\documentclass{amsart}
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\begin{document}
\title{Chapter 7 Calculus of Attraction}
\begin{abstract} According to the last chapter, if $F$ is a
minimal threshold
transformation, then $\bar{\neg}F$ is PDNN-definable. Moreover,
an attractor of
$F$ is easily converted to that of $\bar{\neg}F$. In this
chapter, various
attractors of minimal threshold transformations are constructed
from one-to-one
transformations that we have already obtained in chapter 4. 
Further, given a
threshold transformation having an expected attractor, some
computational
processes of proving the attractiveness are developed. For that
purpose,
extended representations of self-dual transformations are
introduced from their [
]-representations. In some cases, attractiveness can be proved
by decomposition
of the transformation. 
\end{abstract}
\maketitle

\section*{7.1 Extended representations and neighborhood
functions}

We have already shown that there exist an attractive loop,
$3$-cycle, and $4$-cycle in our PDNNs. In this chapter we seek
the existence of more general attractors. Note that if $F$ is a
minimal threshold transformation, then $\bar{\neg}F$ is PDNN-
definable for spontaneous firing rate $1/2$ by Corollary 6.4.8 of
Chapter 6. Moreover, an attractor of $F$ is easily converted to
that of $\bar{\neg}F$ by Proposition 6.6.1 of Chapter 6. 
Therefore, we limit ourselves to minimal transformations. Note
that all these transformations and their attractors have their
isometrically similar counterparts.

Before we go further, we need a tool for systematic analysis
of attractiveness for various transformations. This tool is the
extended representation of a Boolean transformation.

Let $d_{SH}(x,S)$ be the \emph{signed Hamming distance}
between a point $x$ and a non-empty proper subset $S$
of $\mathbf{Q}^{n}$ defined by
$$
d_{SH}(x,S) = \left \{
\begin{array}{ll}
d_{H}(x,S) & \mbox{if $x \notin S$,}\\
1-d_{H}(x,S^{c}) & \mbox{if $x \in S$.}
\end{array}
\right.
$$

\textbf{Definition 7.1.1} Let $x$ be an element of
$\mathbf{Q}^{n}$ and $F = [f_{1},...,f_{n}]$. Then
the extended representation $F^{\#}$ of $F$ is a
function from $\mathbf{Q}^{n}$ to $\mathbf{Z}^{n}$ defined by
$$
(F^{\# }x)_{i} = \left \{
\begin{array}{ll}
 d_{SH}(P_{\mathbf{N}\backslash i}x,
P_{\mathbf{N}\backslash i}f_{i})& 
\mbox{if $x_{i} = 1$}\\ 
 d_{SH}(P_{\mathbf{N}\backslash i}x,
P_{\mathbf{N}\backslash i}\bar{\neg}f_{i}))&
\mbox{if $x_{i} = 0$.}
\end{array}
 \right.
$$
 Clearly $|(F^{\# }x)_{i}| \leq n - 1$ for every $i$ for
every $x$. In general,
$x \in f_{i}$ or $x \in \bar{\neg}f_{i}$, if and only if
$(F^{\# }x)_{i} \leq 0$. That is,
$$
x_{i} \ne (Fx)_{i} \quad \mbox{iff $(F^{\# }x)_{i}
\leq 0$.}
\eqno (7.1.1)
$$
For example, let
$$
f = p_{1}\cdot S_{4}\{ p_{2},p_{3},p_{4},\neg
p_{6},\neg p_{7},\neg p_{8}\} , 
$$
and $F = \langle f\rangle $ be a transformation of
$\mathbf{Q}^{8}$. Let $c =
11110000$. Then $F^{\# }c = (-2,0,2,4,-2,0,2,4)$. 
Therefore, $Fc = 00111100$.

By Proposition 2.4.3, we have
$$
Fk^{-} = [k^{-}f_{1},..,p_{k}\cdot (\neg
\bar{\neg}(f_{k}|1)),..,k^{-}f_{n}].
\eqno (7.1.2)
$$
Therefore, 
$$
(F^{\#}x)_{i} - 1 \leq ((Fk^{-})^{\#}x)_{i} \leq 
(F^{\#}x)_{i} + 1, \quad
\mbox{if $i \ne k$.}
\eqno (7.1.3)
$$
For $i = k$, if $|f_{k}| \leq 2^{n-2}$, then by Proposition
4.3.8, $(f_{k}|1) \subseteq \neg \bar{\neg}(f_{k}|1)$, so that
by (7.1.2),
$$
((Fk^{-})^{\#}x)_{k} \leq (F^{\#}x)_{k}, \quad \mbox{if
$|f_{k}| \leq 2^{n-2}$}.
\eqno (7.1.4)
$$

The following example is obtained as $\bar{\neg}G$ for (6.5.3)
through (6.5.6) of Chapter 6 and is a basis of our various
constructions.\\

\textbf{Example 7.1.2} Let $2 \leq k \leq [n/2]$,
$$
f = p_{1}\cdot S_{n-k}\{ p_{2},p_{3},..,p_{n}\} 
$$
and $F = \langle f\rangle $ be a transformation of
$\mathbf{Q}^{n}$. The $2$-cycle $C = (1...1,0...0)$ is the
unique attractor, and $U_{k-1}\mathbf{C}$ is the basin of
attraction.\\

\textbf{Theorem 7.1.3} Let $F = [f_{1},...,f_{n}]$ be a self-
dual minimal threshold transformation, and let $\gamma i =
d_{H}(f_{i}|1,\bar{\neg}(f_{i}|1))$. If $(F^{\#}x)_{i} \leq -1$
or $(F^{\#}x)_{i} \geq \mathrm{max}(2,\gamma i)$ for every $i$
for any point $x$ on a cycle of $F$, then that cycle is a strong
attractor of $F$.

\begin{proof} Assume $(F^{\#}x)_{i} \leq -1$ or $(F^{\#}x)_{i}
\geq \mathrm{max}(2,\gamma i)$ for every $i$ for any point $x$
on a cycle of $F$.

Let $(Fx)_{i} = \neg x_{i}$ for a point $x$ on the cycle,
say $x \in f_{i}$. Then $d_{SH}(x,f_{i}) = (F^{\#}x)_{i} \leq 
-1$. $((Fi^{-})^{\#}x)_{i} \leq (F^{\#}x)_{i} \leq -1$ by
(7.1.4). Therefore, $((Fi^{-})x)_{i} = \neg x_{i} = (Fx)_{i}$. 
Also, $((Fk^{-})^{\#}x)_{i} \leq 0$ for every $k \ne i$ by
(7.1.3), so that $((Fk^{-})x)_{i} = \neg x_{i} = (Fx)_{i}$.

Let $(Fx)_{i} = x_{i} = 1$. Then
$$
d_{SH}(P_{\mathbf{N}\backslash
i}x,P_{\mathbf{N}\backslash i}f_{i}) = (F^{\#}x)_{i} \geq 
d_{H}(P_{\mathbf{N}\backslash i}f_{i},P_{\mathbf{N}\backslash
i}(\bar{\neg}f_{i})),
$$
so that
$$
d_{SH}(P_{\mathbf{N}\backslash i}x,P_{\mathbf{N}\backslash
i}(p_{i}\cdot \bar{\neg}(f_{i}|1))) =
d_{SH}(P_{\mathbf{N}\backslash i}x,P_{\mathbf{N}\backslash
i}(\bar{\neg}f_{i})) \leq 0.
$$
Therefore, 
$$
((Fi^{-})^{\#}x)_{i} = d_{SH}(P_{\mathbf{N}\backslash
i}x,P_{\mathbf{N}\backslash i}(p_{i}\cdot \neg
\bar{\neg}(f_{i}|1))) \geq 1.
$$
Therefore, $((Fi^{-})x))_{i} = x_{i} = (Fx)_{i}$ by (7.1.2). 
Also, $(F^{\#}x)_{i} \geq 2$, so that $((Fk^{-})^{\#}x)_{i} \geq 
1$ for every $k \ne i$
by (7.1.3). Therefore, $((Fk^{-})x)_{i} = x_{i} = (Fx)_{i}$. 
Therefore,
$((Fk^{-})x)_{i} = (Fx)_{i}$ for every $i$ for every $k$. 
Therefore, the cycle is an strong attractor.
\end{proof} 

In Example 7.1.2, $F$ is minimal, and $(F^{\#}l)_{i} \leq -1$
for every $i$. $(l,o)$ is obviously a strong attractor but also
confirmed by Theorem 7.1.3.
 
Analogously to Example 7.1.2, we can construct
transformations having attractors by modifying some of the one-
to-one transformations listed in Chapter 4.4. First we give
the following general definition. Let $f$ be a
function from $\mathbf{Q}^{n}$ to $\mathbf{Q}$. Then we
identify the $\epsilon$-neighborhood $U_{\epsilon }f$
of the set $f$ with the function under which the inverse
image of $1$ is the set $U_{\epsilon }f$. That is,
$$
(U_{\epsilon }f)^{-1}1 =
U_{\epsilon }f = U_{\epsilon }(f^{-1}1).
$$
The function $U_{\epsilon }f$ is called a \emph{neighborhood
function} of $f$. Then clearly
$$
U_{\epsilon }(f\vee g) = (U_{\epsilon }f)
\vee (U_{\epsilon }g).
$$
For example, if $f$ is a one-term function,
$f = q_{k1}\cdot ...\cdot q_{km}$, where $(k1,..,km)$ is a
subsequence of $(1,2,..,n)$ and $q_{ki} = p_{ki}$ or
$\neg p_{ki}$, then
$$
U_{\epsilon }f = S_{m-\epsilon }\{q_{k1},..,q_{km}\} .
$$

We consider hereafter only the case $\epsilon = 1$ for
simplicity. Let $F = [f_{1},...,f_{n}]$ be a self-dual
transformation of $\mathbf{Q}^{n}$. Then let
$G = [g_{1},...,g_{n}]$ be the transformation defined by
$$
g_{i} = p_{i}\cdot U_{1}(f_{i}|1).
\eqno (7.1.5)
$$
Then
$$ 
(G^{\# }x)_{i} = (F^{\# }x)_{i} - 1 \quad \mbox{for
every $i$ for every $x$,}
\eqno (7.1.6)
$$
The following proposition immediately follows from (7.1.6).\\

\textbf{Proposition 7.1.4} A cycle of $F = [f_{1},...,f_{n}]$
is also a cycle of $G = [g_{1},...,g_{n}]$ defined by (3.5),
if and only if $(F^{\# }x)_{i} \ne 1$ for every $i$ for any
point $x$ on the cycle.
\begin{proof} $x_{i} \ne (Gx)_{i}$ iff $(G^{\# }x)_{i}
\leq 0$ by (7.1.1). $(G^{\# }x)_{i} = (F^{\# }x)_{i} - 1$
by (7.1.6). Therefore, $Gx = Fx$ iff $(F^{\# }x)_{i} \ne 1$
for every $i$.  
\end{proof}

Recall that a term of degree $m$ is a conjunction $f_{1}\cdot
f_{2}\cdot \cdot \cdot f_{m}$ of Boolean functions $f_{i}:
\mathbf{Q}^{n} \rightarrow \mathbf{Q}$, such that there exists
an injection $\varphi : \mathbf{N}_{m} \rightarrow \mathbf{N}$
such that $f_{i} = p_{\varphi i}$ or $\neg p_{\varphi i}$ for
each $i$. For a special case where $f$ consists of one term we
obtain the following theorem.\\

\textbf{Theorem 7.1.5} Let $F = [f_{1},...,f_{n}]$ be a self-dual
minimal threshold transformation, and let $f_{i}|1$ consist of
one term of degree $r_{i}$ for $r_{i} \geq 3$ for every $i$. 
Then, if $(F^{\#}x)_{i} \leq 0$ or $(F^{\#}x)_{i} = r_{i}$ for
every $i$ for any point $x$ on a cycle of $F$, then that cycle is
an strong attractor of $G = [g_{1},...,g_{n}]$ defined by
(7.1.5).

\begin{proof} Assume that $(F^{\#}x)_{i} \leq 0$ or
$(F^{\#}x)_{i} = r \geq 3$ for every $i$ for any point $x$ on a
cycle of $F$. Then that cycle is also a cycle of $G$ by
Proposition 7.1.4.

