# Counting Rotational Sets for Laminations of the Unit Disk from First
Principles

John C. Mayer  ,  Michael J. Moorman  ,  Gabriel B. Quijano  and  Matthew C.
Williams

(Date: September 2023)

###### Abstract.

By studying laminations of the unit disk, we can gain insight into the
structure of Julia sets of polynomials and their dynamics in the complex
plane. The polynomials of a given degree, $d$, have a parameter space. The
hyperbolic components of such parameter spaces are in correspondence to
rotational polygons, or classes of “rotational sets”, which we study in this
paper. By studying the count of such rotational sets, and therefore the
underlying structure behind these rotational sets and polygons, we can gain
insight into the interrelationship among hyperbolic components of the
parameter space of these polynomials.

These rotational sets are created by uniting rotational orbits, as we define
in this paper. The number of such sets for a given degree $d$, rotation number
$\frac{p}{q}$, and cardinality $k$ can be determined by analyzing the
potential placements of pre-images of zero on the unit circle with respect to
the rotational set under the $d$-tupling map. We obtain a closed-form formula
for the count. Though this count is already known based upon some
sophisticated results, our count is based upon elementary geometric and
combinatorial principles, and provides an intuitive explanation.

##  1\. Introduction

###  1.1. Motivation

What are “rotational sets” for laminations for the unit disk under the action
of the angle $d$-tupling map, and why count them? Laminations are a
topological and combinatorial model of the connected Julia sets of polynomials
considered as functions of the complex numbers, modeled by the plane. Such
models are used both to understand specific types of polynomials and their
Julia sets, and to study the parameter spaces of polynomials. For example, the
well-known Mandelbrot set [12] is the parameter space of quadratic polynomials
of the form $P_{c}(z)=z^{2}+c$ with parameter $c$ with connected Julia set.
The so-called hyperbolic components of that parameter space are of interest,
including how they are connected to each other, how they are arranged in the
Mandelbrot set, and how many components there are that are associated with
attractive orbits (of the associated polynomials) of a given period, rotation
number, and the like. These terms are defined below. Our research is concerned
with polynomials of higher degree ($d>2$), about which much less is currently
understood.

Laminations are composed of leaves (chords of the unit circle) which form a
closed collection of non-crossing segments that are forward and backward
invariant under a natural extension of the angle $d$-tupling map (the angular
part or argument of the complex power function $z\mapsto z^{d}$, where
$z=re^{2\pi it}$ and the argument is the exponent of $e$) on the unit circle.
Leaves connecting points of a rotational set in circular order form polygons
in the lamination. There is a correspondence between rotational polygons in
laminations and fixed points of polynomials that have a non-zero infinitesimal
rotation number (determined by the argument of the derivative of the
polynomial at the fixed point). Such polygons are in correspondence to a
fundamental class of hyperbolic components of the parameter space of degree
$d$ polynomials with connected Julia set.

For example, in the Mandelbrot set for the hyperbolic component marked star in
Figure 1, all the Julia sets have a (repelling) fixed point (the marked point
in the Julia set) which is represented in the lamination for that Julia set by
a rotational triangle (marked star). The Julia set is actually the boundary of
the shaded blue region, which contains all the points running off to infinity
under iteration of the polynomial. The white regions in the lamination
correspond to the black regions in the “filled-in” Julia set. The filled-in
Julia set consists of all points whose orbits under iteration of the
polynomial are bounded.

“Counting … from First Principles” in our title indicates that we will use the
most fundamental geometric and combinatorial properties of the angle
$d$-tupling map to make the count. By studying laminations in the abstract
without reference to a particular polynomial or Julia set, we aim to reverse
the process by which a Julia set leads to a lamination. By understanding what
is possible for laminations, we can constrain what is possible for locally
connected Julia sets. Our main result is Theorem 21. For a preview of where
that theorem takes us with Julia sets, skip ahead to view Figure 5.

Figure 1. Hyperbolic component of the Mandelbrot set (marked star) with
repelling fixed point (marked disk) in Julia set of
$P(z)=z^{2}+(-0.117+0.743i)$ modeled by a rotational triangle (marked star) in
the lamination for that Julia set.

###  1.2. Orbits, the Angle $d$-tupling Map, and Itineraries

Most of the following definitions are adapted from [1] and [2]. Elementary
proofs of some propositions are left to the reader.

