# Unfolding numbers for interval maps

Sourav Bhattacharya  Department of Mathematics, Visvesvaraya National
Institute Of Technology Nagpur, Nagpur, Maharashtra 440010, India
<EMAIL_ADDRESS>
###### Abstract.

We propose a new invariant (_rotation number_) for a _cycle_ of an _interval
map_ , called its _unfolding number_ to circumvent some difficulties faced in
the _over-rotation theory_ for _interval maps_. We corroborate the
_completeness_ of the theory and exhibit that the set of _unfolding numbers_
of points under a given interval map $f$ forms an _interval_ of the form
$[u_{f},\frac{1}{2}]$ where $u_{f}$ can be _computed_ by a simple _algorithm_
for any _interval_ map $f$. Further, $u_{f}$ is realized on a _closed
invariant_ subset $Z_{f}\subset[0,1]$ such that $f|_{Z_{f}}$ is _strictly
ergodic_. In the end we forge connection between _unfolding numbers_ and
_over-rotation numbers_ : we reveal that while disparate in essence, there
exists _special patterns_ $\pi_{s}$ called _sheer patterns_ for whom
_unfolding numbers_ equals _over-rotation numbers_.

###### Key words and phrases:

Sharkovsky Theorem, rotation numbers, unfolding numbers, unfolding twists

###### 2010 Mathematics Subject Classification:

Primary 37E05, 37E15; Secondary 37E45

##  1\. Introduction

An important problem in the field of _dynamical systems theory_ revolves
around investigating the _co-existence_ of diverse “ _types_ ” of _cycles_ ,
or _periodic orbits_ , of a given _continuous interval map_. However, it is
imperative to grasp the precise meaning of a _cycle’s_ “ _type_ ”. If we
designate the _period_ of a _cycle_ as its defining “ _type_ ”, then the
results prove to be robust and resilient.

###  1.1. Sharkovsky Theorem

The _celebrated_ _Sharkovksy Theorem_ [21] offers a comprehensive account of
every conceivable collection of _periods_ for the _periodic points_ of a
_continuous interval map_. To understand this theorem, it is necessary to
introduce a special ordering of the set $\mathbb{N}$ of natural numbers called
the _Sharkovsky ordering_ :

$3\succ\mkern-14.0mu_{s}\;5\succ\mkern-14.0mu_{s}\;7\succ\mkern-14.0mu_{s}\;\dots\succ\mkern-14.0mu_{s}\;2\cdot
3\succ\mkern-14.0mu_{s}\;2\cdot 5\succ\mkern-14.0mu_{s}\;2\cdot
7\succ\mkern-14.0mu_{s}\;\dots$ $\succ\mkern-14.0mu_{s}\;\dots 2^{2}\cdot
3\succ\mkern-14.0mu_{s}\;2^{2}\cdot 5\succ\mkern-14.0mu_{s}\;2^{2}\cdot
7\succ\mkern-14.0mu_{s}\;\dots\succ\mkern-14.0mu_{s}\;8\succ\mkern-14.0mu_{s}\;4\succ\mkern-14.0mu_{s}\;2\succ\mkern-14.0mu_{s}\;1$

We denote by $Sh(k)$ the set of all natural numbers $m$ such that
$k\succ\mkern-14.0mu_{s}\;m$, together with $k$, and by $Sh(2^{\infty})$ the
set $\\{1,2,4,8,\dots\\}$ which includes all powers of $2$. For a continuous
map $f$, let us denote the set of _periods_ of its cycles by $Per(f)$.

######  Theorem 1.1 ([21]).

If $f:[0,1]\to[0,1]$ is a continuous map, $m\succ\mkern-14.0mu_{s}\;n$ and
$m\in Per(f)$, then $n\in Per(f)$. Therefore, there exists
$k\in\mathbb{N}\cup\\{2^{\infty}\\}$ such that $Per(f)=Sh(k)$. Conversely, if
$k\in\mathbb{N}\cup\\{2^{\infty}\\}$ then there exists a continuous map
$f:[0,1]\to[0,1]$ such that $Per(f)=Sh(k)$.

Theorem 1.1 unveils an intricate and concealed combinatorial framework that
governs the disposition of _cycles_ of a _continuous interval_ map. In this
context, certain “ _types_ ” of _dynamical behaviour_ (here _period_), once
exhibited by a map, _forces_ some other “ _type_ ” and thus, the question of “
_co-existence_ ” becomes a question of “ _forcing_ ”. However, _periods_ offer
only a coarse depiction of a _cycle_ , as numerous _cycles_ share the same
_period_. A more nuanced approach to characterizing _cycles_ involves
examining their _cyclic permutations_ — _cyclic permutation_ we get when we
look at how the map acts on the points of the _cycle_ , ordered from the _left
to the right_ ; we _identify_ _cycles_ having the _same_ underlying _cyclic
permutations_ and call the corresponding _equivalence classes_ : _patterns_.

###  1.2. Patterns

A _pattern_ $A$ _forces_ _pattern_ $B$ if every continuous map having a
_cycle_ of _pattern_ $A$ has a _cycle_ of _pattern_ $B$. By [3] _forcing among
patterns_ is a _partial ordering_. A useful _algorithm_ allows one to describe
all _patterns_ _forced_ by a given _pattern_ $A$. Namely, consider a _cycle_
$P$ of _pattern_ $A$; assume that _leftmost_ and _rightmost_ _points_ of $P$
are $a$ and $b$ respectively. Every _component_ of $[a,b]\setminus P$ is said
to be a _$P$ -basic interval_. Extend the map from $P$ to the interval $[a,b]$
by defining it _linearly_ on each $P$-_basic interval_ and call the resulting
map $f_{P}$ the _$P$ -linear map_. Then the _patterns_ of all cycles of
$f_{P}$ are exactly the _patterns_ forced by the _pattern_ of $P$ (see [3] and
[1]).

As it turns out (see [3]), this _forcing relation_ among _patterns_ is rather
too detailed and doesn’t allow for a transparent description. This motivates
one to look for another, middle-of-the-road way of describing cycles, a way
not as _crude_ as _periods_ but not as fine as _permutations_ , which would
still allow for a transparent description.

###  1.3. Rotation Theory in a general setting

The concept of _rotation numbers_ was historically first introduced by
Poincaré for _circle homeomorphisms_ ([20]). In general _rotation numbers_ can
be defined as follows ([18, 22]): Let $X$ be a _compact metric space_ with a
Borel $\sigma$-algebra, $\phi:X\to\mathbb{R}$ be a _bounded measurable
function_ (often called an _observable_) and $f:X\to X$ be a _continuous_ map.
Then for any $x\in X$ the set $I_{f,\phi}(x)$, of all _sub-sequential limits_
of the sequence $\left\\{{\frac{1}{n}}\sum^{n-1}_{i=0}\phi(f^{i}(x))\right\\}$
is called the $\phi$-rotation set of $x$. If
$I_{f,\phi}(x)=\\{\rho_{\phi}(x)\\}$ is a singleton, then the number
$\rho_{\phi}(x)$ is called the $\phi$-rotation number of $x$. It is easy to
see that the $\phi$-_rotation set_ , $I_{f,\phi}(x)$ is a closed interval for
all $x\in X$. The union of all $\phi$-_rotation sets_ of all points of $X$ is
called the _$\phi$ -rotation set_ of the map $f$ and is denoted by
$I_{f}(\phi)$. If $x$ is an $f$-_periodic point_ of period $n$ then its
$\phi$-rotation number $\rho_{\phi}(x)$ is well-defined, and a related concept
of the _$\phi$ -rotation pair_ of $x$ can be introduced: the pair
$({\frac{1}{n}}\sum^{n-1}_{i=0}\phi(f^{i}(x)),n)$ is the _$\phi$ -rotation
pair_ of $x$.

The use of _rotation theory_ in the study of _dynamical systems_ is useful
when our _rotation theory_ is _complete_. This means that: _rotation set_ is
_convex_ ; _rotation number_ of a _cycle_ of _period_ $n$ is of the form
$\frac{m}{n}$ where $m,n\in\mathbb{N}$; if $\frac{m}{n}$ belongs to the
_interior_ of the _rotation set_ , then there exists a _cycle_ of _period_
$kn$ and _rotation pair_ $(km,kn)$ for every $k\in\mathbb{N}$. Another way of
looking at _rotation theory_ is through _ergodic invariant measures_ ([13],
[1]).

###  1.4. Ergodic Invariant Measures

Let $(X,\mathcal{B})$ be a _measure space_ where $\mathcal{B}$ is a
$\sigma$-_algebra_ on _Borel subsets_ of $X$. A _probability measure_ $\mu$ on
$X$ is said to be _invariant_ under a map $f:X\to X$ if
$\mu(f^{-1}(A))=\mu(A)$ for every $A\in\mathcal{B}$. An _invariant measure_
$\mu$ is called _ergodic_ if every _invariant Borel subset_ of $X$ has
_measure_ $0$ or $1$. For _Borel measures_ on _compact spaces_ we consider
their _weak topology_ defined by the _continuous functions_ [13].

If $P$ is a _cycle_ of $f:[0,1]\to[0,1]$, we can look at the _ergodic measure_
$\mu_{P}$, _equidistributed_ on $P$, that is, such that
$\mu_{P}(\\{x\\})=\frac{1}{n}$ for all $x\in P$, where $n$ is the _period_ of
$P$. Let us call $\mu_{P}$, a _CO-measure_ [13] (comes from “closed orbit”).
The following result follows:

######  Theorem 1.2 ([7]).

Suppose that $f:[0,1]\to[0,1]$ is a continuous interval map or a circle map
with non-empty set of periodic points. Then any invariant probability measure
$\mu$ for whom there exists a point $x$ with $\mu(\omega_{f}(x))=1$ where
$\omega_{f}(x)$ denotes omega limit set of $x$ under $f$ can be approximated
by CO-measures arbitrary well. In particular, CO-measures are dense in all
ergodic invariant measures of $f$.

If the _dynamics_ of $f$ and the _observable_ $\phi$ are related we can
further deduce additional informations about $\phi$-_rotation sets_ , for
instance, in the case of _circle degree one_ case (see [16], [1]).

###  1.5. Rotation numbers for Circle Maps

Consider the circle $\mathbb{S}=\mathbb{R}/\mathbb{Z}$ with the _natural
projection_ $\pi:\mathbb{R}\to\mathbb{S}$. If $f:\mathbb{S}\to\mathbb{S}$ is
_continuous_ , then there is a _continuous map_
$F_{f}:\mathbb{R}\to\mathbb{R}$ such that $F_{f}\circ\pi=\pi\circ f$. Such a
map $F_{f}$ is called a _lifting_ of $f$. It is unique up to translation by an
integer. An integer $d$ with $F_{f}(x+1)=F_{f}(x)+d$ for all $x\in\mathbb{R}$
is called the _degree_ of the map $f$ and is independent of the choice of $F$.
Conversely, maps $G:\mathbb{R}\to\mathbb{R}$ such that $G(x+1)=G(x)+d$ for
every real $x$ are said to be _maps of the real line of degree $d$_, can be
defined independently, and are semi-conjugate by the same map
$\pi:\mathbb{R}\to\mathbb{S}$ to the circle maps of degree $d$. In this paper
we consider both maps of the circle and maps of the real line _of degree one_.
Denote by $\mathcal{L}_{1}$ the set of all _liftings_ of _continuous degree
one_ self-mappings of $\mathbb{S}$ endowed with the _sup norm_.

Given, a _continuous map_ , $f:\mathbb{S}\to\mathbb{S}$, consider its
_lifting_ $F_{f}\in\mathcal{L}_{1}$. Define an _observable_
$\phi_{f}:\mathbb{S}\to\mathbb{R}$ so that $\phi_{f}(x)=F_{f}(u_{x})-u_{x}$
for any $x\in\mathbb{S}$ and $u_{x}\in\pi^{-1}(x)\in\mathbb{R}$; then
$\phi_{f}$ is well-defined, the _classical rotation set_ of a point
$x\in\mathbb{S}$ is simply the $\phi_{f}$ -_rotation set_
$I_{f,\phi_{f}}(x)=I_{f}(x)$. It is easy to see that
$I_{f}(x)=\left\\{\overline{\rho_{F_{f}}}(u_{x}),\underline{\rho_{F_{f}}}(u_{x})\right\\}$
where
$\overline{\rho_{F_{f}}}(u_{x})=\displaystyle\lim\sup_{n\to\infty}\frac{F_{f}^{n}(u_{x})-u_{x}}{n}$
and
$\underline{\rho_{F_{f}}}(u_{x})=\displaystyle\lim\inf_{n\to\infty}\frac{F_{f}^{n}(u_{x})-u_{x}}{n}$.
The _numbers_ : $\overline{\rho_{F_{f}}}(u_{x})$ and
$\underline{\rho_{F_{f}}}(u_{x})$ are respectively called _upper_ and _lower_
_rotation numbers_ of $u_{x}\in\mathbb{R}$ under $F_{f}$ (equivalently of
$x\in\mathbb{S}$ under $f$). If
$\overline{\rho_{F_{f}}}(u_{x})=\underline{\rho_{F_{f}}}(u_{x})=\rho_{F_{f}}(u_{x})$,
we write $\rho_{F_{f}}(u_{x})$, the _rotation number_ of $u_{x}$ under $F_{f}$
(equivalently of $x$ under $f$).