Let $x$ be a point on a cycle of $F$. Then, $(G^{\#}x)_{i} 
= -1$ or $(G^{\#}x)_{i} = r_{i}-1 \geq 2$ for any point $x$ on
the cycle. Further, we have
$$
\gamma i = d_{H}((g_{i}|1),\bar{\neg}(g_{i}|1)) = r_{i}-2
$$
Therefore, $(G^{\#}x)_{i} \leq -1$ or $(G^{\#}x)_{i} \geq 
\mathrm{max}(2,\gamma i)$, so that the cycle is a strong
attractor of $G$ by Theorem 7.1.3.
\end{proof}



\section*{7.2 Multipe attractors} 

In this section as in other parts, $1^{m}$ denotes the $m$-
vector whose every coordinate is $1$, and $0^{m}$ denotes the
$m$-vector whose every coordinate is $0$.\\ 

\textbf{Example 7.2.1} Let $n = 2m$ for $m \geq 2$, $f =
p_{1}\cdot p_{m}\cdot \neg p_{2m}$, and $F = \langle f\rangle $
be a transformation of $mathbf{Q}^{2m}$.
Let 
$$
\begin{array}{lll}
D &=& \{ x\mid x \in \mathbf{Q}^{2m}, \rho ^{m}x =
\bar{\neg}x\} \\
\Psi &=& \mathrm{Orb}_{\rho }D.
\end{array}
\eqno (7.2.1)
$$
Clearly $\rho D = D$, so that $\mathrm{Im}\Psi = D$.

Let $x \in D$. Then,
$$
(F^{\# }x)_{i} =\left\{
\begin{array}{ll}
0 & \mbox{if $x_{i} = \neg x_{i-1}$,} \\
2 & \mbox{if $x_{i} = x_{i-1}$,}
\end{array} 
\right. 
\eqno (7.2.2)
$$
for every $i$. Therefore, $Fx = \rho x$ for every $x \in D$.
Therefore, $\mathrm{Im}\Psi $ is an invariant set of $F$. We now
show that $\Psi $ is an attractor.

Let $x \in D$ and $k \in \mathbf{N}$. It suffices to show
$F(k^{-}x) \in D$. First, by (7.1.4),
$$
 ((Fk^{-})^{\#}x)_{k} \leq 0 \quad \mbox{if $x_{k} =
\neg x_{k-1}$.}
$$
If $x_{k} = x_{k-1}$, say $x_{k} = x_{k-1} = 1$, then $x_{k-1+m}
= 0$.
By (7.1.2), 
\begin{eqnarray*}
 (Fk^{-}) &=& [k^{-}f_{1},..,p_{k}\cdot (\neg
\bar{\neg}(f_{k}|1)),..,k^{-}f_{n}].\\
p_{k}\cdot (\neg \bar{\neg}(f_{k}|1)) &=& p_{k}\cdot \neg
\bar{\neg}(p_{k+m-1}\cdot \neg p_{k-1})\\
&=& p_{k}\cdot \neg (\neg p_{k+m-1}\cdot p_{k-1})\\
&=& p_{k}\cdot (p_{k+m-1}\vee \neg p_{k-1}).
\end{eqnarray*}
Therefore, 
$$
 ((Fk^{-})^{\#}x)_{k} = 1 \quad \mbox{if $x_{k} = x_{k-1}$.}
$$
Therefore, $(F(k^{-}x))_{k} = (Fx)_{k}$.

Let $k \ne i$. If $x_{i} = x_{i-1}$, then by (7.2.2) and
(7.1.3), 
$$
 ((Fk^{-})^{\#}x)_{i} \geq 1.
$$
Assume $x_{i} = \neg x_{i-1}$. Then, if $k \notin \{i-1,i+m-1\}$,
then
$$
 ((Fk^{-})^{\#}x)_{i} = (F^{\#}x)_{i}.
$$
Further, if $k \in \{i-1,i+m-1\}$, then
$$
 ((Fk^{-})^{\#}x)_{i} = 1, \quad \mbox{while
$(F^{\#}x)_{i} = 0$,}
$$
but also,
$$
 ((Fk^{-})^{\#}x)_{i+m} = 1, \quad
\mbox{while $(F^{\#}x)_{i+m} = 0$.}
$$
Therefore, $F(k^{-}x) \in D$. 

Therefore, $\Psi $ is a strong attractor. We have obtained
the following theorem.\\

\textbf{Proposition 7.2.2} $\Psi $ defined by (7.2.1) is a
strong attractor of $F$ in Example 7.2.1.

However, $\Psi $ is not a minimal attractor. Let $c =
l^{m}0^{m}$. Then $c \in D$. we now show that
$$
 C = \mathrm{Orb}_{\rho }c.
$$
is a (minimal) attractor. Let $x \in U_{1}\mathbf{C}$. It
suffices to consider $x = 1^{m}(k^{-}0^{m})$ for
$1 \leq k \leq m-1$. Then $Fx = \rho c$, so that
$Fx \in \mathbf{C}$.  Therefore, $C$ is a strong attractor.

\textbf{Example 7.2.3} Let $n = 2m$ for $m \geq 2$, $f =
p_{1}\cdot p_{m-1}\cdot \neg p_{2m-1}$, and
$F = \langle f\rangle $ be a transformation of $\mathbf{Q}^{2m}$.

In Example 7.2.3, computational results expect that there exists
an attractor corresponding to the above $\Psi $, but the
attractor and the orbits starting at its $1$-neighborhood are not
simple. Corresponding to the above $C$, we define

$$
c\prime = 1^{m}0^{m-1},\; c\prime\prime = 1^{m-1}0^{m},
\quad \Phi = \mathrm{Orb}_{\rho }\{ c\prime ,c\prime\prime \}.
\eqno (7.2.3)
$$
$Fc\prime = \rho c\prime\prime \in \mathrm{Im}\Phi $ and
$F^{2}c\prime = \rho c\prime \in \mathrm{Im}\Phi $, so that
$\mathrm{Im}\Phi $ is invariant. We will show that 
$CY(F|\Phi )$ is a (minimal) attractor of $F$. 
Let $x \in U_{1}(\mathrm{Im}\Phi )$. It suffices to consider 
$x = 1^{m}(k^{-}0^{m-1})$ for some $1 \leq k \leq m-2$. 
If $k = 1$ then $Fx = \rho c\prime $. If $1 < k \leq m-2$ then
$Fx = \rho c\prime\prime $. Therefore, 
$Fx \in \mathrm{Im}\Phi $, so that $CY(F|\mathrm{Im}\Phi )$
is an attractor. We have obtained the following proposition.\\


\textbf{Proposition 7.2.4} $\mathrm{CY}(F|\mathrm{Im}\Phi )$
is a minimal attractor in the FSDS generated by $F$ in Example
7.2.3, where $\Phi $ is defined by (7.2.3).\\


\textbf{Example 7.2.5} Let $n = 3m$ for $m \geq 2$ and
$$
f= p_{1}\cdot \neg p_{m}\cdot \neg p_{2m}\cdot \neg
p_{3m}. 
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{3m}$ defined by
$$
g = p_{1}\cdot S_{2}\{\neg p_{m},\neg p_{2m},\neg p_{3m}\}. 
$$
Let $x$ be a point such that $\rho ^{m}x = x$. Then
$$
(F^{\# }x)_{i} = \left\{
\begin{array}{ll}
0 & \mbox{if $x_{i} = \neg x_{i-1}$} \\
3 & \mbox{if $x_{i} = x_{i-1}$}. 
\end{array} 
\right.
$$
for every $i$. Therefore, $(Fx)_{i} = \neg x_{i}$ if and only if
$x_{i} = \neg x_{i-1}$, that is, $(Fx)_{i} = x_{i-1}$ for every
$i$, so that
$Fx = \rho x$. Therefore, $\mathrm{Orb}_{\rho }x$ is a cycle of
$F$. Further,
$\mathrm{Orb}_{\rho }x$ is also a strong attractor of $G$ by
Theorem 7.1.5.\\

\textbf{Example 7.2.6} Let $n = 4m$ for $m \geq 2$, and
$$
f= p_{1}\cdot p_{m}\cdot \neg p_{2m}\cdot p_{3m}\cdot
\neg p_{4m}. 
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{4m}$ defined by
$$
g = p_{1}\cdot S_{3}\{p_{m},\neg p_{2m},p_{3m},\neg
p_{4m}\}. 
$$
Let $x$ be a point such that $\rho ^{m}x = \bar{\neg}x$. Then
$$
(F^{\# }x)_{i} = \left\{
\begin{array}{ll}
0 & \mbox{if $x_{i} = \neg x_{i+m-1}$} \\
4 & \mbox{if $x_{i} = x_{i+m-1}$}. 
\end{array} 
\right. 
$$
for every $i$. Therefore, $(Fx)_{i} = x_{i-1}$ for every $i$, so
that $Fx = \rho x$, that is, $\mathrm{Orb}_{\rho }x$ is a cycle
of $F$. $\mathrm{Orb}_{\rho }x$ is also a strong attractor
of $G$ by Theorem 7.1.5.\\

\textbf{Example 7.2.7} Let $n = 4m$ for $m = 4$, and let
$$
f = 1\cdot 4\cdot \neg 5\cdot \neg 8\cdot 12\cdot \neg
13\cdot \neg 16. 
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{n}$ defined by
$$
g = 1\cdot S_{5}\{ 4,\neg 5,\neg 8,12,\neg 13,\neg 16\}
. 
$$
Let $a = (11110000)^{2}$ and $b = (11010010)^{2}$. Then
\begin{eqnarray*}
F^{\# }a &=& (0,2,2,2,0,2,2,2,0,2,2,2,0,2,2,2),\\
F^{\# }b &=& (0,2,0.0,0,2,0,0,0,2,0.0,0,2,0,0).
\end{eqnarray*}
Therefore, $\mathrm{Orb}_{\rho }a$ and $\mathrm{Orb}_{\rho }b$
are non-loop cycles
of $F$. Also, $\mathrm{Orb}_{\rho }a$ and $\mathrm{Orb}_{\rho
}b$ are non-loop cycles of $G$ by Proposition 7.2.2. Further
calculation
proves that $\mathrm{Orb}_{\rho }a$ and $\mathrm{Orb}_{\rho }b$
are attractors of $G$.\\


\section*{7.3 Multi-cycle attractors}

In this section and later in this chapter, $[a]$ for an
element $a$ in $\mathbf{Q}^{n}$ denotes the orbit of $\langle
\bar{\neg},\rho \rangle $ containing $a$ if the operating
transformation is self-dual and circular; $[A]$ for a subset $A$
of $\mathbf{Q}^{n}$ denotes the union of the orbits of $\langle
\bar{\neg}, \rho \rangle $containing $a \in A$. Similarly, $[a]$
and $[A]$ respectively denote the orbit of$\langle \rho
n^{-}\rangle $ containing $a$ andthe union of orbits of $\langle
\rho n-\rangle $ for $a \in A$, if the operating transformation
is skew-circular. Note that $\langle \bar{\neg},\rho
n^{-}\rangle = \langle \rho n^{-}\rangle $, since $(\rho
n^{-})^{n} = \bar{\neg}$.
Then, a transformation $F^{\sim }$ of the orbit set
$ \{ [x] \mid x \in \mathbf{Q}^{n}\}$ is naturally defined
by $F^{\sim }[x] = [Fx]$. \\

The condition in Theorem 7.1.3 is very strong. In fact,
transformations have attractors under weaker conditions as shown
in the following Theorems. As in Section 7.2, $G = \langle
g\rangle $ is defined by (7.1.5) from a transformation $F =
\langle f\rangle $ of $\mathbf{Q}^{n}$.

First we consider a circular transformation belonging to the
class determined by Theorem 4.4.3. Specifically, for any $h-1$
relatively prime with odd $n$ and $0 < h-1 < n$, there exists $F
= \langle f\rangle $ of $\mathbf{Q}^{n}$,
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},
$$
such that
$$
F = \bar{\neg}\rho ^{h-1} \quad \mbox{on
$\mathrm{Car}F$.}
$$
Let $f = \{c\}$. Then, $C = \mathrm{Orb}_{\bar{\neg}\rho h-1}c$
is a cycle of $F$.

Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{n}$ defined by
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}. 
$$
Referring to (4.4.3), we have
\begin{eqnarray*}
(F^{\#}c)_{i} &=& 0 \quad \mbox{iff $i = 1$;}\\ 
(F^{\#}c)_{i} &=& 1 \quad \mbox{iff $i = h$, since $1^{-}c =
\bar{\neg}\rho ^{h-1}c$;}\\
(F^{\#}c)_{i} &=& 2 \quad \mbox{iff $i = 2h-1$;}\\
(F^{\#}c)_{i} &\geq & 3 \quad \mbox{for every other $i$.}
\end{eqnarray*}
Therefore, by (7.1.6),
$$
\begin{array}{l}
(G^{\#}c)_{1} = -1, \qquad (G^{\#}c)_{h} = 0, \qquad
(G^{\#}c)_{2h-1} = 1,\\ 
(G^{\#}c)_{i} \geq 2 \quad \mbox{for every other $i$.}
\end{array}
$$
In particular,
$$
Gc = \{1,h\}^{-}c = \rho ^{2(h-1)}c.
\eqno (7.3.1)
$$ 
Let $k \ne 1$. Then if $k \ne h$, then
\begin{eqnarray*}
((Gk^{-})^{\#}c)_{1} &\leq & 0 \quad \mbox{by (7.1.3);}\\ 
((Gk^{-})^{\#}c)_{h} &=& 1, \quad \mbox{since $(\bar{\neg}\rho
)^{h-1}c = 1^{-}c$ and $k \ne 1$;}\\
((Gk^{-})^{\#}c)_{2h-1} &=& 2, \quad \mbox{since
$(\bar{\neg}\rho )^{2h-2}c = \{1,h\}^{-}c$ and $k \notin 
\{1,h\}$;}\\
((Gk^{-})^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other $i$
such that $i \ne k$ by (7.1.3).}
\end{eqnarray*}
For $k = h$, 
\begin{eqnarray*}
((Gh^{-})^{\#}c)_{1} &\leq & 0;\\ 
((Gh^{-})^{\#}c)_{h} &\leq & (G^{\#}c)_{h} = 0 \quad
\mbox{by (7.1.4);}\\
((Gh^{-})^{\#}c)_{2h-1} &=& 0, \quad \mbox{since
$(\bar{\neg}\rho )^{2h-2}c = \{1,h\}^{-}c$ and $h \in 
\{1,h\}$;}\\
((Gh^{-})^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other
$i$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
G(k^{-}c) = \bar{\neg}\rho ^{h-1}c,
$$
or
$$
G(k^{-}c) = \bar{\neg}\rho ^{h-1}((\rho ^{-(h-1)}k)^{-}c),
$$
or
\begin{eqnarray*}
G(k^{-}c) &=& \bar{\neg}\rho ^{h-1}
((\rho ^{-(h-1)}\{h,2h-1\})^{-}c)\\ 
&=& \{h,2h-1\}^{-}(\bar{\neg}\rho ^{h-1}c)\\
&=& \{1,h,2h-1\}^{-}c\\
&=& \rho ^{3(h-1)}c.
\end{eqnarray*}
Therefore, under $G^{\sim }$, 
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [c],
$$
or
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [1^{-}c].
$$
On the other hand, by (7.3.1),
\begin{eqnarray*}
G(1^{-}c) &=& G(\bar{\neg}\rho ^{h-1}c)\\
& =& \bar{\neg}\rho ^{h-1}Gc\\
&=& \bar{\neg}\rho ^{h-1}\rho ^{2(h-1)}c\\
& =& \bar{\neg}\rho ^{3(h-1)}c.
\end{eqnarray*}
Thus we obtained:\\ 

\textbf{Theorem 7.3.1} For any $h-1$ relatively prime with odd
$n$ and $0 < h-1 < n$, let $F = \langle f\rangle $ of
$\mathbf{Q}^{n}$,
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},
$$
be a transformation determined by Theorem 4.4.3. Let 
$G = \langle g\rangle $ be defined by
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}.
$$
Let $f = \{c\}$ and 
$$
\Phi = \mathrm{Orb}_{\rho 2(h-1)}\{c,\bar{\neg}\rho
^{h-1}c\}.
$$
Then $\Phi $ is an attractor of $G$.\\

We also get a double-cycle attarctor, if we take $F$ from the
class described in Theorem 4.5.3. Specifically, for any $h-1$
relatively prime with $2n$ and $0 < h-1 < 2n$, there exists $F =
\langle\langle f\rangle\rangle $ of $\mathbf{Q}^{n},$
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},
$$
such that
$$
F = (\rho n^{-})^{h-1} \quad \mbox{on $\mathrm{Car}F$.}
$$
Let $f = \{c\}$. Then, $C = \mathrm{Orb}_{(\rho n-)h-1}c$
is a cycle of $F$.

Then we get the transformation $G = \langle\langle
g\rangle\rangle $ of $\mathbf{Q}^{n}$ defined by
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}. 
$$
Referring to (4.5.3), we have
\begin{eqnarray*}
(F^{\#}c)_{i} &=& 0 \quad \mbox{iff $i = 1$;}\\
(F^{\#}c)_{i} &=& 1 \quad \mbox{iff $i = h$, since
$(\rho n^{-})^{h-1}c = 1^{-}c$;}\\
(F^{\#}c)_{i} &=& 2 \quad \mbox{iff $i = 1+2(h-1) = 2h-1$;}\\
(F^{\#}c)_{i} &\geq & 3 \quad \mbox{for every other $i$}.
\end{eqnarray*}
Therefore, by (7.1.6),
$$
\begin{array}{l}
(G^{\#}c)_{1} = -1,\qquad (G^{\#}c)_{h} = 0,\qquad
(G^{\#}c)_{2h-1} = 1,\\ 
(G^{\#}c)_{i} \geq 2 \quad \mbox{for every other $i$.}

\end{array}
$$
In particular,
$$
Gc = \{1,h\}^{-}c = (\rho n^{-})^{2(h-1)}c.
\eqno (7.3.2)
$$
Let $k \ne 1$. Then if $k \ne h$, then
\begin{eqnarray*}
((Gk^{-})^{\#}c)_{1} &\leq & 0 \quad \mbox{by (7.1.3);}\\ 
((Gk^{-})^{\#}c)_{h} &=& 1, \quad \mbox{since 
$(\rho n^{-})^{h-1}c = 1^{-}c$ and $k \ne 1$;}\\
((Gk^{-})^{\#}c)_{2h-1} &=& 2, \quad \mbox{since
$(\rho n^{-})^{2h-2}c = \{1,h\}^{-}c$ and $k \notin \{1,h\}$;}\\ 
((Gk^{-})^{\#}c)_{i} &\geq& 1 \quad \mbox{for every other $i$
such that $i \ne k$ by (7.1.3).}
\end{eqnarray*}
For $k = h$, 
\begin{eqnarray*}
((Gh^{-})^{\#}c)_{1} &\leq & 0 \quad \mbox{by (7.1.3);}\\ 
((Gh^{-})^{\#}c)_{h} &\leq & (G^{\#}c)_{h} = 0 \quad
\mbox{by (7.1.4);}\\
((Gh^{-})^{\#}c)_{2h-1} &=& 0, \quad \mbox{since
$(\rho n^{-})^{2h-2}c = \{1,h\}^{-}c$ and $h \in \{1,h\}$;}\\
((Gh^{-})^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other
$i$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
G(k^{-}c) = (\rho n^{-})^{h-1}c,
$$
or
$$
G(k^{-}c) = (\rho n^{-})^{h-1}((\rho ^{-(h-1)}k)^{-}c),
$$
or
\begin{eqnarray*}
G(k^{-}c) &=& (\rho n^{-})^{h-1}((\rho ^{-(h-1)}
\{h,2h-1\})^{-}c)\\ 
&=& \rho ^{h-1}\rho ^{-(h-1)}\{h,2h-1\})^{-}
(\rho n^{-})^{2(h-1)}c\\
&=& \{1,h,2h-1\}^{-}c\\
&=& (\rho n^{-})^{3(h-1)}c.
\end{eqnarray*}
Therefore, under $G^{\sim }$, 
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [c],
$$
or
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [1^{-}c].
$$
On the other hand, by (7.3.2)
\begin{eqnarray*}
G(1^{-}c) &=& G((\rho n^{-})^{(h-1)}c)\\
&=& (\rho n^{-})^{(h-1)}Gc\\
&=& (\rho n^{-})^{(h-1)}(\rho n^{-})^{2(h-1)}c\\
& =& (\rho n^{-})^{3(h-1)}c 
\end{eqnarray*}
Thus we obtained:\\ 

\textbf{Theorem 7.3.2} For any $h-1$ relatively prime with $2n$
and $0 < h-1 < 2n$, let $F = \langle\langle f\rangle\rangle $ of
$\mathbf{Q}^{n}$,
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot
\alpha _{n}p_{n},
$$
be a transformation determined by Theorem 4.5.3.
Let $G = \langle g\rangle $ be defined by
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}.
$$
Let $f = \{c\}$ and
$$ 
\Phi = \mathrm{Orb}_{(\rho n-)2(h-1)}
     \{c,(\rho n^{-})^{h-1}c\}.
$$
Then $\Phi $ is an attractor of $G$.\\

\textbf{Example 7.3.3} Let $n = 2m$ for $m \geq 2$,
and $F = \langle f\rangle $ for
$$
f = p_{1}\cdot S_{m-2}\{p_{2},p_{3},..,p_{m}\}\cdot
\neg p_{2m}.
$$

\textbf{Proposition 7.3.4} Let $a = 1^{m+1}0^{m-1}$,
$b = 1^{m-1}0^{m+1}$, and $c = 1^{m}0^{m}$. Then,
$\Phi = \mathrm{Orb}_{\rho }\{a,b,c\}$ is an attractor of
$F$ in Example 7.3.3.

\begin{proof} Note that $[a] = [b]$ and
$\bar{\neg}\mathrm{Im}\Phi = \mathrm{Im}\Phi $. we have
\begin{eqnarray*} 
F^{\#}a &=& (0,1,1,2,3,..,m-1,0,2,3,..,m-1),\\
F^{\#}b &=& (0,2,3,..,m-1,0,1,1,2,3,..,m-1),\\
F^{\#}c &=& (0,1,2,..,m-1,0,1,2,..,m-1).
\end{eqnarray*}
Therefore, $\mathrm{Orb}_{\rho }a$, $\mathrm{Orb}_{\rho }b$,
and $\mathrm{Orb}_{\rho }c$ are cycles of $F$.

Let $2 \leq k \leq m-1$. Then
\begin{eqnarray*}
((Fk^{-1})^{\#}c)_{1} &=& 0, \\
((Fk^{-})^{\#}c)_{2} &=& 2 \quad \mbox{if $k \ne 2$,}\\
((Fk^{-})^{\#}c)_{m+1} &=& 0,\\
(Fk^{-})^{\#}c)_{m+2} &=& 1,\\
((Fk^{-})^{\#}c))_{i} &\geq & 1 \quad \mbox{for every
other $i$ such that $i \ne k$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
k^{-}c \rightarrow \rho ((k-1)^{-}c) \quad \mbox{or}
\quad k^{-}c \rightarrow \rho c.
$$
Also $1^{-}c = \rho b$. Therefore,
$$
[k^{-}c] \rightarrow ...\rightarrow [a] \quad
\mbox{or} \quad [k^{-}c] \rightarrow ...\rightarrow [c].
\eqno (7.3.3)
$$
Since $[k^{-}c] = [(k-m)^{-}c]$, (7.3.3) is also true for
$m+2 \leq k \leq 2m-1$.

Let $3 \leq k \leq m$. Then
\begin{eqnarray*}
((Fk^{-})^{\#}a)_{1} &=& 0,\\
((Fk^{-})^{\#}a))_{2} &=& 1,\\
((Fk^{-})^{\#}a)_{3} &=& 1,\\
((Fk^{-})^{\#}a)_{m+2} &=& 0.\\
((Fk^{-})^{\#}a)_{i} &\geq & 1 \quad \mbox{for every other $i$
such that $i \ne k$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
k^{-}a \rightarrow \rho ((k-1)^{-}a) \quad \mbox{or}
\quad k^{-}a \rightarrow \rho a.
$$
\begin{eqnarray*}
((F2^{-})^{\#}a)_{1} &=& 0\\
(F^{\#}(2^{-}a))_{2} &=& m-2,\\
(F^{\#}(2^{-}a))_{3} &=& 0,\\
(F^{\#}(2^{-}a))_{m+2} &=& 0.\\
(F^{\#}(2^{-}a))_{i} &\geq & 1 \quad \mbox{for every
other $i$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
2^{-}a \rightarrow \rho ^{3}b.
$$
Therefore, for $2 \leq k \leq m$,
$$
[k^{-}a] \rightarrow ...\rightarrow [a],
$$
and $(m+1)^{-}a = c.$ 
\begin{eqnarray*}
(F((m+2)^{-})^{\#}a)_{1} &=& 0,\\
(F((m+2)^{-})^{\#}a)_{2} &=& 1,\\
(F((m+2)^{-})^{\#}a)_{3} &=& 1.\\
(F((m+2)^{-})^{\#}a)_{m+2} &\leq & (F^{\#}a)_{m+2} = 0
\quad \mbox{by (7.1.4).}\\
(F((m+2)^{-})^{\#}a)_{i} &\geq & 1 \quad
\mbox{for every other $i$ by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
(m+2)^{-}a \rightarrow \rho a.
$$
Let $m+3 \leq k \leq 2m-1$. Then
\begin{eqnarray*}
((Fk^{-})^{\#}a)_{1} &=& 0,\\
((Fk^{-})^{\#}a)_{2} &=& 1,\\
((Fk^{-})^{\#}a)_{3} &=& 2,\\
((Fk^{-})^{\#}a)_{m+2} &=& 1.\\
((Fk^{-})^{\#}a)_{i} &\geq & 1 \quad \mbox{for every
other $i \ne k$.}
\end{eqnarray*}
Therefore,
$$
k^{-}a \rightarrow \rho ((k-1)^{-}c) \quad \mbox{or}
\quad k^{-}a \rightarrow \rho c.
$$
Therefore, by (7.3.3),
$$
k^{-}a \rightarrow ... \rightarrow [a] \quad
\mbox{or} \quad k^{-}a \rightarrow ... \rightarrow [c].
$$
Therefore, if $x \in U_{1}\mathrm{Im}\Phi $ then
$x \in U_{1}\mathrm{Im}\Phi $ and
$\omega _{F}x \in \mathrm{Im}\Phi $. 
Therefore, $\Phi $ is an attractor of $F$.
\end{proof}

