# Density of periodic measures and large deviation principle for generalized
($\alpha,\beta$)- transformations

Mao Shinoda  Department of Mathematics
Ochanomizu University
2-1-1, Otsuka, Bunkyo-ku,Tokyo 112-8610, JAPAN<EMAIL_ADDRESS>and
Kenichiro Yamamoto  Department of General Education
Nagaoka University of Technology
Nagaoka 940-2188, JAPAN<EMAIL_ADDRESS>
###### Abstract.

We introduce generalized $(\alpha,\beta)$-transformations, which include all
$(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all
transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2
large deviation principle with a unique measure of maximal entropy. A crucial
step in our proof is to establish density of periodic measures in the set of
ergodic measures.

###### 2020 Mathematics Subject Classification:

Primary 37A50, 37E05, 37B10; Secondary 60F10

##  1\. Introduction

In this paper, we consider piecewise monotonic maps on the unit interval
$[0,1]$. We say that $T\colon[0,1]\to[0,1]$ is a piecewise monotonic map if
there exist integer $k>1$ and $0=c_{0}<c_{1}<\cdots<c_{k}=1$, which we call
the critical points, such that $T|_{(c_{i-1},c_{i})}$ is strictly monotonic
and continuous for each $1\leq i\leq k$. Throughout this paper, we further
assume the following conditions for a piecewise monotonic map $T$.

  * •

The topological entropy $h_{\rm top}(T)$ of $T$ is positive (see [1, Ch. 9]
for the definition of topological entropy for piecewise monotonic maps).

  * •

$T$ is transitive, i.e., there exists a point $x\in[0,1]$ whose forward orbit
$\\{T^{n}(x):n\geq 0\\}$ is dense in $[0,1]$.

Under these conditions, it is proved in [10, Theorem 4] that there exists a
unique measure of maximal entropy for $T$, that is, a $T$-invariant measure
whose metric entropy coincides with $h_{\rm top}(T)$. The aim of this paper is
to investigate whether the large deviation principle holds for a piecewise
monotonic map with the unique measure of maximal entropy as a reference.

Let $\mathcal{M}([0,1])$ be the set of all Borel probability measures on
$[0,1]$ endowed with the weak∗-topology. We say that $([0,1],T)$ satisfies the
(level-2) large deviation principle with the unique measure of maximal entropy
$m$ as a reference if there exists a lower semi-continuous function
$\mathcal{J}\colon\mathcal{M}([0,1])\to[0,\infty]$, called a rate function,
such that

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log
m\left(\left\\{x\in[0,1]:\frac{1}{n}\sum_{j=0}^{n-1}\delta_{T^{j}(x)}\in\mathcal{K}\right\\}\right)\leq-\inf_{\mathcal{K}}\mathcal{J}$

holds for any closed set $\mathcal{K}\subset\mathcal{M}([0,1])$ and

$\liminf_{n\rightarrow\infty}\frac{1}{n}\log
m\left(\left\\{x\in[0,1]:\frac{1}{n}\sum_{j=0}^{n-1}\delta_{T^{j}(x)}\in\mathcal{U}\right\\}\right)\geq-\inf_{\mathcal{U}}\mathcal{J}$

holds for any open set $\mathcal{U}\subset\mathcal{M}([0,1])$. Here
$\delta_{y}$ signifies the Dirac mass at a point $y\in[0,1]$. We refer to [6]
for a general theory of large deviations and its background in statistical
mechanics.

In this literature, the case that $T$ has a constant slope is important
because this condition implies that the unique measure of maximal entropy is
absolutely continuous to the Lebesgue measure. Hence we focus our attention to
piecewise monotonic maps with a constant slope $\beta>1$. A familiar example
included in this family is a $\beta$-transformation.

  * •

$\beta$-transformations.
The $\beta$-transformation $T_{\beta}\colon[0,1]\to[0,1]$ with $\beta>1$ was
introduced by Rényi [21] and defined by

$T_{\beta}(x)=\left\\{\begin{array}[]{ll}\beta x\ (\text{mod}\
1)&(x\not=1),\\\ \displaystyle\lim_{y\rightarrow 1-0}(\beta y\ (\text{mod}\
1))&(x=1).\end{array}\right.$

This family has recently attracted attention in the setting beyond
specification, since $T_{\beta}$ does not satisfy the specification property
for Lebesgue almost parameter $\beta>1$ ([2, 22]). Inspired by
$\beta$-transformations, many authors have considered various generalizations
of $T_{\beta}$: $(\alpha,\beta)$-transformations ([4, 5, 7, 9, 20]),
$(-\beta)$-transformations ([15, 16, 17, 23]), generalized
$\beta$-transformations ([7, 8, 24, 26]).

