# Tower multiplexing and slow weak mixing

Terrence M. Adams  Terrence M. Adams is with the Department of Defense, 9161
Sterling Drive Laurel, MD 20723 United States of America

###### Abstract

A technique is presented for multiplexing two ergodic measure preserving
transformations together to derive a third limiting transformation. This
technique is used to settle a question regarding rigidity sequences of weak
mixing transformations. Namely, given any rigidity sequence for an ergodic
measure preserving transformation, there exists a weak mixing transformation
which is rigid along the same sequence. This establishes a wide range of
rigidity sequences for weakly mixing dynamical systems.

##  1 Introduction

Fix a Lebesgue probability space. Endow the set of invertible measure
preserving transformations with the weak topology. It is well known that both
the properties of weak mixing and rigidity are generic properties in this
topological space [16]. This is interesting since the behaviors of these two
properties contrast greatly. Weak mixing occurs when a system equitably
spreads mass throughout the probability space for most times. Rigidity occurs
when a system evolves to resemble the identity map infinitely often. Since
both of these behaviors exist simultaneously in a large class of
transformations, it is natural to ask what types of rigidity sequences are
realizable by weak mixing transformations. Here we resolve this question to
show all rigidity sequences are realizable by the class of weak mixing
transformations.

######  Theorem 1.1.

Given any ergodic measure preserving transformation $R$ on a Lebesgue
probability space, and any rigid sequence $\rho_{n}$ for $R$, there exists a
weak mixing transformation $T$ on a Lebesgue probability space such that $T$
is rigid on $\rho_{n}$.

Prior to proving this main result, we present a new and direct method for
combining two invertible ergodic finite measure preserving transformations to
obtain a third limiting transformation. The technique iteratively utilizes the
Kakutani-Rokhlin lemma ([19],[24]). A measure preserving transformation $T$ on
a separable probability space $(X,\mathbb{B},\mu)$ is ergodic if any invariant
measurable set $A$ has measure 0 or 1. In particular, $TA=A$ implies
$\mu(A)=\mbox{0 or 1}$.

######  Lemma 1.2.

(Kakutani 1943, Rokhlin 1948) Let $T:X\to X$ be an ergodic measure preserving
transformation on a nonatomic probability space $(X,\mathcal{B},\mu)$, $h$ a
positive integer and $\epsilon>0$. There exists a measurable set $B\subset X$
such that $B,TB,\ldots,T^{h-1}B$ are pairwise disjoint and
$\mu(\bigcup_{i=0}^{h-1}T^{i}(B))>1-\epsilon$. The collection
$\\{B,TB,\ldots,T^{h-1}B\\}$ is referred to as a Rokhlin tower of height $h$
for transformation $T$.

Clearly, this lemma demonstrates that any ergodic measure preserving
transformation can be approximated arbitrarily well by periodic
transformations in an appropriate topology (i.e. uniform topology). See Halmos
[17], [16], Rokhlin [25], Katok and Stepin [22]. Much of the early work in
this regard focuses on the topological genericity of specific properties of
measure perserving transformations. In [22], results are presented on rates of
approximation by periodic transformations, and connections with dynamical
properties. Recent research of Kalikow demonstrates the utility of developing
a general theory of Rokhlin towers [20]. Also, it is clear from the Kakutani-
Rokhlin lemma that any ergodic measure preserving tranformation can be
approximated arbitrarily well by another ergodic measure preserving
transformation from any isomorphism class. This observation is utilized
repeatedly in this work.

Two input transformations $R$ and $S$ are multiplexed together to derive an
output transformation $T$ with prescribed properties. The multiplexing
operation is defined using an infinite chain of measure theoretic
isomorphisms. In the case where $R$ is ergodic and rigid, and $S$ weak mixing,
we present a method for unbalanced multiplexing of $R$ and $S$. Over time,
transformations isomorphic to $R$ are used on a higher proportion of the
measure space, as the action by $S$ dissipates over time. We refer to this
process informally as slow weak mixing.

A measure preserving transformation $T:X\to X$ is weak mixing, if for all
measurable sets $A$ and $B$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}|\mu(T^{i}A\cap
B)-\mu(A)\mu(B)|=0.$

Clearly, if $T$ is weak mixing, then $T$ is ergodic. Also, $T$ is weak mixing,
if and only if $T$ has only 1 as an eigenvalue, and all eigenfunctions are
constants almost everywhere. An ergodic measure preserving transformation $R$
is rigid on a sequence $\rho_{n}\to\infty$, if for any measurable set $A$,

$\lim_{n\to\infty}\mu(T^{\rho_{n}}A\triangle A)=0.$

The sequence $\rho_{n}$ is called a rigidity sequence for $R$.

Several forms of rigidity have been studied in both ergodic theory and
topological dynamics. In the case of topological dynamics, both rigidity and
uniform rigidity are considered. Uniform rigidity was introduced in [14] and
given a specific generic characterization. In [18], it is shown the notion of
uniform rigidity is mutually exclusive from measurable weak mixing on a Cantor
set. In particular, every finite measure preserving weak mixing transformation
has a representation that is not uniformly rigid. Weak mixing and rigidity
have been studied for interval exchange transformations. See [7] and [3] for
recent results in this regard. Rigid, weak mixing transformations have been
studied in the setting of infinite measure preserving transformations, as well
as nonsingular transformations. Mildly mixing transformations are finite
measure preserving transformations that do not contain a rigid factor. These
are the transformations which yield ergodic products with any infinite ergodic
transformation [13]. See works [1], [2], [4] and the references therein for
results related to notions of weak mixing and rigidity for infinite measure
preserving or nonsingular transformations. The notion of IP-sequences was
introduced by Furstenberg and Weiss in connection with rigid transformations.
There has been recent research on IP-rigidity sequences (i.e. IP-sequences
which form a rigid sequence) for weak mixing transformations. See [4] and [15]
for results on IP-rigidity.

The notion of rigidity was extended to $\alpha$-rigidity by Friedman [10].
Transformations are constructed which are $\alpha$-rigid and
$(1-\alpha)$-partial mixing for any $0<\alpha<1$. See [12] and [9] for further
research on $\alpha$-rigid transformations. Many of these notions have been
studied for more general group actions. See [5] for a survey of weak mixing
group actions. Since our results depend mainly on the use of Lemma 1.2 which
extends to more general groups (i.e. amenable, abelian), there should exist an
extension of techniques provided in this work to a wider class of groups.
Since some of the principles provided in this work appear new, we focus
exclusively on the case of measure preserving $\mathbb{Z}$-actions on $[0,1)$
with Lebesgue measure.

For a recent comprehensive account on rigidity sequences, we recommend recent
publications [4] and [8]. Both of these works provide much detail on the
current understanding of rigidity for weak mixing transformations.

##  2 Towerplex Constructions

The main result is established constructively using Lemma 1.2. Given two
transformations $R$ and $S$, we define a third transformation $T$ which is
constructed as a blend of $R$ and $S$, such that $T$ acts more like $R$,
asymptotically. We will define a sequence of positive integers
$h_{n},n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, and a sequence of real numbers
$\epsilon_{n}>0$ such that $\sum_{n=1}^{\infty}{1}/{h_{n}}<\infty$ and
$\sum_{n=1}^{\infty}\epsilon_{n}<\infty$. Also, let $r_{n}$ and $s_{n}$ for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ be sequences of real numbers
satisfying: $0\leq r_{n},s_{n}\leq 1$.

###  2.1 Initialization

Suppose $R$ and $S$ are ergodic measure preserving transformations defined on
a Lebesgue probability space $(X,\mu,\mathbb{B})$. Partition $X$ into two
equal sets $X_{1}$ and $Y_{1}$ (i.e. $\mu(X_{1})=\mu(Y_{1})=1/2$). Initialize
$R_{1}$ isomorphic to $R$ and $S_{1}$ isomorphic to $S$ to operate on $X_{1}$
and $Y_{1}$, respectively. Define $T_{1}(x)=R_{1}(x)$ for $x\in X_{1}$ and
$T_{1}(x)=S_{1}(x)$ for $x\in Y_{1}$. Produce Rohklin towers of height $h_{1}$
with residual less than $\epsilon_{1}/2$ for each of $R_{1}$ and $S_{1}$. In
particular, let $I_{1},J_{1}$ be the base of the $R_{1}$-tower and
$S_{1}$-tower such that
$\mu(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}I_{1})>1/2(1-\epsilon_{1})$ and
$\mu(\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}J_{1})>1/2(1-\epsilon_{1})$. Let
$X_{1}^{*}=X_{1}\setminus\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1})$ and
$Y_{1}^{*}=Y_{1}\setminus\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1})$ be the
residuals for the $R_{1}$ and $S_{1}$ towers, respectively. Choose
$I^{\prime}_{1}\subset I_{1}$ and $J^{\prime}_{1}\subset J_{1}$ such that

$\mu(I^{\prime}_{1})=r_{1}\mu(I_{1})\mbox{ and
}\mu(J^{\prime}_{1})=s_{1}\mu(J_{1}).$

Set
$X_{2}=X_{1}\setminus[\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{\prime})]\cup[\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{\prime})]$
and
$Y_{2}=Y_{1}\setminus[\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{\prime})]\cup[\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{\prime})]$.
We will define second stage transformations $R_{2}:X_{2}\to X_{2}$ and
$S_{2}:Y_{2}\to Y_{2}$. First, it may be necessary to add or subtract measure
from the residuals so that $X_{2}$ is scaled properly to define $R_{2}$, and
$Y_{2}$ is scaled properly to define $S_{2}$.

###  2.2 Tower Rescaling

In the case where $\mu(I_{1}^{\prime})\neq\mu(J_{1}^{\prime})$, we give a
procedure for transferring measure between the towers and the residuals. This
is done in order to consistently define $R_{2}$ and $S_{2}$ on the new
inflated or deflated towers. Let
$a=\mu(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}I_{1})$ and
$b=h_{1}(\mu(J^{\prime}_{1})-\mu(I^{\prime}_{1}))$. Let $c$ be the scaling
factor and $d$ the amount of measure transferred between
$\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J^{\prime}_{1})$ and $X_{1}^{*}$. Thus,
$a+b-d=ca$ and $1/2-a+d=c(1/2-a)$. The goal is to solve two unknowns $d$ and
$c$ in terms of the other values. Hence, $d=(1-2a)b$ and $c=1+2b$.

####  2.2.1 $R$ Rescaling

If $d>0$, define $I_{1}^{*}\subset J^{\prime}_{1}$ such that
$\mu(I_{1}^{*})=d/h_{1}$. Let
$X_{1}^{\prime}=X_{1}^{*}\cup(\bigcup_{i=0}^{h_{1}-1}R_{1}^{i}(I_{1}^{*}))$.
If $d=0$, set $X_{1}^{\prime}=X_{1}^{*}$. If $d<0$, transfer measure from
$X_{1}^{*}$ to the tower. Choose disjoint sets
$I_{1}^{*}(0),I_{1}^{*}(1),\ldots,I_{1}^{*}(h_{1}-1)$ contained in $X_{1}^{*}$
such that $\mu(I_{1}^{*}(i))=d/h_{1}$. Denote $I_{1}^{*}=I_{1}^{*}(0)$. Begin
by defining $\mu$ measure preserving map $\alpha_{1}$ such that
$I_{1}^{*}(i+1)=\alpha_{1}(I_{1}^{*}(i))$ for $i=0,1,\ldots,h_{1}-2$. In this
case, let
$X_{1}^{\prime}=X_{1}^{*}\setminus[\bigcup_{i=0}^{h_{1}-1}I_{1}^{*}(i)]$.

####  2.2.2 $S$ Rescaling

The direction mass is transferred depends on the sign of $b$ above. If $d>0$,
then $\mu(J^{\prime}_{1})>\mu(I^{\prime}_{1})$ and mass is transferred from
the residual $Y_{1}^{*}$ to the $S_{1}$-tower. Choose disjoint sets
$J_{1}^{*}(0),J_{1}^{*}(1),\ldots,J_{1}^{*}(h_{1}-1)$ contained in $Y_{1}^{*}$
such that $\mu(J_{1}^{*}(i))=d/h_{1}$. Denote $J_{1}^{*}=J_{1}^{*}(0)$. Begin
by defining $\mu$ measure preserving map $\beta_{1}$ such that
$J_{1}^{*}(i+1)=\beta_{1}(J_{1}^{*}(i))$ for $i=0,1,\ldots,h_{1}-2$. In this
case, let
$Y_{1}^{\prime}=Y_{1}^{*}\setminus[\bigcup_{i=0}^{h_{1}-1}J_{1}^{*}(i)]$. If
$d=0$, set $Y_{1}^{\prime}=Y_{1}^{*}$. If $d<0$, transfer measure from the
$S_{1}$-tower to the residual $Y_{1}^{*}$. Define $J_{1}^{*}\subset
J_{1}\setminus J^{\prime}_{1}$ such that $\mu(J_{1}^{*})=d/h_{1}$. Let
$Y_{1}^{\prime}=Y_{1}^{*}\cup(\bigcup_{i=0}^{h_{1}-1}S_{1}^{i}(J_{1}^{*}))$.

Note, if $d\neq 0$, then both $\epsilon_{1}$ and $\mu(X_{1}^{*})$ may be
chosen small enough (relative to $r_{1}$) to ensure the following solutions
lead to well-defined sets and mappings. For subsequent stages, assume
$\epsilon_{n}$ is chosen small enough to force well-defined rescaling
parameters, transfer sets and mappings $R_{n}$, $S_{n}$.

###  2.3 Stage 2 Construction

We have specified three cases: $d>0$, $d=0$ and $d<0$. The case $d=0$, can be
handled along with the case $d>0$. This gives two essential cases. Note the
case $d<0$ is analogous to the case $d>0$, with the roles of $R_{1}$ and
$S_{1}$ reversed. However, due to a key distinction in the handling of the
$R$-rescaling and the $S$-rescaling, it is important to clearly define $R_{2}$
and $S_{2}$ in both cases.

