] Joel H Shapiro.  Composition Operators and Classical Function Theory.  Springer, 1993.
  * [3] Carl C Cowen, Eungil Ko, Derek Thompson, and Feng Tian.  Spectra of some weighted composition operators on ${H}^{2}$.  Acta Sci. Math.(Szeged), 82:221–234, 2016.
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