######  Definition 1.

Let $f:X\to X$ be a function. By $f^{q}(x)$ we denote the composition
$f(f(\dots f(x)\dots))$, where $f$ is composed with itself $q$ times.

By convention, $f^{0}(x)=x$.

######  Definition 2.

Given a point $x\in X$, the orbit of $x$ is the set
$\mathcal{O}=\\{f^{n}(x)\\}^{\infty}_{n=0}$.

We denote the unit circle in the Cartesian plane by $S^{1}=\\{(x,y)\mid
x^{2}+y^{2}=1\\}$. We will describe points in $S^{1}$ by their central angle
and we will measure angles in revolutions rather than radians or degrees. A
full circle is 1 revolution. We say we measure angles mod 1, such that angles
which differ by a full revolution are the same.

We now consider the specific function $\sigma_{d}$ in whose orbits we are
interested.

######  Definition 3.

Let $d\in\mathbb{Z}$ with $d>1$. Define the angle $d$-tupling map
$\sigma_{d}:S^{1}\to S^{1}$ by $\sigma_{d}(x)=dx\pmod{1}$.

We represent points on the circle coordinatized by $[0,1)$ by their base $d$
expansion. In base $d=2$, the notation $\\_001$ denotes for us that the digits
$001$ repeat infinitely which, as the reader can check, is the point
$\frac{1}{3}\in[0,1)$. However, in base $d=3$, $\frac{1}{3}$ is the point
$1\\_0$, which in our notation means that the initial digit $1$ does not
repeat, but the digit $0$ repeats. We can use the tools of symbolic dynamics,
particularly the forgetful shift. Because points under $\sigma_{d}$ are
multiplied by $d$ and are taken modulo $1$, the first digit of the base $d$
expansion becomes the integer part which goes away after taking the modulus.
So we can quickly calculate the next point in an orbit by simply “forgetting”
the leading digit (when written as a base $d$ expansion). For convenience, we
will define $\mathbb{Z}_{+}$ to be $\\{i\in\mathbb{Z}\mid i\geq 0\\}$.

We denote the set of pre-images under $\sigma_{d}^{q}$ of a point $x$ by
$\sigma_{d}^{-q}(x)=\\{y\in S^{1}\mid\sigma_{d}^{q}(y)=x\\}$. If there is a
$q\in\mathbb{Z}_{+}$ such that $\sigma_{d}^{q}(x)=x$, then the orbit of $x$ is
finite, and we say it is a periodic orbit and $x$ is a periodic point. If $q$
is least for which $\sigma_{d}^{q}(x)=x$, then we say $q$ is the period of the
point (respectively, orbit). The set of points visited on those $q$ iterations
make up the orbit $\mathcal{O}$ for that given point. If this $q$ exists, then
for the set ${\mathcal{O}}=\\{\sigma_{d}^{n}(x)\mid 0\leq n<q\\}$ it is true
that $\sigma_{d}:\mathcal{O}\to\mathcal{O}$ and
$\sigma_{d}({\mathcal{O}})={\mathcal{O}}$.

######  Proposition 4.

For a given degree $d$, the pre-images of $0$ are

(1)
$\sigma_{d}^{-1}(0)=\left\\{0,\frac{1}{d},\frac{2}{d},\dots,\frac{d-1}{d}\right\\}$

or when written in base $d$ expansion

(2) $\sigma_{d}^{-1}(\\_0)=\left\\{\\_0,1\\_0,2\\_0,\dots,(d-1)\\_0\right\\}$

These pre-images serve as the border between neighboring intervals of length
$\frac{1}{d}$ in $S^{1}$.

######  Definition 5.

Fix $d>1$. Define intervals $I_{0}=[0,\frac{1}{d})$, and in general
$I_{j}=[\frac{j}{d},\frac{j+1}{d})$ for $1\leq j\leq d-1$. Recall $0$ and $1$
are identical in $S^{1}$. Then $S^{1}$ is the disjoint union
$\bigcup_{j=0}^{d-1}I_{j}$.

For instance, the smallest non-zero pre-image of $0$, $\frac{1}{d}$
(equivalently, $1\\_0$), is the border between $I_{0}$ and $I_{1}$. Note that
each interval $I_{j}$ maps one-to-one in counterclockwise order from $\\_0$
onto $S^{1}$. Consequently, within each $I_{j}$ there is a preimage of every
$I_{k}$ consecutively in order, but of length $\frac{1}{d^{2}}$. With these
tools in hand, we can now express orbits in a much more useful manner.