######  Case 2.1 ($d\geq 0$).

Define $\tau_{1}:X_{1}^{\prime}\to X_{1}^{*}$ as a measure preserving map
between normalized spaces $(X_{1}^{\prime},\mathbb{B}\cap
X_{1}^{\prime},\frac{\mu}{\mu(X_{1}^{\prime})})$ and
$(X_{1}^{*},\mathbb{B}\cap X_{1}^{*},\frac{\mu}{\mu(X_{1}^{*})})$. Extend
$\tau_{1}$ to the new tower base,

$\tau_{1}:[I_{1}\setminus I_{1}^{\prime}]\cup[J^{\prime}_{1}\setminus
I_{1}^{*}]\to I_{1}$

such that $\tau_{1}$ preserves normalized measure between

$\frac{\mu}{\mu([I_{1}\setminus I_{1}^{\prime}]\cup[J^{\prime}_{1}\setminus
I_{1}^{*}])}\mbox{ and }\frac{\mu}{\mu(I_{1})}.$

Define $\tau_{1}$ on the remainder of the tower consistently such that

$\displaystyle\tau_{1}(x)=\left\\{\begin{array}[]{ll}R_{1}^{i}\circ\tau_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ R_{1}^{i}\circ\tau_{1}\circ S_{1}^{-i}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1}\setminus I_{1}^{*})$ for $0\leq
i<h_{1}$}\end{array}\right.$

Define $R_{2}:X_{2}\to X_{2}$ as $R_{2}=\tau_{1}^{-1}\circ
R_{1}\circ\tau_{1}$. Note

$\displaystyle R_{2}(x)=\left\\{\begin{array}[]{ll}S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1}\setminus I_{1}^{*})$ for $0\leq i<h_{1}-1$}\\\
R_{1}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I^{\prime}_{1})$ for $0\leq
i<h_{1}-1$}\end{array}\right.$

Clearly, $R_{2}$ is isomorphic to $R_{1}$ and $R$.

Define $\psi_{1}:Y_{1}^{\prime}\to Y_{1}^{*}$ as a measure preserving map
between normalized spaces $(Y_{1}^{\prime},\mathbb{B}\cap
Y_{1}^{\prime},\frac{\mu}{\mu(Y_{1}^{\prime})})$ and
$(Y_{1}^{*},\mathbb{B}\cap Y_{1}^{*},\frac{\mu}{\mu(Y_{1}^{*})})$. Extend
$\psi_{1}$ to the new tower base,

$\psi_{1}:[J_{1}\setminus J_{1}^{\prime}]\cup J_{1}^{*}\cup I^{\prime}_{1}\to
J_{1}$

such that $\psi_{1}$ preserves normalized measure between

$\frac{\mu}{\mu([J_{1}\setminus J_{1}^{\prime}]\cup J_{1}^{*}\cup
I^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(J_{1})}.$

Define $\psi_{1}$ on the remainder of the tower consistently such that

$\displaystyle\psi_{1}(x)=\left\\{\begin{array}[]{ll}S_{1}^{i}\circ\psi_{1}\circ
S_{1}^{-i}(x)&\mbox{if $x\in S_{1}^{i}(J_{1}\setminus J_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ S_{1}^{i}\circ\psi_{1}\circ R_{1}^{-i}(x)&\mbox{if $x\in
R_{1}^{i}(I^{\prime}_{1})$ for $0\leq i<h_{1}$}\\\
\beta_{1}^{i}\circ\psi_{1}\circ\beta_{1}^{-i}(x)&\mbox{if $x\in J_{1}^{*}(i)$
for $0\leq i<h_{1}$}\end{array}\right.$

In this case, define $S_{2}:Y_{2}\to Y_{2}$ such that
$S_{2}=\psi_{1}^{-1}\circ S_{1}\circ\psi_{1}$. Note

$\displaystyle S_{2}(x)=\left\\{\begin{array}[]{ll}R_{1}(x)&\mbox{if $x\in
R_{1}^{i}I^{\prime}_{1}$ for $0\leq i<h_{1}-1$}\\\ S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J_{1}\setminus J^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\
\beta_{1}(x)&\mbox{if $x\in J_{1}^{*}(i)$ for $0\leq i<h_{1}-1$}\\\
\psi_{1}^{-1}\circ S_{1}\circ\psi_{1}(x)&\mbox{if $x\in Y_{1}^{\prime}\cup
S_{1}^{h_{1}-1}(J_{1}\setminus J_{1}^{\prime})\cup
R_{1}^{h_{1}-1}I^{\prime}_{1}\cup\beta_{1}^{h_{1}-1}J_{1}^{*}$}\end{array}\right.$

and $S_{2}$ is isomorphic to $S_{1}$ and $S$.

######  Case 2.2 ($d<0$).

Define $\tau_{1}:X_{1}^{\prime}\to X_{1}^{*}$ as a measure preserving map
between normalized spaces $(X_{1}^{\prime},\mathbb{B}\cap
X_{1}^{\prime},\frac{\mu}{\mu(X_{1}^{\prime})})$ and
$(X_{1}^{*},\mathbb{B}\cap X_{1}^{*},\frac{\mu}{\mu(X_{1}^{*})})$. Extend
$\tau_{1}$ to the new tower base,

$\tau_{1}:[I_{1}\setminus I_{1}^{\prime}]\cup I_{1}^{*}\cup J^{\prime}_{1}\to
I_{1}$

such that $\tau_{1}$ preserves normalized measure between

$\frac{\mu}{\mu([I_{1}\setminus I_{1}^{\prime}]\cup I_{1}^{*}\cup
J^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(I_{1})}.$

Define $\tau_{1}$ on the remainder of the tower consistently such that

$\displaystyle\tau_{1}(x)=\left\\{\begin{array}[]{ll}R_{1}^{i}\circ\tau_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I_{1}\setminus I_{1}^{\prime})$ for
$0\leq i<h_{1}$}\\\ R_{1}^{i}\circ\tau_{1}\circ S_{1}^{-i}(x)&\mbox{if $x\in
S_{1}^{i}(J^{\prime}_{1})$ for $0\leq i<h_{1}$}\\\
\alpha_{1}^{i}\circ\tau_{1}\circ\alpha_{1}^{-i}(x)&\mbox{if $x\in
I_{1}^{*}(i)$ for $0\leq i<h_{1}$}\end{array}\right.$

In this case, define $R_{2}:X_{2}\to X_{2}$ such that

$\displaystyle R_{2}(x)=\left\\{\begin{array}[]{ll}S_{1}(x)&\mbox{if $x\in
S_{1}^{i}J^{\prime}_{1}$ for $0\leq i<h_{1}-1$}\\\ R_{1}(x)&\mbox{if $x\in
R_{1}^{i}(I_{1}\setminus I^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\
\alpha_{1}(x)&\mbox{if $x\in I_{1}^{*}(i)$ for $0\leq i<h_{1}-1$}\\\
\tau_{1}^{-1}\circ R_{1}\circ\tau_{1}(x)&\mbox{if $x\in X_{1}^{\prime}\cup
R_{1}^{h_{1}-1}(I_{1}\setminus I_{1}^{\prime})\cup
S_{1}^{h_{1}-1}J^{\prime}_{1}\cup\alpha_{1}^{h_{1}-1}I_{1}^{*}$}\end{array}\right.$

Clearly, $R_{2}$ is isomorphic to $R_{1}$ and $R$.

Define $\psi_{1}:Y_{1}^{\prime}\to Y_{1}^{*}$ as a measure preserving map
between normalized spaces $(Y_{1}^{\prime},\mathbb{B}\cap
Y_{1}^{\prime},\frac{\mu}{\mu(Y_{1}^{\prime})})$ and
$(Y_{1}^{*},\mathbb{B}\cap Y_{1}^{*},\frac{\mu}{\mu(Y_{1}^{*})})$. Extend
$\psi_{1}$ to the new tower base,

$\psi_{1}:[J_{1}\setminus(J_{1}^{\prime}\cup J_{1}^{*})]\cup I^{\prime}_{1}\to
J_{1}$

such that $\psi_{1}$ preserves normalized measure between

$\frac{\mu}{\mu([J_{1}\setminus(J_{1}^{\prime}\cup J_{1}^{*})]\cup
I^{\prime}_{1})}\mbox{ and }\frac{\mu}{\mu(J_{1})}.$

Define $\psi_{1}$ on the remainder of the tower consistently such that

$\displaystyle\psi_{1}(x)=\left\\{\begin{array}[]{ll}S_{1}^{i}\circ\psi_{1}\circ
S_{1}^{-i}(x)&\mbox{if $x\in S_{1}^{i}(J_{1}\setminus[J_{1}^{\prime}\cup
J_{1}^{*}])$ for $0\leq i<h_{1}$}\\\ S_{1}^{i}\circ\psi_{1}\circ
R_{1}^{-i}(x)&\mbox{if $x\in R_{1}^{i}(I^{\prime}_{1})$ for $0\leq
i<h_{1}$}\end{array}\right.$

Define $S_{2}:Y_{2}\to Y_{2}$ such that $S_{2}=\psi_{1}^{-1}\circ
S_{1}\circ\psi_{1}$. Note

$\displaystyle S_{2}(x)=\left\\{\begin{array}[]{ll}R_{1}(x)&\mbox{if $x\in
R_{1}^{i}(I^{\prime}_{1})$ for $0\leq i<h_{1}-1$}\\\ S_{1}(x)&\mbox{if $x\in
S_{1}^{i}(J_{1}\setminus[J^{\prime}_{1}\cup J_{1}^{*}])$ for $0\leq
i<h_{1}-1$}\end{array}\right.$

Transformation $S_{2}$ is isomorphic to $S_{1}$ and $S$.

Define $T_{2}$ as

$\displaystyle T_{2}(x)=\left\\{\begin{array}[]{ll}R_{2}(x)&\mbox{if $x\in
X_{2}$}\\\ S_{2}(x)&\mbox{if $x\in Y_{2}$}\end{array}\right.$

Clearly, neither $T_{1}$ nor $T_{2}$ are ergodic. For $T_{1}$, $X_{1}$ and
$Y_{1}$ are ergodic components, and $X_{2}$, $Y_{2}$ are ergodic components
for $T_{2}$. See the appendix for a pictorial of the multiplexing operation
used to produce $R_{2}$ and $S_{2}$ from $R_{1}$, $S_{1}$ and the intermediary
maps defined in this section.

###  2.4 General Multiplexing Operation

For $n\geq 1$, suppose that $R_{n}$ and $S_{n}$ have been defined on $X_{n}$
and $Y_{n}$ respectively. Construct Rohklin towers of height $h_{n}$ for each
$R_{n}$ and $S_{n}$, and such that $I_{n}$ is the base of the $R_{n}$ tower,
$J_{n}$ is the base of the $S_{n}$ tower, and
$\mu(\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n})+\mu(\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n})>1-\epsilon_{n}$.
Let $I^{\prime}_{n}\subset I_{n}$ be such that
$\mu(I^{\prime}_{n})=r_{n}\mu(I_{n})$. Similarly, suppose
$J^{\prime}_{n}\subset J_{n}$ such that $\mu(J^{\prime}_{n})=s_{n}\mu(J_{n})$.

We define $R_{n+1}$ and $S_{n+1}$ by switching the subcolumns

$\\{I^{\prime}_{n},R_{n}(I^{\prime}_{n}),R_{n}^{2}(I^{\prime}_{n}),\ldots,R_{n}^{h_{n}-1}(I^{\prime}_{n})\\}$

and

$\\{J^{\prime}_{n},S_{n}(J^{\prime}_{n}),S_{n}^{2}(J^{\prime}_{n}),\ldots,S_{n}^{h_{n}-1}(J^{\prime}_{n})\\}.$

Let

$\displaystyle X_{n+1}$ $\displaystyle=$
$\displaystyle[\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}\setminus
I_{n}^{\prime})]\cup[\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n}^{\prime}]\cup[X_{n}\setminus\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n}]$
$\displaystyle Y_{n+1}$ $\displaystyle=$
$\displaystyle[\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}(J_{n}\setminus
J_{n}^{\prime})]\cup[\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}I_{n}^{\prime}]\cup[Y_{n}\setminus\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}J_{n}].$

As in the initial case, it may be necessary to transfer measure between each
column and its respective residual. We can follow the same algorithm as above,
and define maps $\tau_{n},\alpha_{n},\psi_{n}$ and $\beta_{n}$. Thus, we get
the following definitions:

######  Case 2.3 ($d\geq 0$).

$\displaystyle\tau_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}^{i}\circ\tau_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ R_{n}^{i}\circ\tau_{n}\circ S_{n}^{-i}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n}\setminus I_{1}^{*})$ for $0\leq
i<h_{n}$}\end{array}\right.$ $\displaystyle R_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n}\setminus I_{n}^{*})$ for $0\leq i<h_{n}-1$}\\\
R_{n}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I^{\prime}_{n})$ for $0\leq
i<h_{n}-1$}\\\ \tau_{n}^{-1}\circ R_{n}\circ\tau_{n}(x)&\mbox{if $x\in
X_{n}^{\prime}\cup R_{n}^{h_{n}-1}(I_{n}\setminus I_{n}^{\prime})\cup
S_{n}^{h_{n}-1}(J^{\prime}_{n}\setminus I_{n}^{*})$}\end{array}\right.$

and $R_{n+1}=\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}$.

$\displaystyle\psi_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}^{i}\circ\psi_{n}\circ
S_{n}^{-i}(x)&\mbox{if $x\in S_{n}^{i}(J_{n}\setminus J_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ S_{n}^{i}\circ\psi_{n}\circ R_{n}^{-i}(x)&\mbox{if $x\in
R_{n}^{i}(I^{\prime}_{n})$ for $0\leq i<h_{n}$}\\\
\beta_{n}^{i}\circ\psi_{n}\circ\beta_{n}^{-i}(x)&\mbox{if $x\in J_{n}^{*}(i)$
for $0\leq i<h_{n}$}\end{array}\right.$ $\displaystyle S_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}(x)&\mbox{if $x\in
R_{n}^{i}I^{\prime}_{n}$ for $0\leq i<h_{n}-1$}\\\ S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J_{n}\setminus J^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\
\beta_{n}(x)&\mbox{if $x\in J_{n}^{*}(i)$ for $0\leq i<h_{n}-1$}\\\
\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}(x)&\mbox{if $x\in Y_{n}^{\prime}\cup
S_{n}^{h_{n}-1}(J_{n}\setminus J_{n}^{\prime})\cup
R_{n}^{h_{n}-1}I^{\prime}_{n}\cup\beta_{n}^{h_{n}-1}J_{n}^{*}$}\end{array}\right.$

and $S_{n+1}=\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}$.

######  Case 2.4 ($d<0$).

$\displaystyle\tau_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}^{i}\circ\tau_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I_{n}\setminus I_{n}^{\prime})$ for
$0\leq i<h_{n}$}\\\ R_{n}^{i}\circ\tau_{n}\circ S_{n}^{-i}(x)&\mbox{if $x\in
S_{n}^{i}(J^{\prime}_{n})$ for $0\leq i<h_{n}$}\\\
\alpha_{n}^{i}\circ\tau_{n}\circ\alpha_{n}^{-i}(x)&\mbox{if $x\in
I_{n}^{*}(i)$ for $0\leq i<h_{n}$}\end{array}\right.$ $\displaystyle
R_{n+1}(x)=$ $\displaystyle\left\\{\begin{array}[]{ll}S_{n}(x)&\mbox{if $x\in
S_{n}^{i}J^{\prime}_{n}$ for $0\leq i<h_{n}-1$}\\\ R_{n}(x)&\mbox{if $x\in
R_{n}^{i}(I_{n}\setminus I^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\
\alpha_{n}(x)&\mbox{if $x\in I_{n}^{*}(i)$ for $0\leq i<h_{n}-1$}\\\
\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}(x)&\mbox{if $x\in X_{n}^{\prime}\cup
R_{n}^{h_{n}-1}(I_{n}\setminus I_{n}^{\prime})\cup
S_{n}^{h_{n}-1}J^{\prime}_{n}\cup\alpha_{n}^{h_{n}-1}I_{n}^{*}$}\end{array}\right.$

and $R_{n+1}=\tau_{n}^{-1}\circ R_{n}\circ\tau_{n}$.