Suppose $f^{n}(y)=y$, $y\in\mathbb{S}$, that is, $y$ is a _periodic point_ of
$f:\mathbb{S}\to\mathbb{S}$ of period $n$. Then, for any
$v_{y}\in\pi^{-1}(y)$, $F_{f}^{n}(v_{y})=v_{y}+m$ for some $m\in\mathbb{N}$
and hence
$\overline{\rho_{F_{f}}}(v)=\underline{\rho_{F_{f}}}(v)=\frac{m}{n}$. We call
$\frac{m}{n}$, the _rotation number_ of $v_{y}\in\mathbb{R}$ (equivalently of
$y\in\mathbb{S}$) and $(m,n)$ the _rotation pair_ of $v_{y}\in\mathbb{R}$
(equivalently of $y\in\mathbb{S}$). If $P\subset\mathbb{S}$ is the _orbit_ of
$y\in\mathbb{S}$, then the set $P^{\prime}=\pi^{-1}(P)\subset\mathbb{R}$ is
called its _lifted periodic orbit_ or _lifted cycle_ under $F_{f}$. In such a
case, each _point_ of $P$ under $f$ has _rotation number_ $\frac{m}{n}$
(equivalently of $P^{\prime}$ under $F_{f}$). We call $\frac{m}{n}$ and
$(m,n)$ respectively _rotation number_ and _rotation pair_ of
$P\subset\mathbb{S}$ under $f$ (equivalently of $P^{\prime}$ under $F_{f}$).

Conversely, _lifted periodic orbits_ can also defined independently: a set
$Q^{\prime}\subset\mathbb{R}$ is a _lifted periodic orbit_ of a _degree one
map_ $F$ with _period_ $q$ and _rotation number_ $\frac{p}{q}$ if for any
$z\in Q^{\prime}$, $F^{q}(z)=z+p$. In such a case, there always exists a
_periodic orbit_ $Q\subset\mathbb{S}$ of $f:\mathbb{S}\to\mathbb{S}$ _period_
$q$ and _rotation number_ $\frac{p}{q}$ such that: $Q^{\prime}=\pi^{-1}(Q)$.

Figure 1. Representation of _modified rotation pairs_ (_mrp_) using real line
and _prongs_ attached to it

###  1.6. Modified rotation pairs

_Rotation pairs_ can be represented in an interesting way using the notations
used in [9]. We represent the _rotation pair_ $rp(x)=(mp,mq)$ with $p,q$
coprime and $m$ being some natural number as a pair $(t,m)$ where
$t=\frac{p}{q}$. We call the later pair a _modified-rotation pair_(_mrp_) and
write $mrp(x)=(t,m)$. We then think of the real line with a _prong_ attached
at each rational point and the set $\mathbb{N}\cup\\{2^{\infty}\\}$ marked on
this prong in the _Sharkovsky ordering_ $\succ\mkern-14.0mu_{s}\;$ with $1$
_closest_ to the _real line_ and $3$ _farthest_ from it. All points of the
_real line_ are marked $0$; at irrational points we can think of degenerate
prongs with only $0$ on them. The union of all _prongs_ and the _real line_ is
denoted by $\mathbb{M}$. Thus, a _modified rotation pair_ $(t,m)$ corresponds
to the specific element of $\mathbb{M}$, namely to the number $m$ on the
_prong_ attached at $t$ (See Figure 1). However, no _rotation pair_
corresponds to $(t,2^{\infty})$ or to $(t,0)$. Then, for $(t_{1},m_{1})$ and
$(t_{2},m_{2})$, in $\mathbb{M}$, the _convex hull_ $[(t_{1},m_{1}),$
$(t_{2},m_{2})]$ consists of all _modified rotation pairs_ $(t,m)$ with $t$
strictly between $t_{1}$ and $t_{2}$ or $t=t_{i}$ and $m\in Sh(m_{i})$ for
$i=1,2$. (See Figure 1 for a intuitive visualization of
$[(t_{1},m_{1}),(t_{1},m_{2})]$).

Let $F\in\mathcal{L}_{1}$. Let $mrp(F)$ be the set of _modified rotation
pairs_ of all _lifted cycles_ of $F$. Clearly, $mrp(F)\subset\mathbb{M}$. The
following theorem holds:

######  Theorem 1.3 ([16]).

Let $F\in\mathcal{L}_{1}$. Then there are elements $(t_{1},m_{1})$ and
$(t_{2},m_{2})$ of $\mathbb{M}$ such that
$mrp(F)=[(t_{1},m_{1}),(t_{2},m_{2})]$ and if $t_{i}$ is rational, then
$m_{i}\neq 0$ for $i=1,2$. Moreover, for any set of the above form there
exists $F\in\mathcal{L}_{1}$ with $mrp(F)$ equal to this set.

###  1.7. Over-rotation numbers

Results similar to Theorem 1.3 were also proven for maps of intervals in [8]
when a version of _rotation numbers_ for _interval maps_ called _over-rotation
numbers_ was introduced using a special _observable_ $\chi$. Let
$f:[0,1]\to[0,1]$ be a _continuous interval map_ , $Per(f)$ be its set of
_periodic points_ , and $Fix(f)$ be its set of _fixed points_. It is easy to
see that if $Per(f)=Fix(f)$, then the _omega limit set_ , $\omega(y)$ is a
_fixed point_ for any $y$. So, we can assume that $Per(f)\neq Fix(f)$. Now,
define:

$\displaystyle\chi(x)=\begin{cases}\frac{1}{2}&\text{if
$(f(x)-x)(f^{2}(x)-f(x))\leqslant 0$,}\\\ {0}&\text{if
$(f(x)-x)(f^{2}(x)-f(x))>0$.}\end{cases}$

For any non-fixed periodic point $y$ of period $p(y)$, the integer
$l(y)=\displaystyle\sum^{p(y)-1}_{i=0}\chi(f^{i}(y))$ is at most
$\frac{p(y)}{2}$ and is the same for all points from the orbit of $y$. The
quantity $\frac{l(y)}{p(y)}$ is called the _over-rotation number_ of $y$ and
the pair $orp(y)=(l(y),p(y))$ is called the over-rotation pair of $y$. In an
_over-rotation pair_ $(p,q)$, $p$ and $q$ are integers and
$0<\frac{p}{q}\leqslant\frac{1}{2}$.

Like before we can transform all _over-rotation pairs_ of _cycles_ of a given
map $f$ into _modified over-rotation pairs_ of $f$ and denote the set of all
_modified over-rotation pairs_ of _cycles_ of $f$ by $mrp(f)$. Then, again
$mrp(f)\subset\mathbb{M}$ and the following theorem holds:

######  Theorem 1.4 ([8]).

If $f:[0,1]\to[0,1]$ is a continuous interval map with a non-fixed periodic
point, then $mrp(f)=[(t_{1},m_{1}),(\frac{1}{2},3)]$ for some
$(t_{1},m_{1})\in\mathbb{M}$. Moreover, for every
$(t_{1},m_{1})\in\mathbb{M}$, there exists a continuous map $f:[0,1]\to[0,1]$
with $mrp(f)=[(t_{1},m_{1}),(\frac{1}{2},3)]$.

Given an interval map $f$, denote by $I_{f}$ the _closure of the union of
over-rotation numbers_ of $f$-_periodic points_ , and call $I_{f}$ the _over-
rotation interval_ of $f$. By Theorem 1.1, any map $f$ with _non-fixed
periodic points_ has a _cycle_ of _period_ 2 and such as cycle has _over-
rotation number_ $\frac{1}{2}$; by Theorem 1.4, if $\rho(P)=\frac{p}{q}$ for a
cycle $P$ then $[\frac{p}{q},\frac{1}{2}]\subset I_{f}$; hence for any
interval map $f$ there exists a number $r_{f},0\leqslant r_{f}<\frac{1}{2},$
such that $I_{f}=[r_{f},\frac{1}{2}]$. A _cycle_ $P$ is called _over-twist_ if
it does not _force_ any other _cycle_ with the same _over-rotation number_.
They are the analogues of _twist cycles_ for _interval maps_. By [8] and by
_properties of forcing relation_ , for any
$\frac{p}{q}\in(r_{f},\frac{1}{2})$, $f$ has an _over-twist cycle_ of _over-
rotation number_ $\frac{p}{q}$. We call a _pattern_ $\pi$, an _over-twist
pattern_ if any _cycle_ which _exhibits_ it is an _over-twist_ cycle. Now we
discuss our plans for the paper.

###  1.8. Need for a new invariant

In the papers [5] and [11], an _algorithm_ to find out _over-rotation
interval_ of an _interval map_ of _modality_ less than or equal to $2$ was
concocted and the complete description of the dynamics of _unimodal_ and
_bimodal_ _over-twist patterns_ was corroborated. Further, in [6] it was shown
that the _over-rotation interval_ grows _monotonically_ in the space of all
_bimodal interval maps_. However, the task of determining the _over-rotation
interval_ for an _interval map_ $f:[0,1]\to[0,1]$ with _modality greater than
or equal to_ $3$, along with elucidating _over-twist patterns_ of the same
_modality_ , presents significant challenges. The methodologies presented in
[5], [11] and [6] for devising an algorithm to identify the _over-rotation
interval_ of any _unimodal_ or _bimodal interval map_ were relatively
straightforward due to their reliance on the existence of a simple
_discontinuous conjugacy_ $\sigma_{f}:[0,1]\to[0,1]$, defined by:

$\sigma_{f}(x)=\begin{cases}x&\text{if $0\leq x\leq a_{f}$}\\\
a_{f}+1-x&\text{if $a_{f}\leq x\leq 1$}\\\ \end{cases}$

where $a_{f}$ is the _unique fixed point_ of the map $f$ such that the map
$g_{f}=\sigma_{f}\circ f\circ\sigma_{f}^{-1}$ can be _lifted_ to a _degree
one_ _discontinuous map_ $F_{f}:\mathbb{R}\to\mathbb{R}$ of the real line (see
[5]). Although the classical tools for _degree one circle maps_ outlined in
[16] doesn’t apply to $F_{f}$ since it is _discontinuous_ , the intricacies of
the construction interestingly implies the existence of a _continuous
monotonically increasing degree one map_ $G_{f}:\mathbb{R}\to\mathbb{R}$ such
that the _classical rotation number_ of the _degree one map_ $G_{f}$ gives us
the _left end point_ of the _over-rotation interval_ of the map $f$ (see [5]).
However, the _conjugacy_ $\sigma_{f}$ fails for a general _interval map_ of
_modality_ strictly greater than $2$, rendering the construction and
associated arguments ineffectual. It seems that some more intricate
_conjugacy_ $\sigma_{f}$ might work in such cases but the the ensuing
construction would inevitably become exceedingly convoluted! Consequently,
this formidable obstacle prompted us to develop a new invariant, called “
_unfolding numbers_ ” for _periodic orbits_ of _interval maps_ , offering a
promising pathway for navigating through the hurdles inherent in the realm of
_over-rotation theory_.