###  1.3. Itineraries

We have given the points in $S^{1}$ when considered under the map $\sigma_{d}$
their base $d$ expansion. This will allow us to describe the orbit of a point
in the circle under $\sigma_{d}$ in terms the visits of its orbit to the
distinguished intervals in Definition 5. The itinerary of a point $s\in S^{1}$
is the ordered list of its visits in its orbit to the distinguished intervals.

The proof of the following Proposition is left to the reader.

######  Proposition 6.

Under $\sigma_{d}$, the itinerary of a point is exactly its base $d$
expansion. Consequently, two points $s$ and $t$ in $S^{1}$ under $\sigma_{d}$
have the same itinerary if, and only if, $s=t$.

Now, the points of an orbit are defined by their relative placement to pre-
images of zero. The utility in this is found in how orbits can be defined by
the location of pre-images relative to the points of the orbit. This will be
essential to counting rotational sets from first principles.

Figure 2. See if you can identify this orbit for $\sigma_{4}$ based on the
position of its points relative to pre-images of $\\_0$.

Because any given itinerary for a point in a periodic orbit can be shifted to
find the itineraries of every other point in that orbit, periodic orbits can
be clearly referenced with the itinerary of any given point it contains. For
consistency we will define orbits to have the same itinerary as their smallest
(compared to $\\_0$) point.

######  Definition 7.

Let ${\mathcal{O}}=\\{0\leq x_{1}<x_{2}<x_{3}\dots<x_{q}<1\\}$ be a periodic
orbit. Define $Itin(\mathcal{O})=Itin(x_{1})$.

######  Definition 8.

A gap is a complementary interval in $S^{1}\setminus\mathcal{O}$.

###  1.4. Spatial and Temporal Order of Orbits

The count of rotational sets is fundamentally based upon the comparison of
spatial and temporal orders of points in an orbit with reference to the pre-
images of $\\_0$. See Figure 3.

######  Definition 9.

Spatial order refers to the ordering of points in an orbit by value, from
smallest to largest in $[0,1)$. In terms of a map onto $S^{1}$, this order is
given by starting at $\\_0$ and following our points counterclockwise.

######  Definition 10.

Temporal order orders the points in an orbit by starting with the lowest
valued point in our orbit (or closest to $\\_0$ spatially, measuring from
$\\_0$ counterclockwise) and applying $\sigma_{d}$ repeatedly. So, temporal
ordering is based on how our itinerary is followed by repeated applications of
$\sigma_{d}$.

Figure 3. Spatial ordering left; Temporal ordering right.

##  2\. Rotational Sets

Consider $\sigma_{d}:S^{1}\to S^{1}$ for a particular $d>1$.

######  Definition 11.

Let ${\mathcal{O}}=\\{x_{1}<x_{2}<x_{3}\dots<x_{q}\\}$ be a periodic orbit
under $\sigma_{d}$. If, and only if, there exists a $p\in\mathbb{Z}^{+}$ such
that for all $i\in\\{1,2,\dots q\\}$, $\sigma_{d}(x_{i})=x_{i+p\pmod{q}}$,
then we say that $\mathcal{O}$ is a rotational periodic orbit with rotation
number $\frac{p}{q}$ (in lowest terms).

######  Remark 12.

Note that the numerator $p$ of the rotation number $p/q$ of a rotational
periodic orbit is sufficient to determine the temporal order of its points.
Along with how many points of the orbit are in each interval, this is enough
to determine its itinerary.

It follows from Proposition 6 that if, and only if, two rotational periodic
orbits have the same $p/q$ and each of their corresponding points are in the
same intervals, then they have the same itinerary and are the same orbit.

As for the practical generation of this itinerary, it can be found by reading
off the interval of each point by starting with $x_{1}$ and “jumping” forwards
$p$ points counter-clockwise along $S^{1}$, repeating until getting back to
that initial point. Consider what $p$ is in the rotational orbit in Figure 3.