$\displaystyle\psi_{n}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}S_{n}^{i}\circ\psi_{n}\circ
S_{n}^{-i}(x)&\mbox{if $x\in S_{n}^{i}(J_{n}\setminus[J_{n}^{\prime}\cup
J_{n}^{*}])$ for $0\leq i<h_{n}$}\\\ S_{n}^{i}\circ\psi_{n}\circ
R_{n}^{-i}(x)&\mbox{if $x\in R_{n}^{i}(I^{\prime}_{n})$ for $0\leq
i<h_{n}$}\end{array}\right.$ $\displaystyle S_{n+1}(x)=$
$\displaystyle\left\\{\begin{array}[]{ll}R_{n}(x)&\mbox{if $x\in
R_{n}^{i}(I^{\prime}_{n})$ for $0\leq i<h_{n}-1$}\\\ S_{n}(x)&\mbox{if $x\in
S_{n}^{i}(J_{n}\setminus[J^{\prime}_{n}\cup J_{n}^{*}])$ for $0\leq
i<h_{n}-1$}\\\ \psi_{n}^{-1}\circ S_{n}\circ\psi_{n}(x)&\mbox{if $x\in
Y_{n}^{\prime}\cup S_{n}^{h_{n}-1}(J_{n}\setminus[J_{n}^{\prime}\cup
J_{n}^{*}])\cup R_{n}^{h_{n}-1}(I^{\prime}_{n})$}\end{array}\right.$

and $S_{n+1}=\psi_{n}^{-1}\circ S_{n}\circ\psi_{n}$.

###  2.5 The Limiting Transformation

Define the transformation $T_{n+1}:X_{n+1}\cup Y_{n+1}\to X_{n+1}\cup Y_{n+1}$
such that

$\displaystyle T_{n+1}(x)=\left\\{\begin{array}[]{ll}R_{n+1}(x)&\mbox{if $x\in
X_{n+1}$}\\\ S_{n+1}(x)&\mbox{if $x\in Y_{n+1}$}\end{array}\right.$

The set where $T_{n+1}\neq T_{n}$ is determined by the top levels of the
Rokhlin towers, the residual and the transfer set. Note the transfer set has
measure $d$. Since this set is used to adjust the size of the residuals
between stages, it can be bounded below a constant multiple of $\epsilon_{n}$.
Thus, there is a fixed constant $\kappa$, independent of $n$, such that
$T_{n+1}(x)=T_{n}(x)$ except for $x$ in a set of measure less than
$\kappa(\epsilon_{n}+{1}/{h_{n}})$. Since
$\sum_{n=1}^{\infty}(\epsilon_{n}+{1}/{h_{n}})<\infty$,
$T(x)=\lim_{n\to\infty}T_{n}(x)$ exists almost everywhere, and preserves
normalized Lebesgue measure. Without loss of generality, we may assume
$\kappa$ and $h_{n}$ are chosen such that if

$E_{n}=\\{x\in X|T_{n+1}(x)\neq T_{n}(x)\\}$

then $\mu(E_{n})<\kappa\epsilon_{n}$ for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. In the following section, additional
structure and conditions are implemented to ensure that $T$ inherits
properties from $R$ and $S$, and is also ergodic.

For the remainder of this paper, assume the parameters are chosen such that

  1. 1.

$\lim_{n\to\infty}r_{n}=0$;

  2. 2.

$\sum_{n=1}^{\infty}r_{n}=\sum_{n=1}^{\infty}s_{n}=\infty$;

  3. 3.

$\lim_{n\to\infty}\mu(Y_{n})=0$;

  4. 4.

$\sum_{n=1}^{\infty}\epsilon_{n}<\infty$.

###  2.6 Isomorphism Chain Consistency

In the following sections, rigidity and ergodicity will be established on sets
from a refining sequence of partitions. For
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, let $P_{n}$ be a refining sequence
of finite partitions which generates the sigma algebra. By refining $P_{n}$
further if necessary, assume $X_{n},Y_{n},X_{n}^{*},Y_{n}^{*}\in P_{n}$. Also,
assume $R_{n}^{i}(I^{\prime}_{n}),R_{n}^{i}(I_{n}\setminus
I^{\prime}_{n}),S_{n}^{i}(J^{\prime}_{n}),S_{n}^{i}(J_{n}\setminus
J^{\prime}_{n})$ are elements of $P_{n}$ for $0\leq i<h_{n}$. Finally, assume
for $0\leq i<h_{n}-1$, if $p\in P_{n}$ and $p\subset R_{n}^{i}(I_{n})$ then
$R_{n}(p)\in P_{n}$. Likewise, assume for $0\leq i<h_{n}-1$, if $p\in P_{n}$
and $p\subset S^{i}(J_{n})$ then $S_{n}(p)\in P_{n}$. Previously, we required
that $\tau_{n}$ map certain finite orbits from the $R_{n}$ and $S_{n}$ towers
to a corresponding orbit in the $R_{n+1}$ tower. In this section, further
regularity is imposed on $\tau_{n}$ relative to $P_{n}$ to ensure dynamical
properties of $R_{n}$ are inherited by $R_{n+1}$.

Let $P_{n}^{\prime}=\\{p\in
P_{n}|p\subset\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}\setminus
I_{n}^{\prime})\\}$. For each of the following three cases, impose the
corresponding restriction on $\tau_{n}$:

  1. 1.

for $d_{R}=0$ and $p\in P_{n}^{\prime}$, $\tau_{n}$ is the identity map (i.e.
$\tau_{n}(p)=p$);

  2. 2.

for $d_{R}>0$ and $p\in P_{n}^{\prime}$, $\tau_{n}(p)\subset p$;

  3. 3.

for $d_{R}<0$ and $p\in P_{n}^{\prime}$, $p\subset\tau_{n}(p)$.

This can be accomplished by uniformly distributing the appropriate mass from
the sets $R_{n}^{i}(I_{n}^{*})$ using $\tau_{n}$. Note that $\tau_{n}$ either
preserves Lebesgue measure in the case $d_{R}=0$, or $\tau_{n}$ contracts sets
relative to Lebesgue measure in the case $d_{R}>0$, or it inflates measure in
the case $d_{R}<0$. In all three cases, for $p\in P_{n}^{\prime}$,

$\frac{\mu(p)}{\mu(\tau_{n}(p))}=\frac{\mu(X_{n+1})}{\mu(X_{n})}.$

It is straightforward to verify for any set $A$ measurable relative to
$P_{n}^{\prime}$,

$\mu(A\triangle\tau_{n}A)<|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|.$

The properties of $\tau_{n}$ allow approximation of $R_{n+1}$ by $R_{n}$
indefinitely over time. This is needed to establish our rigidity sequence for
the limiting transformation $T$. This lemma is not required for establishing
ergodicity, but for convenience we will reuse it to prove our limiting $T$ is
ergodic.

######  Lemma 2.5.

Suppose $\delta>0$ and $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ is chosen such
that

$\displaystyle|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\frac{\delta}{7},$
$\displaystyle r_{n}+\epsilon_{n}+\mu(Y_{n})<\frac{\delta}{7}.$

Then for $A,B\in P_{n}$ and $i\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, the
following holds:

  1. 1.

$|\mu(R_{n+1}^{i}A\cap B)-\mu(A)\mu(B)|<|\mu(R_{n}^{i}A\cap
B)-\mu(A)\mu(B)|+\delta$;

  2. 2.

$\mu(R_{n+1}^{i}A\triangle A)<\mu(R_{n}^{i}A\triangle A)+\delta$.

###### Proof.

For $A,B\in P_{n}$, let $A^{\prime}=\bigcup_{p\in P_{n}^{\prime}}p\cap A$ and
$B^{\prime}=\bigcup_{p\in P_{n}^{\prime}}p\cap B$. Since
$\mu(\bigcup_{j=0}^{h_{n}-1}R_{n}^{j}(I_{n}^{\prime}))=h_{n}\mu(I_{n}^{\prime})<r_{n}$
and $\mu(X_{n}^{*})<\epsilon_{n}$, then $\mu(A\triangle
A^{\prime})<r_{n}+\epsilon_{n}<\frac{\delta}{7}$. Likewise, $\mu(B\triangle
B^{\prime})<\frac{\delta}{7}$. Since
$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\frac{\delta}{7}$, then
$\mu(A\triangle\tau_{n}A)<\frac{\delta}{7}$. By applying the triangle
inequality several times, we can get our approximations. Below is a sequence
of quantities to chain through such that consecutive values in the chain are
less than $\delta/7$ apart.

$\displaystyle\mu(R_{n+1}^{i}A\cap B)\rightarrow\mu(R_{n+1}^{i}A\cap
B^{\prime})\rightarrow\mu(R_{n+1}^{i}A^{\prime}\cap
B^{\prime})=\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\cap B^{\prime})$
$\displaystyle\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\cap
B^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\cap\tau_{n}B^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\cap
B^{\prime})$ $\displaystyle\rightarrow\mu(R_{n}^{i}A^{\prime}\cap
B^{\prime})\rightarrow\mu(R_{n}^{i}A^{\prime}\cap
B)\rightarrow\mu(R_{n}^{i}A\cap B)$

Each arrow in the chain signifies less than $\frac{\delta}{7}$ difference.
Hence,

$|\mu(R_{n+1}^{i}A\cap B)-\mu(R_{n}^{i}A\cap B)|<\delta$

which implies

$|\mu(R_{n+1}^{i}A\cap B)-\mu(A)\mu(B)|<|\mu(R_{n}^{i}A\cap
B)-\mu(A)\mu(B)|+\delta.$

The second part of the lemma can be proven in a similar fashion using the
triangle inequality, or chaining through the following six approximations.

$\displaystyle\mu(R_{n+1}^{i}A\triangle A)\rightarrow\mu(R_{n+1}^{i}A\triangle
A^{\prime})\rightarrow\mu(R_{n+1}^{i}A^{\prime}\triangle
A^{\prime})=\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\triangle A^{\prime})$
$\displaystyle\mu(\tau_{n}^{-1}R_{n}^{i}\tau_{n}A^{\prime}\triangle
A^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\triangle\tau_{n}A^{\prime})\rightarrow\mu(R_{n}^{i}\tau_{n}A^{\prime}\triangle
A^{\prime})$ $\displaystyle\rightarrow\mu(R_{n}^{i}A^{\prime}\triangle
A^{\prime})\rightarrow\mu(R_{n}^{i}A^{\prime}\triangle
A)\rightarrow\mu(R_{n}^{i}A\triangle A)$

Since each arrow indicates a difference less than $\frac{\delta}{7}$, then

$|\mu(R_{n+1}^{i}A\triangle A)-\mu(R_{n}^{i}A\triangle A)|<\delta.$

This completes the proof of the lemma. ∎

##  3 Establishing Rigidity

Suppose that $\rho_{n}$ is a rigidity sequence for $R$. In this section, we
define parameters such that $T$ is rigid on $\rho_{n}$.

###  3.1 Waiting for Rigidity

Let $\delta_{n}$ be a sequence of positive real numbers such that
$\lim_{n\to\infty}\delta_{n}=0$. Since $T_{n}|_{X_{n}}=R_{n}$ is rigid, choose
natural number $M_{n}^{1}>\max{\\{h_{n-1},M_{n-1}^{1}\\}}$ such that for
$N\geq M_{n}^{1}$, and $A\in P_{n-1}\cap X_{n}$,

$\mu(R_{n}^{\rho_{N}}A\triangle A)<\delta_{n}.$

Choose $\epsilon_{n}$ such that

$\displaystyle\frac{\epsilon_{n-1}}{M_{n}^{1}}<\epsilon_{n}.$ (19)

Also, without loss of generality, assume $h_{n}>M_{n}^{1}$. Below we show this
choice of $\epsilon_{n}$ is sufficient to produce
$T(x)=\lim_{n\to\infty}T_{n}(x)$ rigid on $\rho_{n}$. First, we provide a
diagram and heuristic description of our method for establishing rigidity on
$\rho_{n}$.

###  3.2 The Key Idea

$h_{n-1}$$M_{n}^{1}$$h_{n}$$M_{n+1}^{1}$$h_{n+1}$

To establish rigidity of $T$, we can focus on the asymptotic rigidity of $T$
on the intervals $(M_{n}^{1},M_{n+1}^{1}]$. We have chosen $M_{n}^{1}$
sufficiently large such that rigidity ”kicks in” for $R_{n}$ and
$\rho_{i}>M_{n}^{1}$. Lemma 2.5 allows us to approximate $R_{n}$ by $R_{n+1}$
as $\rho_{i}$ becomes closer to $M_{n+1}^{1}$. The fact that we can choose
$\epsilon_{n+1}$ arbitrarily small compared to ${1}/{M_{n+1}^{1}}$ allows us
to carry over the approximation to $T$. A precise proof is given below.

###  3.3 Rigidity Proof

If $E_{n+1}=\\{x\in X:T_{n+2}(x)\neq T_{n+1}(x)\\}$ and

$E_{n+1}^{1}=\bigcup_{i=0}^{M_{n+1}^{1}-1}[T_{n+2}^{-i}E_{n+1}\cup
T_{n+1}^{-i}E_{n+1}]$

then $\mu(E_{n+1}^{1})<2M_{n+1}^{1}\kappa\epsilon_{n+1}$. For $x\notin
E_{n+1}^{1}$, $T_{n+2}^{i}(x)=T_{n+1}^{i}(x)$ for $0\leq i\leq M_{n+1}^{1}$.
Let $\hat{E}_{n+1}^{1}=\bigcup_{k=n+1}^{\infty}E_{k}^{1}$. For
$x\notin\hat{E}_{n+1}^{1}$ and $0\leq i\leq M_{n+1}^{1}$,
$T^{i}(x)=T_{n+1}^{i}(x)$. Also,

$\mu(\hat{E}_{n+1}^{1})<\sum_{k=n+1}^{\infty}2M_{k}^{1}\kappa\epsilon_{k}<\sum_{k=n+1}^{\infty}2\kappa\epsilon_{k-1}\to
0$

as $n\to\infty$.

###### Proof of rigidity.

Let $A$ be a set in $P_{n_{1}}$ for some $n_{1}$, and let $\delta>0$. Choose
$n_{2}\geq n_{1}$ such that for $n\geq n_{2}$,

  1. 1.

$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\delta/28$;

  2. 2.