###  1.9. Unfolding numbers

The basic idea is as follows. Given a _continuous map_ $f:[0,1]\to[0,1]$, we
construct the map $g_{f}:[0,\frac{1}{2}]\to[0,\frac{1}{2}]$ which is
_conjugate_ to the map $f$ via the via the _linear contraction_
$\lambda:[0,1]\to[0,\frac{1}{2}]$ defined by $\lambda(x)=\frac{x}{2}$. We call
$g_{f}$ the _miniature model map_ corresponding to the map $f$. We _fold_ the
_unit square_ $[0,\frac{1}{2}]\times[0,\frac{1}{2}]$ containing the _graph_ of
$g_{f}$ _vertically_ across the line $y=\frac{1}{2}$. This yields a map
$p_{f}$ in $[0,\frac{1}{2}]\times[\frac{1}{2},1]$. The _unit square_
$[0,\frac{1}{2}]\times[\frac{1}{2},1]$ is now _folded_ _horizontally_ across
$x=\frac{1}{2}$ to obtain a map $q_{f}$ in
$[\frac{1}{2},1]\times[\frac{1}{2},1]$. Finally, the _unit square_
$[\frac{1}{2},1]\times[\frac{1}{2},1]$ is _folded_ _vertically_ across $y=1$
to obtain a map $r_{f}$ in $[\frac{1}{2},1]\times[1,\frac{3}{2}]$. Selecting
part of _graph_ of the _maps_ : $g_{f}$ , $p_{f}$, $q_{f}$ and $r_{f}$ yields
a _degree one map_ $F_{f}:\mathbb{R}\to\mathbb{R}$ of the real line (_for
details see Section_ 3). The _unfolding number_ of a _periodic orbit_ $P$ of
the map $f:[0,1]\to[0,1]$ is defined to to be _rotation number_ of the _lifted
periodic orbit_ $P^{\prime}$ of $F_{f}$ corresponding to $P$.

We begin study of _unfolding numbers_ for _interval maps_ in this paper. As we
show in Section 3, the _completeness_ of the theory of _unfolding numbers_ can
be deduced from the _completeness_ of _rotation theory_ for _degree one circle
maps_ established in [16, 17]. We further portray that, that the _closure_ of
the set of _unfolding numbers_ of _periodic orbits_ of a given _interval map_
$f$ forms an interval of the form $[u_{f},\frac{1}{2}]$ for some
$u_{f}\geqslant 0$, called its _unfolding interval_. In this way, a _single
number_ , $u_{f}$, encapsulates concealed information about the _dynamical
complexity_ of $f$, just like in _over-rotation theory_. However, we validate
that, in contrast to the latter theory, definition of _unfolding number_
naturally enables us to evade complications faced earlier and compute $u_{f}$
using a straightforward algorithm for interval maps of any _modality_. We
depict that there exists a _point_ $z\in[0,\frac{1}{2}]$ such that the map $f$
_restricted on_ the _omega limit set_ $Z_{f}=\omega_{f}(z)$ of $z$ is
_strictly ergodic_ and that for any _point_ $y\in Z_{f}$, the _unfolding
number_ of $y$ under $f$ _exists_ and _equals_ $u_{f}$.

It is natural question to ask: can we characterize the _dynamics_ of a
_periodic orbit_ of $f$ given its _unfolding number_? For this, at the outset
it is vital to see whether _unfolding number_ of $P$ can be read directly from
the _combinatorics_ of $P$ without the need to construct the map $F_{f}$.
Next, it is reasonable to wonder about the _connection_ between _over-rotation
numbers_ and _unfolding numbers_. We take precisely these questions in Section
4. We show that for any _pattern_ with _modality_ _less than or equal to $2$_,
in fact, _over-rotation numbers, pairs_ equals _unfolding numbers, pairs_. We
give a _counter-example_ to display that this is not true in general, for a
_pattern_ of _modality greater than or equal to $3$_. However, we discover a
_class_ of _patterns of any modality_ for whom _over-rotation numbers, pairs_
_equals_ _unfolding numbers, pairs_. We name these: _sheer patterns_.

The ultimate _goal_ of this paper is to establish _unfolding numbers_ as a
“new ” means of computing the _dynamical complexity_ of an _interval map_
which can be _effectively computed_ and which is _invariant_ under
_topological conjugacy_.

The paper is divided into $4$ _simple sections_. The organization of this
manuscript unfolds as follows:

  1. (1)

Section 1 serves as the _introductory segment_ of the paper, providing an
initial _overview_ and context for the subsequent content.

  2. (2)

Section 2 lays the groundwork with essential _preliminaries_.

  3. (3)

Section 3 is the _main section_ of the paper, where the _foundational_ tenets
of the theory of _unfolding numbers_ are laid bare, _consistency_ of the
_theory_ is enacted and the structure of the _unfolding interval_ and its
_realization_ is expounded.

  4. (4)

In Section 4 we establish _connection_ between _unfolding numbers_ and _over-
rotation numbers_.

##  2\. Preliminaries

###  2.1. Notations

The notation conventions adopted in this paper are as follows. We specifically
consider _continuous maps_ $f$ characterized by a _unique point_ of _absolute
maximum_ denoted by $M_{f}$ and _unique point_ of _absolute minimum_ denoted
by $m_{f}$. Moreover, for any _periodic orbit_ $P$ of $f$, its _leftmost
endpoint_ is represented by $le(P)$ and its _right end point_ is denoted by
$ri(P)$. Furthermore, for any _point_ $x$ and a map $g$, the _trajectory_ of
$x$ under $g$, that is the set $\\{x,g(x),g^{2}(x),\dots\\}$ is symbolized as
$T_{g}(x)$. For a set $A$, let $\overline{A}$ denotes its _closure_.
Additionally, given a map $f:[0,1]\to[0,1]$, $g_{f}$ and $F_{f}$ respectively
signify _miniature model map_ and _heaved map_ corresponding to the map $f$
(_as defined in_ Section 3).

###  2.2. Results on Degree one circle maps

Let’s examine some properties of _degree one circle maps_ that are pertinent
to our discussion in this paper. Let $\mathcal{L}_{1}$ represents the
_collection_ of all _liftings_ of _continuous degree one self-mappings_ of
$\mathbb{S}$ under the _sup norm_ , while
$\mathcal{L}_{1}^{\prime}\subset\mathcal{L}_{1}$ consists of _non-decreasing
elements_ of $\mathcal{L}_{1}$. Subsequently, the ensuing result follows:

######  Theorem 2.1 ([1, 16]).

If $F\in\mathcal{L}_{1}^{\prime}$ is a lifting of a circle map $f$ then
$\rho_{F}(x)$ exists for all $x\in\mathbb{R}$ and is independent of $x$.
Moreover, it is rational if and only if $f$ has a periodic point.

This number is called the _rotation number_ of $F$ and denoted by $\rho(F)$.

######  Lemma 2.2 ([1, 16]).

The function $\chi:\mathcal{L}_{1}^{\prime}\to\mathbb{R}$ defined by
$\chi(F)=\rho(F)$ is continuous.

######  Definition 2.3 ([1, 16]).

For $F\in\mathcal{L}_{1}$ we define maps
$F_{l},F_{u}\in\mathcal{L}_{1}^{\prime}$ by “ _pouring water_ ” from _below_
and _above_ respectively on the _graph_ of $F$ respectively as follows:
$F_{\ell}(x)=\inf\\{F(y):y\geqslant x\\}$ and $F_{u}(x)=\sup\\{F(y):y\leqslant
x\\}$. We call the _maps_ : $F_{\ell}$ and $F_{u}$, _lower bound map_ and
_upper bound map_ corresponding to the _degree one map_ $F$ respectively.

Let Const($G$) be the union of all _open intervals_ on which $G$ is
_constant_.

######  Theorem 2.4 ([1, 16]).

Then the following statements are true:

  1. (1)

Let $F\in\mathcal{L}_{1}$. Then, $F_{\ell}(x)\leqslant F(x)\leqslant F_{u}(x)$
for every $x\in\mathbb{R}$.

  2. (2)

Let $F,G\in\mathcal{L}_{1}$ with $F\leqslant G$, then $F_{\ell}\leqslant
G_{\ell}$ and $F_{u}\leqslant G_{u}$.

  3. (3)

Let $F\in\mathcal{L}_{1}^{\prime}$. Then, $F_{\ell}=F_{u}=F$.

  4. (4)

The maps $F\mapsto F_{\ell}$ and $F\mapsto F_{u}$ are Lipschitz continuous
with constant $1$ in the sup norm.

  5. (5)

If $F_{\ell}(x)\neq F(x)$, then $x\in Const(F_{\ell})$; similarly for $F_{u}$,

  6. (6)

Const($F$) $\subset$ Const($F_{\ell}$) $\cap$ Const($F_{u}$)

######  Theorem 2.5 ([1, 16]).

If $F\in\mathcal{L}_{1}^{\prime}$ is a lifting of a circle map $f$ and
$\rho(F)=\frac{k}{n}$ for coprime integers $k,n$, then $f$ has a $n$-cycle $P$
such that $\pi^{-1}(P)$ is disjoint from Const($F$).

######  Theorem 2.6 ([1, 16]).

Let $F\in\mathcal{L}_{1}$. Define:

$F_{\mu}=(min(F,F_{\ell}+\mu))_{u}$ for $\mu\in[0,\lambda]$,
$\lambda=sup_{x\in\mathbb{R}}(F-F_{\ell})(x)$. Then:

  1. (1)

$F_{\mu}\in\mathcal{L}_{1}^{\prime}$ for all $\mu$,

  2. (2)

$F_{0}=F_{\ell}$ and $F_{\lambda}=F_{u}$

  3. (3)

the map $\mu\mapsto F_{\mu}$ is Lipschitz continuous with constant 1,

  4. (4)

the map $\mu\mapsto\rho(F_{\mu})$ is continuous,

  5. (5)

if $\mu\leqslant\lambda$, then $F_{\mu}\leqslant F_{\lambda}$,

  6. (6)

each $F_{\mu}$ coincides with $F$ outside $Const(F_{\mu})$,

  7. (7)

$Const(F)\subset Const(F_{\mu})$ for each $\mu$.

######  Definition 2.7 ([1, 16]).

Let $F\in\mathcal{L}_{1}$. Then, define _upper-rotation set_ of $F$ by
$\overline{Rot(F)}=\\{\overline{\rho_{F}(x)}:x\in\mathbb{R}\\}$; _lower-
rotation set_ of $F$ by
$\underline{Rot(F)}=\\{\underline{\rho_{F}(x)}:x\in\mathbb{R}\\}$. Also,
define _rotation set_ of $F$ by
$Rot(F)=\displaystyle\\{\rho_{F}(x):x\in\mathbb{R}$ such that
$\overline{\rho_{F}(x)}=\underline{\rho_{F}(x)}=\rho_{F}(x)\\}$.

######  Theorem 2.8 ([1, 16]).

Let $F\in\mathcal{L}_{1}$. Then,
$\overline{Rot(F)}=\underline{Rot(F)}=Rot(F)=[\rho(F_{\ell}),\rho(F{{}_{u}})]$
where $F_{\ell}(x)=\inf\\{F(y):y\geqslant x\\}$ and
$F_{u}(x)=\sup\\{F(y):y\leqslant x\\}$

######  Corollary 2.9 ([1, 16]).

The endpoints of $Rot(F)$ depend continuously on $F\in\mathcal{L}_{1}$.

##  3\. Main Section

We begin by defining _unfolding numbers_ in details. Consider a _continuous
map_ $f:[0,1]\to[0,1]$ with a _unique point_ of _absolute_ _maximum_ $M_{f}$
and a _unique point_ of _absolute minimum_ $m_{f}$ with $f(M_{f})=1$ and
$f(m_{f})=0$; assume for definiteness that $M_{f}<m_{f}$. Observe that from
the standpoint of the _co-existence_ of _cycles_ of _interval maps_ this is
the most important case. Indeed, let $P\subset[0,1]$ be a _cycle_ of a
_continuous_ map $f$ with $\\{0,1\\}\subset P$. Then, $m_{f}>M_{f}$ implies
that there exist two disjoint _open intervals_ whose _closures_ contain their
_union_ in their images(“$1$-_dimensional horseshoe_ ”), $f$ has the _entropy_
at least $\ln 2$, _cycles_ of all _periods_. So, $f$ is a _chaotic map_. Thus,
the case $M_{f}<m_{f}$ is the only interesting case (_even if our construction
stated below works in the case $M_{f}>m_{f}$ too_).

Consider the map $g_{f}:[0,\frac{1}{2}]\to[0,\frac{1}{2}]$ which is
_conjugate_ to the map $f:[0,1]\to[0,1]$ via the _linear contraction_
$\lambda:[0,1]\to[0,\frac{1}{2}]$ defined by $\lambda(x)=\frac{x}{2}$. We call
$g_{f}$ the _miniature model map_ corresponding to the map $f$. Let
$g_{f}(0)=c_{f},g_{f}(\frac{1}{2})=d_{f}$.