\textbf{Example 7.3.5} Let $m \geq 4$, and let
$$
f = p_{1}\cdot p_{2}\cdot p_{3}\cdot ..\cdot p_{m-
1}\cdot \neg p_{m+2}\cdot ..\cdot \neg p_{2m-1}. 
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{2m}$ defined by
$$
g = p_{1}\cdot S_{2m-5}\{ p_{2},p_{3},..,p_{m-1},
\neg p_{m+2},..,\neg p_{2m-1}\} . 
$$
Let $c = 1^{m}0^{m}$. Then,
$$
F^{\# }c = (0,0,2,4,..,2m-4,0,0,2,4,..,2m-4).
$$
Therefore, $Fc = \rho ^{2}c$, so that $\mathrm{Orb}\rho _{2}c$
is a cycle of $F$. Further, $\mathrm{Orb}\rho _{2}c$ is a cycle
of $G$ by Proposition 7.1.4.
Let 
$$
\Phi = \mathrm{Orb}_{\rho _{2}}\{ c,\rho c\} .
$$
We have
\begin{eqnarray*}
\neg \bar{\neg}(f|1) &=& \neg (\neg p_{2}\cdot \neg
p_{3}\cdot ..\cdot \neg p_{m-1}\cdot p_{m+2}\cdot ..\cdot
p_{2m-1})\\
&=& p_{2}\vee p_{3}\vee ..\vee p_{m-1}\vee \neg
p_{m+2}\vee ..\vee \neg p_{2m-1}.\\
(F1^{-})^{\# }c &=& (-,0,1,..,2m-5,0,0,1,3,..,2m-5),\\
(F2^{-})^{\# }c &=& (1,-,2,3,..,2m-5,1,0,2,3,..,2m-5),\\
(F3^{-})^{\# }c &=& (1,1,-,5,..,2m-5,1,1,2,4,5,..,2m-5),\\
&&...\\
(F(m-1)^{-})^{\# }c &=& (1,1,3,..,2m-7,-,2m-4,1,1,3,..,
2m-7,2m-6,2m-4).
\end{eqnarray*}
Therefore, by (7.1.6)
\begin{eqnarray*}
(G1^{-})^{\# }c &=& (-,-1,0,..,2m-6,-1,-1,0,..,2m-6),\\
(G2^{-})^{\# }c &=& (0,-,1,2,..,2m-6,0,-1,1,2,..,2m-6),\\
(G3^{-})^{\# }c &=& (0,0,-,4,..,2m-6,0,0,1,3,4,..,2m-6),\\
&&...\\
(G(m-1)^{-})^{\# }c &=& (0,0,2,..,2m-8,-,2m-5,0,0,2,..,2m-8,
2m-7,2m-5).
\end{eqnarray*}
Therefore, under $G$,
\begin{eqnarray*}
1^{-}c &\rightarrow & \rho ^{3}c,\\
2^{-}c &\rightarrow & \rho ^{2}c,\\
3^{-}c &\rightarrow & \rho ^{2}(1^{-}c),\\
&&. . . .\\
(m-1)^{-}c &\rightarrow & \rho ^{2}((m-3)^{-}c),\\
m^{-}c &=& \rho ^{-1}(1^{-}c) = \rho ^{-1}(\rho ^{3}c)
= \rho ^{2}c. 
\end{eqnarray*} 
Therefore, if $x \in U_{1}\mathrm{Im}\Phi $ then $x \in 
U_{1}\mathrm{Im}\Phi $ and $\omega _{G}x = \mathrm{Im}\Phi $.
Therefore, $\Phi $ is an attractor of $G$.


\textbf{Example 7.3.6} Let $m \geq 2$, and consider a special
case of $G = F^{2}$ for transformations described in Theorem
4.4.3, where $G = \langle g\rangle $,
$$
g = p_{1}\cdot p_{2}\cdot \cdot \cdot p_{m}\cdot \neg
p_{m+2}\cdot \cdot \cdot \neg p_{2m+1}.
$$
From $G$ we construct $H = \langle h\rangle $, 
$$
h = p_{1}\cdot S_{2m-2}\{p_{2},p_{3},..,p_{m},\neg
p_{m+2},..,\neg p_{2m+1}\}, 
$$
and $H = \langle h\rangle $ be a transformation of
$\mathbf{Q}^{2m+1}$.

Let
\begin{eqnarray*}
h^{(2)} &=& 1\cdot \neg 2\cdot 3\cdot ..\cdot (m+1)\cdot \neg
(m+2)\cdot ..\cdot \neg (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot m\cdot \neg (m+1)\cdot
(m+2)\cdot \neg (m+3)\cdot ..\cdot \neg (2m+1),\\
..&&...\\
h^{(m)} &=& 1\cdot 2\cdot ..\cdot (m-1)\cdot \neg m\cdot
(m+1)\cdot \neg (m+2)..\cdot \neg (2m-1)\\
&&\vee 1\cdot 2\cdot ..\cdot m\cdot \neg (m+1)\cdot
..\cdot \neg (2m-1)\cdot 2m\cdot \neg (2m+1),\\
h^{(m+1)} &=& 1\cdot 2\cdot ..\cdot m\cdot \neg (m+1)..\cdot
\neg (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot (m+1)\cdot \neg
(m+2)\cdot ..\cdot \neg (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot m\cdot \neg (m+1)\cdot
..\cdot \neg 2m\cdot (2m+1),\\
h^{(m+2) }&=& 1\cdot 2\cdot ..\cdot (m+2)\cdot \neg
(m+3)\cdot ..\cdot \neg (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot (m+1)\cdot \neg
(m+2)\cdot ..\neg (2m)\cdot (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot (m-1)\cdot \neg m\cdot
..\cdot (2m+1),\\
h^{(m+3)} &=& 1\cdot 2\cdot ..\cdot (m+1)\cdot \neg
(m+2)\cdot (m+3)\cdot \neg (m+4)\cdot ..\cdot \neg (2m+1)\\
&&\vee 1\cdot \neg 2\cdot 3\cdot ..\cdot m\cdot \neg
(m+1)\cdot ..\cdot \neg (2m+1),\\
..&&...\\
h^{(2m)} &=& 1\cdot 2\cdot ..\cdot (m+1)\cdot \neg
(m+2)\cdot ..\cdot \neg (2m-1)\cdot (2m)\cdot \neg (2m+1)\\
&&\vee 1\cdot 2\cdot ..\cdot (m-2)\cdot
\neg (m-1)\cdot m\cdot \neg (m+1)\cdot ..\cdot \neg (2m-1).
\end{eqnarray*}
Then 
$$
h = h^{(2)} \vee . . . \vee h^{(2m)},
$$
and
$$
h^{(i)}\cdot h^{(k)} = 0 \quad \mbox{for every $i \ne k$.}
$$
Let $H^{(i)} = \langle h^{(i)}\rangle $. Let $c = 1^{m+1}0^{m}$. 
Then
$$
H = H^{(2)} + . . . + H^{(2m)},
$$
and a flow graph of $H$ is
$$
\begin{array}{ccccccccc}
\, [h^{(2m)}] &\rightarrow & [h^{(2m-1)}]&\rightarrow &..&
\rightarrow & [h^{(m+1)}]& =& [c]\partial .\\
& & & & & & \uparrow& & \\
\, [h^{(m)}] &\rightarrow & [h^{(m-1)}]& \rightarrow &..&
\rightarrow & [h^{(2)}]& &
\end{array}
$$
Further, $Hc = \rho ^{-(m-1)}\bar{\neg}c$, so that 
$H^{2}c = \rho ^{-2(m-1)}c$. We have\\ 
$\mathrm{gcd}(2m+1,2(m-1)) = \mathrm{gcd}(2(m-1),3) =
\mathrm{gcd}(m-1,3)$. If $m-1$ is a multiple of 3, then let
$$
\Phi = \mathrm{Orb}_{\rho^{-(m-2)}\bar{\neg}}\{ c,
\rho c, \rho ^{2}c\} ;
$$
otherwise, let
$$
C = \mathrm{Orb}_{\rho^{-(m-2)}\bar{\neg}}c.
$$
Since $\mathrm{Im}\Phi = [c]$,$\mathbf{C} = [c]$,
$U_{1}(\mathrm{Im}\Phi ) = [h]$, and $U_{1}\mathbf{C} = [h]$,
$\Phi $ and $C$ are attractors of $H$.\\


\section*{7.4 Single-cycle attractors I}

In order to construct a transformation having a single-cycle
attractor, we consider the expansion described in Example 5.2.6
deribed from a circular transformation belonging to the class
determined by Theorem 4.4.3. Specifically, for any $h-1$
relatively prime with odd $n$ and $0 < h-1 < n$, there exists $E
= \langle e\rangle $ of $\mathbf{Q}^{2n}$,
$$
e = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}\cdot p_{n+1}\cdot \alpha _{2}p_{n+2}\cdot ..\cdot
\alpha _{n}p_{2n,}
$$
such that
$$
E = \bar{\neg}\rho ^{h-1} \quad \mbox{on}
\mathrm{Car}E.
$$
Then we get the transformation $G = \langle g\rangle $ defined by
$$
g = p_{1}\cdot S_{2n-3}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n},p_{n+1},\alpha _{2}p_{n+2},..,\alpha _{n}p_{2n}\}. 
$$
Let $e = \{c\}$ Then,
$C = \mathrm{Orb}_{\bar{\neg}\rho h-1}c$ is a cycle of $E$. 
$(E^{\#}c)_{i}$ is even for every $i$, since $\rho ^{n}x = x$
for every $x \in \mathrm{Car}E$. This is a nice effect of an
expansion. Therefore, $\mathrm{Orb}_{\rho h-1}c$ is a cycle of
$G$ by Proposition 7.1.4.

We have $(E^{\#}c)_{i} \leq 0$ iff $i = 1$ or $n+1$, since
$Ec = \{1,n+1\}^{-}c$. Since $\bar{\neg}\rho ^{h-1}c = Ec =
\{1,n+1\}^{-}c$, $c = \{1,n+1\}^{-}\bar{\neg}\rho ^{n-1}e$. 
Therefore, referring to (4.4.3), $(E^{\#}c)_{i} = 2$ iff $i = h$
or $n+h$. Then, by (7.1.6),
$$
\begin{array}{lll}
(G^{\#}c)_{1} &\leq & -1, \qquad (G^{\#}c)_{n+1}\leq -1,\\
(G^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other $i$,}\\
(G^{\#}c)_{i} &=& 1 \quad \mbox{only if $i = h$ and $n+h$.}
\end{array}
\eqno (7.4.1)
$$
Let $k \notin \{1, n+1\}$. Then, referring to (4.4.3),
\begin{eqnarray*}
((Gk^{-})^{\#}c)_{h} &=& 2 \quad \mbox{if $k \ne h$, \quad
by (7.1.2),}\\
((Gk^{-})^{\#}c)_{n+h} &=& 2 \quad \mbox{if $k \ne n+h$
\quad by (7.1.2).}
\end{eqnarray*}
Also
$$
\begin{array}{l}
((Gk^{-})^{\#}c)_{1} \leq 0 \quad \mbox{and} \quad
((Gk^{-})^{\#}c)_{n+1} \leq 0, \quad \mbox{by (7.1.3).} \\
((Gk^{-})^{\#}c)_{i} \geq 1 \quad \mbox{for every other $i$ such
that $i \ne k$ \quad by (7.1.3).}
\end{array}
$$
Therefore,
$$
G(k^{-}c) = \bar{\neg}\rho ^{h-1}c,
$$
or
$$
G(k^{-}c) = \bar{\neg}\rho ^{h-1}((\rho ^{-(h-1)}k)^{-}c).
$$
Therefore, under $G^{\sim }$, 
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [c],
$$
or
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [1^{-}c].
$$
On the other hand, referring to (7.4.1), we have
$$
((G1^{-})^{\#}c)_{1} \leq (G^{\#}c)_{1} \leq -1 \quad
\mbox{by (7.1.4).}
$$
$$
((G1^{-})^{\#}c)_{n+1} \leq 0 \quad
      \mbox{by (7.1.3).}
$$
Further, since $1^{-}(\bar{\neg}\rho ^{h-1}c)$ and $c$ differ
only at the 1st coordinate, (7.1.2) implies
$$
((G1^{-})^{\#}c)_{h} = 0 \quad \mbox{and} \quad
      ((G1^{-})^{\#}c)_{n+h} = 0.
$$
Also $((G1^{-})^{\#}c)_{i} > 0$ for every other $i$ by (7.1.3)
and (7.4.1). Therefore,
$$
G(1^{-}c) = \rho ^{2(h-1)}c.
$$
Therefore, if $x \in U_{1}\mathbf{C}$ then $x \in 
U_{1}\mathbf{C}$ and $\omega _{G}x = \mathbf{C}$.
Therefore, $C$ is an attractor of $G$. Thus we obtained\\ 