$r_{n}+\epsilon_{n}+\mu(Y_{n})<\delta/28$;

  3. 3.

$\delta_{n}<\delta/6$;

  4. 4.

$\sum_{i=n_{2}}^{\infty}2\kappa\epsilon_{i}<\delta/12$.

For $n>n_{2}$, let $M_{n}^{1}<N\leq M_{n+1}^{1}$,
$A_{1}=A\setminus\hat{E}_{n+1}^{1}$ and $A_{2}=A\cap X_{n}$. Thus,

$\displaystyle\mu(T^{\rho_{N}}A\triangle A)$ $\displaystyle\leq$
$\displaystyle\mu(T^{\rho_{N}}A\triangle
T^{\rho_{N}}A_{1})+\mu(T^{\rho_{N}}A_{1}\triangle A)$ $\displaystyle=$
$\displaystyle\mu(A\triangle A_{1})+\mu(R_{n+1}^{\rho_{N}}A_{1}\triangle A)$
$\displaystyle<$
$\displaystyle\frac{\delta}{4}+\mu(R_{n+1}^{\rho_{N}}A_{1}\triangle
R_{n+1}^{\rho_{N}}A)+\mu(R_{n+1}^{\rho_{N}}A\triangle A)$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\mu(R_{n+1}^{\rho_{N}}A\triangle A).$

By Lemma 2.5,

$\displaystyle\mu(T^{\rho_{N}}A\triangle A)$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\mu(R_{n+1}^{\rho_{N}}A\triangle
A)<\frac{3\delta}{4}+\mu(R_{n}^{\rho_{N}}A\triangle A)$ $\displaystyle\leq$
$\displaystyle\frac{3\delta}{4}+\mu(R_{n}^{\rho_{N}}A\triangle
R_{n}^{\rho_{N}}A_{2})+\mu(R_{n}^{\rho_{N}}A_{2}\triangle
A_{2})+\mu(A_{2}\triangle A)$ $\displaystyle<$
$\displaystyle\frac{3\delta}{4}+2\mu(Y_{n})+\delta_{n}<\delta.$

Therefore, $\rho_{n}$ is a rigidity sequence for $T$. ∎

##  4 Ergodicity

A measure preserving transformation $T$ on a Lebesgue space is ergodic if any
invariant set has measure zero or one. It is well known this is equivalent to
the mean and pointwise ergodic theorem. For our purposes, we use the following
equivalent condition of ergodicity: for all measurable sets $A$ and $B$,

$\lim_{N\to\infty}\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)=\mu(A)\mu(B).$

Let $P_{n},n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ be a sequence of finite
refining partitions as defined in the previous section. Using approximation,
$T$ is ergodic if the previous condition holds for all natural numbers $n$ and
sets $A$ and $B$ from $P_{n}$.

###  4.1 Ergodic Parameter Choice

Let $\delta_{n}$ be a sequence of positive real numbers such that
$\lim_{n\to\infty}\delta_{n}=0$. Since $T_{n}|_{X_{n}}=R_{n}$ is ergodic,
choose natural number $M_{n}=M_{n}^{2}$ such that for $N\geq M_{n}$, and sets
$A,B\in P_{n}\cap X_{n}$,

$|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})^{2}}|<\delta_{n}.$

Note that

$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T_{n}^{i}A\cap B)$
$\displaystyle-$ $\displaystyle\mu(A)\mu(B)|$ $\displaystyle=$
$\displaystyle\mu(X_{n})|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})}|$ $\displaystyle\leq$
$\displaystyle\mu(X_{n})|\frac{1}{N}\sum_{i=0}^{N-1}\frac{\mu(T_{n}^{i}A\cap
B)}{\mu(X_{n})}-\frac{\mu(A)\mu(B)}{\mu(X_{n})^{2}}|$ $\displaystyle+$
$\displaystyle|\frac{\mu(A)\mu(B)}{\mu(X_{n})}-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\delta_{n}+\frac{\mu(Y_{n})}{\mu(X_{n})}$

Choose $\epsilon_{n+1}$ such that

$\displaystyle\frac{\epsilon_{n}}{M_{n}}<\epsilon_{n+1}.$ (20)

###  4.2 Approximation

As previously, set $E_{n+1}=\\{x\in X:T_{n+2}(x)\neq T_{n+1}(x)\\}$. Let

$E_{n+1}^{2}=\bigcup_{i=0}^{M_{n+1}-1}[T_{n+2}^{-i}E_{n+1}\cup
T_{n+1}^{-i}E_{n+1}]$

Thus, $\mu(E_{n+1}^{2})<2M_{n+1}\kappa\epsilon_{n+1}$. For $x\notin
E_{n+1}^{2}$, $T_{n+2}^{i}(x)=T_{n+1}^{i}(x)$ for $0\leq i\leq M_{n+1}$. Let
$\hat{E}_{n+1}^{2}=\bigcup_{k=n+1}^{\infty}E_{k}^{2}$. For
$x\notin\hat{E}_{n+1}^{2}$ and $0\leq i\leq M_{n+1}$,
$T^{i}(x)=T_{n+1}^{i}(x)$. Also,

$\mu(\hat{E}_{n+1}^{2})<\sum_{k=n+1}^{\infty}2M_{k}\kappa\epsilon_{k}<\sum_{k=n+1}^{\infty}2\kappa\epsilon_{k-1}\to
0$

as $n\to\infty$.

###### Proof of ergodicity.

Let $A$ and $B$ be sets in $P_{n_{1}}$ for some $n_{1}$, and let $\delta>0$.
Choose $n_{2}\geq n_{1}$ such that for $n\geq n_{2}$,

  1. 1.

$|\frac{\mu(X_{n+1})}{\mu(X_{n})}-1|<\delta/28$;

  2. 2.

$r_{n}+\epsilon_{n}+\mu(Y_{n})<\delta/28$;

  3. 3.

$\delta_{n}+\frac{\mu(Y_{n})}{\mu(X_{n})}<\delta/4$;

  4. 4.

$\sum_{i=n_{2}}^{\infty}2\kappa\epsilon_{i}<\delta/12$.

For $n>n_{2}$, let $M_{n}<N\leq M_{n+1}$, $A_{1}=A\setminus\hat{E}_{n+1}^{2}$
and $B_{1}=B\setminus\hat{E}_{n+1}^{2}$.

$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)$ $\displaystyle-$
$\displaystyle\mu(A)\mu(B)|$ $\displaystyle\leq$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap
B)-\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A_{1}\cap B)|$ $\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A_{1}\cap B)-\mu(A)\mu(B)|$
$\displaystyle\leq$
$\displaystyle\frac{1}{N}\sum_{i=0}^{N-1}|\mu(T^{i}A)-\mu(T^{i}A_{1})|$
$\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A_{1}\cap
B)-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{4}+|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A_{1}\cap
B)-\mu(R_{n+1}^{i}A\cap B)|$ $\displaystyle+$
$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A\cap
B)-\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+|\frac{1}{N}\sum_{i=0}^{N-1}\mu(R_{n+1}^{i}A\cap
B)-\mu(A)\mu(B)|.$

Since $A,B\in P_{n}$, then by Lemma 2.5,

$\displaystyle|\frac{1}{N}\sum_{i=0}^{N-1}\mu(T^{i}A\cap B)$ $\displaystyle-$
$\displaystyle\mu(A)\mu(B)|$ $\displaystyle<$
$\displaystyle\frac{\delta}{2}+\frac{1}{N}\sum_{i=0}^{N-1}|\mu(R_{n+1}^{i}A\cap
B)-\mu(A\cap B)|$ $\displaystyle<$
$\displaystyle\frac{3\delta}{4}+\frac{1}{N}\sum_{i=0}^{N-1}|\mu(R_{n}^{i}A\cap
B)-\mu(A\cap B)|<\delta.$

Since $\delta$ is chosen arbitrarily, and the above holds for any $n>n_{2}$
and $M_{n}<N\leq M_{n+1}$, then $T$ is ergodic. ∎

##  5 Weak Mixing

Since the weak mixing component is dissipative, and the resulting
transformation inherits its rigidity properties from $R$, we do not focus on
multiplexing with general weak mixing transformations. Instead, we set $S$
equal to the famous Chacon transformation. It is defined via cutting and
stacking, and considered the earliest construction demonstrated to be weak
mixing and not mixing. See [11] for a precise definition. For the remainder of
this paper, assume both $R$ and $S$ are defined on $([0,1),\mu,\mathbb{B})$
where $\mu$ is Lebesgue measure. In this section, we further specify $h_{n}$
and switching sets $C_{n}=\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}^{\prime})$
for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. As in previous sections, all
conditions imposed are easily satisfied by choosing a faster growing sequence
of tower heights $h_{n}$. No upper bounds are imposed on the growth rate of
$h_{n}$.

###  5.1 Switching Set Definition

For each $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ and $n>k$, denote
$U_{k}^{n}=\bigcup_{j=k}^{n-1}C_{j}$, $V_{k}^{n}=(U_{k}^{n})^{c}$ and
$\dot{V}_{k}^{n}=V_{k}^{n}\cap X_{n}$. Since $R_{n}$ is ergodic on $X_{n}$,
$r_{n}$ is fixed, and $C_{n}$ predominantly represents long orbits of $R_{n}$,
then $h_{n}$ may be chosen sufficiently large such that $C_{n}$ is near
conditionally independent of $\dot{V}_{k}^{n}$ for each $k<n$.

Precisely, define $h_{n}$ and $C_{n}$ such that

$\displaystyle|\frac{\mu(C_{n}\cap\dot{V}_{k}^{n})}{\mu(X_{n})}-\frac{\mu(C_{n})\mu(\dot{V}_{k}^{n})}{\mu(X_{n})^{2}}|\leq\frac{1}{2}\mu(C_{n})\mu(\dot{V}_{k}^{n}).$
(21)

######  Lemma 5.1.

For each $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$,
$\lim_{n\to\infty}\mu(V_{k}^{n})=0$.

###### Proof.

Suppose the claim is not true, and there exists
$k_{0}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that

$\lim_{n\to\infty}\mu(V_{k}^{n})>0.$

Since $\lim_{n\to\infty}\mu(Y_{n})=0$, we can choose $k_{1}>k_{0}$ such that
$\mu(Y_{j})<\frac{1}{2}\mu(V_{k_{1}}^{k_{1}+n})$ for $j\geq k_{1}$ and
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. Thus,

$\displaystyle\frac{\mu(C_{k_{1}+1}\cap\dot{V}_{k_{1}}^{k_{1}+1})}{\mu(X_{k_{1}+1})}$
$\displaystyle\geq$
$\displaystyle\frac{\mu(C_{k_{1}+1})\mu(\dot{V}_{k_{1}}^{k_{1}+1})}{\mu(X_{k_{1}+1})^{2}}-\frac{1}{2}\mu(C_{k_{1}+1})\mu(\dot{V}_{k_{1}}^{k_{1}+1})$
$\displaystyle\mu(C_{k_{1}+1}\cap\dot{V}_{k_{1}}^{k_{1}+1})$
$\displaystyle\geq$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})\frac{\mu(V_{k_{1}}^{k_{1}+1}\cap
X_{k_{1}+1})}{\mu(V_{k_{1}}^{k_{1}+1})}\mu(C_{k_{1}+1})[\frac{1}{\mu(X_{k_{1}+1})}-\frac{\mu(X_{k_{1}+1})}{2}]$
$\displaystyle>$
$\displaystyle\frac{1}{4}\mu(C_{k_{1}+1})\mu(V_{k_{1}}^{k_{1}+1}).$

Hence,

$\displaystyle\mu(V_{k_{1}}^{k_{1}+2})$ $\displaystyle=$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})-\mu(C_{k_{1}+1}\cap
V_{k_{1}}^{k_{1}+1})$ $\displaystyle<$
$\displaystyle\mu(V_{k_{1}}^{k_{1}+1})[1-\frac{1}{4}\mu(C_{k_{1}+1})]$
$\displaystyle<$
$\displaystyle(1-\frac{1}{4}\mu(C_{k_{1}}))(1-\frac{1}{4}\mu(C_{k_{1}+1})).$

Extending this inductively produces

$\mu(V_{k_{1}}^{k_{1}+n})<\prod_{i=0}^{n-1}(1-\frac{1}{4}\mu(C_{k_{1}+i})).$

Note that

$\mu(C_{n})=\mu(I_{n}^{\prime})h_{n}=\frac{\mu(I_{n}^{\prime})}{\mu(I_{n})}\mu(I_{n})h_{n}=r_{n}\mu(X_{n}).$

Since $\sum_{n=1}^{\infty}r_{n}=\infty$ and $\lim_{n\to\infty}\mu(X_{n})=1$,
then both $\sum_{n=1}^{\infty}\mu(C_{n})=\infty$ and
$\sum_{n=1}^{\infty}\frac{1}{4}\mu(C_{n})=\infty$. This is sufficient to force
$\lim_{n\to\infty}\prod_{i=0}^{n-1}(1-\frac{1}{4}\mu(C_{k_{1}+i}))=0$ which
proves our claim by contradiction. ∎

The previous claim establishes the following property that almost every point
falls in infinitely many sets $C_{n}$.

######  Property 5.2.

$\mu(\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}C_{i})=1$.

###  5.2 Multiplexing Chacon’s Transformation

Chacon’s transformation $S$ is typically defined using cutting and stacking
[11]. Initialize $I_{1}^{0}=[0,2/3)$ and $\mathcal{C}_{1}=I_{1}^{0}$. Cut
$I_{1}$ into 3 pieces of equal width,
$I_{2}^{0}=[0,2/9),I_{2}^{1}=[2/9,4/9),I_{2}^{3}=[4/9,2/3)$, and add a single
spacer $I_{2}^{2}=[2/3,8/9)$ above interval $I_{2}^{1}$. Stack into a single
column $\mathcal{C}_{2}=<I_{2}^{0},I_{2}^{1},I_{2}^{2},I_{2}^{3}>$. Define $S$
as the linear map from $I_{2}^{i}$ to $I_{2}^{i+1}$ for $i=0,1,2$. Let
$H_{n}={(3^{n}-1)}/{2}$ be the height of column $\mathcal{C}_{n}$. Obtain
$\mathcal{C}_{n+1}$ by cutting $\mathcal{C}_{n}$ into 3 subcolumns of equal
width, $\mathcal{C}_{n}^{0},\mathcal{C}_{n}^{1},\mathcal{C}_{n}^{2}$, adding
one spacer above the second subcolumn and stacking left to right. Again, $S$
maps each level linearly to the level directly above it. Also, notice the
height of $\mathcal{C}_{n+1}$ equals $H_{n+1}=3H_{n}+1=\frac{3^{n+1}-1}{2}$.
The main property we utilize in this work is related to one of its limit
joinings.