A consequence of how the rotation number of an orbit can describe the forward
orbits of its points is that it can also describe their pre-images. Since
$\\_0$ lies between $x_{1}$ and $x_{q}$, at least one pre-image of $\\_0$ must
lie between $x_{q-p}\in\sigma_{d}^{-1}(x_{q})$ and
$x_{1+q-p}\in\sigma_{d}^{-1}(x_{1})$. We call such pre-images the Principal
Pre-image of their respective orbits.

######  Definition 13.

Let the principal preimage for a rotational orbit $\mathcal{O}$ be a pre-image
of $\\_0$ lying between $x_{q-p}\in\sigma_{d}^{-1}(x_{q})$ and
$x_{1+q-p}\in\sigma_{d}^{-1}(x_{1})$.

The reader can verify that there must always be a principal preimage.

###  2.1. Rotational Sets Containing Multiple Orbits

Not only can points rotate together while maintaining order, but so too can
multiple orbits together, forming a rotational set.

######  Definition 14.

Let ${\mathcal{P}}=\\{x_{i}\mid 0\leq x_{1}<x_{2}<x_{3}<\dots<x_{qk}<1\\}$ be
a finite set in consecutive counterclockwise order in $S^{1}$. We say
$\mathcal{P}$ is a rotational set containing $k$ orbits with rotation number
$\frac{p}{q}$ for $\sigma_{d}$ if, and only if,

  1. (1)

$\sigma_{d}({\mathcal{P}})={\mathcal{P}}$

  2. (2)

$x_{i},x_{i+1},\dots,x_{i+k-1\pmod{qk}}$ for $i\in[1,2,\dots qk]$ are in
different orbits

  3. (3)

$x_{i}$ moves to $x_{i+pk\pmod{qk}}$

######  Definition 15.

Let $G_{i}=\\{x_{(i-1)k+1},x_{(i-1)k+2},\dots,x_{ik}\\}$ where
$i\in[1,2,\dots,q]$. We say $G_{i}$ is the $i$th group and $G_{i}$ is the set
of the $i$th points spatially of each orbit.

######  Remark 16.

Items (2) and (3) of Definition 2.3 also show that $G_{i}$ moves together
preserving spatial order to $G_{i+p\pmod{q}}$.

######  Definition 17.

Let a principal preimage for a rotational set ${\mathcal{P}}$ be the pre-image
of $\\_0$ that lies between $G_{q-p}\subset\sigma_{d}^{-1}(G_{q})$ and
$G_{q-p+1}\subset\sigma_{d}^{-1}(G_{1})$.

######  Remark 18.

For the principal pre-image of a set, it must lie between
$x_{(q-p)k}\in\sigma_{d}^{-1}(x_{qk})$ and
$x_{1+(q-p)k}\in\sigma_{d}^{-1}(x_{1})$. $x_{1+(q-p)k}$ is the smallest point
in group $G_{q-p+1}$ and $x_{(q-p)k}$ is the largest point in group $G_{q-p}$,
therefore the principal pre-image lies between those two groups.

Figure 4. These two rotational orbits for $\sigma_{4}$ form a rotational set.
The high-lighted inter-group gap contains the principal pre-image of this
rotational set. See if you can identify these orbits, as practice.

In order to aid with the identification and counting of rotational sets, we
need to differentiate gaps between points that are within groups from those
outside groups. This distinction is necessary as pre-images that lie in the
former are what differentiate orbits within a rotational set from each other.
In other words, the pre-images that are within the groups are such that if
they were removed, the orbits would no longer be different.

######  Definition 19.

Let intra-group gaps be gaps that lie within a group, or in other words are
gaps that are in between two points from two different orbits that aren’t the
first and last orbits spatially in their group.

######  Definition 20.

Let inter-group gaps be gaps that lie outside a group (or between groups).
These are all of the gaps that are not intra-group gaps.

Note that with these definitions, the principal pre-image must always lie
within an inter-group gap. See if you can build intuition for this by
considering Figure 4 through this new lens.