\textbf{Theorem 7.4.1} Let $E = \langle e\rangle $ of
$\mathbf{Q}^{2n}$,
$$
e = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}\cdot p_{n+1}\cdot \alpha _{2}p_{n+2}\cdot ..\cdot
\alpha _{n}p_{2n.}
$$
be the expansion described in Example 5.2.6 and derived from a
transformation determined by Theorem 4.4.3. Let 
$G = \langle g\rangle $,
$$
g = p_{1}\cdot S_{2n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n},p_{n+1},\alpha _{2}p_{n+2},..,\alpha _{n}p_{2n}\}.
$$
Then the $2n$-cycle of $E$ is an attractor of $G$.\\

Theorem 4.5.3 determined a class of one-to-one skew-circular
threshold transformations. Example 5.2.4 gave their circular
expansions. Such a transformation $E$ of $\mathbf{Q}^{2n}$
exists for any $h$ such that $0 < h-1 < 2n$ and $h-1$ is
relatively prime with $2n$, that is, $E = \langle e\rangle $,
\begin{eqnarray*}
e &=& p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}\cdot \neg p_{n+1}\cdot \neg \alpha _{2}p_{n+2}\cdot
..\cdot \neg \alpha _{n}p_{2n},\\
E &=& \rho ^{h-1} \quad \mbox{on $\mathrm{Car}E$.}
\end{eqnarray*}
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{2n}$ defined by
$$
g = p_{1}\cdot S_{2n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n},\neg p_{n+1},\neg \alpha _{2}p_{n+2},..,\neg \alpha
_{n}p_{2n}\}. 
$$
Let $e = \{c\}$ Then, 
$$
C = \mathrm{Orb}_{\rho h-1}c
$$
is a cycle of $E$. $(E^{\#}c)_{i}$ is even for every $i$, since
$\rho ^{n}x = \bar{\neg}x$ for every $x \in \mathrm{Car}E$. 
Therefore,
$\mathrm{Orb}_{\rho h-1}c$ is a cycle of $G$ by Proposition
7.1.4.

We have
$$
(E^{\#}c)_{i} \leq 0 \quad \mbox{iff $i = 1$ or
$n+1$,}
$$
since $Ec = \{1,n+1\}^{-}c$. Since $\rho ^{h-1}c = Ec =
\{1,n+1\}^{-}c$,
$$
c = \{1,n+1\}^{-}\rho ^{h-1}c.
$$
Therefore, referring to (4.5.4),
$$
(E^{\#}c)_{i} = 2 \quad \mbox{iff $i = h$ or $n+h$.}
$$
Then, by (7.1.6),
$$
\begin{array}{lll}
(G^{\#}c)_{1} &\leq & -1, \qquad (G^{\#}c)_{n+1}\leq -1,\\
(G^{\#}c)_{i} &\geq & 1 \quad \mbox{for every other $i$,}\\
(G^{\#}c)_{i} &=& 1 \quad \mbox{only if $i = h$ and $n+h$.}
\end{array}
\eqno (7.4.2)
$$
Let $k \notin \{1, n+1\}$. Then, referring to (4.5.4),
$$
\begin{array}{llll}
((Gk^{-})^{\# }c)_{h} &=& 2 \quad \mbox{if $k \ne h$,}
&\quad \mbox{by (7.1.2)}\\
((Gk^{-})^{\# }c)_{n+h} &=& 2 \quad \mbox{if $k \ne n+h$,}
&\quad \mbox{by (7.1.2)}\\
((Gk^{-})^{\# }c)_{1} &\leq & 0, &\quad \mbox{by (7.1.3)}\\
((Gk^{-})^{\# }c)_{n+1} &\leq & 0, &\quad \mbox{by (7.1.3)}\\
((Gk^{-})^{\# }c)_{i} &\geq & 1 \quad \mbox{for every other
$i \ne k$.} & \quad \mbox{by (7.1.3)}
\end{array}
$$
Therefore,
$$
G(k^{-}c) = \rho ^{h-1}c,
$$
or
$$
G(k^{-}c) = \rho ^{h-1}((\rho ^{-(h-1)}k)^{-}c).
$$
Therefore, under $G^{\sim }$, 
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [c],
$$
or
$$
[k^{-}c] \rightarrow ... \rightarrow [i^{-}c]
\rightarrow ... \rightarrow [1^{-}c].
$$
On the other hand, referring to (7.4.2), we have
$$
\begin{array}{llll}
((G1^{-})^{\# }c)_{1} &\leq & (G^{\# }c)_{1} = -1,
& \quad \mbox{by (7.1.4)}\\
((G1^{-})^{\# }c)_{n+1} &\leq & 0.
& \quad \mbox{by (7.1.3)}
\end{array}
$$
Further, since $1^{-}(\rho ^{h-1}c)$ and $c$ differ only
at their $1$st coordinates, (7.1.2) implies 
$$
((G1^{-})^{\#}c)_{h} = 0 \quad \mbox{and} \quad
((G1^{-})^{\#}c)_{n+h} = 0.
$$
Also $((G1^{-})^{\#}c)_{i} \geq 1$ for every other $i$ by
(7.1.3) and (7.4.2). Therefore,
$$
G(1^{-}c) = \rho ^{2(h-1)}c.
$$
Therefore, if $x \in U_{1}\mathbf{C}$ then $x \in 
U_{1}\mathbf{C}$ and $\omega _{G}x = \mathbf{C}$.
Therefore, $C$ is an attractor of $G$. Thus we obtained\\ 

\textbf{Theorem 7.4.2} Let $E = \langle e\rangle $ of
$\mathbf{Q}^{2n}$,
$$
e = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}\cdot \neg p_{n+1}\cdot \neg \alpha _{2}p_{n+2}\cdot
..\cdot \neg \alpha _{n}p_{2n.}
$$
be the expansion described in Example 5.2.4 and derived from a
transformation determined by Theorem 4.5.3. Let 
$G = \langle g\rangle $,
$$
g = p_{1}\cdot S_{2n-2}\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\cdot \neg p_{n+1}\cdot \neg \alpha _{2}p_{n+2}\cdot
..\cdot \neg \alpha _{n}p_{2n}\}.
$$
Then the $2n$-cycle of $E$ is an attractor of $G$.\\

\textbf{Example 7.4.3} Let $n = 2m$ for $m \geq 2$, and
$$
f = p_{1}\cdot p_{2}\cdot p_{3}\cdot ..\cdot p_{m}\cdot
\neg p_{m+2}\cdot ..\cdot \neg p_{2m}. 
$$
Then we get the transformation $G = \langle g\rangle $ of
$\mathbf{Q}^{2m}$ defined by
$$
g = p_{1}\cdot S_{2m-3}\{p_{2},p_{3},..,p_{m},\neg
p_{m+2},..,\neg p_{2m}\}. 
$$

\textbf{Proposition 7.4.4} Let $c = 1^{m}0^{m}$. Then, $C =
\mathrm{Orb}_{\rho }c$ is an attractor of $G$ in Example 7.4.3.

\begin{proof} We have 
$$
F^{\#}c = (0,2,4,..,2m-2,0,2,4,..,2m-2).
$$
Therefore, $Fc = \rho c$, so that $\mathrm{Orb}_{\rho }c$ is a
cycle of $F$. Further, $\mathrm{Orb}_{\rho }c$ is a cycle of $G$
by Proposition 7.2.2.

Further, by (7.1.6),
$$
G^{\#}c = (-1,1,3,..,2m-3,-1,1,3,..,2m-3).
$$
Let $2 \leq k \leq m-1$. Then,
\begin{eqnarray*}
((Gk^{-})^{\#}c)_{1} &\leq & 0 \quad \mbox{by (7.1.3)},\\
((Gk^{-})^{\#}c)_{m+1} &\leq & 0 \quad \mbox{by (7.1.3)},\\
((Gk^{-})^{\#}c)_{2} &=& 2 \quad \mbox{if $k \ne 2$,
\quad by (7.1.2),}\\
(Gk^{-})^{\#}c)_{m+2} &=& 2 \quad \mbox{by (7.1.2),}\\
((Gk^{-})^{\#}c)_{i} &\geq& 2 \quad \mbox{for every
other $i$ such that $i \ne k$ \quad by (7.1.3).}
\end{eqnarray*}
Therefore,
$$
k^{-}c \rightarrow \rho ((k-1)^{-}c) \quad \mbox{or}
\quad k^{-}c \rightarrow \rho c.
$$
Therefore,
$$
[k^{-}c] \rightarrow ... \rightarrow [1^{-}c] \quad
\mbox{or} \quad [k^{-}c] \rightarrow ... \rightarrow [c].
$$
We have
\begin{eqnarray*}
((G1^{-})^{\#}c)_{1} &\leq & (G^{\#}c)_{1} = -1 \quad
\mbox{by (7.1.4),}\\
((G1^{-})^{\#}c)_{m+1} &\leq & 0 \quad \mbox{by (7.1.3),}\\
((G1^{-})^{\#}c)_{2} &=& 0, \quad \mbox{by (7.1.2),}\\
((G1^{-})^{\#}c)_{m+2} &=& 0, \quad \mbox{by (7.1.2),}\\
((G1^{-})^{\#}c)_{i} &\geq & 1 \; \mbox{for every other $i$
\quad by (7.1.3).}
\end{eqnarray*}
Therefore, $1^{-}c \rightarrow \rho ^{2}c$.
Therefore, if $x \in U_{1}\mathbf{C}$ then $x \in 
U_{1}\mathbf{C}$ and $\omega _{G}x = \mathbf{C}$.
Therefore, $C$ is an attractor of $G$.
\end{proof}


\section*{7.5 Single-cycle attractors II} 

Another method of constructing single-cycle attractors is to
apply partial neighborhood functions. First we consider a
circular transformation belonging to the class determined by
Theorem 4.4.3. Let $h-1$ be relatively prime with odd $n$
and $0 < h-1 < n$, and let $F = \langle f\rangle $ of
$\mathbf{Q}^{n}$, 
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},
$$
such that
$$
F = \bar{\neg}\rho ^{h-1} \quad \mbox{on
$\mathrm{Car}F$.}
$$
Let $f = \{c\}$. Let $G = \langle g\rangle $,
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},\alpha
_{3}p_{3},...,\alpha _{n}p_{n}\}.
$$
Then $Gc = \{1,h\}^{-}c = \rho ^{2(h-1)}c$ by (7.3.1),
and $G$ has two cycles
$\mathrm{Orb}_{\rho 2(h-1)}\{c,\bar{\neg}\rho ^{h-1}c\}$.

In order to preserve the original one cycle such that $Gc =
1^{-}c$ we remove $c$ from $\bar{\neg}\rho ^{h-1}g$,
i.e. remove $\bar{\neg}\rho ^{-(h-1)}c$ from $g$.  Note 
$$
(\rho n^{-})^{-(h-1)}c = (1-(h-1))^{-}c = (2-h)^{-}c,
$$
referring to (4.4.3). $(2-h)^{-}c$ is the only element $x \ne 
c$ in g such that $x \in [c]$, otherwise $G^{\#}$ would have
more than two non-positive coordinates.
Further we consider the set 
$$
\{2^{-}c,,,n^{-}c\}\backslash (2-h)^{-}c 
$$
and find out some elements $i^{-}c \ne j^{-}c$ such that
$$
[i^{-}c] = [j^{-}c].
$$
We have
$$
(\bar{\neg}\rho ^{h-1})^{k}i^{-}c = 
      (\rho ^{k(h-1)}i)^{-}(\bar{\neg}\rho ^{h-1})^{k}c,
$$
by (2.1.3), and 
$$
(\bar{\neg}\rho ^{h-1})^{k}c = (1 + (k-1)(h-1))^{-}
(1+(k-2)(h-1))^{- }...(1+(h-1))^{-}1^{-}c
$$
by (4.5.4). Therefore, $(\bar{\neg}\rho ^{h-1})^{k}i^{-}c =
j^{-}c$ implies
$$
(i+k(h-1))^{-}(1 + (k-1)(h-1))^{-}(1+(k-2)(h-1))^{-}
...(1+(h-1))^{-}1^{-} = j^{-},
$$
so that $k = 2$, and $(i+k(h-1))^{-}h^{-}1^{-} = j^{- }$.
Therefore,
$$
(i+k(h-1)) = 1, \quad j = h;
$$
or
$$
(i+k(h-1)) = h, \quad j = 1. 
$$
Since $j > 1$, $j = h$ and $i = 3-2h$. We want to remove one of
$(3-2h)-c$ and $h^{-}c$ from $g$, but we must decide which to
remove. If we remove $(3-2h)-c$ then $Gh^{-}c^{ }= G^{2}c$.
If we remove $h^{-}c$, then $G(3-2h)-c = (3-2h)^{-}Gc$.
The former is better in view of of continuity. 
Thus we obtained $V = \langle v\rangle $ defined by
$$
v = p_{1}\cdot S_{n-4}(\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}\backslash \{\alpha _{2-h}p_{2-h},
\alpha _{3-2h}p_{3-2h}\})
\cdot \alpha _{2-h}p_{2-h}\cdot \alpha _{3-2h}p_{3-2h}.
\eqno (7.5.1)
$$
By the above removal, if $x \in v$, then $x \notin
\{\rho ^{i}v,\bar{\neg}\rho ^{i}v\}$ for every $i \ne 0$
mod $n$.