######  Lemma 5.3.

Let $S$ be Chacon’s transformation. Given any two measurable sets, $A$ and
$B$,

$\lim_{n\to\infty}\mu(S^{H_{n}}A\cap B)=(\mu(A\cap B)+\mu(S^{-1}A\cap B))/2.$

###### Proof.

Each column $\mathcal{C}_{n}$, $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, has a
single level of spacer above precisely half the mass of the top level of
$\mathcal{C}_{n}$. This includes the spacers added when $\mathcal{C}_{n}$ is
cut into 3 subcolumns, as well as the infinitely many spacers added when
$\mathcal{C}_{n+1},\mathcal{C}_{n+2},\ldots$ are cut into 3 subcolumns and
stacked. Thus, $S^{H_{n}}$ maps half of each level to the same level, and maps
the other half to the level directly below itself. This establishes the lemma
for sets consisting of a finite union of levels. Since the levels of the
columns form a refining sequence of partitions which generate the sigma
algebra, the lemma follows by approximation. ∎

###  5.3 Weak Mixing Stage

Now we define $S_{n}$ inductively to ensure the final transformation $T$ is
weak mixing. Let $S_{1}$ be the Chacon transformation defined on $Y_{1}$.
Suppose $S_{n}\simeq S$ has been defined on $Y_{n}$. Now we specify the manner
in which $S_{n+1}$ should be defined.

####  5.3.1 Local Approximation of Switching Sets

Choose natural number $k_{n}>n$ such that for each $i=0,1,\ldots,h_{n}-1$,
there exists a finite collection of indices $\hat{K}_{n}^{i}$ and dyadic
intervals $K_{n}^{i}(j)$, $j\in\hat{K}_{n}^{i}$, such that
$\mu(K_{n}^{i}(j))=\frac{1}{2^{k_{n}}}$ and
$K_{n}^{i}=\bigcup_{j\in\hat{K}_{n}^{i}}K_{n}^{i}(j)$ satisfies
$\mu(R_{n}^{i}I_{n}^{\prime}\triangle
K_{n}^{i})<(\frac{\epsilon_{n}}{h_{n}})^{2}\mu(I_{n}^{\prime})$. Let
$\hat{G}_{n}^{i}=\\{j\in\hat{K}_{n}^{i}:\mu(R_{n}^{i}I_{n}^{\prime}\cap
K_{n}^{i}(j))>(1-\frac{\epsilon_{n}}{h_{n}})\mu(K_{n}^{i}(j))\\}$. It is not
difficult to show
$\mu(\bigcup_{j\in\hat{G}_{n}^{i}}K_{n}^{i}(j))>(1-\frac{\epsilon_{n}}{h_{n}})\mu(I_{n}^{\prime})$.
Set $G_{n}^{i}=\bigcup_{j\in\hat{G}_{n}^{i}}K_{n}^{i}(j)$. For each
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, define

$D_{n}=\bigcup_{\ell=0}^{h_{n}-1}G_{n}^{\ell}.$

Note that

$\mu(C_{n}\setminus
D_{n})<\sum_{\ell=0}^{h_{n}-1}\frac{\epsilon_{n}}{h_{n}}=\epsilon_{n}.$

Next, we show almost every point falls in infinitely many $D_{n}$.

######  Property 5.4.

$\mu(\overline{D})=1$ where
$\overline{D}=\bigcap_{n=1}^{\infty}\bigcup_{i=n}^{\infty}D_{i}$.

###### Proof.

Given $\epsilon>0$, choose
$N=N(\epsilon)\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\sum_{n=N}^{\infty}\epsilon_{n}<\epsilon$. Thus,

$\displaystyle\mu(\bigcup_{n=N}^{\infty}D_{n})$ $\displaystyle\geq$
$\displaystyle\mu(\bigcup_{n=N}^{\infty}C_{n})-\sum_{n=N}^{\infty}\mu(C_{n}\setminus
D_{n})$ $\displaystyle>$ $\displaystyle
1-\sum_{n=N}^{\infty}\epsilon_{n}>1-\epsilon.$

Since $\epsilon$ is arbitrarily small, then
$\mu(\bigcup_{n=N}^{\infty}D_{n})=1$, and Property 5.4 is establshed. ∎

####  5.3.2 Weak Mixing Component

The main goal in this work is to demonstrate how properties of a given ergodic
transformation can be transferred to produce a tailored weak mixing
transformation. Since the weak mixing component will dissipate over time, we
do not focus on introducing general properties using $S$. Instead, we set $S$
to the Chacon transformation inside our towerplex construction. Thus, $S_{n}$
will be isomorphic to Chacon’s transformation. By Lemma 5.3, for each
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, there exists
$m_{n}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that for each
$i=0,1,\ldots,h_{n}-1$, $j\in\hat{K}_{n}^{i}$ and $A=K_{n}^{i}(j)$,

$|\mu(S_{n+1}^{H_{m_{n}}}A\cap A)-\frac{1}{2}\mu(A)|<\epsilon_{n}\mu(A)$

and

$|\mu(S_{n+1}^{H_{m_{n}}}A\cap
S^{-1}(A)-\frac{1}{2}\mu(A)|<\epsilon_{n}\mu(A).$

Let $w_{n}=\min{\\{\mu(K_{\ell}^{i}(j))>0:1\leq\ell\leq n,0\leq i\leq
h_{n}-1,j\in\hat{K}_{\ell}^{i}\\}}$. Choose $h_{n+1}$ such that

$\displaystyle h_{n+1}>\frac{H_{m_{n}}}{\epsilon_{n}w_{n}}.$ (22)

##  6 Slow Weak Mixing Theorem

In this final section, we prove our main result using the towerplex
constructions. First, we give explicit parameters $r_{n}$ and $s_{n}$ that can
be used to generate our rigid weak mixing examples. Let
$r_{n}=\frac{\mu(I_{n}^{\prime})}{\mu(I_{n})}=\frac{1}{2(n+2)}$ and
$s_{n}=\frac{\mu(J_{n}^{\prime})}{\mu(J_{n})}=\frac{1}{2}$. Thus, each of the
switching sets have measure
$\mu(\bigcup_{i=0}^{h_{n}-1}R_{n}^{i}(I_{n}^{\prime}))={(\mu(X_{n})-\mu(X_{n}^{*}))}/{2(n+2)}$
and
$\mu(\bigcup_{i=0}^{h_{n}-1}S_{n}^{i}(J_{n}^{\prime}))={(\mu(Y_{n})-\mu(Y_{n}^{*}))}/2$
for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. This implies

$\mu(Y_{n+1})=\frac{1}{2(n+2)}[(n+1)\mu(Y_{n})+1]+\kappa_{n}\epsilon_{n}$

where $|\kappa_{n}|$ is bounded for all
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. If all residuals had zero mass, then
$\kappa_{n}\epsilon_{n}=0$ and by induction:

$\mu(X_{n})=\frac{n}{n+1}\mbox{ and }\mu(Y_{n})=\frac{1}{n+1}.$

In the case the residuals are not null, the next lemma obtains

$\lim_{n\to\infty}\mu(X_{n})=1\mbox{, }\lim_{n\to\infty}\mu(Y_{n})=0.$

Parameters given here are called the canonical towerplex settings.

######  Lemma 6.1.

If real numbers $\epsilon_{n}>0$ are chosen sufficiently small for
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, then a canonical towerplex
construction, given by $r_{n}=\frac{1}{2(n+2)}$ and $s_{n}=\frac{1}{2}$ has
the property, for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$,

$\displaystyle\frac{1}{n+2}<\mu(Y_{n})<\frac{1}{n}.$ (23)

###### Proof.

The function $f(y)=({1}/{2(n+2)})[(n+1)y+1]$ has a fixed point at
$y={1}/{(n+3)}$. If $y>{1}/{(n+3)}$, then $f(y)>{1}/{(n+3)}$. Thus, if
$\epsilon_{n}$ is sufficiently small, and $\mu(Y_{n})>{1}/{(n+2)}$, then
$\mu(Y_{n+1})>{1}/{(n+3)}$. This establishes the first inequality from (23).

To prove the second inequality, assume $y=\mu(Y_{n})<{1}/{n}$ for fixed
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$. Thus,

$\displaystyle f(y)$ $\displaystyle<$
$\displaystyle\frac{1}{2(n+2)}[(n+1)(\frac{1}{n})+1]=\frac{1}{2(n+2)}[2+\frac{1}{n}]$
$\displaystyle=$
$\displaystyle\frac{1}{n+2}+\frac{1}{2n(n+2)}=\frac{1}{n+1}+\frac{1-n}{2n(n+1)(n+2)}\leq\frac{1}{n+1}.$

Therefore, if $\epsilon_{n}$ is sufficiently small,
$\mu(Y_{n+1})<\frac{1}{n+1}$. ∎

Now, we are ready to prove our main theorem.

######  Theorem 6.2.

Given an ergodic measure preserving transformation $R$ on a Lebesgue
probability space, and a rigid sequence $\rho_{n}$ for $R$, there exists a
weak mixing transformation $T$ on a Lebesgue probability space such that $T$
is rigid on $\rho_{n}$.

###### Proof.

Much of the details have been established in the previous sections. In
particular, the conditions imposed in each of the sections on ergodicity,
rigidity and weak mixing, are consistent. Essentially, $\epsilon_{n}\to 0$
arbitrarily fast which is possible since only the extra mass from successive
Rokhlin towers is bounded by $\epsilon_{n}$. Also, each section imposes a
lower bound on the growth rate of the tower heights $h_{n}$, but no upper
bound. Appendix B lists conditions that can be used to support the explicit
proofs. Below, we need to complete the argument that $T$ is weak mixing.

Suppose $f\neq 0$ is an eigenfunction for $T$ with eigenvalue $\lambda$. Since
we established that $T$ is ergodic, we may assume $|f|$ is a constant. Without
loss of generality, assume $|f|=|\lambda|=1$. Given $\delta>0$, there exists a
set $\Lambda_{\delta}$ of positive measure such that for
$x,y\in\Lambda_{\delta}$, $|f(x)-f(y)|<\delta$. Let
$\Lambda_{\delta}^{\prime}$ be the set of Lebesgue density points of
$\Lambda_{\delta}$. In particular, if
$\Lambda_{\delta}^{\prime}=\\{x\in\Lambda_{\delta}:\lim_{\eta\to
0}\frac{\mu(\Lambda_{\delta}\cap(x-\eta,x+\eta))}{2\eta}=1\\}$, then
$\mu(\Lambda_{\delta}^{\prime})=\mu(\Lambda_{\delta})>0$. Choose
$x\in\Lambda_{\delta}^{\prime}\cap\overline{D}$. Choose $\eta^{\prime}>0$ such
that for $\eta<\eta^{\prime}$,
$\frac{\mu(\Lambda_{\delta}\cap(x-\eta,x+\eta))}{2\eta}>1-\delta$. Choose
$n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that
$\frac{1}{2^{k_{n}}}<\eta^{\prime}$, $\sum_{i=n}^{\infty}\epsilon_{i}<\delta$
and $x\in D_{n}$. There exists $i=i(x)$ such that $x\in G_{n}^{i}$, and
subsequently $j=j(x)$ such that $x\in K_{n}^{i}(j)$. Let
$\eta_{x}=\max{\\{|y-x|:y\in K_{n}^{i}(j)\\}}$. Note $\eta_{x}<\eta^{\prime}$,
and
$\frac{\mu(\Lambda_{\delta}\cap(x-\eta_{x},x+\eta_{x}))}{2\eta_{x}}>1-\delta$.
Thus,

$\displaystyle\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))$ $\displaystyle>$
$\displaystyle\mu(K_{n}^{i}(j))-2\eta_{x}\delta\mu(K_{n}^{i}(J))-2\delta\mu(K_{n}^{i}(j))$
(24) $\displaystyle\geq$ $\displaystyle(1-2\delta)\mu(K_{n}^{i}(j)).$ (25)

Hence,

$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle\leq$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\mu(S_{n+1}^{H_{m_{n}}}(K_{n}^{i}(j))\cap K_{n}^{i}(j))|$
$\displaystyle+$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(K_{n}^{i}(j))\cap(K_{n}^{i}(j)))-\frac{1}{2}\mu(K_{n}^{i}(j))|$
$\displaystyle+$
$\displaystyle|\frac{1}{2}\mu(K_{n}^{i}(j))-\frac{1}{2}\mu(\Lambda_{\delta}\cap
K_{n}^{i}(j))|$ $\displaystyle<$ $\displaystyle
4\delta\mu(K_{n}^{i}(j))+\epsilon_{n}\mu(K_{n}^{i}(j))+\delta\mu(K_{n}^{i}(j))=(5\delta+\epsilon_{n})\mu(K_{n}^{i}(j)).$

We wish to establish that $T$ is weak mixing, and $T$ does not equal $S_{n+1}$
everywhere. In particular, $T$ may differ from $S_{n+1}$ on the top levels of
the towers of height $h_{n+1},h_{n+2},\ldots$, on the accompanying residuals,
and on the transfer sets. However, we have chosen the growth of the tower
heights sufficient to ensure the set where $T$ and $S_{n+1}$ may differ will
be small relative to interval, $K_{n}^{i}(j)$. Thus,

$\displaystyle\mu(\\{x\in Y_{n+1}:Tx\neq S_{n+1}x\\})$ $\displaystyle<$
$\displaystyle\sum_{i=n+1}^{\infty}[\frac{1}{h_{i}}+4\epsilon_{i}]$ (26)
$\displaystyle<$
$\displaystyle\sum_{i=n}^{\infty}[\frac{5\epsilon_{i}w_{n}}{H_{m_{i}}+1}].$
(27)

This implies

$\displaystyle\mu(\\{x\in Y_{n+1}$ $\displaystyle:$ $\displaystyle T^{i}x\neq
S_{n+1}^{i}x,i=1,2,\ldots,H_{m_{n}}+1\\})$ $\displaystyle<$ $\displaystyle
w_{n}(H_{m_{n}}+1)\sum_{i=n}^{\infty}\frac{5\epsilon_{i}}{H_{m_{i}}+1}<5w_{n}\sum_{i=n}^{\infty}\epsilon_{i}<5\delta
w_{n}.$

Hence,

$\displaystyle|\mu(T^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle\leq$ $\displaystyle|\mu(T^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap K_{n}^{i}(j)))$ $\displaystyle-$
$\displaystyle\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap K_{n}^{i}(j)))|$ $\displaystyle+$
$\displaystyle|\mu(S_{n+1}^{H_{m_{n}}}(\Lambda_{\delta}\cap
K_{n}^{i}(j))\cap(\Lambda_{\delta}\cap
K_{n}^{i}(j)))-\frac{1}{2}\mu(\Lambda_{\delta}\cap K_{n}^{i}(j))|$
$\displaystyle<$ $\displaystyle 5\delta
w_{n}+(5\delta+\epsilon_{n})\mu(K_{n}^{i}(j))\leq(10\delta+\epsilon_{n})\mu(K_{n}^{i}(j)).$

For $\delta$ and $\epsilon$ sufficiently small, there exists
$x_{1}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$ such that
$T^{H_{m_{n}}}x_{1}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$, and
$x_{2}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$ such that
$T^{H_{m_{n}}+1}x_{2}\in\Lambda_{\delta}\cap K_{n}^{i}(j)$. Thus,

$\displaystyle|\lambda^{H_{m_{n}}}f(x_{1})-f(x_{1})|=|f(T^{H_{m_{n}}}x_{1})-f(x_{1})|<\delta,$
(28)

and

$\displaystyle|\lambda^{H_{m_{n}}+1}f(x_{2})-f(x_{2})|=|f(T^{H_{m_{n}}+1}x_{2})-f(x_{2})|<\delta.$
(29)

Hence,

$|\lambda^{H_{m_{n}}}-1|<\frac{\delta}{|f(x_{1})|}=\delta\mbox{ and
}|\lambda^{H_{m_{n}}+1}-1|<\frac{\delta}{|f(x_{2})|}=\delta.$

Therefore,

$|\lambda-1|=|\lambda^{H_{m_{n}}+1}-\lambda^{H_{m_{n}}}|\leq|\lambda^{H_{m_{n}}+1}-1|+|\lambda^{H_{m_{n}}}-1|<2\delta.$

Since $\delta>0$ may be chosen arbitrarily small, then $\lambda=1$. Since it
was established that $T$ is ergodic in an earlier section, then $f$ must be a
constant. Therefore, $T$ is weak mixing. ∎

Our theorem establishes the following corollaries which answer questions
raised in the ground-breaking works [4] and [8].