Define $p_{f}:[0,\frac{1}{2}]\to[\frac{1}{2},1]$ by $p_{f}(x)=1-g_{f}(x)$.
Clearly, $p_{f}(0)=1-c_{f}$, $p_{f}(M_{f})=\frac{1}{2}$, $p_{f}(m_{f})=1$,
$p_{f}(\frac{1}{2})=1-d_{f}$. The _graph_ of $p_{f}(x)$ is obtained by
_reflecting vertically_ , the _graph_ of $g_{f}(x)$ with the respect with the
line $y=\frac{1}{2}$.

Define $q_{f}:[\frac{1}{2},1]\to[\frac{1}{2},1]$ by
$q_{f}(x)=p_{f}(1-x)=1-f(1-x)$. Clearly, $q_{f}(\frac{1}{2})=1-d_{f}$,
$q_{f}(1-m_{f})=1$, $q_{f}(1-M_{f})=\frac{1}{2}$, $q_{f}(1)=1-c_{f}$. The
_graph_ of $q_{f}(x)$ is obtained by _reflecting horizontally_ , the _graph_
of $p_{f}(x)$ across the line $x=\frac{1}{2}$.

Define $r_{f}:[\frac{1}{2},1]\to[1,\frac{3}{2}]$ by
$r_{f}(x)=2-q_{f}(x)=1+f(1-x)$. Clearly, $r_{f}(\frac{1}{2})=1+d_{f}$,
$r_{f}(1-m_{f})=1$, $r_{f}(1-M_{f})=\frac{3}{2}$, $r_{f}(1)=1+c_{f}$. Thus,the
_graph_ of $r_{f}$ is obtained from the _graph_ of $q_{f}(x)$ by _reflecting
vertically_ across the line $y=1$.

We now _construct_ a map $F_{f}:[0,1]\to[1,\frac{3}{2}]$ as follows:

$F_{f}(x)=\begin{cases}g_{f}(x)&\text{if $0\leqslant x\leqslant M_{f}$}\\\
p_{f}(x)&\text{if $M_{f}\leqslant x\leqslant\frac{1}{2}$.}\\\
q_{f}(x)&\text{if $\frac{1}{2}\leqslant x\leqslant 1-m_{f}$.}\\\
r_{f}(x)&\text{if $1-m_{f}\leqslant x\leqslant 1$.}\\\ \end{cases}$

Observe that $F_{f}(1)=F_{f}(0)+1$. Extend $F_{f}:\mathbb{R}\to\mathbb{R}$ by
$F_{f}(x+1)=F_{f}(x)+1$ for all $x\in\mathbb{R}$. We shall call the map
$F_{f}$ the _heaved map_ corresponding to the map $f$. By construction
$F_{f}(x)$ is a _degree one map_ of the real line (See Figure 2).

Figure 2. Graph of $F_{f}$ shown in _dotted line_

######  Definition 3.1.

For $x\in[0,1]$, the _upper unfolding number_ of $x$ under $f$, denoted by
$\overline{u_{f}(x)}$, is defined as the _upper rotation number_ of
$\lambda(x)=\frac{x}{2}$ under $F_{f}$, that is,
$\overline{u_{f}(x)}=\overline{\rho_{F_{f}}(x)}=\displaystyle\lim_{n\to\infty}\mathrm{sup}\displaystyle\frac{F^{n}(\lambda(x))-\lambda(x)}{n}$.

Similarly, the _lower unfolding number_ of $x$ under $f$, denoted by
$\underline{u_{f}(x)}$, is defined as the _lower rotation number_ of
$\lambda(x)$ under $F_{f}$, that is,
$\underline{u_{f}(x)}=\underline{\rho_{F_{f}}(x)}=\displaystyle\lim_{n\to\infty}\mathrm{inf}\displaystyle\frac{F^{n}(\lambda(x))-\lambda(x)}{n}$.

Clearly, $\underline{u_{f}(x)}\leqslant\overline{u_{f}(x)}$. The set
$\mathcal{U}_{f}(x)=\\{\underline{u_{f}(x)},\overline{u_{f}(x)}\\}$ is called
the _unfolding set_ of $x$ _under_ $f$. If
$\underline{u_{f}(x)}=\overline{u_{f}(x)}=u_{f}(x)$, we call: $u_{f}(x)$, the
_unfolding number_ of $x$ _under_ $f$.

######  Definition 3.2.

Let $f:[0,1]\to[0,1]$. Then, define _upper-unfolding set_ of $f$ by
$\overline{Unf(f)}=\\{\overline{u_{f}(x)}:x\in[0,1]\\}$; _lower-unfolding set_
of $f$ by $\underline{Unf(f)}=\\{\underline{u_{f}(x)}:x\in[0,1]\\}$. Also,
define _unfolding set_ of $f$ by $Unf(f)=\\{u_{f}(x):x\in\mathbb{R}$ such that
$\overline{u_{f}(x)}=\underline{u_{f}(x)}=u_{f}(x)\\}$.

We now establish the _consistency_ of the theory of _unfolding numbers_. We
begin with a simple lemma:

######  Lemma 3.3.

Given any $z\in\mathbb{R}$, there exists $x_{z}\in[0,\frac{1}{2}]$ such that
$F_{f}^{n_{z}}(x_{z})=z$ for some $n_{z}\in\mathbb{N}$.

###### Proof.

Since, $F_{f}$ is a _degree-one_ map, there exists $y\in[0,1]$ such that
$F_{f}^{m}(y)=z$. If $y\in[0,\frac{1}{2}]$ then $y=x_{z}$ and $n_{z}=m$ and we
are done. Otherwise if $y_{z}\in[\frac{1}{2},1]$, then it follows from
construction of $F_{f}$ that $F_{f}^{-1}(y_{z})\in[0,\frac{1}{2}]$. In the
latter case, $x_{z}=F_{f}^{-1}(y)$ and $n_{z}=m+1$. ∎

######  Theorem 3.4.

Let $f:[0,1]\to[0,1]$ be a given continuous interval map. Then,
$\overline{Unf(f)}=\underline{Unf(f)}=Unf(f)=[\rho({(F_{f})}_{\ell}),\rho({(F_{f})}_{u})]$,
where ${(F_{f})}_{\ell}$ and $(F_{f})_{u}$ are respectively the lower bound
map and upper bound map corresponding to the degree one map $F_{f}$.

###### Proof.

Choose $x\in[0,1]$ and $\underline{u_{f}(x)}\in\underline{Unf(f)}$. Then,
$\lambda(x)\in[0,\frac{1}{2}]$ and clearly by definition 3.1,
$\underline{u_{f}(x)}=\underline{\rho_{F_{f}}(\lambda(x))}$ and hence
$\underline{u_{f}(x)}\in\underline{Rot(F_{f})}$. Thus,
$\underline{Unf(f)}\subseteq\underline{Rot(F_{f})}$. Now, take
$y\in\mathbb{R}$ and $\underline{\rho_{F_{f}}(y)}\in\underline{Rot(F_{f})}$.
Lemma 3.3 guarantees the existence of a point $z\in[0,\frac{1}{2}]$ such that
$F_{f}^{n}(z)=y$ for some $n\in\mathbb{N}$. Let $\lambda^{-1}(z)=x\in[0,1]$.
Then, $\underline{u_{f}(x)}=\underline{\rho_{F_{f}}(y)}$ and hence the reverse
inclusion follows as well implying
$\underline{Unf(f)}=\underline{Rot(F_{f})}$. Similarly, it can be shown that:
$\overline{Unf(f)}=\overline{Rot(F_{f})}$. The result then follows from
Theorem 2.8.

∎

######  Theorem 3.5.

If $P$ is a periodic orbit of $f:[0,1]\to[0,1]$ of period $q$, then the
unfolding number of each point of $P$ is same and is of the form $\frac{p}{q}$
for $p,q\in\mathbb{N}$.

###### Proof.

Let $x\in P$. Then, $\lambda(x)\in[0,\frac{1}{2}]$ is _periodic_ under
$g_{f}:[0,\frac{1}{2}]\to[0,\frac{1}{2}]$ with _period_ $q$. Construction of
the map $F_{f}:\mathbb{R}\to\mathbb{R}$ then implies that
$F_{f}^{q}(\lambda(x))=\lambda(x)+p$ for some $p\in\mathbb{N}$, that is, the
_trajectory_ $T_{F_{f}}(\lambda(x))$ of $\lambda(x)$ under $F_{f}$ is a
_lifted periodic orbit_ of $F_{f}$ of _period_ $q$ and the _classical rotation
number_ of $\lambda(x)$ under $F_{f}$ is $\frac{p}{q}$. Thus, from Definition
3.1, the _unfolding number_ of $x$ under $f$ is $\frac{p}{q}$. ∎

######  Definition 3.6.

We call $\frac{p}{q}$, the _unfolding number_ of the periodic orbit
$P=T_{f}(x)$ of $f$ and the ordered pair $(p,q)$ is called the _unfolding
pair_ of $P$. We call the _unfolding pair_ $(p,q)$ _coprime_ if $p$ and $q$
are _coprime_. If $\frac{p}{q}=\frac{r}{s}$ where $r$ and $s$ are _coprime_
and $\frac{p}{r}=\frac{q}{s}=m$ then we call the ordered pair
$(\frac{r}{s},m)$, the _modified unfolding pair_ of $P$ (_similar to rotation
theory for degree one circle maps, see Section_ 1.6). Let $mup(f)$ be the set
of all _modified unfolding pairs_ of _cycles_ of $f$.

######  Theorem 3.7.

The set of modified unfolding pairs of cycles of $f:[0,1]\to[0,1]$ is equal to
the set of modified rotation pairs of lifted cycles of the degree one circle
map $F_{f}:\mathbb{R}\to\mathbb{R}$, that is, $mup(f)=mrp(F_{f})$.

###### Proof.

Let $P$ be a _cycle_ of $f:[0,1]\to[0,1]$ with _period_ $mq$ with _rotation
number_ $\frac{p}{q}$, $p,q$ are _co-prime_ and _modified unfolding pair_
$(\frac{p}{q},m)$ for some $m\in\mathbb{N}$. Let $x\in P$. Then, $\lambda(x)$
is a _periodic point_ of $g_{f}:[0,\frac{1}{2}]\to[0,\frac{1}{2}]$ of period
$mq$. Construction of the map $F_{f}:\mathbb{R}\to\mathbb{R}$ then stipulates
that $F_{f}^{mq}(\lambda(x))=\lambda(x)+mp$, that is, the _trajectory_
$T_{F_{f}}(\lambda(x))$ of $\lambda(x)$ under $F_{f}$ is a _lifted periodic
orbit_ of $F_{f}$ of _period_ $mq$ and _rotation number_ $\frac{p}{q}$.
Definition 3.1 then entails that _rotation number_ of $\lambda(x)$ under
$F_{f}$ is $\frac{p}{q}$. Hence, the _modified rotation pair_ of the _lifted
cycle_ $T_{F_{f}}(\lambda(x))$ of $F_{f}$ is $(\frac{p}{q},m)$. Hence,
$(\frac{p}{q},m)\in mrp(F_{f})$.

Conversely, suppose $(\frac{p}{q},m)$ be the _modified rotation pair_ of a
_lifted cycle_ $P^{\prime}$ of $F_{f}$. Choose $z\in P^{\prime}$. Then,
$F_{f}^{mq}(z)=z+mp$. By Lemma 3.3, there exists $x_{z}\in[0,\frac{1}{2}]$
such that $F_{f}^{n}(x_{z})=z$ for some $n\in\mathbb{N}$. Connection between
the maps $g_{f}$ and $F_{f}$, now warrants that $x_{z}$ is a _periodic point_
of $g_{f}$ of period $mq$ and hence $\lambda^{-1}(x_{z})=y_{z}$ is a periodic
point of $f$ of period $mq$. From Definitions 3.1 and 3.6, the _modified
unfolding pair_ of the _periodic orbit_ $T_{f}(y_{z})$ of $y_{z}$ under $f$ is
$(\frac{p}{q},m)$ and hence the result follows. ∎

######  Theorem 3.8.

There exists $t_{1},t_{2}\in\mathbb{R}$ and $m_{1},m_{2}\in\mathbb{N}$ such
that $mup(f)=[(t_{1},m_{1}),(t_{2},m_{2})]$.