We prove the attractiveness of the cycle by decomposition of
the transformations instead of calculating the extended
representations of the transformations. For the notion of the
sum of transformations, refer to Chapter 1.1. 
Let 
\begin{eqnarray*}
v^{(1)} &=& p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n},\\
v^{(i)} &=& v\cdot \neg \alpha _{i}p_{i} \quad \mbox{for $i
\in \mathbf{N}\backslash \{1,2-h,3-2h\}$.}
\end{eqnarray*}
Then 
\begin{eqnarray*}
v &=& v^{(1)} \vee . . . \vee v^{(n)},\\
v^{(i)}\cdot v^{(j)} &=& 0 \quad \mbox{for every $i \ne 
j$,}
\end{eqnarray*}
as clear from the above process of removing $(2-h)^{-}c$
and $(3-2h)^{-}c$ from $g$.

Let $V^{(i)} = \langle v^{(i)}\rangle $. Then, if $x \in 
v^{(1)}$, then $x = c$ and
$$
Vc = 1^{-}c \in [v^{(1)}].
$$
If $x \in v^{(i)}$ for $i \in \mathbf{N}\backslash \{1,2-h,3-
2h\}$, then $x = i^{-}c$, and
$$
V(i^{-}c) = V^{(i)}(i^{-}c) = \{1,i\}^{-}c,
$$
since $x \notin \{\rho ^{i}v,\bar{\neg}\rho ^{i}v\}$
for every $i \ne 0$ mod $n$. Therefore,
$$
V(i^{-}c) = \bar{\neg}\rho ^{h-1}
((\rho ^{-(h-1)}i)^{-}c \in [v^{(i-h+1)}]. 
$$
Therefore, 
$$
V = V^{(1)} + V^{(2)} + ... + V^{(n)},
$$
and a flow graph of $V$ is
$$
[v^{(4-3h)}] \rightarrow [v^{(5-4h)}] \rightarrow ...
\rightarrow [v^{(1)}] \rightarrow [v^{(1)}] = [c]. 
$$
Let $C = \mathrm{Orb}_{\bar{\neg}\rho h-1}c$. Then
$C$ is a cycle of $V$ and $\mathbf{C} = [c]$.  Further,
$1^{-}c \in [v^{(1)}]$, $(2-h)^{-}c \in v^{(1)}$,
and $(3-2h)^{-}c \in [v^{(h)}]$. Therefore,
$U_{1}\mathbf{C} = [v]$. Therefore, $C$
is an attractor of $V$. Thus we obtained the following
theorem.\\

\textbf{Theorem 7.5.1} Let $V = \langle v\rangle $ of
$\mathbf{Q}^{n}$ be the transformation defined by (7.5.1)
from a transformation determined by Theorem 4.4.3. Then
$G$ has a $2n$-cycle attractor.

Next, let $0 < h-1 < 2n$ and $h-1$ be relatively prime with
$2n$, let $F = \langle\langle f\rangle\rangle $,
$$
f = p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot \alpha
_{n}p_{n}
$$
be a transformation determined by Theorem 4.5.3,
and let $f = \{c\}$. Let $G = \langle\langle g\rangle\rangle $,
$$
g = p_{1}\cdot S_{n-2}\{\alpha _{2}p_{2},\alpha
_{3}p_{3},...,\alpha _{n}p_{n}\}.
$$
Then $Gc = \{1,h\}^{-}c = (\rho n^{-})^{2(h-1)}c$ by (7.3.2),
and $G$ has two cycles $\mathrm{Orb}_{(\rho n-)2(h-1)}
\{c,(\rho n^{-})^{h-1}c\}$. In order to preserve the original
one cycle such that $Gc = 1^{-}c$ we remove $c$ from
$(\rho n^{-})^{h-1}g$, i.e. remove $(\rho n^{-})^{-(h-1)}c$
from $g$. Note 
$$
(\rho n^{-})^{-(h-1)}c = (1-(h-1))^{-}c = (2-h)^{-}c,
$$
referring to (4.5.4). $(2-h)^{-}c$ is the only element $x \ne 
c$ in $g$ such that $x \in [c]$, otherwise $G^{\#}$ would have
more than two non-positive coordinates.

Further we consider the set 
$$
\{2^{-}c,,,n^{-}c\}\backslash (2-h)^{-}c 
$$
and find out some elements $i^{-}c \ne j^{-}c$ such that
$$ [i^{-}c] = [j^{-}c].
$$
We have
$$
(\rho n^{-})^{k(h-1)}i^{-}c = (\rho ^{k(h-1)}i)^{-}(\rho
n^{-})^{k(h-1)}c,
$$
by (2.1.3), and 
$$
(\rho n^{-})^{k(h-1)}c = (1 + (k-1)(h-1))^{-}
(1+(k-2)(h-1))^{-}...(1+(h-1))^{-}1^{-}c
$$
by (4.5.4). Therefore, $(\rho n^{-})^{k(h-1)}i^{-}c = j^{-}c$
implies
$$
(i+k(h-1))^{-}(1 + (k-1)(h-1))^{-}(1+(k-2)(h-1))^{-}
...(1+(h-1))^{-}1^{-} = j^{-},
$$
so that $k = 2$, and $(i+k(h-1))^{-}h^{-}1^{-} = j^{-}$. 
Therefore,
$$
(i+k(h-1)) = 1, \quad j = h;
$$
or
$$
(i+k(h-1)) = h, \quad j = 1. 
$$
Since $j > 1$, $j = h$ and $i = 3-2h$. We want to remove one of
$(3-2h)-c$ and $h^{-}c$ from $g$, but we must decide which to
remove. If we remove $(3-2h)-c$ then $Gh^{-}c^{ }= G^{2}c$.
If we remove $h^{-}c$, then $G(3-2h)-c = (3-2h)^{-}Gc$.
The former is better in view of of continuity. Thus we obtained
$V = \langle\langle v\rangle\rangle $ defined by
$$
v = p_{1}\cdot S_{n-4}(\{\alpha _{2}p_{2},..,\alpha
_{n}p_{n}\}\backslash \{\alpha _{2-h}p_{2-h},
\alpha _{3-2h}p_{3-2h}\})
\cdot \alpha _{2-h}p_{2-h}\cdot \alpha _{3-2h}p_{3-2h}.
\eqno (7.5.2)
$$
By the above removal, if $x \in v$, then $x \notin (\rho
n^{-})^{i}v$ for every $i \ne 0$ mod $2n$.

We prove the attractiveness of the cycle by decomposition of
the transformations instead of calculating the extended
representations of the transformations. For the notion of the
sum of transformations, refer to Chapter 1.1. 
Let 
\begin{eqnarray*}
v^{(1)} &=& p_{1}\cdot \alpha _{2}p_{2}\cdot ..\cdot
\alpha _{n}p_{n},\\
v^{(i)} &=& v\cdot \neg \alpha _{i}p_{i} \quad
\mbox{for $i \in \mathbf{N}\backslash \{1,2-h,3-2h\}$.}
\end{eqnarray*}
Then 
\begin{eqnarray*}
v &=& v^{(1)} \vee . . . \vee v^{(n)},\\
v^{(i)}\cdot v^{(j)} &=& 0 \quad \mbox{for every $i \ne j$,}
\end{eqnarray*}
as clear from the above process of removing $(2-h)^{-}c$
and $(3-2h)^{-}c$ from $g$.

Let $V^{(i)} = \langle\langle v^{(i)}\rangle\rangle $. 
Then, if $x \in v^{(1)}$, then $x = c$ and
$$
Vc = 1^{-}c \in [v^{(1)}].
$$
If $x \in v^{(i)}$ for $i \in \mathbf{N}\backslash \{1,2-h,3-
2h\}$, then $x = i^{-}c$, and
$$
V(i^{-}c) = V^{(i)}(i^{-}c) = \{1,i\}^{-}c,
$$
since $1^{-}c \in (\rho n-)^{j}v$ for only $j = 0$ mod $2n$. 
Therefore,
$$
V(i^{-}c) = (\rho n^{-})^{h-1}((\rho ^{-(h-1)}i)^{-}c
\in [v^{(i-h+1)}]. 
$$
Therefore, 
$$
V = V^{(1)} + V^{(2)} + ... + V^{(n)},
$$
and a flow graph of $V$ is
$$
[v^{(4-3h)}] \rightarrow [v^{(5-4h)}] \rightarrow ...
\rightarrow [v^{(1)}] \rightarrow [v^{(1)}] = [c]. 
$$
Let $C = \mathrm{Orb}_{(\rho n-)h-1}c$. Then $C$ is a cycle of
$V$ and $\mathbf{C} = [c]$. Further, $1^{-}c \in [v^{(1)}]$,
$(2-h)^{-}c \in v^{(1)}$, and $(3-2h)^{-}c \in [v^{(h)}]$. 
Therefore, $U_{1}\mathbf{C} = [v]$. Therefore, $C$ is an
attractor of $V$. Thus we obtained the following theorem.\\

\textbf{Theorem 7.5.2} Let $V = \langle v\rangle $ of
$\mathbf{Q}^{n}$ be the transformation defined by (7.5.2)
from a transformation determined by Theorem 4.5.3. Then
$G$ has a $2n$-cycle attractor.


\section*{7.6 Attractors derived from polynomials}

So far we have shown attractors in transformations obtained by
modifying one-to-one transformations $F = \langle f\rangle $ or
$F = \langle\langle f\rangle\rangle$ such that $f$ consists of
one term. In this section, we generalize the construction to one-
to-one transformations $F = \langle f\rangle $ such that $f$
consists of more than one term. For this purpose, we start with
the one-to-one transformation of Example 4.4.7 defined by
$$
f = 1\cdot 2\cdot \neg 3\cdot (4\vee 5)\cdot \neg 6.
$$
$\mathrm{GRAPH}(F)$ consists of the three $6$-cycles:
$$
\begin{array}{ccccc}
110110& \rightarrow & 010010& \rightarrow & 011011 \\ 
\uparrow & & & & \downarrow \\
100100& \leftarrow & 101101& \leftarrow & 001001 \\
110100& \rightarrow & 010110& \rightarrow &010011\\
\uparrow & & & & \downarrow \\
100101& \leftarrow & 001101& \leftarrow & 011001\\
110010& \rightarrow & 011010& \rightarrow & 001011\\
\uparrow & & & &\downarrow \\
100110& \leftarrow & 101100& \leftarrow & 101001
\end{array}
$$
We modify $f$ by expanding it to
\begin{eqnarray*}
g &=& 1\cdot U_{1}(2\cdot \neg 3\cdot (4\vee 5)\cdot \neg
6)\\
&=& 1\cdot (2\cdot \neg 3\cdot \neg 6 \vee S_{2}\{
2,3,\neg 6\} \cdot (4\vee 5)).
\end{eqnarray*}
Here, $2\cdot \neg 3\cdot \neg 6 \vee S_{2} \{ 2,3,\neg 6\}
\cdot (4\vee 5)$ is a threshold function by Proposition 4.1.6,
so that $g$ is a threshold function by Proposition 4.1.2. 

For the point $110110$ in the first cycle of $F$,
$$
F^{\# } (110110) = (0,2,1,0,2,1).
$$
Therefore, the condition in Proposition 7.1.4 is violated, so
that the cycle is not preserved in $G = \langle g\rangle $.
In fact, by (7.1.6), we have
$$
G^{\# }(110110) = (-1,1,0,-1,1,0),
$$
so that $110110 \rightarrow 011011$ under $G$. We have
thus obtained the two 3-cycles
$$
\begin{array}{cccccc}
110110& \rightarrow & 011011& \qquad 001001&
\rightarrow & 100100\\
\nwarrow & &\swarrow& \nwarrow & 
& \swarrow \\
& 101101& && 0100101& 
\end{array}
$$ 
Further calculation shows that the set of these two cycles
is an attractor of G. 