  * •

$(\alpha,\beta)$-transformations.
The $(\alpha,\beta)$-transformation $T_{\alpha,\beta}\colon[0,1]\to[0,1]$ with
$\beta>1$ and $0\leq\alpha<1$ was introduced by Parry ([18]) and defined by

$T_{\alpha,\beta}(x)=\left\\{\begin{array}[]{ll}\beta x+\alpha\ (\text{mod}\
1)&(x\not=1),\\\ \displaystyle\lim_{y\rightarrow 1-0}(\beta y+\alpha\
(\text{mod}\ 1))&(x=1).\end{array}\right.$

  * •

Generalized $\beta$-transformations.
Let $\beta>1$ and $k$ be a smallest integer not less than $\beta$, fix
$E=(E_{1},\ldots,E_{k})\in\\{+1,-1\\}^{k}$, and consider $k$ intervals

$I_{1}:=[0,1/\beta),\ I_{2}:=[1/\beta,2/\beta),\ldots,I_{k}:=[k-1/\beta,1].$

The generalized $\beta$-transformation $T_{\beta,E}\colon[0,1]\to[0,1]$ was
introduced by Góra ([8]) and defined by

$T_{\beta,E}(x)=\left\\{\begin{array}[]{ll}\beta x-i+1&(x\in I_{i},\
E(i)=+1),\\\ -\beta x+i&(x\in I_{i},\ E(i)=-1).\end{array}\right.$

If $E=(-1,\ldots,-1)$, then we call $T_{\beta,E}$ a $(-\beta)$-transformation.

Pfister and Sullivan ([19]) established the large deviation principle for
$\beta$-transformations for any $\beta>1$, which is the first work on the
large deviation principle in this family without the specification property.
In [5], Chung and the second author proved that the large deviation principle
holds for $(\alpha,\beta)$ and generalized $\beta$-transformations in the
following parameters:

  * •

$(\alpha,\beta)$-transformations for $0\leq\alpha<1$ and $\beta>2$.

  * •

Generalized $\beta$-transformations for $\frac{1+\sqrt{5}}{2}<\beta<2$ and
$E=(-1,-1)$.

Then it is natural to ask whether does the large deviation principle hold for
the other parameters. In this paper not only we get an affirmative answer to
the all parameters, but also we show that it holds for more general class of
piecewise monotonic maps with constant slope, which we call generalized
$(\alpha,\beta)$-transformations.

  * •

Generalized $(\alpha,\beta)$-transformations. Let $0\leq\alpha<1$, $\beta>1$,
$k$ be a smallest integer not less than $\alpha+\beta$, fix
$E=(E_{1},\ldots,E_{k})\in\\{+1,-1\\}^{k}$, and consider $k$ intervals

$\ \ \ \ \ \ \ \ I_{1}:=\left[0,\frac{1-\alpha}{\beta}\right),\
I_{2}:=\left[\frac{1-\alpha}{\beta},\frac{2-\alpha}{\beta}\right),\ldots,I_{k}:=\left[\frac{k-1-\alpha}{\beta},1\right].$

We define the generalized $(\alpha,\beta)$-transformation
$T_{\alpha,\beta,E}\colon[0,1]\to[0,1]$ by

$T_{\alpha,\beta,E}(x)=\left\\{\begin{array}[]{ll}\alpha+\beta x-i+1&(x\in
I_{i},\ E(i)=+1),\\\ -\alpha-\beta x+i&(x\in I_{i},\
E(i)=-1).\end{array}\right.$

Generalized $(\alpha,\beta)$-transformations are clearly a generalization of
both $(\alpha,\beta)$-transformations and generalized $\beta$-transformations.
The graphs of $T_{\beta}$, $T_{\alpha,\beta}$, $T_{\beta,E}$ and
$T_{\alpha,\beta,E}$ are plotted in Figure 1.

Now we state our first main result of this paper.

Figure 1. Graphs of $T_{\beta}$, $T_{\alpha,\beta}$, $T_{\beta,E}$ and
$T_{\alpha,\beta,E}$.

######  Theorem A.

Let $T\colon[0,1]\to[0,1]$ be a transitive generalized $(\alpha,\beta)$\-
transformation. Then $([0,1],T)$ satisfies the level-2 large deviation
principle with the unique measure of maximal entropy.

A crucial step to prove Theorem A is to show density of periodic measures. For
a metrizable space $X$ and a Borel measurable map $f\colon X\to X$, denote by
$\mathcal{M}(X)$ the set of all Borel probability measures on $X$ endowed with
the weak∗-topology, by $\mathcal{M}_{f}(X)\subset\mathcal{M}(X)$ the set of
$f$-invariant ones, and by $\mathcal{M}^{e}_{f}(X)\subset\mathcal{M}_{f}(X)$
the set of ergodic ones. We say that $\mu\in\mathcal{M}(X)$ is a periodic
measure if there exist $x\in X$ and $n>0$ such that $f^{n}(x)=x$ and
$\mu=\delta_{n}^{f}(x):=1/n\sum_{j=0}^{n-1}\delta_{f^{j}(x)}$ hold. Then, it
is clear that $\mu\in\mathcal{M}_{f}^{e}(X)$. We denote by
$\mathcal{M}_{f}^{p}(X)\subset\mathcal{M}_{f}^{e}(X)$ the set of all periodic
measures on $X$. It is established by Chung and the second author that the
level-2 large deviation principle for a piecewise monotonic map is followed by
density of periodic measures in the set of ergodic measures with the
irreducibility of its Markov diagram (see [5, Theorem A]). Although the
irreducibility is slightly stronger than the transitivity, we show in this
paper that this result remains true if the irreducibility of a Markov diagram
is replaced by the transitivity of a map.