##  3\. Algorithms and resulting Formulas

###  3.1. Counting Rotational Sets

Goldberg [3] counted the rotational orbits with a given rotation number
$\frac{p}{q}$ for $\sigma_{d}$ and indicated that rotational sets containing
multiple orbits with that rotation number could also be counted, providing an
example count for $\sigma_{3}$, but no general formula. As a corollary to her
characterization of rotational orbits in terms of their temporal and spatial
placement with respect to the $d-1$ fixed points of $\sigma_{d}$, she showed
that the maximal number of orbits in a rotational set for $\sigma_{d}$ was
$d-1$. This also follows as a corollary to our main theorem, Theorem 21.
McMullen ([5], Section 2) built upon Goldberg to provide a criterion for two
orbits for $\sigma_{d}$ with the same rotation number to be compatible in one
rotational set. Tan [6] used an algorithm based upon the Goldberg/McMullen
criterion to count the number of rotational sets containing $k$ orbits for
$\sigma_{d}$ with a given rotation number $\frac{p}{q}$. So, while the content
of our main theorem is known, the proof here is new and more elementary.

######  Theorem 21 (Identifying and Counting Rotational Sets).

Consider the collection $\mathcal{B}$ of all rotational sets for a given
degree $d$, rotation number $\frac{p}{q}$ in lowest terms, and number $k$ of
distinct orbits per set. The cardinality of $\mathcal{B}$ is given by

(3)
$|\mathcal{B}|=\sum_{i=k-1}^{l}\left[\binom{d+q-2}{d-2-i}\sum_{j=0}^{k-1}\left[(-1)^{j}\binom{k-1}{j}\binom{q(k-1-j)}{i}\right]\right]$

where

(4) $l=\begin{cases}q(k-1)&d-2>q(k-1)\\\ d-2&otherwise\end{cases}$

###### Proof.

For any given rotational set $B$ within $\mathcal{B}$, there are $q$ points in
each of the $k$ orbits within $B$. The count of each way to place pre-images
between these neighboring points is the same as the count of rotational sets
because the placement of pre-images dictates the digits for the itinerary of
each point as can be seen in Remark 12. However, placing a pre-image between
$\\_0$ and the smallest point in the set is different from placing a pre-image
between the largest point and $\\_0$. The former would increase the digits in
all the itineraries while the latter would not. Therefore, we need to count
the number of ways to place pre-images within the gaps distinguished by $\\_0$
and the points within $B$.

Let the range of values from $0$ to $qk$ correspond with the gaps between
neighboring points in the set of points that contain $\\_0$ and the points
within each orbit in $B$. $0$ corresponds with the gap between $\\_0$ and the
smallest point in $B$, $1$ with the gap between the smallest and second
smallest, and so on.

There are $d$ pre-images to place. The first is $\\_0$, which is its own pre-
image. The next is the principal pre-image, whose position is already
determined (as can be seen in Remark 18). This leaves us with $d-2$ pre-images
to place.

Now, we must concern ourselves with where these pre-images can be placed to
form a valid rotational set with $k$ orbits.

######  Lemma 22.

Label gaps with their congruence class modulo $k$. Rotational sets in
$\mathcal{B}$ must have at least one pre-image in each non-zero congruence
class. Therefore, the cardinality of $\mathcal{B}$ is equivalent to that of
$P$ when $P$ is defined to be the set of all sets composed of $d-2$ non-
negative integers less than or equal to $qk$, such that each one contains at
least one element from each non-zero congruence class modulo $k$.

###### Proof.

In order to ensure differentiation between orbits, each orbit must be
differentiated from its intra-group neighbors (the groups are differentiated
by the principal pre-image and $\\_0$). The first spatial orbit is not a
neighbor with the last as they lie on opposite sides of any given group. So
for each intra-group neighbor, an orbit must have at least one pre-image
between one of its points and that neighbor’s points. Therefore, this
requirement can only be fulfilled by placing pre-images in intra-group gaps.

Here is an example to provide clarity. The reader is invited to draw their own
illustration for this example. For the first orbit (spatially) to be distinct
from the second orbit, there must be a pre-image either in the gap between
their smallest points respectively, second smallest, or any other pair of
corresponding points. This particular restriction for the first and second
orbits can be restated as the requirement for a pre-image to exist in a gap
with label $n$ such that $n$ is in the congruence class of $1$ modulo $k$.
This rule can be generalized for all intra-group gaps by separating them into
congruence classes. For the $n$th and $(n+1)$th orbits to be differentiated,
there must be at least one pre-image in the set of gaps with labels in
$\\{x\in[0\dots qk]\mid x\in[n]_{k}\\}$. Therefore, all non-zero congruence
classes require at least one pre-image for the rotational set to be valid.