Next, we take the one-to-one transformation of Example
4.4.17 in Chapter 4. Therefore, we start with $f$ defined
recursively as follows,
$$ 
\begin{array}{lll}
f^{(4)} &=& p_{1}\cdot p_{2}\cdot \neg p_{3}\cdot \neg
p_{4},\\
f^{(5)} &=& p_{1}\cdot p_{2}\cdot \neg p_{4}\cdot \neg
p_{5},\\
f^{(n)} &=& p_{1}\cdot p_{2}\cdot \neg p_{n-1}\cdot
\neg p_{n}\vee f^{(n-2)}\cdot \neg p_{n} \\
&=& p_{1}\cdot p_{2}\cdot \neg p_{n}\cdot (\neg
p_{n-1}\vee (f^{(n-2)}|11)),
\end{array}
\eqno (7.6.1)
$$
where
$$
(f^{(n-2)}|11)(x_{3},..,x_{n-2}) =
f^{(n-2)}(1,1,x_{3},..,x_{n-2})
$$
for every $(x_{3},..,x_{n-2})$.

We modify $f^{(n)}$ by expanding it to
$$
g^{(n)} = p_{1}\cdot U_{1}(p_{2}\cdot \neg p_{n}\cdot
(\neg p_{n-1}\cdot \vee (f^{(n-2)}|11))).
$$
Then
\begin{eqnarray*}
g^{(n)} &=& U_{1}(p_{2}\cdot \neg p_{n}\cdot \neg p_{n-
1})\vee U_{1}(p_{2}\cdot \neg p_{n}\cdot (f^{(n-2)}|11)))\\
&=& p_{2}\cdot \neg p_{n}\vee \cdot (p_{2}\vee \neg
p_{n})\cdot \neg p_{n-1}\\
& & \quad \vee p_{2}\cdot \neg p_{n}\cdot U_{1}(f^{(n-
2)}|11))\vee (p_{2}\vee \neg p_{n})(f^{(n-2)}|11)\\
&=& p_{2}\cdot \neg p_{n}\vee \cdot (p_{2}\vee \neg
p_{n})\cdot \neg p_{n-1}\vee (p_{2}\vee \neg p_{n})(f^{(n-2)}|
11)\\
&=& p_{2}\cdot \neg p_{n}\vee \cdot (p_{2}\vee \neg
p_{n})\cdot (\neg p_{n-1}\vee (f^{(n-2)}|11))
\end{eqnarray*}
Thus we have obtained
$$
g = g^{(n)} = p_{1}\cdot (p_{2}\cdot \neg p_{n}\vee
(p_{2}\vee \neg p_{n})\cdot (\neg p_{n-1}\cdot \vee (f^{(n-
2)}|11))). 
\eqno (7.6.2)
$$ 

and the transformation $G = \langle g\rangle $ of $Q^{n}$ for
$n \geq 6$.\\

\textbf{Proposition 7.6.1} $g$ defined by (7.6.1) and (7.6.2) is
a threshold function.
\begin{proof} $f^{(n-2)}$ is a threshold function by Proposition
4.4.18. Therefore, $f^{(n-2)}|11$ is a threshold
transformation by Proposition 4.1.7. Therefore,
$\neg p_{n-1}\cdot \vee (f^{(n-2)}|11)$ is a threshold
function by Proposition 4.1.2. Therefore, $p_{2}\cdot \neg
p_{n}\vee (p_{2}\vee \neg p_{n})\cdot (\neg p_{n-1}\cdot \vee
(f^{(n-2)}|11))$ is a threshold function by Proposition
4.1.6. Therefore, $g$ is a threshold function by Proposition
4.1.2.
\end{proof}

Computational results expect that the transformations $G$ have
non-trivial attractors. However, at present I can give only the
following weaker result Proposition 7.6.4. First, we generalize
the $\epsilon $-neighborhood of a non-empty proper subset $S$ of
$Q^{n}$, $U_{\epsilon }S$, for any integer $\epsilon $ as
$$
U_{\epsilon }S = \{ x\mid d_{SH}(x,S) \leq \epsilon \} .
$$
Here, $d_{SH}(x,S)$ is the signed Hamming distance between the
point $x$ and the set $S$ defined in Section 7.1.

As before, $o$ is the $n$-vector whose every coordinate is
$0$, $l$ is the $n$-vector whose every coordinate is $1$.
Further, $(10)^{n/2} = 1010...10$ is the concatenation of $n/2$
$10$, and $(10)^{n/2} = 0101...01$ is the concatenation of $n/2$
$01$.\\

\textbf{Lemma 7.6.2} Let $A = \{o, l\}$ and $B = \{(10)^{n/2},
(01)n/2\}$. Then $\mathrm{Car}G = A^{C}$ for odd $n$, and
$\mathrm{Car}G = (A\cup B)^{c}$ for even $n$. 
\begin{proof} If $x$ is an element of $A\cup B$, then clearly $x
\notin g$ and $x \notin \bar{\neg}g$, so that $x \notin \rho ^{i-
1}g$ and $x \notin \bar{\neg}\rho ^{i-1}g$ for every $i$, so that
$x \notin \mathrm{Car}G$.

Let $x \in (A\cup B)^{c}$. Then there exists some $i$ such
that $(x_{i},x_{i+1},x_{i+2})$ is $011$ or $100$. Let $x_{1} =
x_{2} = 1$, and $x_{n} = 0$ without loss of generality. Then $x
\in g$. Therefore, $x \in \mathrm{Car}G$.
\end{proof} 

\textbf{Lemma 7.6.3} Let $Gx = y$. Then if
$(x_{i},x_{i+1},x_{i+2}) = 110$ then $(y_{i+1},y_{i+2}) = 11$. 
If $(x_{i},x_{i+1},x_{i+2},x_{i+3}) = 0111$ then
$(y_{i+1},y_{i+2})= 00$. If $(x_{k},x_{k+1},x_{k+2}) = 011$
then $y_{k+1} = 0$.
\begin{proof}
The proof is clear from the definition of $g$.
\end{proof}

\textbf{Proposition 7.6.4} Let $G = \langle g\rangle $ be the
transformation of $Q^{n}$, where $g$ is defined by (7.6.1) and
(7.6.2). Then there exists a subset $\Phi $ of $CY(G)$ such that
\begin{eqnarray*}
(1)&& G(\mathrm{Car}G) \subseteq \mathrm{Car}G,\\
(2)&& \omega _{G}(\mathrm{Car}G) = \mathrm{Im}\Phi ,\\
(3)&& \mathrm{Im}\Phi \subseteq U_{-1}(\mathrm{Car}G).
\end{eqnarray*}
\begin{proof}
Assume $x \in \mathrm{Car}G$, and $y = Gx$. Then there exists
some $i$ such that $(x_{i},x_{i+1},x_{i+2}) = 110$ or
$(x_{i},x_{i+1},x_{i+2}) = 001$. Let $(x_{1},x_{2},x_{3}) = 110$
without loss of generality. Then $(y_{2},y_{3}) = 11$. Assume
there exists some $j$ such that $(x_{j},x_{j+1},x_{j+2}) = 001$. 
Then $(y_{j+1},y_{j+2}) = 00$, so that $d_{H}(y,
(\mathrm{Car}G)^{c}) \geq 2$.

Assume no $j$ such that $(x_{j},x_{j+1},x_{j+2}) = 001$
exists. Assume there exists some $j$ such that
$(x_{j},x_{j+1},x_{j+2},x_{j+3}) = 0111$, then
$(y_{j+1},y_{j+2})= 00$, so that $d_{H}(y,(\mathrm{Car}G)^{c})
\geq 2$. Assume no such $j$ exists. Then $x_{n} = 0$, so that
$y_{1} = 0$. Assume there exists no $j \ne n$ such that
$(x_{j},x_{j+1},x_{j+2}) = 011$. Then $x = 11(01)^{r}0$, so that
$y_{n} = 0$. Therefore, $(y_{n},y_{1},y_{2},y_{3}) = 0011$, so
that $d_{H}(y,(\mathrm{Car}G)^{c}) \geq 2$. Assume there exists
some $j \ne n$ such that $(x_{j},x_{j+1},x_{j+2}) = 011$, and
let $(x_{j},x_{j+1},x_{j+2})$ be the first such coordinates. 
Then $x = 11(01)^{r}011...0$. Therefore, $y_{j+1} = 0$, so that
$(y_{1},y_{2},y_{3},y_{j+1}) = 0110$. Since $j+1$ is even,
$d_{H}(y,(\mathrm{Car}G)^{c}) \geq 2$. 

Therefore, $d_{SH}(y,\mathrm{Car}G) \leq 1-2 = -1$ in every
case, Therefore, $G(\mathrm{Car}G) \subseteq U_{-
1}(\mathrm{Car}G)$. Let $\Phi $ be the set of all non-loop
cycles. Then the desired properties (1), (2), and (3) are clear. 
\end{proof}


\section*{7.7 Orbit modification}

We have shown that orbit modification provided a powerful tool
for constructing new desired
transformations in Chapters 3 and 4. In particular, we
constructed a McCulloch and Pitts
network (6.2.1) that has an attractive unique $k$-cycle for any
$k \leq 2^{n}$ (Theorems 5.5.2 and 6.6.3). In that network, any
state converges to a
unique cycle regardless of the initial state. Here we further
explore the possibilities of the
method to construct some transformations that have the same
property. 

(1) We start with the skew circular transformation for $n = 2$,
$F = [f_{1}, f_{2}]$,
\begin{eqnarray*}
f_{1} & =& {11} = 1\cdot 2,\\
f_{2} &=& {01} = \neg 1\cdot 2.
\end{eqnarray*}
$\mathrm{GRAPH}(F)$ is
$$
11 \rightarrow 01 \rightarrow 00 \rightarrow 10
\rightarrow 11. 
$$
(2) For $n = 3$, we have the 3rd face copies of (1) defined by $F
= [f_{1},f_{2},f_{3}]$,
\begin{eqnarray*}
f_{1} &=& \{111, 110\} = 1\cdot 2,\\
f_{2} &=& \{011, 010\} = \neg 1\cdot 2,\\
f_{3} &=& \emptyset.
\end{eqnarray*}
$\mathrm{GRAPH}(F)$ is
$$
\begin{array}{ccccccccc}
111& \rightarrow& 011& \rightarrow& 001& \rightarrow& 101&
\rightarrow& 111,\\
110& \rightarrow& 010& \rightarrow& 000& \rightarrow& 100&
\rightarrow& 110.
\end{array}
$$
(3) Modify (2) by first letting $f_{3} = \{001\} = \neg 1\cdot
\neg 2\cdot 3$, and delete the
complement of $001$, i.e. $110$ from $f_{1}$. We now have $F =
[f_{1},f_{2},f_{3}]$,
\begin{eqnarray*}
f_{1} &=& 1\cdot 2\cdot 3,\\
f_{2} &=& \neg 1\cdot 2,\\
f_{3} &=& \neg 1\cdot \neg 2\cdot 3.
\end{eqnarray*}
$\mathrm{GRAPH}(F)$ is
$$
\begin{array}{ccccccccc}
111& \rightarrow& 011& \rightarrow& 001& & 101& \rightarrow& 
111.\\
& & & & \downarrow& & & & \uparrow\\ 
& & 010& \rightarrow& 000& \rightarrow& 100& \rightarrow& 110
\end{array}
$$
This has a unique $6$-cycle, $\mathrm{Orb}_{\rho 3^{-}}111$.

(4) For $n = 4$, we have the $4$th face copies of (3) $F =
[f_{1},f_{2},f_{3},f_{4}]$,
\begin{eqnarray*}
f_{1} &=& 1\cdot 2\cdot 3,\\
f_{2} &=& \neg 1\cdot 2,\\
f_{3} &=& \neg 1\cdot \neg 2\cdot 3,\\
f_{4} &=& \emptyset .
\end{eqnarray*}
$\mathrm{GRAPH}(F)$ is
$$
\begin{array}{ccccccccc}
1111& \rightarrow& 0111& \rightarrow& 0011& & 1011&
\rightarrow& 1111,\\
& & & & \downarrow& & & & \uparrow\\ 
& & 0101& \rightarrow& 0001& \rightarrow& 1001& \rightarrow& 
1101,\\
1110& \rightarrow& 0110& \rightarrow& 0010& & 1010&
\rightarrow& 1110.\\
& & & & \downarrow& & & & \uparrow\\ 
& & 0100& \rightarrow& 0000& \rightarrow& 1000& \rightarrow& 
1100
\end{array}
$$

(5) Modify (2) by first letting $f_{4} = \{0001\} = \neg 1\cdot
\neg 2\cdot \neg 3\cdot 4$,
and delete the complement of $0001$, i.e. $1110$ from $f_{1}$. 
We now have $F =
[f_{1},f_{2},f_{3},f_{4}]$,
\begin{eqnarray*}
f_{1} &=& 1\cdot 2\cdot 3\cdot 4,\\
f_{2} &=& \neg 1\cdot 2,\\
f_{3} &=& \neg 1\cdot \neg 2\cdot 3,\\
f_{4} &=& \neg 1\cdot \neg 2\cdot \neg 3\cdot 4.
\end{eqnarray*}
$\mathrm{GRAPH}(F)$ is
$$
\begin{array}{ccccccccccc}
1111& \rightarrow & 0111& \rightarrow & 0011& & && 1011&
\rightarrow & 1111,\\
&&&& \downarrow &&&&&&\uparrow \\
&& 0101& \rightarrow & 0001& \rightarrow & 0000,& & 1001&
\rightarrow & 1101,\\
1110& \rightarrow & 1111,& & 0110& \rightarrow & 0010& & 1010&
\rightarrow &
1110.\\
&&&&&& \downarrow &&&& \uparrow \\
&&&&0100& \rightarrow & 0000& \rightarrow & 1000& \rightarrow &
1100
\end{array}
$$
This has a unique $8$-cycle, $\mathrm{Orb}_{\rho 4^{-}}1111$.