######  Corollary 6.3.

Given any ergodic measure preserving transformation $R$ on a Lebesgue
probability space with discrete spectrum, and a rigidity sequence $\rho_{n}$
for $R$, there exists a weak mixing transformation $T$ with rigidity sequence
$\rho_{n}$. In particular, for any
$k\in\hbox{\rm\hbox{I}\kern-1.62498ptN},k\geq 2$, there exists a weak mixing
transformation with $k^{n}$, $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, as a
rigidity sequence.

The next corollary gives an explicit characterization of ”large” rigid
sequences for weak mixing transformations. While this corollary appears known
in [2], our characterization gives a general concrete method for establishing
”large” rigidity sequences of weak mixing transformations. Given a sequence
$\mathcal{A}$, define the density function
$g_{\mathcal{A}}:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to[0,1]$ such that
$g_{\mathcal{A}}(k)={\\#(\mathcal{A}\cap\\{1,2,\ldots k\\})}/{k}$.

######  Corollary 6.4.

Given any real-valued function
$f:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to(0,\infty)$ such that

$\lim_{n\to\infty}f(n)=0,$

there exists a weak mixing transformation with rigidity sequence $\mathcal{A}$
such that

$\lim_{n\to\infty}\frac{f(n)}{g_{\mathcal{A}}(n)}=0.$

Also, there exist weak mixing transformations with rigidity sequences
$\rho_{n}$ satisfying

$\lim_{n\to\infty}\frac{\rho_{n+1}}{\rho_{n}}=1.$

###### Proof.

Let $\alpha$ be an irrational number and $R_{\alpha}$ the rotation by
$2\pi\alpha$ on the unit circle. Given $\epsilon>0$, define
$\mathcal{A}(\epsilon)=\\{j\in\hbox{\rm\hbox{I}\kern-1.62498ptN}:|\exp{(2\pi\alpha
j)}-1|<\epsilon\\}$, and for $n\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, define
$\mathcal{A}(\epsilon,n)=\mathcal{A}(\epsilon)\cap\\{1,2,\ldots n\\}$. For
$\bar{\epsilon}=\\{\epsilon_{1}<\epsilon_{2}<\ldots\\}$, let
$\mathcal{A}(\bar{\epsilon})=\bigcup_{n=1}^{\infty}\mathcal{A}(\epsilon_{n},n)$.
If $\lim_{n\to\infty}\epsilon_{n}=0$ and $\mathcal{A}(\bar{\epsilon})$ is
infinite, then $\mathcal{A}(\bar{\epsilon})$ forms a rigidity sequence for
$R_{\alpha}$. Let $f:\hbox{\rm\hbox{I}\kern-1.62498ptN}\to(0,\infty)$ be such
that $\lim_{n\to\infty}f(n)=0$. Since $\mathcal{A}(1/2^{i})$ has positive
density for $i\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, there exists
$j_{i}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that

$\frac{|\mathcal{A}({1}/{2^{i}},j)|}{j}>f(j).$

for all $j\geq j_{i}$. For $k\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$, choose
$i=i_{k}\in\hbox{\rm\hbox{I}\kern-1.62498ptN}$ such that $j_{i}+1\leq k\leq
j_{i+1}$. Set $\epsilon_{k}=\frac{1}{2^{i}}$ and let
$\mathcal{A}=\mathcal{A}(\bar{\epsilon})$. Thus,

$\displaystyle g_{\mathcal{A}}(k)$ $\displaystyle=$
$\displaystyle\frac{|\mathcal{A}\cap\\{1,2,\ldots,k\\}|}{k}$ (30)
$\displaystyle\geq$
$\displaystyle\frac{|\mathcal{A}(\epsilon_{k},k)|}{k}>2^{i}f(k).$ (31)

Hence,

$\displaystyle\frac{f(k)}{g_{\mathcal{A}}(k)}$ $\displaystyle>$
$\displaystyle\frac{1}{2^{i}}.$ (32)

This confirms that $\lim_{k\to\infty}{f(k)}/{g_{\mathcal{A}}(k)}=0$ for the
rigidity sequence $\mathcal{A}$. Therefore, by Theorem 6.2, $\mathcal{A}$ is a
rigidity sequence for a weak mixing transformation. The second assertion of
Corollary 6.4 can be established in a similar manner. Since ergodic rotations
on the unit circle have rigid sequences $\rho_{n}$ such that
$\lim_{n\to\infty}{\rho_{n+1}}/{\rho_{n}}=1$, then weak mixing transformations
admit these rigid sequences as well. ∎

Previously, it was established that denominators from convergents of continued
fractions serve as rigidity sequences for weak mixing transformations. A
partial result was provided in [8] for restricted convergents, and then a
general result was established in [4]. In this paper, we extend these results
to show that any rigidity sequence for an ergodic rotation on the unit circle
is also a rigidity sequence for a weak mixing transformation. This includes
sequences $q_{n}$ formed from the denominators of convergents
${p_{n}}/{q_{n}}$ of an irrational $\alpha$.

######  Corollary 6.5.

Let $\alpha\in(0,1)$ be any irrational number, and let $\rho_{n}$ be a
sequence of natural numbers satisfying

$\lim_{n\to\infty}|\exp{(2\pi i\alpha\rho_{n})}-1|=0.$

Then there exists a weak mixing transformation $T$ such that $\rho_{n}$ is a
rigidity sequence for $T$.

Acknowledgements
The author wishes to thank Joseph Rosenblatt, Andrew Parrish, Ayse Sahin, Karl
Petersen, Nathaniel Friedman, Cesar Silva, Keri Kornelson and Tatjana Eisner
for feedback on a previous version of this article.

##  Appendix A Towerplex Pictorial

This appendix provides an illustration of towers for $R_{1}$, $S_{1}$, and the
multiplexing operation applied to obtain towers for $R_{2}$ and $S_{2}$. The
picture below represents only the case where $d_{R}>0$ and $d_{S}<0$. The
other cases are handled as described in the section on towerplex
constructions. Also, the general case of deriving $R_{n+1}$ and $S_{n+1}$ from
$R_{n}$ and $S_{n}$ is analogous to the initial multiplexing operation for
deriving $R_{2}$ and $S_{2}$.

$I_{1}$$R_{1}$$r_{1}$$I_{1}^{\prime}$$X_{1}^{*}$$X_{1}$$J_{1}$$S_{1}$$s_{1}$$J_{1}^{\prime}$$d_{R}/h_{1}$$\supset
I_{1}^{*}$$Y_{1}^{*}$$Y_{1}$

Transformations $R_{2}$ and $S_{2}$ are derived from $R_{1}$ and $S_{1}$ by
switching the red subcolumn with the green subcolumn. We refer to these sets
as the switching sets, and are the main multiplexing operation. In order to
preserve maps isomorphic to $R$ and $S$, and avoid redefining $R_{1}$ or
$S_{1}$ on most of the probability space, it may be necessary to transfer
measure between the towers and residuals. This is a rescaling operation, and
these sets are referred to as transfer sets. In the case where $d_{R}>0$, the
blue colored subcolumn $I_{1}^{*}$ from $J_{1}^{\prime}\subset Y_{1}$ is
absorbed into $X_{1}^{\prime}$. For $d_{S}<0$, mass is removed from
$Y_{1}^{*}$ and added as a blue subcolumn to define $S_{2}$.

$I_{1}$$R_{1}$$R_{2}$$s_{1}$$J_{1}^{\prime}\setminus{I_{1}^{*}}$$S_{1}$$X_{1}^{\prime}$$X_{2}$$d_{S}/h_{1}$$J_{1}^{*}$$J_{1}$$S_{1}$$S_{2}$$r_{1}$$I_{1}^{\prime}$$R_{1}$$Y_{1}^{\prime}$$Y_{2}$

##  Appendix B Towerplex Conditions

Below is a list of explicit conditions that can be used to prove theorem 6.2.

  1. 1.

$\lim_{n\to\infty}r_{n}=0$;

  2. 2.

$\sum_{n=1}^{\infty}r_{n}=\sum_{n=1}^{\infty}s_{n}=\infty$;

  3. 3.

$\lim_{n\to\infty}\mu(Y_{n})=0$;

  4. 4.

${\epsilon_{n}}/{\max{\\{M_{n}^{1},M_{n}^{2}\\}}}<\epsilon_{n+1}$;

  5. 5.

$h_{n-1}<M_{n}^{1},M_{n}^{2}<h_{n}$;

  6. 6.

$h_{n}\mbox{ sufficiently large such that equation \ref{SwitchingSetInd}
holds}$;

  7. 7.

$h_{n+1}\epsilon_{n}w_{n}>H_{m_{n}}+1$;

  8. 8.

$\epsilon_{n+1}(H_{m_{n}}+1)<\epsilon_{n}w_{n}$;

  9. 9.

$H_{m_{n+1}}\geq H_{m_{n}}$.

If $r_{n}={1}/{2(n+2)}$ and $s_{n}=1/2$, and $\epsilon_{n}$ is sufficiently
small such that Lemma 6.1 holds, then we have a canonical towerplex
construction.

## References

  * [1] Aaronson, J., Rational ergodicity, bounded rational ergodicity and some continuous measures on the circle, Israel Journal of Mathematics, 33:3 (1979), 181-197.
  * [2] Aaronson, J., Hosseini, M. and Lemanczyk, M, IP-rigidity and eigenvalue groups, arXiv:1203.2257 (25 May 2012).
  * [3] Avila, A. and Forni, G., Weak mixing for interval exchange transformations and translation flows, Annals of mathematics, 165:2 (2007), 637-664.
  * [4] Bergelson, V., del Junco, A., Lemańczyk, M, Rosenblatt, J., Rigidity and non-recurrence along sequences, arXiv 1103.0905 (4 Mar 2011).
  * [5] Bergelson, V. and Gorodnik, A., Weakly mixing group actions: a brief survey and an example, Modern dynamical systems and applications (2005), 3-25.
  * [6] Chacon, R. V., Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc., 22 (1969), 559-562.
  * [7] Chaika, J., Every ergodic transformation is disjoint from almost every interval exchange transformation, Annals of Mathematics, 175 (2012), 237-253.
  * [8] Eisner, T. and Grivaux, S., Hilbertian Jamison sequences and rigid dynamical systems, J. Funct. Anal. 261:7 (2011), 2013-2052.
  * [9] El Abdalaoui and El Houcein, On the spectrum of $\alpha$-rigid maps, Journal of dynamical and control systems, 15.4 (2009), 453-470.
  * [10] Friedman, N., Partial mixing, partial rigidity, and factors, Contemp. Math, 94 (1989), 141-145.
  * [11] Friedman, N. A., Introduction to ergodic theory. Vol. 29. Van Nostrand Reinhold, New York (1970).
  * [12] Friedman, N., Gabriel, P. and King, J., An invariant for rigid rank-1 transformations, Ergodic Theory and Dynamical Systems, 8 (1988), 53-72. doi:10.1017/S0143385700004314.
  * [13] Furstenberg, H. and Weiss, B., The finite multipliers of infinite ergodic transformations, The structure of attractors in dynamical systems (1978), 127-132.
  * [14] Glasner, S. and Maon, D., Rigidity in topological dynamics, Ergodic Theory Dynam. Systems, 9:2 (1989), 309-320.
  * [15] Grivaux, S., IP-Dirichlet measures and IP-rigid dynamical systems: an approach via generalized Riesz products, arXiv:1209.2884 (13 Sep 2012).
  * [16] Halmos, P., Lectures on Ergodic Theory, Math. Soc. Japan, (1956) MR0097489
  * [17] Halmos, P., Approximation Theories for Measure Preserving Transformations, Transactions of the American Mathematical Society, (1944) MR0009703
  * [18] James, Jennifer, Thomas Koberda, Kathryn Lindsey, Cesar E. Silva, and Peter Speh, On ergodic transformations that are both weakly mixing and uniformly rigid, New York J. Math, 15 (2009), 393-403.
  * [19] Kakutani, S., Induced Measure Preserving Transformations, Proc. Japan Acad., 19 (1943), 65-41.
  * [20] Kalikow, S., Infinite partitions and Rokhlin towers, Ergodic Theory and Dynami. Systems, 32:2 (2012), 707-738. doi:10.1017/S0143385711000381.
  * [21] Katok, A. and Lemanczyk, M., Some new cases of realization of spectral multiplicity function for ergodic transformations, Fundamenta Mathematicae, 206 (2009), 185-215.
  * [22] Katok, A.B. and Stepin, A.M. Approximations in Ergodic Theory, Russian Math. Surveys , 22:5 (1967), 77-102. MR0219697
  * [23] Petersen, K. E. Ergodic theory. Vol. 2. Cambridge University Press (1989).
  * [24] Rokhlin, V.A., A General Measure Preserving Transformation is Not Mixing, Dokl. Akad. Nauk, 60 (1948), 349-51.
  * [25] Rokhlin, V.A., Selected Topics from the Metric Theory of Dynamical Systems, Amer. Math. Soc. Transl. Series, 2:49 (1949), 171-240.


draft color=1.0 1.0 0.0, open=true

# High-Q localized states in finite arrays of subwavelength resonators

Danil F. Kornovan1 Roman S. Savelev1 Yuri S. Kivshar1,2 Mihail I. Petrov1
1Department of Physics and Engineering, ITMO University, St.-Petersburg
197101, Russia  2Nonlinear Physics Center, Research School of Physics,
Australian National University, Canberra ACT 2601, Australia

###### Abstract

We introduce a novel physical mechanism for achieving giant quality factors
($Q$-factors) in finite-length periodic arrays of subwavelength optical
resonators. The underlying physics is based on interference between the band-
edge mode and another standing mode in the array, and the formation of
spatially localized states with dramatically suppressed radiative losses. We
demonstrate this concept for an array of $N$ dipoles with simultaneous
cancellation of multipoles up to $N$-th order and the $Q$ factor growing as
$Q\propto N^{\alpha}$, where $\alpha\gtrsim 6.88$. Based on this finding, we
propose a realistic array of Mie-resonant nanoparticles ($N\lesssim 29$) with
a dramatic enhancement of the Purcell factor (up to $\sim$3400) achieved by
tuning of the array parameters.