The set of rotational sets, with the restrictions articulated above, will have
the same cardinality as $P$ when defined as follows. $P$ is the set of all
sets composed of $d-2$ non-negative integers less than or equal to $qk$, such
that each one contains at least one element from each non-zero congruence
class modulo $k$.

∎

We will now define sets that correspond with the choice of gaps in which to
put pre-images. As of now, we are only placing one pre-image per gap even
though more than one can be placed for a valid rotational set. This is done to
make counting simpler later on.

We will refer to the range of labels for gaps as $\lambda$, and define it as:

(5) $\lambda=\\{x\in\mathbb{Z}_{+}\mid x\leq qk\\}$

In other words, $\lambda$ is the set of non-negative integers less than or
equal to $qk$. We will refer to the set of all labels for intra-group gaps as
$\psi$ and define $\psi$ as

(6) $\psi=\\{x\in\lambda\mid x\ (\mathrm{mod}\ k)\neq 0\\}$

There are multiple possible values for how many pre-images may be in intra-
group gaps for any given rotational set. We will call the number of pre-images
in intra-group gaps $i$.

Define $T_{i}$ as the set of all sets that contain $i$ elements from $\psi$.
We also know that $|T_{i}|=\binom{|\psi|}{i}=\binom{q(k-1)}{i}$ since
$|\psi|=q(k-1)$ because there are $k-1$ non-zero congruence classes with $q$
integers in each.

(7) $|T_{i}|=\binom{q(k-1)}{i}$

######  Lemma 23.

The range for the possible number of pre-images in intra-group gaps varies
from $k-1$ to $l$ where

(8) $l=\begin{cases}q(k-1)&d-2>q(k-1)\\\ d-2&otherwise\end{cases}$

###### Proof.

Only $k-1$ pre-images are required for differentiation. The minimal value of
$i$ is the minimal number of pre-images required for differentiation, $k-1$.
As for the maximum value of $i$, it can be limited by either the number of
empty gaps, $q(k-1)$, or the number of pre-images to place, $d-2$. Therefore,
the maximal value of $i$ is $l$ where

(9) $l=\begin{cases}q(k-1)&d-2>q(k-1)\\\ d-2&otherwise\end{cases}$

∎

The sets in $T_{i}$ for $i\in[k-1,k,\dots,l]$ that follow our requirement of
differentiation are valid (yet may not be complete as these only correspond to
$i$ pre-images of the $d-2$ that need to be placed). So we will define
$\gamma_{i}$ as the subset of $T_{i}$ that contains all the sets where there
is at least one element from each non-zero congruence class modulo $k$.

######  Lemma 24.

The count of sets in $\mathcal{B}$ such that each $B\in\mathcal{B}$ has $i$
pre-images in its intra-group gaps is given by

(10) $\sum_{j=0}^{k-1}(-1)^{j}\binom{k-1}{j}\binom{q(k-1-j)}{i}$

###### Proof.

Each set within $P$ that has $i$ elements in non-zero congruence classes
corresponds with an element in $\gamma_{i}$. Similarly, each rotational set in
$\mathcal{B}$ that has $i$ pre-images in intra-group gaps corresponds with an
element in $\gamma_{i}$. For counting purposes, we found it easier to find
$T_{i}\setminus\gamma_{i}$ than $\gamma_{i}$, so we will define
$W_{i}=T_{i}\setminus\gamma_{i}$.

We can identify and count the sets in $W_{i}$ (as opposed to the valid sets in
$\gamma$) by noticing the equivalence of the problem with finding the union of
sets. The goal is to find every possible way to leave at least one congruence
class unfilled. Categorize each placement as belonging to a series of sets
$(C_{1},C_{2},\dots,C_{k-1})$ where $C_{j}$ is defined as the set of elements
of $T_{i}$ where the $j$th congruence class is not represented. With this
definition, a singular placement can belong to multiple sets $C_{j}$, and we
seek the union of each of these sets. In other words,
$\bigcup\limits_{j=1}^{k-1}C_{j}=W_{i}$ because it contains every set where at
least 1 non-zero congruence class is not represented. This is a well known
problem, and the solution utilizes the inclusion-exclusion principle [9] with
the count given by