###### Proof.

By Theorem 1.3, there exists _modified rotation pairs_ $(t_{1},m_{1})$ and
$(t_{2},m_{2})$ such that the set of all _modified rotation pairs_ of cycles
of the degree one map $F_{f}$ is given by
$mrp(F_{f})=[(t_{1},m_{1}),(t_{2},m_{2})]$. Now, the result follows from
Theorem 3.7.

∎

Theorems 3.5 and 3.8 guarantees the _completeness_ of the _unfolding numbers_.

######  Theorem 3.9.

Let $f:[0,1]\to[0,1]$ be a given interval map. Then
$Unf(f)=[u_{f},\frac{1}{2}]$, where $u_{f}=\rho((F_{f})_{\ell})$ where
${(F_{f})}_{\ell}$ represents the lower bound map corresponding to the degree
one map $F_{f}$.

###### Proof.

By Theorem 3.4, $Unf(f)=[\rho({(F_{f})}_{\ell}),\rho({(F_{f})}_{u})]$, where
${(F_{f})}_{\ell}(x)=\inf\\{F_{f}(y):y\geqslant x\\}$ and
${(F_{f})}_{u}(x)=\sup\\{F_{f}(y):y\leqslant x\\}$. It remains to show
$\rho((F_{f})_{u})=\frac{1}{2}$. It is easy to see that
$(F_{f})_{u}=\frac{1}{2}$ in $[0,M_{f}]$ and $(F_{f})_{u}=1$ in the interval
$[m_{f},1-m_{f}]$ ($m_{f}$ and $M_{f}$ are respectively the points of
_absolute minimum_ and _absolute maximum_ of $f$). It follows that
$(F_{f})_{u}(0)=\frac{1}{2}$ and $(F_{f})_{u}(\frac{1}{2})=1$ and hence
$T_{(F_{f})_{u}}(0)$ has _rotation number_ $\frac{1}{2}$. Now, the result
follows from Theorem 2.1. ∎

######  Definition 3.10.

The _interval_ $Unf(f)=[u_{f},\frac{1}{2}]$ from Theorem 3.9 is called the
_unfolding interval_ of the interval _map_ $f:[0,1]\to[0,1]$.

######  Definition 3.11.

Given a map $f:[0,1]\to[0,1]$, a _set_ $A\subset[0,1]$ and a _point_
$y\in[0,1]$, we define:

  1. (1)

$\displaystyle\eta_{A}(y)=\mathrm{sup}\\{x\in A:f(x)=f(y)\\}$

  2. (2)

$\xi_{A}(y)=\mathrm{inf}\\{x\in A:f(x)=f(y)\\}$

  3. (3)

$Y_{f}=\eta_{[0,M_{f}]}([0,M_{f}])$ $\cup\eta_{[0,1]}([M_{f},m_{f}])$
$\cup\xi_{[0,1]}([M_{f},m_{f}])\cup\eta_{[m_{f},1]}([m_{f},1])$.

Let us denote by $\mathcal{C}$, the set of _open subsets_ of $\mathbb{R}$
where $(F_{f})_{\ell}(x)$ is _constant_ and let $\mathcal{U}$ to be the
_union_ of _open intervals_ from $\mathcal{C}$.

######  Theorem 3.12.

Let $f:[0,1]\to[0,1]$ be a continuous map with unfolding interval
$[u_{f},\frac{1}{2}]$. Then for any point $y\in[0,1]$ whose trajectory is
contained in $Y_{f}$, the unfolding set $\mathcal{U}_{f}(y)=\\{u_{f}\\}$.

###### Proof.

Construction of the maps $g_{f}$ and $F_{f}$ reveals that for $y\in[0,1]$, if
the _trajectory_ $T_{f}(y)$ of $y$ under $f$ is _disjoint_ from $Y_{f}$, then
the _trajectory_ $T_{F_{f}}(\lambda(y))$ of $\lambda(y)$ under $F_{f}$ is
_disjoint_ from $\mathcal{U}$. This means the map $F_{f}$ restricted on the
_trajectory_ of $\lambda(y)$ and the _lower bound map_ $(F_{f})_{\ell}$
restricted on the _trajectory_ of $\lambda(y)$ are precisely the same. So,
from Theorem 3.9, _rotation number_ of $\lambda(y)$ under $F_{f}$ is equal to
$u_{f}$: the _rotation number_ of $\lambda(y)$ under the _lower bound map_
$(F_{f})_{\ell}$. Finally from definition 3.1, this means that the _unfolding
set_ $\mathcal{U}_{f}$ of $y$ is equal to $\\{u_{f}\\}$. ∎

######  Theorem 3.13.

Let $f:[0,1]\to[0,1]$ be a continuous map with unfolding interval
$[u_{f},\frac{1}{2}]$. Then, there exists a minimal $f$-invariant set
$Z_{f}\subset Y_{f}$ such that for any point $y\in Z_{f}$, we have
$\mathcal{U}_{f}(y)=\\{u_{f}\\}$ and there are two possibilities :

  1. (1)

$u_{f}$ is rational, $Z_{f}$ is a periodic orbit, and $f|_{Z_{f}}$ is
canonically conjugate to the circle rotation by $u_{f}$ restricted on one of
its cycles.

  2. (2)

$u_{f}$ is irrational, $Z_{f}$ is a Cantor set, and $f|_{Z_{f}}$ is
canonically semi-conjugate to the circle rotation by $u_{f}$.

###### Proof.

The _natural projection_ map $\pi:\mathbb{R}\to\mathbb{S}$ _monotonically
semi-conjugates_ $(F_{f})_{\ell}$ to a _degree one map_
$\tau_{f}:\mathbb{S}\to\mathbb{S}$. Further, since $(F_{f})_{\ell}$ is
_monotonically non-decreasing_ , $\tau_{f}:\mathbb{S}\to\mathbb{S}$
_preserves_ _cyclic orientation_ in the _non-strict sense_. From Theorem 2.5,
it follows that there exists a _point_ $z\in[0,1)$ whose _orbit_ is _disjoint_
from $\mathcal{U}$ (that is, the $(F_{f})_{\ell}$-orbit of $z$ is the same as
the $F_{f}$-orbit of $z$) and such that the _classical rotation number_ of $z$
under $(F_{f})_{\ell}$ is $u_{f}$. We have two cases here:

  1. (1)

If $u_{f}=\frac{p}{q}$ is _rational_ , then $(F_{f})_{\ell}^{q}(z)=z+p$,
$\pi(z)\in\mathbb{S}$ is _periodic of period_ $q$. If $P_{f}$ be the _orbit_
of $\pi(z)$, then the map $\tau$ _restricted_ on $P_{f}$ acts as _circle
rotation_ by _angle_ $\frac{p}{q}$.

  2. (2)

If $u_{f}$ is _irrational_ , then
$\pi(z)\in\omega_{\tau_{f}}\left(\pi(z)\right)=Q_{f}\subset\mathbb{S}$. It is
_minimal_ (that is, $\tau_{f}$-_orbits_ of all points of $Q_{f}$ are _dense_
in $Q_{f}$ such that _collapsing arcs_ of $\mathbb{S}$ complementary to
$Q_{f}$, we can _semi-conjugate_ $\tau_{f}:\mathbb{S}\to\mathbb{S}$ to
_irrational rotation_ of $\mathbb{S}$ by the _angle_ $u_{f}$. In this case
$Q_{f}$ is a _cantor set_.

We need to now find the appropriate $f$-_invariant set_ $Z_{f}\subset[0,1]$
whose existence is claimed in the theorem. The _correspondence_ between the
maps involved in our construction implies the existence of a _continuous
conjugacy_ $\psi_{f}:\mathbb{S}\to[0,1]$ between the maps
$\tau:\mathbb{S}\to\mathbb{S}$ and $f:[0,1]\to[0,1]$ applicable outside
$\pi(\overline{\mathcal{U}})\subset\mathbb{S}$.

If $u_{f}$ is _irrational_ , then we can easily a point $y\in Q_{f}$ such that
the _trajectory_ of $y$ under $\tau_{f}$ on $\mathbb{S}$ is _disjoint_ from
$\pi(\overline{\mathcal{U}})$. Then, it follows that there is a _continuous
conjugacy_ between map: $\tau:\mathbb{S}\to\mathbb{S}$ on the _trajectory_ of
$y$ and $f|_{Z_{f}}$ where $Z_{f}$ is the _omega limit set_
$\omega_{f}(\psi_{f}(y))$ of $\psi_{f}(y)$ under $f$. It is then easy to see
that $Z_{f}$ is a _Cantor Set_ and has all the desired properties.

Suppose $u_{f}=\frac{p}{q}$ is _rational_. By Theorem 2.5, $P_{f}$ is
_disjoint_ from $\pi(\mathcal{U})$, but, it is not guaranteed that $P_{f}$
will be also be _disjoint_ from $\pi(\overline{\mathcal{U}})$. However, as we
now show the specifics of the _construction_ helps us to elude this problem.
If $P_{f}\cap\pi(\overline{\mathcal{U}})=\phi$, then we are done by the
arguments similar to the ones from the previous paragraph. Suppose that
$P_{f}$ passes through an _endpoint_ $b$ of a _flat spot_ of $\tau_{f}$.
Choose a point $y$ very _close_ to $b$ avoiding _flat spots_ of $\tau_{f}$.
Choose a _point_ $y$ very _close_ to $b$ avoiding _flat spots_ of $\tau_{f}$.
Then, the _finite segment_ of the $\tau_{f}$ orbit of $y$ consisting of
_points_ $y,\tau_{f}(y),\dots\tau_{f}^{q}(y)\approx y$ is _transformed_ by
$\psi_{f}$ into a _finite segment_ $\psi_{f}(y),f(\psi_{f}(y)),\dots
f^{q}(\psi_{f}(y))\approx\psi_{f}(y)$ which _converges_ to an $f$-_periodic
orbit_ as $y\to b$. The _limit periodic orbit_ $Z_{f}$ consisting of
$\psi_{f}(y),f(\psi_{f}(y)),\dots f^{q}(\psi_{f}(y))\approx\psi_{f}(y)$ has
all the desired properties.

∎

######  Theorem 3.14.

Let $f:[0,1]\to[0,1]$ be a continuous map with unfolding interval
$[u_{f},\frac{1}{2}]$. Then, there exists a point $z\in[0,1]$ such that the
map $f|_{Z_{f}}$ where $Z_{f}=\omega_{f}(z)$ is strictly ergodic.

###### Proof.

It follows from Theorem 3.13, that there exists a point $z\in[0,1]$ such that
the map $f|_{Z_{f}}$ where $Z_{f}=\omega_{f}(z)$ has a unique _invariant
measure_ $\mu_{f}$ such that the _measure_ $\mu_{f}$ can be _transformed_ in a
canonical fashion to a specific _invariant measure_ related to _circle
rotation_ by angle $u_{f}$. If $u_{f}$ is _rational_ , the _measure_ $\mu_{f}$
is simply the CO-_measure_ xz concentrated on the $f$-_periodic orbit_ of $z$
whereas if $u_{f}$ is _irrational_ , $\mu_{f}$ corresponds in a canonical
fashion to the _Lebesgue measure_ on $\mathbb{S}$ _invariant_ under
_irrational rotation_ on angle $u_{f}$. Hence, the result follows.

∎

##  4\. Connection with Over-rotation numbers

In this section we ascertain a connection between _unfolding numbers_ and
_over-rotation numbers_. For this we first portray that _unfolding number_ of
$P$ can be read directly from the _combinatorics_ of $P$ without the need to
construct the map $F_{f}$. We commence by introducing a few notions. Let $P$
be a _periodic orbit_ of _period_ $q$. Let $f:[0,1]\to[0,1]$ be a $P$-_linear_
map such that the _left-end point_ and _right-end point_ of $P$ are $le(P)=0$
and $ri(P)=1$ respectively. Let $m_{f}$ and $M_{f}$ be points of _absolute
minimum_ and _absolute maximum_ of $f$ such that $f(m_{f})=0$ and
$f(M_{f})=1$.

######  Definition 4.1.

A point $x\in P$ will be called an _unfolding point_ of $P$ if
$f^{-1}(x)<m_{f}$ and $f^{-2}(x)>M_{f}$.