Subwavelength trapping and guiding of light has attracted a great attention
because it provides unique opportunities for miniaturization of the optical
interconnect technology. Diverse implementations of subwavelength-engineered
structures in integrated optics have been discussed for a design of integrated
photonic platforms Cheben2018  and for engineering the mode dispersion and
waveguide anisotropy to achieve subwavelength propagation of slow light
Figotin2011 ; Ding2020 .

For many years, plasmonics was considered as the only available platform for
nanoscale optics, but the recently emerged novel field of Mie resonant
metaphotonics provides practical alternatives for nanoscale localization of
light by employing resonances in high-index dielectric nanoparticles and
structures Koshelev2020_2 ; Won2019 . Future technologies underpinning high-
performance optical communications, ultrafast computations and compact
biosensing will rely on densely packed reconfigurable optical circuitry based
on nanophotonics. Importantly, in the last few years we observe novel trends
in achieving high values of the quality factor (Q-factor) in dielectric
structures Koshelev2019  for spatial and temporal control of light by
employing multipolar resonances and their interference.

High-index dielectric nanoantennas supporting multipolar Mie resonances
represent a novel class of building blocks of metamaterials for generating
Rutckaia2017a ; Tiguntseva2020 ; Mylnikov2020 , manipulating Kamali2018 , and
modulating Ding2020  light. By combing both electric and magnetic multipolar
modes, one can not only modify far-field radiation patterns but also localize
the electromagnetic energy in open resonators by employing the physics of
bound states in the continuum Hsu2016  to achieve destructive interference of
two (or more) leaky modes Rybin2017 ; Koshelev2020 .

Figure 1: An inflection point in the dispersion curve of an infinite one-
dimensional dipole array of Mie-resonant particles (an upper inset) enables
interference between the staggered band-edge mode and another standing mode of
a finite-extent array, thus suppressing radiative losses with the formation of
a high-$Q$ localized state.

The further increase of the $Q$-factor of resonant systems can be achieved by
combining several subwavelength resonators which can provide additional degree
of freedom in suppressing radiative losses. One of the simplest geometries is
a finite-extend array of coupled dielectric resonant structures which supports
a set of standing waves Savelev2014 ; Krasnok2016 ; Bakker2017 , as shown
schematically in the insert of Fig. 1. Importantly, at the frequencies close
to the band edge the phases of the fields in the neighbour resonators become
close to $\pi$, which leads to destructive interference between the field
scattered by individual resonators and consequently to suppressed radiation
from the whole structure. Such band-edge (BE) localized states have been
explored in applications to photonic crystals Othman2016 , and they have
already been utilized for light emitting Agrawal1986 ; Dowling1994 ; Kim2011 ;
Hoang2020 ; Rutckaia2020  and sensing Cusano2016 ; Yang2020  applications.

Figure 2: Radiation suppression in arrays of $N$ dipoles. (a) The $Q$-factor
of the resonant modes with different separation distances $a$. Blue dashed
line corresponds to the band-edge mode with $a_{BE}/\lambda_{0}=0.35$, while
three solid lines show the $Q$-factor dependence for three values of
separation distance $a/\lambda_{0}=0.238,0.239,\text{and }0.240$ with maximal
$Q$-factors at $N=19,22,\text{and }29$, respectively. Brown dash-dot line
denotes the asymptotic dependence of the maximal $Q$-factors. (b) The total
and partial radiative losses correspondent to different multipoles orders $j$
for the high-$Q$ state as a function of the spacing $a$ for $N=10$.

Here, we introduce and explore a novel physical mechanism for achieving giant
$Q$-factors in dielectric structures of subwavelength optical resonators. We
consider an array of resonators and assume that, due to their long-range
interaction, the dispersion curve can have a maximum, as schematically shown
in Fig. 1. Then, for the frequency selected at the band edge, a staggered mode
can couple resonantly with another standing mode supported by the array, so
that we observe the formation of a hybrid localized state with dramatically
suppressed radiative losses due to several interference mechanisms. First, we
demonstrate this general concept for an array of $N$ dipoles that can support
the modes with the $Q$-factor growing as $Q\propto N^{\alpha}$, where
$\alpha\sim 6.88$. Second, we show the formation of a high-$Q$ state
accompanied by simultaneous resonant suppression of high-order radiation
multipoles. Finally, we consider a realistic structure based on silicon-on-
insulator resonator arrays and demonstrate a dramatic Purcell enhancement up
to $\sim$ 3400 achieved by tuning parameters of the array.

Dipole arrays. Our starting point is a simple model describing a one-
dimensional periodic array of resonant dipoles with polarizability
$\alpha(\omega)$ and lattice spacing $a$. The electromagnetic coupling between
the dipoles leads to a self-consistent system of equations that describes
collective dipolar oscillations supported by the structure [see Eq. (S1) in
Supplemental Materials SuppMat ]. Solution of these equations provides us with
a set of complex eigenfrequencies $\omega=\omega^{\prime}-i\gamma$, where
$\gamma$ determines the decay rate of the corresponding mode and the ratio
$\omega^{\prime}/(2\gamma)$ is known as the $Q$ factor.

The $Q$ factor calculated within the quasi-resonant approximation (see
Supplemental Materials SuppMat ), normalized by the $Q$ factor of an
individual dipole $Q_{0}$, is shown in Fig. 2(a) as a function of the number
of dipoles $N$ and for different values of the period $a$. For the period
$a_{BE}=0.35\lambda_{0}$, where $\lambda_{0}$ is the free space wavelength, we
observe a typical growth $Q\propto N^{3}$ inherent to the regular band-edge
states. Such dependence is dictated by the type of the dispersion near the
edge of the Brillouine zone Zhang2020 , which in general case is quadratic.
Decrease of the period leads to modification of the dispersion curve and, for
a specific value of the period (see analytical expression in Supplemental
Materials SuppMat ) the group velocity dispersion vanishes and the dispersion
curve becomes quartic. Such a type of dispersion is known to exist in
different systems, e.g. in anisotropic crystals Figotin2005  or coupled
waveguides Burr2015 ; Othman2016 ; Veysi2018 , and it leads to a faster growth
of $Q$-factor as $Q\propto N^{5}$ Figotin2005 ; Zhang2020 . Interestingly,
even further decrease of the period results in a change of the sign of the
group velocity dispersion, as shown schematically in Fig. 1. As a result, the
dependence $Q(N)$ becomes nonmonotonic and preserves for arbitrarily large $N$
(see Fig. 1, solid curves). Three values of the period $a_{1},a_{2},a_{3}$ are
chosen so that the calculated $Q(N)$ curves exhibit the first maximum for
$N=19$, $N=22$, and $N=29$, respectively. However, we notice that by fine
tuning of the period it is possible to shift the first maximum to arbitrary
number $N$. The maximal $Q$-factor, obtained separately for each value of $N$
then becomes approximately proportional to $N^{\sim 6.88}$, see upper dashed
line in Fig. 2(a). Such a seemingly slight modification allows increasing the
$Q$ factor by several orders of magnitude even for short arrays, $N\approx
10$.

Spherical multipole expansion. One of the most powerful and universal tools
used in nanophotonics for describing radiative properties of finite structures
is based on the multipolar content analysis. Despite that the considered
system consists of dipolar emitters each with a single multipolar channel of
radiation, the overall multipolar content of the collective modes can be
arbitrary complex. However, it turns out, that the observed suppression of
radiation has a clear signature in the multipolar picture, similar to the
quasi-bound states in the continuum resonances Rybin2017 ; Bulgakov2020 ,
where lower-order multipole moments are suppressed. To elaborate more on that,
we perform the multipolar decomposition of the eigenmodes of the array of
$N=10$ dipoles in terms of the vector spherical harmonics (VSH) basis as
described in Supplemental Materials SuppMat . The total radiative decay rate
$\Gamma$ normalized to the radiative decay rate of an individual dipole
$\gamma_{0}$ can be expanded into a sum containing all multipole channels
labeled by the subscript $j$ (dipole $j=1$, quadrupole $j=2$, octupole $j=3$,
etc.): ${\Gamma}=\gamma/\gamma_{0}=\sum\limits_{j=1}^{\infty}{\Gamma}_{j}$,
where ${\Gamma}_{j}$ can be obtained as follows:

Figure 3: (a) Normalized radiative losses rate $\gamma/\gamma_{0}$ as function
of the distance $a$ for $N$ and $N-2$. Solid lines correspond to exact full
solution, while dashed lines correspond to approximate coupled mode model. (b)
Decomposition amplitudes of the high-$Q$ band-edge state showing that the
radiation cancellation occurs due to the interference of $N$ and $N-2$ modes
onle ($C_{N-4}=0$ at $a=\tilde{a}$) illustrated in the inset.

$\displaystyle\Gamma_{j}=\dfrac{3}{8\pi}\dfrac{1}{|\mathbf{d}|^{2}k_{0}^{6}}\sum\limits_{m=-j}^{+j}\left(|a_{j,m}|^{2}+|b_{j,m}|^{2}\right),$
(1)

and $a_{j,m}$ and $b_{j,m}$ are electric and magnetic multipolar amplitudes
[see Supplemental Materials SuppMat , Eq. (S10)], $d$ is the amplitude of an
individual dipole, and $k_{0}$ is the wavenumber corresponding to resonant
frequency $\omega_{0}$. The total radiative decay rate and its multipole
decomposition for the band edge states are shown in Fig. 2(b) as a function of
the period $a$. The reported high-$Q$ state formed for the optimal distance
$\tilde{a}/\lambda_{0}\sim 0.24$ (indicated by dashed vertical line)
corresponds to the strongest deep in total radiative decay rate. Surprisingly,
it turns out that all the contributions of VSH with order $j$ up to
approximately $N$ are also resonantly suppressed at this optimal distance. We
notice that for other separation distances there are dips in the radiative
emission rate provided by simultaneous suppression of smaller numbers of
multipoles. The described effect has similarities with other collective effect
such as superscattering Ruan2010  and zero scattering with high-order anapole
states Valero .

Mode coupling approach. The origin of the described cancellation of high-order
multipoles lies in the destructive interference of the standing waves of a
dipolar array which is attributed to their external coupling as schematically
shown in Fig. 1. Such coupling can occur in open systems due to their
interaction via far-field radiation. Initially proposed in quantum mechanics
FriedrichH.1985 , external mode coupling has been actively developed in
nanophotonics since recently Cao2015  for formation of high-$Q$ states in
individual resonators Wiersig2006 ; Rybin2017 ; Koshelev2020  and their
ensembles Bulgakov2020 , including bound states in the continuum that occur in
infinite systems above the light cone.

In the considered finite periodic array of dipoles the eigenmodes can be
represented as two counter propagating polaritonic Bloch modes formed from the
hybridization of dipolar excitation and free space modes Weber2004 . The
solution of the system of equations for coupled dipoles in nearest neighbour
interaction approximation provides us with the distributions of the dipole
moments, which form a full set of functions on a discrete array Malyshev1995 ;
Weber2004 ; Ivchenko2013 :

$d^{(l)}_{k}=\sqrt{\frac{2}{N+1}}\sin\left(\dfrac{k\pi}{N+1}l\right),\quad
k=1..N,$ (2)

where $l$ labels the scatterer, and $k$ is the eigenstate number. In the
general case of long-range interaction, however, such approximation fails, and
that is the major factor that affects the formation of high-Q resonant modes.

The effective Hamiltonian describing the dipole-dipole interaction of the
transverse dipolar excitations in the considered array can be written in the
following form:

$\widehat{H}=-\sum_{n}i\gamma_{0}/2\outerproduct{n}{n}+\sum_{mn,\ m\neq
n}g_{mn}\outerproduct{m}{n}+\text{h.c.}$ (3)

where $\ket{n}$ is the Fock state of a single excitation at the dipolar site
with number $n$, and
$g_{mn}=-3\pi\gamma_{0}k_{0}^{-1}G_{0,tt}(\mathbf{r_{m}},\mathbf{r_{n}},\omega_{0})$
is the dipole-dipole coupling coefficient defined by transverse component of
the dyadic Green’s function. The eigenmodes of the dipolar excitation in the
array can be represented as the solution of Schrodinger equation
$\widehat{H}\ket{\psi}=(\omega-i\Gamma)\ket{\psi}$. In order to construct
them, we use the conventional nearest neighbour basis provided by the
expression (2) and expand the unknown eigenvector over the chosen basis
$\ket{\psi}=\sum_{k}c_{k}\ket{\varphi_{k}}$, where
$\ket{\varphi_{k}}=\sum_{n=1}^{N}d_{k}^{(n)}\ket{n}$. The Schrodinger equation
yields a system of equations for the coupling coefficients:

$(\omega_{n}-i\Gamma_{n})c_{n}=\matrixelement{\varphi_{n}}{\widehat{H}}{\varphi_{n}}c_{n}+\sum_{m\neq
n}\matrixelement{\varphi_{n}}{\widehat{H}}{\varphi_{m}}c_{m}.$ (4)

Figure 4: Dependence of the $Q$-factor of the resonance with the highest $Q$
on the number of the particles $N$ in an array of Si rectangular bars with
height and width equal to 600 and 550 nm, respectively. The period of the
structure is fixed to $a=320$ nm, while thickness of each element $w$ is
varied. Red dashed line corresponds to band-edge states ($w=190$ nm), while
solid black line and circles correspond to the high-$Q$ state ($w_{highQ}=197$
nm). Inset shows the field distribution of electric field intensity in the
array of 19 particles (side view) for the maximal value of $Q$-factor marked
with blue circle. Refractive index of silicon and glass were taken equal to
$n_{Si}=3.5$ and $n_{glass}=1.45$.