For the general dimension $n$, we obtain

\textbf{Example 7.7.1} Let $F = [f_{1},..,f_{n}]$,
\begin{eqnarray*}
f_{1} &=& p_{1}\cdot .. \cdot p_{n},\\
f_{2} &=& \neg p_{1}\cdot p_{2},\\
f_{3} &=& \neg p_{1}\cdot \neg p_{2}\cdot p_{3},\\
&&......\\
f_{n-1} &=& \neg p_{1}\cdot ..\cdot \neg p_{n-2}\cdot p_{n-1},\\
f_{n} &=& \neg p_{1}\cdot ..\cdot \neg p_{n-1}\cdot p_{n},
\end{eqnarray*}
then, $\mathrm{GRAPH}(F)$ consists of the unique $2n$-cycle
$\mathrm{Orb}_{\rho
n^{-}}l$.

For the next example, we consider the transformation $G = \langle
g\rangle$ of Example 7.4.3 for $n = 4$, where $g = 1\cdot (2 \vee
\neg 4)$.
$\mathrm{GRAPH}(G)$ is
$$ 
\begin{array}{ccccccc}
0001&&&&&& 1000\\
&\searrow &&&& \swarrow & \\
1011&\rightarrow & 1100& \rightarrow & 0110& \leftarrow & 1101\\
&& \uparrow && \downarrow && \\
0111& \rightarrow & 1001& \rightarrow & 0011& \leftarrow & 1110\\
& \nearrow &&&& \nwarrow &\\
0010&&&&&& 0100,\\ 
0000& \leftrightarrow & 1111,&& 0101& \leftrightarrow& 
1010.
\end{array}
$$
The 5th face copies are defined by
$G = [g_{1},g_{2},g_{3},g_{4},g_{5}]$,
\begin{eqnarray*}
g_{1} &=& 1\cdot (2\vee \neg 4),\\
g_{2} &=& 2\cdot (3\vee \neg 1),\\
g_{3} &=& 1\cdot (4\vee \neg 2),\\
g_{4} &=& 4\cdot (1\vee \neg 3),\\
g_{5} &=& \emptyset. 
\end{eqnarray*}
$\mathrm{GRAPH}(G)$ is
$$
\begin{array}{ccccccc}
00011&&&&&& 10001\\
&\searrow &&&& \swarrow & \\
10111&\rightarrow & 11001& \rightarrow & 01101& \leftarrow &
11011\\
&& \uparrow && \downarrow && \\
01111& \rightarrow & 10011& \rightarrow & 00111& \leftarrow &
11101\\
& \nearrow &&&& \nwarrow &\\
00101&&&&&& 01001,\\ 
00001& \leftrightarrow & 11111,&& 01011& \leftrightarrow & 
10101,\\
00010&&&&&& 10000\\
&\searrow &&&& \swarrow & \\
10110&\rightarrow & 11000& \rightarrow & 01100& \leftarrow &
11010\\
&& \uparrow && \downarrow &&\\
01110& \rightarrow & 10010& \rightarrow & 00110& \leftarrow &
11100\\
& \nearrow &&&& \nwarrow &\\
00100&&&&&& 01000,\\ 
00000& \leftrightarrow & 11110,&& 01010& \leftrightarrow & 10100.
\end{array}
$$

Modify the above by letting $f_{5} = 1\cdot 2\cdot \neg 3\cdot
\neg 4\cdot 5$. Then
following changes in the arcs of $\mathrm{GRAPH}(G)$ occur, while
the other arcs remain
unchanged.
$$
\begin{array}{ccc}
11001& \rightarrow& 01100,\\
00110& \rightarrow& 10011.
\end{array}
$$
Therefore, 
$$
\begin{array}{ccc}
11001& \rightarrow& 01100\\
\uparrow& &\downarrow\\ 
10011& \leftarrow& 00110 
\end{array}
$$
becomes an attractor.

For the general dimension $n = 2m+1$, we obtain

\textbf{Example 7.7.2} Let $n = 2m+1$ for $m \geq 2$, and
\begin{eqnarray*}
g_{1} &=& p_{1}\cdot S_{2m-3}\{p_{2},p_{3},..,p_{m},\neg
p_{m+2},..,\neg p_{2m}\},\\ 
g_{2} &=& p_{2}\cdot S_{2m-3}\{p_{3},p_{4},..,p_{m+1},\neg
p_{m+3},..,\neg
p_{2m},\neg p_{1}\}, \\
&& ...\\
g_{2m} &=& p_{2m}\cdot S_{2m-3}\{p_{1},p_{2},..,p_{m-1},\neg
p_{m+1},..,\neg
p_{2m-1}\},\\
g_{2m+1} &=& p_{1}\cdot p_{2}\cdot ..\cdot p_{m}\cdot \neg
p_{m+1}\cdot \neg
p_{m+2}\cdot ..\cdot \neg p_{2m-1}\cdot p_{2m}.
\end{eqnarray*}

Then the transformation $G = \langle g\rangle$ of
$\mathbf{Q}^{2m+1}$ has a unique
attractive $2m$-cycle, although there are some non-attractive
loops or cycles.


Next, we consider the direct product of the transformation
$[1\cdot 2\cdot 3,\neg 1\cdot
2,\neg 1\cdot \neg 2\cdot 3]$ of Example 7.7.1 for $n = 3$ and
the skew-circular
transformation $[4\cdot 5,\neg 4\cdot 5]$, which are respectively
transformations of
$\mathbf{Q}^{\{1,2,3\}}$ and $\mathbf{Q}^{\{4,5\}}$. Then
$$
G = [1\cdot 2\cdot 3,\neg 1\cdot 2,\neg 1\cdot \neg 2\cdot
3,4\cdot 5,\neg 4\cdot
5].
$$
$\mathrm{GRAPH}(G)$ is
$$
\begin{array}{ccccccccccc}
10110 &\rightarrow& 11111 &\rightarrow& 01101 &\rightarrow& 00100
&\rightarrow&
00010 &\leftarrow& 01000 \\
&& \uparrow &&&&&& \downarrow && \\
&& 11010 &&&&&& 10011 && \\
&& \uparrow &&&&&&\downarrow && \\ 
&& 10000&&&&&& 11001 &&\\
&& \uparrow &&&&&& \downarrow && \\ 
01011&\rightarrow& 00001 &\leftarrow& 00111 &\leftarrow& 01110
&\leftarrow& 11100
&\leftarrow& 10101, \\
01001 &\rightarrow& 00000 &\rightarrow& 10010 &\rightarrow& 11011
&\rightarrow&
11101 &\leftarrow& 10111 \\
&& \uparrow &&&&&& \downarrow && \\
&& 00101 &&&&&& 01100 && \\
&& \uparrow &&&&&& \downarrow && \\
&& 01111 &&&&&& 00110&& \\
&& \uparrow &&&&&& \downarrow&& \\
10100 &\rightarrow& 11110 &\leftarrow& 11000 &\leftarrow&
10001&\leftarrow& 00011
&\leftarrow& 01010. 
\end{array}
$$ 

We try to unite these two $12$-cycles and construct a
transformation having a unique $24$-
cycle. By try and error, we find that Arimoto's orbit
modification $G$ at point $11011$ is a
threshold transformation. In fact,
\begin{eqnarray*}
1\cdot 2\cdot 3\cup 11011 &=& 1\cdot 2\cdot 3\vee 1\cdot 2\cdot
\neg 3\cdot 4\cdot
5 = 1\cdot 2\cdot (3\vee 4\cdot 5),\\
\neg 1\cdot 2\cup 11011 &=& \neg 1\cdot 2\vee 1\cdot 2\cdot \neg
3\cdot 4\cdot 5
= 2\cdot (\neg 1\vee \neg 3\cdot 4\cdot 5),\\
\neg 1\cdot \neg 2\cdot 3 \backslash 00100 &=& \neg 1\cdot \neg
2\cdot 3\cdot (4\vee \cdot
5),\\
4\cdot 5 \backslash 11011 &= &4\cdot 5\cdot (\neg 1\vee \neg
2\vee 3), \\
\neg 4\cdot 5\cup 11011 &=& \neg 4\cdot 5\vee 1\cdot 2\cdot \neg
3\cdot 4\cdot 5
= (1\cdot 2\cdot \neg 3\vee \neg 4)\cdot 5.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
G &=& [1\cdot 2\cdot (3\vee 4\cdot 5), 2\cdot (\neg 1\vee \neg
3\cdot 4\cdot 5), \neg
1\cdot \neg 2\cdot 3\cdot (4\vee \cdot 5),\\
& & \qquad 4\cdot 5\cdot (\neg 1\vee \neg 2\vee 3),(1\cdot
2\cdot \neg 3\vee \neg
4)\cdot 5]
\end{eqnarray*}
is a threshold transformation. $G$ has a unique $24$-cycle. In
this case, generalization to
the general dimension is impossible. 

Neither $G$ nor $\bar{\neg}G$ is PDNN-definable according to
Proposition 6.4.7. A striking fact is that unlike attractors
obtained by the enhanced Arimoto theorem and the present example,
all of the attractors obtained so far in our PDNN model consist
of one or a few cycles found in the graphs of Boolean isometries. 
Whether this observation generally holds or not is an open
question.\\

\textbf{Open Question} If a self-dual minimal transformation $F$
has an attractor $\Psi$
such that $\mathrm{Im}\Psi \subseteq \mathrm{Car}F$, then is it
true that $\Psi \subseteq
\mathrm{CY}(T)$ for some Boolean isometry $T$?.\\


\section*{7.8 A temporary review}
In this chapter we constructed self-dual minimal threshold
transformations having attractors and developed some systematic
methods and tools for proving the attractiveness.
The transformations and their attractors are easily converted to
those for PDNNS defined in Chapter 6. Seeing from what has been
described so far about non-loop attractors, we can expect
mathematically rich as well as difficult contents as a dynamical
system even in the present simplified PDNN model. 

However, unlike attractors obtained by the enhanced Arimoto
theorem, all non-trivial attractors obtained so far in the class
of minimal self-dual transformations consist of one or a few
cycles found in the graphs of Boolean isometries.
Here, differences from Boolean isometries do not lie in more
complex cycles but in the selection of the few cycles from the
cycles generated by Boolean isometries. Whether this observation
generally holds or not is a mathematical open question.

The present model is autonomous, that is, stable periodic
firing patterns that are represented by attractors are completely
determined by the efficacy matrices of synaptic connections and
the initial states of the neurons at time $t = 0$. However, the
dynamics of any biological system depends on information that
changes at every unit time and that is input from the outside of
the system, from neurons of other nervous systems and/or from
external stimulus. In autonomous models, if a minimal attractor
consists of more than one cycle, then there are some fluidity of
shifting from one pattern to another caused by noise, even with a
change in firing rate in some cases. For example, the attractor
in Example 3.7 of Chapter 6 consists of two completely different
states $1111$ and $-1-1-1-1$, and if the state shifts from 1111
to $11-11$, then it converges to $-1-1-1-1$. This problem may be
solved only in a non-autonomous model with input from outside the
network.

Further, the post-synaptic potential in this model
incorporates only spatial summation and no temporal summation
(See Kalat, 1995). As a result, the firing rate of any neuron
cannot exceed 2 times the spontaneous firing rate. Still
further, the rigid synchronization (alignment) of firing for all
neurons is unrealistic. Some of these problems can be overcome
by modifying the present model and extending these means,
although things become more complex. The trade-off is that we
have a greater variety of stable periodic patterns represented by
attractors. In fact, the present combinatorial approach can be
extended to a modified model with incorporation of temporal
summation in Chapter 8, and some results on non-autonomous
networks will be described in Chapter 9.

\end{document}