(11)
$|W_{i}|=\left|\bigcup\limits_{j=1}^{k-1}C_{j}\right|=\sum_{j=1}^{k-1}(-1)^{j+1}\binom{k-1}{j}\binom{q(k-1-j)}{i}$

Since $\gamma_{i}$ is defined as the set difference of $T_{i}$ and $W_{i}$,

(12)
$|\gamma_{i}|=|T_{i}|-|W_{i}|=\binom{q(k-1)}{i}-\sum_{j=1}^{k-1}(-1)^{j+1}\binom{k-1}{j}\binom{q(k-1-j)}{i}$

which can be rewritten as

(13) $\sum_{j=0}^{k-1}(-1)^{j}\binom{k-1}{j}\binom{q(k-1-j)}{i}$

∎

We have identified and counted all valid placements of intra-group pre-images.
In order to finish constructing a rotational set, the remaining pre-images
must be placed in between groups. For each of these placements in $\gamma$,
the number of pre-images left to place naturally depends on the number of pre-
images already placed. For any given set $g\in\gamma$ with cardinality $i$,
there are $d-2-i$ elements from $\lambda$ (all the labels for the gaps) that
need to be added to construct an element in $\mathcal{B}$.

There are a few limitations on which gaps pre-images can be placed in, due to
the fact that our prior count depends on certain gaps lacking pre-images and
others having at least one. Therefore, we can place pre-images in intra-group
gaps that already have pre-images or any given inter-group gap. In other
words, the added $d-2-i$ elements must either be duplicates of elements
already in $g$ which are also in $\psi$ or elements in $\lambda\setminus\psi$.
There are $\binom{d+q-2}{d-2-i}$ ways to choose these integers, therefore
there are these many elements in $P$ and corresponding orbits for any given
set $g$.

As argued before, $i$ (the cardinality of $g$) ranges from $k-1$ to $l$.

Bringing all of this together, $|\mathcal{B}|$ and $|P|$ are equal to the sum
of $\binom{d+q-2}{d-2-i}|\gamma|$ over possible values of $i$.

(14)
$|P|=|\mathcal{B}|=\sum_{i=k-1}^{l}\left[\binom{d+q-2}{d-2-i}\sum_{j=0}^{k-1}\left[(-1)^{j}\binom{k-1}{j}\binom{q(k-1-j)}{i}\right]\right]$

∎

###  3.2. Identifying Rotational Sets Containing a Given Orbit

This count is certainly insightful, but in order to gain insight as to
specific laminations that one may examine, it would be useful to be able to
count and identify rotational sets that contain a given orbit. Here we state
that this can be done, using a method quite similar to the method used in the
proof of the previous counting theorem.

######  Theorem 25 (Identifying Maximal Rotational Sets that contain a given
orbit).

Consider the orbit $\mathcal{O}$ with degree $d$ and rotation number
$\frac{p}{q}$ in lowest terms. An exhaustive list of all the rotational sets
that contain $\mathcal{O}$ can be found algorithmically, in a way inspired by
the previous proof. The count of maximal rotational sets that contain
$\mathcal{O}$ can be expressed as a closed-form formula in terms of the degree
$d$, the rotation number $\frac{p}{q}$, and the digits of $\mathcal{O}$.

Similar to the count of all rotational sets under certain parameters being
given through the valid placements of pre-images of $\\_0$, the orbits that
belong to rotational sets containing a given orbit $\mathcal{O}$ can be
identified through a similar process. We leave this investigation to the
reader. However, an algorithm for identifying such rotational sets containing
an orbit $\mathcal{O}$ based on the proof strategy of the previous theorem can
be accessed on GitHub [7].

###  3.3. Examples

In order to demonstrate the usefulness of the proposed algorithm, consider the
rotational orbit $[\\_012,\\_120,\\_201]$ under $\sigma_{3}$. Applying the
algorithm, we determine that this orbit can be paired with either the orbit
$[\\_002,\\_020,\\_200]$ or the orbit $[\\_112,\\_121,\\_211]$ (but not both
since a rotational set for $\sigma_{3}$ contains at most two orbits). The
rotational polygons formed by these three sets are in the first row of Figure
5.