Proposition A. Let $T\colon[0,1]\to[0,1]$ be a transitive piecewise monotonic
map with $h_{{\rm top}}(T)>0$. Suppose that $\mathcal{M}^{p}_{T}([0,1])$ is
dense in $\mathcal{M}_{T}^{e}([0,1])$. Then $([0,1],T)$ satisfies the level-2
large deviation principle with the unique measure of maximal entropy.

Hence Theorem A follows from Proposition A and the following theorem, which is
the second main result of this paper:

######  Theorem B.

Let $T\colon[0,1]\to[0,1]$ be a transitive generalized $(\alpha,\beta)$\-
transformation. Then $\mathcal{M}_{T}^{p}([0,1])$ is dense in
$\mathcal{M}_{T}^{e}([0,1])$.

Our proof of Theorem B is based on the work by Hofbauer and Raith ([13]) where
density of periodic measures was proved for a piecewise monotonic map
consisting of two monotonic pieces. Since the family of generalized
$(\alpha,\beta)$-transformations includes maps with more than three monotonic
pieces, we need to improve their method. One key difference between [13] and
ours is Proposition 3.3, which is one of the novelty of this paper (see also
Remark 3.1).

The remainder of this paper is organized as follows. In §2, we establish our
definitions and prepare several facts. Subsequently, we present proofs of
Proposition A and Theorem B in §3.

##  2\. Preliminaries

###  2.1. Symbolic dynamics

Let $\mathbb{N}_{0}$ be the set of non-negative integers. For a finite or
countable set $A$, we denote by $A^{\mathbb{N}_{0}}$ the one-sided infinite
product of $A$ equipped with the product topology of the discrete topology of
$A$. To simplify the notation, given integers $i\leq j$ and
$x_{i},x_{i+1},\ldots,x_{j}\in A$, we set $x_{[i,j]}:=x_{i}x_{i+1}\cdots
x_{j}$. We also write $x_{[i,j)}:=x_{[i,j-1]}$ and similarly for $x_{(i,j]}$
and $x_{(i,j)}$. A sequence $\underline{x}\in A^{\mathbb{N}_{0}}$ will be
defined by all of its coordinates $x_{n}\in A$ with $n\geq 0$. Let $\sigma$ be
the shift map on $A^{\mathbb{N}_{0}}$ (i.e.,
$(\sigma(\underline{x}))_{n}=x_{n+1}$ for each $n\geq 0$ and $\underline{x}\in
A^{\mathbb{N}_{0}}$). When a subset $\Sigma^{+}$ of $A^{\mathbb{N}_{0}}$ is
$\sigma$-invariant and closed, we call it a subshift and call $A$ the alphabet
of $\Sigma^{+}$. For a matrix $M=(M_{ij})_{(i,j)\in A^{2}}$, each entry of
which is $0$ or $1$, we define a subshift $\Sigma^{+}_{M}\subset
A^{\mathbb{N}_{0}}$ by

$\Sigma^{+}_{M}=\\{\underline{x}\in
A^{\mathbb{N}_{0}}:\text{$M_{x_{n}x_{n+1}}=1$ for all $n\geq 0$}\\}$

and call $\Sigma^{+}_{M}$ a Markov shift with an adjacency matrix $M$.

For a subshift $\Sigma^{+}$ on an alphabet $A$, we set
$\mathcal{L}(\Sigma^{+}):=\\{x_{[0,n]}:\underline{x}\in\Sigma^{+},\ n\geq
0\\}$ and $[u]:=\left\\{\underline{x}\in\Sigma^{+}:u=x_{[0,|u|)}\right\\}$ for
each $u\in\mathcal{L}(\Sigma^{+})$, where $|u|$ denotes the length of $u$. A
word $v\in\mathcal{L}(\Sigma^{+})$ is called a subword of $u=u_{0}\cdots
u_{n}\in\mathcal{L}(\Sigma^{+})$ if $v=u_{[i,j]}$ for some $0\leq i\leq j\leq
n$. For $u,v\in\mathcal{L}(\Sigma^{+})$, we use juxtaposition $uv$ to denote
the word obtained by the concatenation and $u^{\infty}$ means a one-sided
infinite sequence $uuu\cdots\in A^{\mathbb{N}_{0}}$. Moreover, we set
$\mathcal{S}(u):=\\{i\in A:ui\in\mathcal{L}(\Sigma^{+})\\}$ for
$u\in\mathcal{L}(\Sigma^{+})$. Finally, we say that $\Sigma^{+}$ is transitive
if for any $u,v\in\mathcal{L}(\Sigma^{+})$, we can find
$w\in\mathcal{L}(\Sigma^{+})$ such that $uwv\in\mathcal{L}(\Sigma^{+})$ holds.
For the rest of this paper, we denote by $h_{\sigma}(\mu)$ the metric entropy
of $\mu\in\mathcal{M}_{\sigma}(\Sigma^{+})$.