Now, the system (4) that describes the coupling of different standing waves in
the dipolar array can be applied for reconstructing the high-$Q$ states in
terms of nearest-neighbour solution (2). It turns out, that though the exact
decomposition should take into account all basis modes, a simplified expansion
involving only three modes is sufficient to describe the considered high-Q
state:
$\ket{\psi_{highQ}}=c_{N}\ket{\phi_{N}}+c_{N-2}\ket{\phi_{N-2}}+c_{N-4}\ket{\phi_{N-4}}$.
In such an expansion, the terms $N-1,N-3$ are missing as they have different
symmetry and are not coupled through the Hamiltonian (3), and matrix (4) is
reduced to $3\times 3$ system. By solving the system numerically one can
derive the radiative losses rates of the states as shown in Fig. 3(a), which
now can be compared with the solution of the exact coupled dipole system [see
Supplemental materials SuppMat )]. We notice that the appearance of radiative
losses repulsion typical for the external coupling scheme Cao2015 . The
relative losses rate of the subradiant mode drop down to $10^{-8}$ for the
specific distance $\tilde{a}$. At the same time, we plot the eigenfrequencies
and losses obtained from the reduced system (4) with dashed and dash-dotted
lines having perfect matching for real and imaginary parts of the high-$Q$
mode energies. Moreover, the expansion amplitudes
$|c_{N}|^{2},|c_{N-2}|^{2},|c_{N-4}|^{2}$ plotted in Fig. 3(b) show that at
the point of minimal losses $C_{N-4}$ vanishes, which means that the high-$Q$
state consists only of two partial modes with number $N$ and $N-2$, as
depicted in Fig. 1. The spectral alignment of the corresponding modes of a
infinite array results in the destructive far-field interference and
suppression of the total radiation rate of the mode of a finite array. A
similar effect has been observed for two-photon states in a closely spaced
arrays of cold atoms placed near a waveguide Poddubny2020  and for the split-
band edge resonances in photonic crystals NohPRA2010 .

Silicon-on-insulator structure. So far, we have discussed an ideal model of
coupled point dipoles. In order to show the application of the described
effect in realistic dielectric optical resonators, we design a nanostructure
with maximized $Q$ factor that can be readily fabricated from the silicon-on-
insulator wafers by means of common fabrication techniques (see inset in Fig.
4). All geometric parameters of the structure are fixed except for the
thickness $w$, which is employed for the fine tuning of the $Q$-factor. We
calculate the $Q$ factors of the magnetic dipole eigenmodes of the finite
arrays with varying $N$ and two values of the thickness $w$. Results presented
in Fig. 4 show that for a specific value $w=w_{highQ}$, the $Q$ factor grows
much faster than that of a regular band-edge state (observed for $w=w_{BE}$),
and it reaches a maximum at $N=19$, similarly to the case of Fig. 2(a). The
$Q$-factor of the high-$Q$ state exceeds that in the band-edge regime by
almost two orders of magnitude reaching the value $\approx 2\cdot 10^{4}$. Due
to interference of two standing waves, the field distribution of the mode
becomes more localized as shown in the inset of Fig. 4. As a result, the mode
volume becomes as small as $\sim 2.5(\lambda/n)^{3}$, implying the Purcell
factor $\sim 560$, which is much larger than the values reported previously
for similar structures Krasnok2016 ; Hoang2020 . Moreover, by tuning the
parameters of the array one can get the optimal conditions for $N=29$ with
higher $Q$-factor of $Q=1.8\cdot 10^{5}$ (see Supplemental Materials SuppMat )
and the Purcell enhancement factor of $\sim 3400$. Note, that the structure
has subwavelength dimensions in two transverse directions and only a few
wavelengths in the longitudinal dimension.

Importantly, this new physical mechanism of the $Q$-factor enhancement can be
compared with interference of two leaky modes supported by isolated dielectric
resonators Koshelev2020 , and also with destructive interference of the dipole
and toroidal modes and the formation of the so-called optical anapoles anapole
. In addition, we should mention some similarities with the physics of the so-
called embedded solitons in nonlinear modes with higher-order dispersion
soliton ; soliton_book .

In summary, we have revealed a new mechanism of the formation of high-$Q$
states in finite-extend subwavelength photonic systems supporting non-
monotonic dispersions. The interference of the band-edge mode and another
standing mode of the finite periodic nanoparticle array leads to the
cancellation of the far-field radiation through simultaneous suppression of
multipole components up to the order $N$, where $N$ is the number of the
elements in the array. We have demonstrated how this general concept can be
realized with an array of Mie-resonant silicon nanoparticles with dramatic
suppression of radiation losses and a giant enhancement of the Purcell factor
by tuning of the array parameters. We believe that our finding will extend the
existing approaches in a design of high-$Q$ photonic resonant subwavelength
structures.

###### Acknowledgements.

The authors acknowledge a financial support from the Russian Foundation of
Basic Research (grant 20-52-12062), the Australian Research Council (grant
DP210101292), the Strategic Grant of the Australian National University. They
also thank A. Bogdanov, P. Belov, and A. Sheremet for useful discussions.

## References

  * (1) P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith,  Subwavelength integrated photonics, Nature 560, 565 (2018).
  * (2) A. Figotin and I. Vitebskiy,  Slow wave phenomena in photonic crystals, Laser & Photon. Rev. 5, 201 (2011).
  * (3) L. Ding, D. Morits, R. Bakker, S. Li, D. Eschimese, S. Zhu, Y. F. Yu, R. Paniagua-Dominguez, and A. I. Kuznetsov,  All-optical modulation in chains of silicon nanoantennas, ACS Photonics 7, 1001 (2020).
  * (4) K. Koshelev and Y. Kivshar,  Dielectric resonant metaphotonics, ACS Photonics (2020).
  * (5) R. Won,  Into the ‘Mie-tronic’ era, Nat. Phot. 13, 585 (2019).
  * (6) K. Koshelev, G. Favraud, A. Bogdanov, Y. Kivshar, and A. Fratalocchi,  Nonradiating photonics with resonant dielectric nanostructures, Nanophotonics 8, 725 (2019).
  * (7) V. Rutckaia, F. Heyroth, A. Novikov, M. Shaleev, M. I. Petrov, and J. Schilling,  Quantum dot emission driven by Mie resonances in silicon nanostructures, Nano Lett. 17, 6886 (2017).
  * (8) E. Tiguntseva, K. Koshelev, A. Furasova, P. Tonkaev, V. Mikhailovskii, E. V. Ushakova, D. G. Baranov, T. Shegai, A. A. Zakhidov, Y. S. Kivshar, and S. V. Makarov,  Room-temperature lasing from Mie-resonant nonplasmonic nanoparticles, ACS Nano 14, 8149 (2020).
  * (9) V. Mylnikov, S. T. Ha, Z. Pan, V. Valuckas, R. Paniagua-Domínguez, H. V. Demir, and A. I. Kuznetsov,  Lasing action in single subwavelength particles supporting supercavity modes, ACS Nano 14, 7338 (2020).
  * (10) S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon,  A review of dielectric optical metasurfaces for wavefront control, Nanophotonics 7, 1041 (2018).
  * (11) C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić,  Bound states in the continuum, Nature Reviews Materials 1, 16048 (2016).
  * (12) M. V. Rybin, K. L. Koshelev, Z. F. Sadrieva, K. B. Samusev, A. A. Bogdanov, M. F. Limonov, and Y. S. Kivshar,  High-Q supercavity modes in subwavelength dielectric resonators, Phys. Rev. Lett. 119, 243901, (2017).
  * (13) K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J. H. Choi, A. Bogdanov, H. G. Park, and Y. Kivshar,  Subwavelength dielectric resonators for nonlinear nanophotonics, Science 367, 288 (2020).
  * (14) R. S. Savelev, A. P. Slobozhanyuk, A. E. Miroshnichenko, Y. S. Kivshar, and P. A. Belov,  Subwavelength waveguides composed of dielectric nanoparticles, Phys. Rev. B 89, 035435 (2014).
  * (15) A. Krasnok, S. Glybovski, M. Petrov, S. Makarov, R. Savelev, P. Belov, C. Simovski, and Y. Kivshar,  Demonstration of the enhanced Purcell factor in all-dielectric structures, Appl. Phys. Lett. 108, 211105 (2016).
  * (16) R. M. Bakker, Y. F. Yu, R. Paniagua-Domínguez, B. Luk’Yanchuk, and A. I. Kuznetsov,  Resonant light guiding along a chain of silicon nanoparticles, Nano Lett. 17, 3458 (2017).
  * (17) M. A. K. Othman, F. Yazdi, A. Figotin, and F. Capolino,  Giant gain enhancement in photonic crystals with a degenerate band edge, Phys. Rev. B 93, 024301 (2016).
  * (18) G. P. Agrawal and N. K. Dutta,  Long-Wavelength Semiconductor Lasers, (Van Nostrand Reinhold Co. Inc., New York, NY, 1986).
  * (19) J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden,  The photonic band edge laser: A new approach to gain enhancement, J. Appl. Phys. 75, 1896 (1994).
  * (20) S. Kim, B.-H. Ahn, J.-Y. Kim, K.-Y. Jeong, K. S. Kim, and Y.-H. Lee,  Nanobeam photonic bandedge lasers, Opt. Express 19, 24055 (2011).
  * (21) T.-X. Hoang, S. T. Ha, Z. Pan, W. K. Phua, R. Paniagua-Dominguez, C. E. Png, H.-S. Chu, and A. Kuznetsov,  Collective Mie resonances for directional on-chip nanolasers, Nano Lett. 20, 5655 (2020).
  * (22) V. Rutckaia, F. Heyroth, G. Schmidt, A. Novikov, M. Shaleev, R. Savelev, J. Schilling, and M. Petrov,  Luminescence enhancement in one-dimensional Mie-resonant arrays, arXiv:2006.13185 (2020).
  * (23) A. Cusano, F. J. Arregui, M. Giordano, and A. Cutolo,  Optochemical Nanosensors, CRC Press (2013).
  * (24) D. Q. Yang, B. Duan, X. Liu, A. Q. Wang, X. G. Li, and Y. F. Ji,  Photonic crystal nanobeam cavities for nanoscale optical sensing: A review, Micromachines 11, 72 (2020).
  * (25) See Supplemental Materials at [URL] for the details of a theoretical model of a periodic array of quasi-resonant point dipoles; the derivation of the transcendental equation on the period allowing for a quartic dispersion at the band-edge for an infinite periodic dipole array; the details on the expansion of a total radiative decay rate $\Gamma$ into contributions of Vector Spherical Harmonics given by Eq. (1); and the characteristics of the SOI-based structure containing the quality factor vs. the mode volume plot along with the field profiles of the eigenmodes. Supplemental Material also contains Ref. Weber2004 ; Petrov2015 ; Novotny2012 ; LoudonJoPBAMOP2006 ; Citrin2006 ; Asenjo2017 ; Kornovan2019 ; Chew1995  .
  * (26) W. H. Weber and G. W. Ford,  Propagation of optical excitations by dipolar interactions in metal nanoparticle chains, Phys. Rev. B 70, 125429 (2004).
  * (27) M. Petrov.  Disorder-induced Purcell enhancement in nanoparticle chains.  Physical Review A, 91(2):023821, feb 2015.
  * (28) L. Novotny and B. Hecht,  Principles of Nano-Optics, 2nd ed., Cambridge University Press, 2012.
  * (29) R. Loudon and S. M. Barnett,  Theory of the linear polarizability of a two-level atom,  J. Phys. B 39, S555 (2006).
  * (30) D. S. Citrin,  Plasmon-polariton transport in metal-nanoparticle chains embedded in a gain medium, Opt. Lett. 31, 98 (2006).
  * (31) A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht, H. J. Kimble, and D. E. Chang,  Exponential improvement in photon storage fidelities using subradiance and “selective radiance” in atomic arrays, Phys. Rev. X 7, 031024 (2017).
  * (32) D. F. Kornovan, N. V. Corzo, J. Laurat, and A. S. Sheremet,  Extremely subradiant states in a periodic one-dimensional atomic array, Phys. Rev. A 100, 063832 (2019).
  * (33) W. C. Chew,  Waves and Fields in Inhomogeneous Media (Electromagnetic Waves), IEEE Computer Society Press, 1995.
  * (34) Y.-X. Zhang.  Universal scaling of decay rates of emitter arrays in the subradiant states, arXiv:2006.06329 (2020).
  * (35) A. Figotin and I. Vitebskiy,  Gigantic transmission band-edge resonance in periodic stacks of anisotropic layers, Phys. Rev. E 72, 036619 (2005).
  * (36) J. R. Burr and R. M. Reano.  Zero-coupling-gap degenerate band edge resonators in silicon photonics, Opt. Express 23, 30933 (2015).
  * (37) M. Veysi, M. A.K. Othman, A. Figotin, and F. Capolino,  Degenerate band edge laser, Phys. Rev. B 97, 195107 (2018).
  * (38) E. N. Bulgakov, K. N. Pichugin, and A. F. Sadreev,  Engineering of the extremely high Q factor in two subwavelength dielectric resonators, arXiv:2005.05554 (2020).
  * (39) Z. Ruan and S. Fan,  Superscattering of light from subwavelength nanostructures, Phys. Rev. Lett. 105, 013901 (2010).
  * (40) A. C. Valero, E. A. Gurvitz, F. A. Benimetskiy, A. Samusev, K. Z. Kamali, A. B. Evlyukhin, A. A. Pavlov, and A. S. Shalin.  Nontrivial pure zero-scattering regime delivered by a hybrid anapole state, arXiv:2004.04659 (2020)
  * (41) H. Friedrich and D. Wintgen,  Interfering resonances and bound states in the continuum, Phys. Rev. A 32, 3231 (1985).
  * (42) H. Cao and J. Wiersig,  Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics, Rev. Mod. Phys. 87, 61 (2015).
  * (43) J. Wiersig,  Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities, Phys. Rev. Lett. 97, 253901 (2006).
  * (44) V. Malyshev and P. Moreno,  Hidden structure of the low-energy spectrum of a one-dimensional localized Frenkel exciton, Phys. Rev. B 51, 14587 (1995).
  * (45) E. L. Ivchenko and A. N. Poddubny,  Resonant diffraction of electromagnetic waves from solids (a review), Phys. Solid State 55, 905 (2013).
  * (46) A. N. Poddubny,  Quasiflat band enabling subradiant two-photon bound states, Phys. Rev. A 101, 043845 (2020).
  * (47) H. Noh, J.-K. Yang, I. Vitebskiy, A. Figotin, and H. Cao,  Giant resonances near the split band edges of two-dimensional photonic crystals, Phys. Rev. A 82, 013801 (2010).
  * (48) K.V. Baryshnikova, D.A. Smirnova, B.S. Lukyanchuk, and Y.S. Kivshar,  Optical anapoles: concepts and applications, Adv. Opt. Mat 7, 1801350 (2019).
  * (49) J. Yang, B. A. Malomed, and D. J. Kaup,  Embedded solitons in second-harmonic-generating Systems, Phys. Rev. Lett.  83, 1958 (1999).
  * (50) Y. S. Kivshar and G. P. Agrawal,  Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).