Let $\mathcal{E}$ be the collection of all _unfolding points_ of $P$. We
define a relation $\sim$ on $\mathcal{E}$ as follows : for $x,y\in\mathcal{E}$
, $x\sim y$ if $f(x)=y$ or $f(y)=x$. For $x\in\mathcal{E}$, the set
$[x]=\\{x\\}\cap\\{y\in\mathcal{E}:y\sim x\\}$ is called _unfolding class_ of
$P$ containing $x$. By construction the _union_ of _unfolding classes_ yields
the set $\mathcal{E}$. It is easy to see that two _unfolding classes_ are
either _coincident_ , _intersect_ at _one point_ or are _disjoint_. In the
later two cases, we call the _unfolding classes_ _distinct_.

######  Definition 4.2.

By _unfolding set_ of $P$ we shall mean the set
$\mathcal{N}=\\{n_{1},n_{2},\dots n_{k}\\}$ of _non-negative integers_
satisfying the following properties:

  1. (1)

$0\leqslant n_{i}<q$, for all $i\in\\{1,2,\dots k\\}$.

  2. (2)

$n_{1}$ is the _smallest non-negative integer_ such that
$f^{n_{1}-1}(0)<m_{f}$ and $f^{n_{1}-2}(0)>M_{f}$.

  3. (3)

$n_{i+1}$ is the _smallest non-negative integer_ _greater than_ $n_{i}+2$ but
_less than_ $q$ and such that $f^{n_{i+1}-1}(0)<m_{f}$ and
$f^{n_{i+1}-2}(0)>M_{f}$, for all $i\in\\{1,2,\dots k-1\\}$.

We call $k$, the _mesh_ of the _unfolding set_ of $P$.

######  Lemma 4.3.

The number of distinct unfolding classes of $P$ equals the mesh of the
unfolding set of $P$.

###### Proof.

It is easy to see that each _exceptional class_ contains a _unique_
$n_{i}\in\mathcal{N}$ and each $n_{i}\in\mathcal{N}$ corresponds to a distinct
_exceptional class_. Hence, the result follows. ∎

######  Theorem 4.4.

If $P$ be a periodic orbit of _period_ $q$ of a $P$-linear map $f$ with mesh
of unfolding set $p$, then the unfolding number of $P$ is $\frac{p}{q}$.

###### Proof.

_Definition_ of $\mathcal{N}$ coupled with the _construction_ of the _maps_
$g_{f}$ and $F_{f}$ reveals that the _trajectory_ of $0$ under $g_{f}$ and
_trajectory_ of $0$ under $F_{f}$ are the _same_ till the first $n_{1}-2$
_iterates_ , that is, $g_{f}^{i}(0)=F_{f}^{i}(0)$, for
$i\in\\{0,1,2,\dots,n_{1}-2\\}$. Furthermore,
$g_{f}^{n_{1}-2}(0)=F_{f}^{n_{1}-2}(0)>M_{f}$. Consequently,
$F_{f}^{n_{1}-1}(0)>\frac{1}{2}$. Moreover, since, $g_{f}^{n_{1}-1}(0)<m_{f}$,
we must have: $2-m_{f}<F_{f}^{n_{1}-1}(0)<1$ and hence,
$F_{f}^{n_{1}}(0)\geqslant 1$. Again, $F_{f}^{i}(0)\in\left(1,2\right)$ for
$n_{1}<i<n_{2}-1$, $F_{f}^{n_{2}}(0)\geqslant 2$.

The above mentioned scheme persists with $F_{f}^{i}(0)\in(2,3)$ for
$n_{2}<i<n_{3}-1$, $F_{f}^{n_{3}}(0)>3$, and so forth, leading to
$F_{f}^{n_{p}}(0)\geqslant p$. We assert that $F_{f}^{q}(0)=0+p$. Indeed,
since $T_{f}(0)=P$ is a _periodic orbit_ of $f$ of period $q$, $T_{F_{f}}(0)$
is a _lifted periodic orbit_ of $0$ of period $q$. Thus, $F_{f}^{q}(0)=0+\ell$
for some $\ell\in\mathbb{N}$. It is evident that $\ell\geqslant p$. If
$\ell\geqslant p+1$, then there exists $m\in\mathbb{N},m>n_{p}$ such that
$F_{f}^{m}(0)>1$. However, this is feasible only if $m\in\mathcal{N}$. Thus,
there exists $m\in\mathcal{N}$ such that $m>n_{p}$ and $n_{p}$, leading to a
contradiction. Thus, $\ell=p$, establishing the result. ∎

Theorem 4.4 enables us to define _unfolding numbers_ in a manner reminiscent
of the traditional approach used for defining _rotation numbers_ , as
expounded in Section 1.3. We define an _observable_ $\Phi:P\to\mathbb{R}$ as
follows. For each $i\in\mathbb{N}$ define:

$\Phi(f^{i}(0))=\begin{cases}1&\text{if $i\in\mathcal{N}$}\\\ 0&\text{if
$i\notin\mathcal{N}$}\\\ \end{cases}$

Then the following easily follows from Theorem 4.4.

######  Corollary 4.5.

The unfolding number of $P$ is $\frac{1}{q}\sum_{i=1}^{q}\Phi(f^{i}(0))$ and
the unfolding pair of $P$ is: $(\sum_{i=1}^{q}\Phi(f^{i}(0)),q)$.

It now follows that if two _cycles_ have the _same cyclic permutation_ , then
they have the _same unfolding pairs_. We define the _unfolding number_ ,
_unfolding pair_ and _modified unfolding pair_ of a _pattern_ $\pi$ to be the
_unfolding pair_ , _unfolding pair_ and _modified unfolding pair_ of any
_cycle_ $P$ which exhibits $\pi$. We say that a _pattern_ $\pi$ _forces_ the
_unfolding interval_ $U_{\pi}$ if $U_{\pi}$ is the _unfolding interval_ of the
$P$-_linear map_ $f$ where $P$ _exhibits_ $\pi$.

######  Definition 4.6.

We call a cycle $P$ _divergent_ (or the _pattern $\pi$ it exhibits_) if there
are points $x,y,z$ of $P$ such that:

  1. (1)

$x<y<z$

  2. (2)

$f(x)<x$, $f(y)\geqslant z$ and $f(z)\leqslant x$.

where $f$ is the $P$-_linear map_. We call a _pattern_ $\pi$ _divergent_ if
any _cycle_ $P$ which _exhibits_ it is _divergent_.

It is well known that if a _pattern_ $\pi$ is _divergent_ then for any cycle
$P$ of this _pattern_ , the $P$-_linear map_ $f$ has a _horseshoe_ and has
_cycles_ of all possible _periods_. We show that in fact, $f$ has _cycles_ of
all possible _unfolding pairs_.

######  Theorem 4.7.

If $f$ has a divergent cycle, then the unfolding interval of $f$ is
$[0,\frac{1}{2}]$.

###### Proof.

Let $x,y,z$ be defined as in Definition 4.6. Then, there exists a _fixed
point_ $a$ between $x$ and $y$ and a _fixed point_ $b$ between $y$ and $z$.
Also, there exists a point $c$ between $a$ and $y$ such that $f(c)=b$. Let
$J=[a,c]$, $K_{1}=[c,y]$, $K_{2}=[y,b]$ and $L=[b,z]$. Then, the interval $J$
$f$-_covers_ the interval $J,K_{1}$ and $K_{2}$; the interval $K_{1}$
$f$-_covers_ $L$ ; the interval $K_{2}$ $f$-_covers_ $L$ and the interval $L$
$f$-_covers_ $J,K_{1},K_{2}$. Consider the _loop_ $\alpha:J\to J\to\dots J\to
K_{2}\to L\to J\to K_{1}\to L\to J\to K_{1}\to L\to J\to K_{1}\to L\to\dots
L\to J\to K_{1}\to L\to J$ consisting of $q-3p-3$ $J^{\prime}s$ followed by
$K_{2}$, then $p$ _blocks_ $L\to J\to K_{1}$, then the _block_ $L\to J$. Let
$Q$ be the _periodic orbit_ associated with $\alpha$. From the _location_ of
the _intervals_ : $J,K_{1},K_{2}$ and $L$, it is clear that the point of
_absolute minimum_ $m_{Q}$ of $Q$ lies in $L$ and the point of _absolute
maximum_ $M_{Q}$ of $Q$ lies in $K_{1}\cup K_{2}$. The interval $J$ lies to
the _left_ of $M_{Q}$ and the interval $L$ lies to the _right_ of $m_{Q}$ and
hence each of the _blocks_ $L\to J\to K_{1}$ in $\alpha$ _contains_ a distinct
_unfolding class_ and no other _blocks_ in $\alpha$ can contain an _unfolding
point_. It follows that that the number of _distinct unfolding classes_ of $P$
is $p$. Thus, from Lemma 4.3 and Theorem 4.4, the _unfolding number_ of $Q$ is
$\frac{p}{q}$ and the _unfolding pair_ of $Q$ is $(p,q)$. Since, $p$ and $q$
are arbitrary , $f$ has _cycles_ of all possible _unfolding pairs_ and hence
the result follows. ∎

A _cycle_ $P$ (or the _pattern_ $\pi$ it _exhibits_) is called _convergent_ if
it is not _divergent_. There is another equivalent way to define _convergent
pattern_. Namely, let $\mathcal{U}$ be the _family_ of all _interval maps_
with a _unique fixed point_. If $f$ is a $P$-linear map for a cycle $P$, then
$P$ is _convergent_ if and only if $f\in\mathcal{U}$. Thus, for studying _co-
existence_ of _cycles_ with different _unfolding pairs_ , it is sufficient to
study _convergent patterns_. Further as discussed in the beginning of Section
3, it is sufficient to consider the case $m_{\pi}>M_{\pi}$ where $m_{\pi}$ and
$M_{\pi}$ are the _points_ of _absolute minimum_ and _absolute maximum_ of
$\pi$.

We now portray the existence of _particular patterns_ , for whom _over-
rotation numbers_ coincides with the _unfolding numbers_. However, before
delving into this, we provide an example to illustrate that, in general,
_over-rotation numbers_ and _unfolding numbers_ are different entities.

######  Example 4.8.

Consider the _trimodal pattern_ characterised by the _permutation_
$\Pi=(1,2,5,7,10,3,6,8,9,4,11)$. It has _over-rotation number_ $\frac{3}{11}$,
but _unfolding number_ $\frac{5}{11}$.

Figure 3. A _trimodal pattern_ having _over-rotation number_ : $\frac{3}{11}$
but _unfolding number_ : $\frac{5}{11}$.

######  Definition 4.9.

A _pattern_ $\pi$ is called _sheer_ if it is _convergent_ and is _decreasing_
in $[M_{\pi},m_{\pi}]$.

######  Theorem 4.10.

For a sheer pattern $\pi$, over-rotation number of $\pi$= unfolding number of
$\pi$.

###### Proof.

Let $P$ be any _cycle_ which _exhibits_ $\pi$ and let $f\in\mathcal{U}$ be the
$P$-_linear_ map with _unique fixed point_ $a_{f}$. Let $m_{f}$ and $M_{f}$ be
points of _absolute minimum_ and _absolute maximum_ of $f$. We assume
$M_{f}<a_{f}<m_{f}$ and thus _points_ of $P$ to the _left_ of $a_{f}$ map to
the _right_ of _itself_ while _points_ of $P$ to the _right_ of $a_{f}$ map to
the _left_ of _itself_. Let the _over-rotation number_ of $P$ be
$\frac{p}{q}$, indicating that there are $p$ points to the _right of the fixed
point_ $a_{f}$ which are mapped to the _left_ of $a_{f}$. Let these $p$ points
be denoted as: $x_{1}>x_{2}>x_{3}>\dots>x_{p}$. It’s noteworthy that for each
$i\in\\{1,2,\dots p\\}$, $f^{2}(x_{i})$ is an _unfolding point_ , since for
all $i\in\\{1,2,\dots p\\}$, $x_{i}>M_{f}$ while $f(x_{i})<m_{f}$. Also, it is
easy to see that the _points_ $f^{2}(x_{i})$ belong to _different_ _unfolding
classes_ indicating that the number of different _unfolding classes_ is at-
least $p$.

We now exhibit that number of different _unfolding classes_ is precisely $p$.
By way of contradiction, let’s entertain the possibility of an _unfolding
point_ $\eta$ lying outside the union $U$ of _unfolding classes_ encompassing
the points $f^{2}(x_{i})$, $i\in\\{1,2,\dots p\\}$. Under this assumption we
can encounter _four_ potential _cases_ :

  1. (1)

$f(m_{f})\leqslant M_{f}\leqslant f^{-2}(\eta)\leqslant a_{f}\leqslant
f^{-1}(\eta)\leqslant m_{f}\leqslant f(M_{f})$.