###  2.2. Markov diagram

Let $X=[0,1]$ and $T\colon X\to X$ be a piecewise monotonic map with critical
points $0=c_{0}<c_{1}<\cdots<c_{k}=1$. Let
$X_{T}:=\bigcap_{n=0}^{\infty}T^{-n}(\bigcup_{j=1}^{k}(c_{j-1},c_{j}))$, and
define the coding map $\mathcal{I}\colon
X_{T}\to\\{1,\ldots,k\\}^{\mathbb{N}_{0}}$ by

$(\mathcal{I}(x))_{n}=j\text{ if and only if }T^{n}(x)\in(c_{j-1},c_{j}),$

which is injective since $T$ is transitive (see [27, Proposition 6.1]). We
denote the closure of $\mathcal{I}(X_{T})$ in $\\{1,\ldots,k\\}^{\mathbb{N}}$
by $\Sigma_{T}^{+}$. Then, $\Sigma_{T}^{+}$ is a subshift, and
$(\Sigma_{T}^{+},\sigma)$ is called the coding space of $(X,T)$. We use the
following notations:

  * •

$\displaystyle\underline{a}^{(i)}:=\lim_{x\rightarrow c_{i}+0}\mathcal{I}(x)$,
$\displaystyle\underline{b}^{(i)}:=\lim_{x\rightarrow c_{i}-0}\mathcal{I}(x)$
for $1\leq i\leq k-1$.

  * •

${\rm adj}(\underline{a}^{(i)}):=\underline{b}^{(i)}$, ${\rm
adj}(\underline{b}^{(i)}):=\underline{a}^{(i)}$ for $1\leq i\leq k-1$.

  * •

$\displaystyle\underline{a}:=\lim_{x\rightarrow+0}\mathcal{I}(x)$,
$\displaystyle\underline{b}:=\lim_{x\rightarrow 1-0}\mathcal{I}(x)$.

We also set $\mathcal{CR}:=\\{\underline{a}^{(i)},\underline{b}^{(i)}:1\leq
i\leq k-1\\}$ and call it a critical set.

In what follows, we define the Markov diagram, introduced by Hofbauer ([10]),
which is a countable oriented graph with subsets of $\Sigma_{T}^{+}$ as
vertices. Let $C\subset\Sigma_{T}^{+}$ be a closed subset with $C\subset[j]$
for some $1\leq j\leq k$. We say that a non-empty closed subset
$D\subset\Sigma_{T}^{+}$ is a successor of $C$ if $D=[l]\cap\sigma(C)$ for
some $1\leq l\leq k$. The expression $C\rightarrow D$ denotes that $D$ is a
successor of $C$. Now, we define a set $\mathcal{D}_{T}$ of vertices by
induction. First, we set $\mathcal{D}_{0}:=\\{[1],\ldots,[k]\\}$. If
$\mathcal{D}_{n}$ is defined for $n\geq 0$, then we set

$\mathcal{D}_{n+1}:=\mathcal{D}_{n}\cup\\{D:{\color[rgb]{0,0,0}D\text{ is a
successor for some }C\in\mathcal{D}_{n}}\\}.$

We note that $\mathcal{D}_{n}$ is a finite set for each $n\geq 0$ since the
number of successors of any closed subset of $\Sigma_{T}^{+}$ is at most $k$
by definition. Finally, we set

$\mathcal{D}_{T}:=\bigcup_{n\geq 0}\mathcal{D}_{n}.$

The oriented graph $(\mathcal{D}_{T},\rightarrow)$ is called the Markov
diagram of $T$. For notational simplicity, we use the expression $\mathcal{D}$
instead of $\mathcal{D}_{T}$ if no confusion arises.

For $\underline{x}\in\Sigma_{T}^{+}$ and $n\geq 0$, we set
$D_{n}^{\underline{x}}:=\sigma^{n}(x_{[0,n]})$. We define a sequence
$\\{R_{m}^{\underline{x}}\\}_{m\geq 0}$ of integers inductively as follows.
First, we set $R_{0}^{\underline{x}}:=0$. If $R_{m}^{\underline{x}}$ is
defined for $m\geq 0$, then let

$R_{m+1}^{\underline{x}}:=\min\\{n>R_{m}:\\#(\mathcal{S}(x_{[0,n-1)}))\geq
2\\}.$

We also set
$r_{m}^{\underline{x}}:=R_{m}^{\underline{x}}-R_{m-1}^{\underline{x}}$. Now,
we summarize properties of Hofbauer’s Markov Diagram, which are appeared in
[12] (see also [13, Page 224]).

######  Proposition 2.1.

Let $(\mathcal{D},\rightarrow)$ be the Markov diagram of $T$.
(1)
$\mathcal{D}=\\{D_{n}^{\underline{x}}:\underline{x}\in\mathcal{CR}\cup\\{\underline{a},\underline{b}\\},\
n\geq 0\\}$.
(2) $(\mathcal{D},\rightarrow)$ has the following arrows:

  * •

For any $\underline{x}\in\mathcal{CR}\cup\\{\underline{a},\underline{b}\\}$
and any $n\geq 0$, $D_{n}^{\underline{x}}\rightarrow D_{n+1}^{\underline{x}}$.