Figure 5. Triangle and two hexagon expansions for $\sigma_{3}$.
Initial rotational data (left to right):
${\\_012}$ orbit, ${\\_012}$ & ${\\_002}$ orbits, ${\\_012}$ & ${\\_112}$
orbits.
Pullbacks laminations (left to right):
3 steps, 2 steps, 2 steps.
Polynomial equations (left to right):
$P(z)=(-0.607786+1.05435i)z-(0.316863+0.325784i)z^{2}+z^{3}/3$
$P(z)=(0.89012-0.512216i)z^{2}+z^{3}/3$
$P(z)=-0.156693-0.682581i+(-0.355147+0.182883i)z^{2}+z^{3}/3$

The second row of the figure shows the first few stages of the pullback
lamination determined by the initial polygons and an appropriate choice of
branches of the inverse of $\sigma_{3}$ [8]. The corresponding Julia sets,
found using Mathematica and FractalStream [1], are displayed in the third row
of the figure. Each polygon in the corresponding lamination represents a
junction point in the Julia set. The central polygon in each case represents a
fixed point of the polynomial for the corresponding Julia set. Note that the
lamination represents reasonably well the geometry of the corresponding Julia
set if you imagine the polygons shrunk to points. In this process it is
important to note that we found the rotational polygons and laminations before
we found the polynomials and their Julia sets.

###  3.4. Future Questions

If you find yourself interested in continuing this work, perhaps consider the
following questions as starting points for areas of research:

  1. (1)

The formula for the count provided in Theorem 21 is quite complicated. Tan
([6] Theorem 3.2) provides an equivalent count, though without a basis in
first principles. How can our count be simplified from first principles?

  2. (2)

Verify that the formula found by Tan and the closed-form formula in our
Theorem 3.1 give the same count of rotational sets.

  3. (3)

Consider the closed form formula given by Theorem 21. The formula implies that
for a fixed $d$ and $k$, you can express the formula as a polynomial equation
in terms of $q$. What is the degree of the polynomial for a given $d$ and $k$?
Is it possible to generate the coefficients for the polynomial for a given $d$
and $k$ in closed form? What can this teach us about the count of rotational
sets for a fixed degree and rotational set size?

  4. (4)

Consider the lattice [11] of rotational sets, partially ordered by subset
inclusion, for a given degree $d$ and rotation number $\frac{p}{q}$, where a
join [10] between sets $A$ and $B$ represents the union of $A$ and $B$. Given
a degree $d$, what can we learn about the underlying structure of these
lattices, and what can it teach us about rotational sets?

## References

  * [1] B. Burdette, C. Falcione, C. Hale, and J. Mayer.  Unicritcal and maximally critical laminations.  arXiv:2303.17668 [math.DS] (2023).
  * [2] A. Blokh, J. Malaugh, J. Mayer, L. Oversteegen, D. Parris.  Rotational subsets of the circle under $z^{d}$.  Topology and its Applications 153 (2006), 1540–1570.
  * [3] L. Goldberg.  Fixed points of polynomial maps, Part 1, Rotation subsets of the circle.  Annales scientifiques de l’É.N.S. 4e série, tome 25, no 6 (1992), p. 679–685.
  * [4] L. Goldberg and J. Milnor.  Fixed points of polynomial maps, Part II. Fixed point portraits.  Annales scientifiques de l’É.N.S. 4e série, tome 26, no 1 (1993), p. 51–98.
  * [5] C. McMullen.  Dynamics on the unit disk: Short geodesics and simple cycles.  Commentarii Mathematici Helvetici, 85 (2010) 723–749.
  * [6] Y. Tan.  Counting rotational subsets of the circle $\mathbb{R}/\mathbb{Z}$ under the angle-multiplying map $t\mapsto dt$.  arXiv:2207.03594v2 [math.CO] (2022).
  * [7] Github: MaxSetGeneratingAlgo. https://github.com/mjmoorman03/MaxSetGeneratingAlgo/tree/main.
  * [8] Falcione, C. Lamination Builder. https://csfalcione.github.io/lamination-builder/.
  * [9] Wikipedia: Inclusion-Exclusion Principle. https://en.wikipedia.org/wiki/Inclusion-exclusion_principle.
  * [10] Wikipedia: Join and Meet. https://en.wikipedia.org/wiki/Join_and_meet.
  * [11] Wikipedia: Lattice (order). https://en.wikipedia.org/wiki/Lattice_(order).
  * [12] Wikipedia: Mandelbrot Set. https://en.wikipedia.org/wiki/Mandelbrot_set.