  2. (2)

$f(m_{f})\leqslant M_{f}\leqslant f^{-2}(\eta)\leqslant f^{-1}(\eta)\leqslant
a_{f}\leqslant m_{f}\leqslant f(M_{f})$.

  3. (3)

$f(m_{f})\leqslant M_{f}\leqslant f^{-1}(\eta)\leqslant a_{f}\leqslant
f^{-2}(\eta)\leqslant m_{f}\leqslant f(M_{f})$.

  4. (4)

$f(m_{f})\leqslant M_{f}\leqslant a_{f}\leqslant f^{-1}(\eta)\leqslant
m_{f}\leqslant f^{-2}(\eta)\leqslant f(M_{f})$

We illustrate that all these _four cases_ leads us to a contradiction. Let’s
first scrutinize the _first_ _case_. If $\eta\leqslant a_{f}$, then $\eta\in
U$, contradicting the _initial assumption_ that $\eta\notin U$. Consequently,
$a_{f}\leqslant\eta\leqslant f^{-1}(\eta)$. But this means, $f$ first
_decreases_ in the interval $[M_{f},a_{f}]$, then _increases_ in the interval
$[a_{f},f^{-1}(\eta)]$ and then _decreases_ in $[f^{-1}(\eta),m_{f}]$ which
contradicts the fact that $\pi$ is a _sheer pattern_.

In the _second case_ , since $\eta\geqslant f^{-1}(\eta)$, $f$ first
_decreases_ in the interval $[M_{f},f^{-2}(\eta)]$, then _increases_ in the
interval $[f^{-2}(\eta),f^{-1}(\eta)]$, and finally _decreases_ in
$[f^{-1}(\eta),m_{f}]$, again contradicting the fact that $\pi$ is a _sheer
pattern_. The _third case_ implies $\eta\in U$, which is not feasible. Lastly,
the arguments for the _fourth case_ mirror those of the _first_. Thus, number
of different _unfolding classes_ is precisely $p$ and hence from Lemma 4.3 and
Theorem 4.4, _unfolding number_ of $P$ is $\frac{p}{q}$. ∎

Notice that every _unimodal_ or _bimodal pattern_ $\pi$ inherently qualifies
as a _sheer pattern_ , leading us to the subsequent corollary:

######  Corollary 4.11.

If $\pi$ has modality less than or equal to $2$, then over-rotation number of
$\pi$= unfolding number of $\pi$.

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# Point Cloud Network: An Order of Magnitude Improvement in Linear Layer
Parameter Count

###### Abstract

This paper introduces the Point Cloud Network (PCN) architecture, a novel
implementation of linear layers in deep learning networks, and provides
empirical evidence to advocate for its preference over the Multilayer
Perceptron (MLP) in linear layers. We train several models, including the
original AlexNet, using both MLP and PCN architectures for direct comparison
of linear layers (Krizhevsky et al., 2012). The key results collected are
model parameter count and top-1 test accuracy over the CIFAR-10 and CIFAR-100
datasets (Krizhevsky, 2009). AlexNet-PCN16, our PCN equivalent to AlexNet,
achieves comparable efficacy (test accuracy) to the original architecture with
a 99.5% reduction of parameters in its linear layers. All training is done on
cloud RTX 4090 GPUs, leveraging pytorch for model construction and training.
Code is provided for anyone to reproduce the trials from this paper.

aMaster of Data Science Student, University of Texas at Austin, Austin, TX,
USA

Keywords— Point cloud network, Low-rank factorization, Linear layer

##  1 Introduction

The Multilayer Perceptron is the simplest type of Artificial Neural Network
(ANN). Since its inception in the mid-20th century, it has held firmly as one
of the most popular structures in deep learning. MLPs were the first networks
used with backpropagation and are relied on heavily in the attention
mechanisms of the popular transformer architectures 11, 12.

Typically, networks that employ MLPs suffer from an extremely large parameter
count. This is because the amount of trainable parameters present in an MLP
scale by $O(n^{2})$ relative to the number of input features. A parameter
count so large that models have to be run across several GPUs because the
parameters alone cannot fit into just one. GPT-3 and GPT-4 are two well-known
models today, both of which rely on MLPs in their transformer architectures,
with GPT-3 holding 175 billion trainable parameters 1, 10.

AlexNet, widely regarded as the catalyst of the modern deep learning boom over
a decade ago, popularized the convolutional operation 8— the key feature of
the convolution being its reduction in parameter count in processing image
data 9.

Despite the value demonstrated by the parameter reduction present in
convolutional networks, MLPs are still prevalent simply because there is
currently no accessible alternative implementation of linear layers. The PCNs
presented in this paper cut the parameter count present in linear layers by an
order of magnitude, $O(n^{2})\rightarrow O(n)$, while still maintaining a
comparable efficacy to their equivalent MLP counterpart.

###  1.1 Related Work

Low-rank compression of ANNs is an emerging area of research which is closely
related to PCNs 3. Most work in this area relies on Singular Value
Decomposition and can be divided into one of two categories (1) finding a low-
rank factorization of a pre-trained network 3, 5, or (2) training a low-rank
network directly 6, 13. The latter is more closely related to a PCN.

###  1.2 Contribution

This work offers a rephrasing of the same problem that low-rank factorization
networks aim to solve. In low-rank factorization, we start with a weight
matrix, $W$, and look to find an optimal compression that maintains efficacy
3. The PCN starts with an already small set of parameters, and looks to find
an optimal expansion of those parameters that will perform with comparable
efficacy to $W$.

We outline a light-weight implementation of the PCN architecture that is
practical and generalizable to most existing deep learning networks, with
source code that makes it trivial to implement.

We also provide a set of key results that demonstrate that a PCN can
substantially reduce the number of parameters in linear layers while still
maintaining a comparable efficacy to an MLP.

##  2 Background– Multilayer Perceptron Architecture

Two terms commonly used in describing ANNs are neurons and weights. In MLPs,
neurons are the space where outputs from one layer and inputs to the next
layer may be found. The weights are the things in between neurons. They are
what processes information from one layer to the next. In most current deep
learning architectures this is where nearly all of the trainable parameters
can be found.

Let’s say we have two layers of neurons in a deep neural network, $l_{i}$, and
$l_{i+1}$, holding $n$ and $m$ neurons, respectively. $l_{i}$ takes input
array $x_{i}$ and processes that through $l_{i+1}$ into $x_{i+1}$. Between
these two layers there will be trainable parameters $W_{i}$, a matrix of size
$n\times m$. There is a also bias term, $b_{i+1}$, an array of size $m$. We
define the MLP forward function as,

$x_{i+1}=x_{i}\cdot W_{i}+b_{i+1}$

noting that this operation contains $O(mn)$ trainable parameters.

Figure 1: visual representation of MLP forward function

##  3 Point Cloud Network Architecture

In contrast to an MLP, the trainable parameters of a PCN are all neuron-
centric. What is learned are features of the neurons themselves, rather than
something in between. In an MLP, we would say that the bias term, $b$ is
neuron-centric, but not $W$ which contains a large majority of MLP parameters.

We will treat the features of neurons as positional information (i.e. each
neuron is a point in space, hence the name). The rest of this section explains
step-by-step how to use these neuron features to process input data in the
same way, and with the same expressiveness, as an MLP.

###  3.1 Distance Matrix

Going back to the prior example network— this time we’ll say $l_{i}$, and
$l_{i+1}$ are actually trainable parameters, where $l_{i}$ is of shape
$n\times d$ and $l_{i+1}$ is of shape $m\times d$. $d$ is a hyperparameter
representing the number of features each of our neurons have, or we can say
this is the dimensionality of the space our neurons exist in. $d$ is an
especially interesting hyperparameter because it allows us to scale up or down
the number of parameters in our network without affecting the number of
features in a given layer. We’ll also use bias term $b_{i+1}$ of size $m$
again.

The $W$ from an MLP is of shape $n\times m$. In this step, we can generate an
equally shaped distance matrix $D(l_{i},l_{i+1})$, where $D_{j,k}$ is the
distance between neurons $l_{i,j}$ and $l_{i+1,k}$.

$D_{j,k}(l_{i},l_{i+1})=\sqrt{\sum_{c=1}^{d}(l_{i,j,c}-l_{i+1,k,c})^{2}}$

The intention is to replace $W$ with $D$ as follows,

$x_{i+1}=x_{i}\cdot D(l_{i},l_{i+1})+b_{i+1}$

However, $D$ only contains nonnegative numbers, whereas
$W\in\mathbb{R}^{n\times m}$. Using $W$, a network can choose to flip and
scale the signal passed forward from one neuron to another, whereas using $D$,
a network can only scale signals. This would make our network using $D$
fundamentally less expressive than one using $W$. $D$ is also prone to
exploding/vanishing gradients.

In this work $D$ is given as the euclidean distance between $d$-dimensional
points, but $D_{j,k}$ has many possible implementations. The important feature
of $D$ is that it outputs an appropriately shaped matrix that facilitates
interaction between every neuron in $l_{i},$ with every neuron in $l_{i+1}$.
An example of an alternate implementation would be to omit the square root in
the definition above. Another example would be the product
$l_{i}l_{i+1}^{\top}$ giving $D$ a similar property to the multiplication of
keys and queries in transformers 12 or $UV^{\top}$ in low-rank factorization
5, 6, 13, 3. There likely exists a more optimal definition of $D$ than the one
defined here.

###  3.2 Distance-Weight-Function

The distance-weight-function, denoted here as $F$, is an element-wise function
to pass $D$ through. The goal of $F$ is to project $D$ into a space that makes
it as expressive as $W$ and to provide regularization properties. In this
paper the triangle wave is selected for $F$. Let $F_{\lambda,\epsilon}$ be an
element-wise triangle wave function centered around $0$ with amplitude
$\lambda$ and period $\epsilon$, with a regularization term included.

$F_{\lambda,\epsilon}(z)=\bm{\frac{1}{\sqrt{n}}}\cdot\frac{\lambda}{\epsilon}\cdot(\epsilon-|z\bmod{2\epsilon}-\epsilon|-\frac{\epsilon}{2})$

There is room for simplification, but the above equation is what is used in
this work. $\bm{\frac{1}{\sqrt{n}}}$ is selected as the regularization term in
order to maintain a stable signal moving forward through the network, agnostic
of layer size and network depth. This term was found through a trial-and-error
approach observing the variance of signal passed through untrained networks,
which can be found in the provided source code. A better regularization term
likely exists.

Selection of The Triangle Wave. The triangle wave is selected for two
desirable properties. Firstly, it takes any number $\in\mathbb{R}$ and clamps
it to the range $[-\lambda,\lambda]$. This provides important control over the
stability of our signal moving forward through the network, ensuring that no
weights are excessively large in magnitude, regardless of how much neurons may
explode away from, or implode into one another during the learning process.
This in turn allows for a steady flow of gradients during backpropagation.

The second property that is specific to the triangle wave is its constant
gradient and continuity— informed by the prevalence of the ReLU shape for
nonlinearities 8. Cos/sin have saddle points where gradients may get stuck.
Square waves’ gradients are flat and saw waves are discontinuous which may
lead to the network pushing or pulling a weight up or down the saw wave’s drop
off.

Figure 2: visual representation of PCN forward function

###  3.3 Forward Function

The PCN forward function is given as follows,

$x_{i+1}=x_{i}\cdot F_{\lambda,\epsilon}(D(l_{i},l_{i+1}))+b_{i+1}$

which has $O(n+m)$ trainable parameters, in contrast to the $O(nm)$ trainable
parameters in an MLP.

##  4 Training

This section details the model architectures implemented as well as the full
training process.

Techniques such as random image augmentation, batch normalization, or residual
connections are refrained from being used in favor of the direct comparison of
linear layer performance of MLPs and PCNs over achieving state of the art
(SOTA) performance. Additionally, a limited compute budget informs several
design choices seen in this section.

###  4.1 Model Definitions

A modest variety of model categories are trained to evaluate the PCNs
performance in different circumstances. For each model category there is a
single baseline model that uses MLPs and one or more equivalent PCN models.
All PCN models use hyperparameters $\lambda=1,\epsilon=0.1$.

Details about the shape and depth of each network can be found in figure 3.

Figure 3: Illustration of LinearNet, ConvNet, and AlexNet architectures. On
the left side of each network are the sizes of the signal passed forward
through the network.