  * •

For any $\underline{x}\in\mathcal{CR}\cup\\{\underline{a},\underline{b}\\}$
and any $m\geq 1$, $D_{R_{m}^{\underline{x}}-1}^{\underline{x}}\rightarrow
D_{r_{m}^{\underline{x}}}^{f_{m}(\underline{x})}$. Here $f_{m}(\underline{x})$
is a unique point in $\mathcal{CR}$ so that
$f_{m}(\underline{x})\not=\sigma^{R_{m-1}^{\underline{x}}}(\underline{x})$ and
$f_{m}(\underline{x})\in D_{R_{m-1}}^{\underline{x}}$ hold.

(3) Let $\underline{x}\in\mathcal{CR}\cup\\{\underline{a},\underline{b}\\}$.

  * •

For any $0\leq n\leq r_{m}^{\underline{x}}-1$, we have
$D^{\underline{x}}_{n+R_{m-1}}\subset D_{n}^{f_{m}(\underline{x})}$. In
particular,
$x_{[R_{m-1}^{\underline{x}},R_{m}^{\underline{x}})}=f_{m}(\underline{x})_{[0,r_{m}^{\underline{x}})}$
holds.

  * •

There exists an integer $q$ such that
$r_{m}^{\underline{x}}=R_{q}^{f_{m}(\underline{x})}$.

(4) If $C\in\mathcal{D}$ and $\\#\mathcal{S}(C)>2$, then
$\mathcal{S}(C)\cap\mathcal{D}_{0}\not=\emptyset$.

For a subset $\mathcal{C}\subset\mathcal{D}$, we define a matrix
$M(\mathcal{C})=(M(\mathcal{C})_{C,D})_{(C,D)\in\mathcal{C}^{2}}$ by

$M(\mathcal{C})_{C,D}=\left\\{\begin{array}[]{ll}1&(\text{$C\rightarrow
D$}),\\\ 0&(\text{otherwise}).\end{array}\right.$

Then,
$\Sigma^{+}_{M(\mathcal{C})}=\\{\underline{C}\in\mathcal{C}^{\mathbb{N}_{0}}:C_{n}\rightarrow
C_{n+1},n\geq 0\\}$ is a one-sided Markov shift with a countable alphabet
$\mathcal{C}$ and an adjacency matrix $M(\mathcal{C})$. For notational
simplicity, we denote $\Sigma_{\mathcal{C}}^{+}$ instead of
$\Sigma^{+}_{M(\mathcal{C})}$.We say that $\mathcal{C}$ is irreducible if for
any $C,D\in\mathcal{C}$, there are finite vertices
$C_{0},\ldots,C_{n}\in\mathcal{C}$ such that $C_{i}\rightarrow C_{i+1}$ for
$0\leq i\leq n-1$ (i.e., $C_{[0,n]}\in\mathcal{L}(\Sigma^{+}_{\mathcal{C}})$),
$C_{0}=C$ and $C_{n}=D$, and if every subset of $\mathcal{D}$, which contains
$\mathcal{C}$ does not have this property. It is clear that
$\Sigma^{+}_{\mathcal{C}}$ is transitive if and only if $\mathcal{C}$ is
irreducible. We define a map
$\Psi\colon\Sigma_{\mathcal{D}}^{+}\to\\{1,\ldots,k\\}^{\mathbb{N}_{0}}$ by

$\Psi((C_{n})_{n\in\mathbb{N}_{0}}):=(x_{n})_{n\in\mathbb{N}_{0}}\text{ for
}(C_{n})_{n\in\mathbb{N}_{0}}\in\Sigma^{+}_{\mathcal{D}},$

where $1\leq x_{n}\leq k$ is a unique integer such that $C_{n}\subset[x_{n}]$
holds for each $n\in\mathbb{N}_{0}$. Then it is not difficult to see that
$\Psi$ is continuous, countable to one and
$\Psi(\Sigma_{\mathcal{D}}^{+})=\Sigma_{T}^{+}$. We say that
$\mu\in\mathcal{M}^{e}_{\sigma}(\Sigma_{T}^{+})$ is liftable if there is
$\overline{\mu}\in\mathcal{M}^{e}_{\sigma}(\Sigma_{\mathcal{D}}^{+})$ such
that $\mu=\overline{\mu}\circ\Psi^{-1}$ holds. It is known that not every
$\mu\in\mathcal{M}^{e}_{\sigma}(\Sigma_{T}^{+})$ is liftable in general
although $\Psi$ is surjective (see [14]). In [10], Hofbauer provided a
sufficient condition for the liftability.

######  Lemma 2.2.

([10, Lemma 3]) If $\mu\in\mathcal{M}_{\sigma}^{e}(\Sigma_{T}^{+})$ has
positive metric entropy, then $\mu$ is liftable.

We recall two important facts for transitive piecewise monotonic maps.

######  Lemma 2.3.

([12, Theorem 11],[13, Page 224]) There exists an irreducible subset
$\mathcal{C}\subset\mathcal{D}$ satisfying the following properties:

  * •

$\Psi(\Sigma_{\mathcal{C}}^{+})=\Sigma_{T}^{+}$.

  * •

$C\in\mathcal{C}$ and $C\rightarrow D$ implies $D\in\mathcal{C}$.

  * •

There exists an integer $n_{0}$ such that
$D_{n_{0}}^{\underline{x}}\in\mathcal{C}$ holds for any
$\underline{x}\in\mathcal{CR}\cup\\{\underline{a},\underline{b}\\}$.

######  Theorem 2.4.

([13, Theorem 1]) Suppose that there are integers $N_{0}$ and $N_{1}$ such
that for any $\underline{x}\in\mathcal{CR}$, and any $j\in\mathbb{N}$ with
$R_{j}^{\underline{x}}>N_{0}$, there exist an integer $1\leq m<j$, a periodic
point $\underline{p}$ with period $l$ and $u\in\mathcal{L}(\Sigma_{T}^{+})$
with $|u|\geq R_{j}^{\underline{x}}-R_{m}^{\underline{x}}-N_{1}$ such that $u$
is a subword of both $x_{[R_{m}^{\underline{x}},R_{j}^{\underline{x}})}$ and
$p_{[0,l)}$. Then $\mathcal{M}_{T}^{p}([0,1])$ is dense in
$\mathcal{M}_{T}^{e}([0,1])$.

Hereafter, let $T\colon[0,1]\to[0,1]$ be a generalized
$(\alpha,\beta)$-transformation and $k$ be a smallest integer not less than
$\alpha+\beta$. To simplifies the notation, we set
$A_{n}:=\sigma^{n}([a_{[0,n]}])$ and $B_{n}:=\sigma^{n}([b_{[0,n]}])$ for each
$n\geq 0$ and set $R_{m}:=R_{m}^{\underline{a}}$,
$S_{m}:=R_{m}^{\underline{b}}$,
$r_{m+1}:=r_{m+1}^{\underline{a}}=R_{m+1}-R_{m}$ and
$s_{m+1}:=r_{m+1}^{\underline{b}}=S_{m+1}-S_{m}$ for each $m\geq 0$. We also
denote

$\mathcal{A}_{1}:=\\{m\in\mathbb{N}:\sigma(f_{m}(\underline{a}))=\underline{a}\\}$,
$\mathcal{A}_{2}:=\\{m\in\mathbb{N}:\sigma(f_{m}(\underline{a}))=\underline{b}\\}$,
$\mathcal{B}_{1}:=\\{m\in\mathbb{N}:\sigma(f_{m}(\underline{b}))=\underline{a}\\}$
and
$\mathcal{B}_{2}:=\\{m\in\mathbb{N}:\sigma(f_{m}(\underline{b}))=\underline{b}\\}$.

Note that $\sigma[i]=\Sigma_{T}^{+}$ for each $2\leq i\leq k-1$ and
$\sigma(\underline{x})\in\\{\underline{a},\underline{b}\\}$ for any
$\underline{x}\in\mathcal{CR}$. These together with Proposition 2.1 and
Theorem 2.4 imply the following:

######  Proposition 2.5.

Let $(\mathcal{D},\rightarrow)$ be the Markov diagram of $T$.
(1) $\mathcal{D}=\\{A_{n},B_{n}:n\geq 0\\}\cup\\{[2],\ldots,[k-1]\\}$.
(2) (i) For any $n\geq 0$, $A_{n}\rightarrow A_{n+1}$ and $B_{n}\rightarrow
B_{n+1}$.
(ii) For $m\geq 1$,

  * •

$A_{R_{m}-1}\rightarrow A_{r_{m}-1}$ and $a_{(R_{m-1},R_{m})}=a_{[0,r_{m}-1)}$
if $m\in\mathcal{A}_{1}$,

  * •

$A_{R_{m}-1}\rightarrow B_{r_{m}-1}$ and $a_{(R_{m-1},R_{m})}=b_{[0,r_{m}-1)}$
if $m\in\mathcal{A}_{2}$.

(iii) For $m\geq 1$,

  * •

$B_{S_{m}-1}\rightarrow A_{s_{m}-1}$ and $b_{(S_{m-1},S_{m})}=a_{[0,r_{m}-1)}$
if $m\in\mathcal{B}_{1}$,

  * •

$B_{S_{m}-1}\rightarrow B_{s_{m}-1}$ and $b_{(S_{m-1},S_{m})}=b_{[0,s_{m}-1)}$
if $m\in\mathcal{B}_{2}$.

(3) There exist two maps $P\colon\mathcal{A}_{2}\to\mathbb{N}_{0}$ and
$Q\colon\mathcal{B}_{1}\to\mathbb{N}_{0}$ such that for any $m\geq 1$,
$r_{m}-1=S_{p(m)}$. and $s_{m}-1=R_{Q(m)}$.

######  Theorem 2.6.

Suppose that there are integers $N_{0}$ and $N_{1}$ satisfying the following
conditions:

  * •

For any $j\in\mathbb{N}$ with $R_{j}>N_{0}$, there is an integer $1\leq m<j$,
a periodic point $\underline{p}$ with period $l$ and
$u\in\mathcal{L}(\Sigma_{T}^{+})$ with $|u|\geq R_{j}-R_{m}-N_{1}$ such that
$u$ is a subword of both $a_{[R_{m},R_{j})}$ and $p_{[0,l)}$.