####  4.1.1 LinearNet

Baseline Network. LinearNet-MLP consists of linear layers followed by ReLUs. A
linear layer with no ReLU is applied to produce the final output.

PCN Network. Four LinearNet-PCNd models are trained, each differing only in
their dimensionality ($d\in[4,8,16,32]$). Each PCN takes the baseline
definition and replaces all MLP layers with equally shaped PCN layers with no
further modification.

####  4.1.2 ConvNet

Baseline Network. ConvNet-MLP has a feature extractor network consisting of
convolutional layers followed by ReLUs. The feature extractor network is then
fed into the classifier network, consisting of linear layers followed by ReLUs
with a linear layer at the end.

PCN Network. ConvNet-PCN16 uses the same feature extractor as ConvNet-MLP. For
the classifier a PCN is used instead of an MLP for linear layers with no
further modifications.

####  4.1.3 AlexNet

Baseline Network. AlexNet-MLP is an untrained replica of the original model
with a single modification made to the last linear layer in order to output
the appropriate number of class predictions for each of CIFAR-10 and
CIFAR-100. Like the previous ConvNet, AlexNet also consists of a convolutional
feature extractor network followed by a linear classifier network. The
classifier network employs dropout=0.5 before each linear layer, besides an
isolated linear layer at the end 8.

PCN Network. AlexNet-PCN16 uses the same feature extractor as AlexNet-MLP. In
the classifier, MLPs are replaced with PCNs for linear layers. AlexNet-PCN16
also features dropout=0.5 layers as used in the original.

###  4.2 Datasets

CIFAR-10 and CIFAR-100 are two popular image classification datasets. Both are
labeled subsets of the tiny images dataset. CIFAR-10 consists of 60000 32x32
images divided into 10 classes, with 6000 images per class. The dataset is
split into 50000 training images and 10000 withheld test images with exactly
1000 images of each class in the test set. CIFAR-100 is the same as CIFAR-10
but with 600 images per class, and follows the same principal for train/test
split. The images and classes used in CIFAR-10 are mutually exclusive from
those in CIFAR-100 7.

The CIFAR datasets are chosen for benchmarks in order to strike balance
between task difficulty and compute required. The MNIST dataset is too easy to
solve— very small networks can achieve close to 100% test accuracy— so it is
difficult to extract conclusive results about a PCN’s efficacy in comparison
to an MLP on this dataset. ImageNet, the dataset AlexNet was originally
trained on, would require too much compute. The CIFAR datasets are
sufficiently difficult tasks, while also being small enough to train the
largest models in a reasonable amount of time given compute constraints.

Although MNIST is not used as a benchmark in this work, it was a valuable
resource in performing rapid preliminary testing of the PCN architecture. The
MNIST dataset was used in making all of the architecture and regularization
choices seen throughout this paper 2.

###  4.3 Preprocessing

For training/validation of LinearNet models, images are scaled down from 32x32
to 16x16, reducing the first linear layer’s input size from $3072\rightarrow
768$. Conversely, All images are scaled up to 227x227 for AlexNet models to
match the original paper 8.

###  4.4 Initialization

MLP and convolutional parameters use default initializations given by
torch.nn.Linear and torch.nn.Conv2d, respectively.

PCN neuron positional values are initialized uniformly over the range $[-1,1]$
and bias terms uniformly over the range $[-0.1,0.1]$.

###  4.5 Loss, Gradient, and Optimizers

Loss for all models are calculated using torch.nn.CrossEntropyLoss, and
parameter gradients are calculate using pytorch’s autograd feature.

MLP and convolutional parameters are updated with stochastic gradient descent
(SGD), via, torch.optim.SGD.

PCN parameters are updated with a slightly modified version of SGD that is
informed by layer size. Given layer size $n$, PCN parameters $l_{i}$, $b_{i}$,
gradients $\Delta l_{i}$, $\Delta b_{i}$, and learning-rate $\gamma$ we
perform a PCN’s SGD update as follows:

$\displaystyle l_{i}\coloneqq l_{i}-\gamma\Delta
l_{i}\bm{\frac{n}{\log_{2}n}}$ $\displaystyle b_{i}\coloneqq
l_{i}-\gamma\Delta b_{i}\bm{10^{5}}$

Both of the terms $\frac{n}{\log_{2}n}$ and $10^{5}$ are used in order to make
parameters throughout the network learn at close to the same rate, agnostic of
layer size. These values were selected in early tests by observing the
variance in gradients during the training process over a variety of network
shapes. This optimization strategy does not account for irregularities in
gradients resulting from network depth and artifacts of this fact may become
pronounced in the loss/accuracy curves when attempting to train deep PCN
networks, although residual connections may alleviate this problem 4. For the
purpose of the trials done in this paper, the above optimization strategy is
sufficient.

###  4.6 Training Loop Details

All models are trained with a batch size of 1024 and learning-rate of 0.0001,
for 3.5k epochs. For each training iteration, we aggregate the loss, and for
each epoch we aggregate both training accuracy and test accuracy, seen in
figure 5.

##  5 Results

This section presents the results of training all LinearNet, ConvNet, and
AlexNet architectures over the CIFAR-10 and CIFAR-100 datasets. Key results
are collected in table 1. Reported train/test accuracies are generated with
the the final models after training. During the training of ANNs, it is normal
for accuracies to fluctuate from epoch-to-epoch which introduces minor
variance into these results. Loss, training accuracy, and test accuracy curves
are collected in figure 5 of the appendix, which display more stable trends.

Discussion focuses on linear parameter count and test accuracy.

| CIFAR-10 | CIFAR-100
---|---|---
model |  # linear params (millions) |  top-1 acc. (train) |  top-1 acc. (test) |  top-1 acc. (train) |  top-1 acc. (test)
LinearNet-PCN4 |  0.024 |  48.4 |  46.0 |  23.5 |  20.9
LinearNet-PCN8 |  0.044 |  53.4 |  48.9 |  26.2 |  22.7
LinearNet-PCN16 |  0.083 |  61.4 |  53.0 |  31.9 |  26.1
LinearNet-PCN32 |  0.161 |  66.9 |  52.8 |  39.1 |  28.1
LinearNet-MLP |  3.957 |  96.8 |  52.1 |  78.7 |  25.2
ConvNet-PCN16 |  0.035 |  88.4 |  60.0 |  72.9 |  26.1
ConvNet-MLP |  1.06 |  98.9 |  58.1 |  99.8 |  27.3
AlexNet-PCN16 |  0.296 |  85.7 |  78.9 |  51.6 |  43.7
AlexNet-MLP |  54.575 |  84.1 |  78.6 |  52.5 |  47.5
Table 1: Train and test accuracies (%) over both CIFAR-10 and CIFAR-100
datasets for each model, along with the parameter counts of their linear
layers.

###  5.1 LinearNet

Four LinearNet-PCNd models and LinearNet-MLP are trained. For $d=4,8$ there is
a degradation in performance relative to the MLP. At $d=16,32$, the PCN
outperforms the MLP on both datasets. As $d$ increases, the PCNs experience
more overfitting. The MLP experiences substantially more overfitting than all
PCNs. LinearNet-PCN32, the largest PCN in this class of models, has 161k
parameters, which is a 95.9% reduction from 3.95 million parameters in the
MLP. Additionally, figure 5 displays a consistent increase in PCN performance
with an increase in $d$.

###  5.2 ConvNet

Both ConvNet-MLP and ConvNet-PCN16 have 5.35 million convolutional parameters.
ConvNet-PCN16 outperforms ConvNet-MLP by 1.9% on CIFAR-10 and underperforms by
1.2% on CIFAR-100. Similarly to LinearNet, ConvNet-MLP experiences more
overfitting than ConvNet-PCN16. ConvNet-PCN16 has 35k linear parameters, which
is a 96.7% reduction from 1.06 million linear parameters in the MLP.

###  5.3 AlexNet

Both AlexNet-MLP and AlexNet-PCN16 have 2.47 million convolutional parameters.
AlexNet-PCN16 outperforms AlexNet-MLP by 0.3% in CIFAR-10, and underperforms
by 3.8% in CIFAR-100. Both models experience similar amounts of overfitting.
AlexNet-PCN16 has 296k linear parameters, which is a 99.5% reduction from 54.6
million linear parameters in the MLP.

##  6 Limitations and Future Work

###  6.1 Memory Requirements

As has been demonstrated by this work, the PCN architecture can substantially
reduce the number of parameters needed to train linear layers. However, the
implementation seen here does not actually reduce the memory requirements.
This is due to my reliance on pytorch’s native autograd feature and
torch.cdist to find $D$. During the forward pass, $D$ in its entirety is
calculated and stored in memory, which is the same size as $W$. A fused kernel
function for calculating $x_{i+1,k}$ that never stores $D$ but instead
calculates $D_{j,k}$ as needed could be used.

$\sigma_{k}(x_{i})=b_{i+1,k}+\sum_{j=1}^{n}x_{i,j}\cdot
F_{\lambda,\epsilon}(D_{j,k}(l_{i,j},l_{i+1,k}))$

Successfully implementing this along with its corresponding gradient functions
on accelerated hardware would reduce memory consumption $O(n^{2})\rightarrow
O(n)$ during training and inference.

###  6.2 Compute Requirements

Two limiting factors of deep learning are memory and compute. The PCN
architecture can alleviate memory consumption, but requires $O(d)$ times more
compute than an MLP.

###  6.3 Network Stability

As has been stated previously, all regularization terms used in the PCNs
presented in this work were found through trial-and-error rather than rigorous
math. Because of this, these PCNs are not resilient to their hyperparameters
and a more robust PCN definition should be investigated.

###  6.4 Applying PCNs Elsewhere

In this work the PCN architecture is applied to linear layers. The same
concept can be applied to the convolutional layers along the channel axis, and
to graph layers along the node-feature axis.

Figure 4: On the left: loss distributed across $\bm{W}$. On the right: loss
distributed across $\bm{W^{*}}$, projected into the space of $\bm{W}$. Blank
space represents where $\bm{W^{*}}$ does not occur.

###  6.5 Conjecture— Why PCNs Work

The concept of a PCN can be boiled down to an MLP where we generate a
plausible $W$, similarly to low-rank factorization 5, 6, 13, 3. Let
$\bm{W}=\mathbb{R}^{n\times m}$ be the set of all possible values for $W$,
$\bm{W^{*}}\subseteq\bm{W}$ be the set of all possible values for $F(D)$, and
$\overline{L}_{\bm{W}}$ be the mean loss w.r.t. $\bm{W}$. If
$\overline{L}_{\bm{W^{*}}}=\overline{L}_{\bm{W}}$, then $F(D)$ should have a
comparable efficacy to $W$. Consequently, if
$\overline{L}_{\bm{W^{*}}}<\overline{L}_{\bm{W}}$ or
$\overline{L}_{\bm{W^{*}}}>\overline{L}_{\bm{W}}$, then $F(D)$ would be
expected to perform better or worse than $W$, respectively.

It may be interesting to investigate $F(D)$ that maximizes
$\overline{L}_{\bm{W}}-\overline{L}_{\bm{W^{*}}}$.

##  7 Ethical Concerns

With the exception of recent high profile publications, it seems a relatively
uncommon practice to include an ethics section in a deep learning paper like
this one. I use this section as a platform to attempt to mindfully outline
some of my concerns. I include this section to advocate for a culture within
academia that normalizes, legitimizes, and prioritizes this conversation—
hoping that a more organized practice forms.

Downstream Consequences. Deep Learning is a unique technology in that it is
largely task-agnostic. Because of this, the set of downstream applications is
uncharacteristically large compared to other technology. Although the PCNs
presented in this paper are applied to test datasets, the intention is to
integrate this into existing deep learning architectures for which there are
existing harmful applications. This makes it important to be cognizant of and
acknowledge these harmful applications.

Mindful Conversations. Having productive conversations about A.I. safety is a
bit paradoxical. It is surely helpful to be aware of potential negative
applications of deep learning, yet it may actually be harmful to indulge in
any unnecessary details that don’t move the conversation forward. For example,
I would consider media outlets echoing unproductive details about harmful
applications to be an unethical practice.

##  8 Acknowledgements

I would like to thank Ryan Schaake for offering fruitful comments, review, and
insight.

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##  Appendix A Supplemental Material

###  A.1 Source Code

Source code and other materials can be found at
https://gitlab.com/cHetterich/pcn-paper-and-materials.

###  A.2 Figures

Figure 5: A collection of training accuracy, test accuracy, and loss curves
for all models.