  * •

For any $j\in\mathbb{N}$ with $S_{j}>N_{0}$, there is an integer $1\leq m<j$,
a periodic point $\underline{p}$ with period $l$ and
$u\in\mathcal{L}(\Sigma_{T}^{+})$ with $|u|\geq S_{j}-S_{m}-N_{1}$ such that
$u$ is a subword of both $b_{[R_{m},R_{j})}$ and $p_{[0,l)}$.

Then $\mathcal{M}^{p}_{T}([0,1])$ is dense in $\mathcal{M}_{T}^{e}([0,1])$.

##  3\. Proofs

###  3.1. Proof of Proposition A

In this subsection, we give a proof of Proposition A with the assumption that
$T$ is transitive. As we mentioned in §1, this theorem appears as [5, Theorem
A] with the stronger assumption that $\mathcal{D}$ is irreducible. In [5], the
hypothesis of the irrecducibility for $\mathcal{D}$ is used only in [5,
Proposition 3.1]. The other part of the proof of [5, Theorem A] can be shown
similarly by using the transitivity of $T$ instead of the irreducibility.
Hence to prove Proposition A, it is sufficient to show the following
proposition, which is analogous to [5, Proposition 3.1].

######  Proposition 3.1.

Let $\mathcal{C}\subset\mathcal{D}$ be as in Lemma 2.3. For any $\epsilon>0$,
any $\mu\in\mathcal{M}_{\sigma}(\Sigma_{T}^{+})$, and any neighborhood
$\mathcal{U}\subset\mathcal{M}(\Sigma_{T}^{+})$ of $\mu$, there exist a finite
set $\mathcal{F}\subset\mathcal{C}$ and
$\rho\in\mathcal{M}_{\sigma}^{e}(\Psi(\Sigma^{+}_{\mathcal{F}}))$ such that
$\rho\in\mathcal{U}$ and
${\color[rgb]{0,0,0}h_{\sigma}(\rho)}\geq{\color[rgb]{0,0,0}h_{\sigma}(\mu)}-2\epsilon$.

###### Proof..

Without loss of generality, we may assume that $h_{\sigma}(\mu)-\epsilon>0$,
otherwise the conclusion is yield by the assumption that
$\mathcal{M}_{\sigma}^{p}(\Sigma_{T}^{+})$ is dense in
$\mathcal{M}_{\sigma}^{e}(\Sigma_{T}^{+})$. By Ergodic Decomposition Theorem
and the affinity of the entropy map, there exists a finite convex combination
of ergodic measures $\nu:=\sum_{i=1}^{p}a_{i}\nu_{i}$ such that
$h_{\sigma}(\nu)\in\mathcal{U}$ and $h_{\sigma}(\nu)\geq
h_{\sigma}(\mu)-\epsilon$. Again by density of
$\mathcal{M}^{p}_{\sigma}(\Sigma_{T}^{+})$ in
$\mathcal{M}_{\sigma}^{e}(\Sigma_{T}^{+})$, we may assume
$\nu_{i}\in\mathcal{M}_{\sigma}^{p}(\Sigma_{T}^{+})$ whenever
$h_{\sigma}(\nu_{i})=0$. Then we need the following lemma.

######  Lemma 3.2.

For each $1\leq i\leq p$, there exists
$\overline{\nu_{i}}\in\mathcal{M}_{\sigma}^{e}(\Sigma_{\mathcal{C}}^{+})$ such
that $\nu_{i}=\overline{\nu_{i}}\circ\Psi^{-1}$ holds. In particular,
$\overline{\nu}:=\sum_{i=1}^{p}a_{i}\overline{\nu_{i}}$ satisfies
$\overline{\nu}\in\mathcal{M}_{\sigma}(\Sigma_{\mathcal{C}}^{+})$ and
$\nu=\overline{\nu}\circ\Psi^{-1}$.

###### Proof..

We divide the proof into two cases.

(Case 1) $h_{\sigma}(\nu_{i})=0$. In this case, $\nu_{i}$ is a periodic
measure. Take a periodic point $\underline{x}\in\Sigma_{T}^{+}$ in the support
of $\nu_{i}$. Then it follows from [12, Theorem 8] and
$\Psi(\Sigma^{+}_{\mathcal{C}})=\Sigma_{T}^{+}$ that there are finite vertices
$C_{0},\ldots,C_{n-1}\in\mathcal{C}$ such that
$(C_{[0,n)})^{\infty}\in\Sigma_{\mathcal{C}}^{+}$ and
$\Psi((C_{[0,n)})^{\infty})=\underline{x}$. Hence if we set
$\overline{\nu_{i}}:=\delta_{n}^{\sigma}((C_{[0,n)})^{\infty}),$ then we have
$\overline{\nu_{i}}\in\mathcal{M}_{\sigma}^{e}(\Sigma_{\mathcal{C}}^{+})$ and
$\nu_{i}=\overline{\nu_{i}}\circ\Psi^{-1}$.

(Case 2) $h_{\sigma}(\nu_{i})>0$. Since
$\nu_{i}\in\mathcal{M}_{\sigma}^{e}(\Sigma_{T}^{+})$ and
$h_{\sigma}(\nu_{i})>0$, it follows from Lemma 2.2 that there exists
$\overline{\nu_{i}}\in\mathcal{M}_{\sigma}^{e}(\Sigma_{\mathcal{D}}^{+})$ such
that $\overline{\nu_{i}}:=\nu_{i}\circ\Psi^{-1}$ holds. Since
$\overline{\nu_{i}}$ is ergodic, there exists an irreducible subset
$\mathcal{C}^{\prime}\subset\mathcal{D}$ such that
$\overline{\nu}_{i}(\Sigma_{\mathcal{C}^{\prime}}^{+})=1$. Assume that
$\mathcal{C}^{\prime}\not=\mathcal{C}$. Since
$\Sigma^{+}_{\mathcal{C}^{\prime}}\subset\Psi^{-1}(\Psi(\Sigma^{+}_{\mathcal{C}^{\prime}}))$,
we have
$\nu_{i}(\Psi(\Sigma^{+}_{\mathcal{C}^{\prime}}))=\overline{\nu_{i}}\circ\Psi^{-1}(\Psi(\Sigma^{+}_{\mathcal{C}^{\prime}}))\geq\overline{\nu_{i}}(\Sigma^{+}_{\mathcal{C}^{\prime}})=1$.
This implies that
$\nu_{i}(\Psi(\Sigma^{+}_{C})\cap\Psi(\Sigma^{+}_{\mathcal{C}^{\prime}}))=\nu_{i}(\Sigma_{T}^{+}\cap\Psi(\Sigma_{\mathcal{C}^{\prime}}^{+}))=1$.
On the other hand, by [11, Theorem 1 (ii)],
$\Psi(\Sigma^{+}_{C})\cap\Psi(\Sigma^{+}_{\mathcal{C}^{\prime}})$ is either
empty or finite (see also [12, Page 385]). Hence by the ergodicity of
$\nu_{i}$, we have $h_{\sigma}(\nu_{i})=0$, which is a contradiction. ∎

Note that $\overline{\nu}$ is an invariant measure on a transitive countable
Markov shift $\Sigma^{+}_{\mathcal{C}}$. Hence by the continuity of $\Psi$ and
[25, Main Theorem], we can find a finite set $\mathcal{F}\subset\mathcal{C}$
and an ergodic measure $\overline{\rho}$ on $\Sigma^{+}_{\mathcal{F}}$ such
that $h_{\sigma}(\overline{\rho})\geq h_{\sigma}(\overline{\nu})-\epsilon$ and
$\rho:=\overline{\rho}\circ\Psi^{-1}\in\mathcal{U}$. Since
$\Psi\colon\Sigma^{+}_{\mathcal{C}}\to\Sigma_{T}^{+}$ is countable to one, by
[3, Proposition 2.8], we have $h_{\sigma}(\nu)=h_{\sigma}(\overline{\nu})$ and
$h_{\sigma}(\rho)=h_{\sigma}(\overline{\rho})$, which prove the proposition. ∎

###  3.2. Proof of Theorem B

The aim of this subsection is to give a proof of Theorem B. Let $n_{0}$ be as
in Lemma 2.3 and $m_{0}$ be a smallest integer such that
$R_{m_{0}},S_{m_{0}}\geq n_{0}$ holds. We set $N_{0}:=\max\\{R_{m_{0}},\
S_{m_{0}}\\}$ and let $n_{1}$ be a smallest number such that for any
$C\in\mathcal{C}\cap\mathcal{D}_{n_{0}}$ and
$D\in\mathcal{C}\cap\mathcal{D}_{N_{0}}$, there exist finite vertices
$C_{0},\ldots,C_{n}$ with $n\leq n_{1}$ such that
$C_{[0,n]}\in\mathcal{L}(\Sigma_{\mathcal{C}}^{+})$, $C_{0}=C$ and $C_{n}=D$.
We note that $n_{1}<\infty$ since $\mathcal{C}$ is irreducible. We set
$N_{1}:=N_{0}+n_{1}$. By Theorem 2.6, to prove Theorem B, it is sufficient to
show the following:

  1. (I)

For any $j\in\mathbb{N}$ with $R_{j}>N_{0}$, there is an integer $1\leq m<j$,
a periodic point $\underline{p}$ with period $l$ and
$u\in\mathcal{L}(\Sigma_{T}^{+})$ with $|u|\geq R_{j}-R_{m}-N_{1}$ such that
$u$ is a subword of both $a_{[R_{m},R_{j})}$ and $p_{[0,l)}$.

  2. (II)

For any $j\in\mathbb{N}$ with $S_{j}>N_{0}$, there is an integer $1\leq m<j$,
a periodic point $\underline{p}$ with period $l$ and
$u\in\mathcal{L}(\Sigma_{T}^{+})$ with $|u|\geq S_{j}-S_{m}-N_{1}$ such that
$u$ is a subword of both $b_{[R_{m},R_{j})}$ and $p_{[0,l)}$.

We only prove the item (I) because (II) can be shown in a similar manner. Take
any $j\in\mathbb{N}$ with $R_{j}>N_{0}$. In what follows we will decompose the
set $\mathbb{N}$ into six sets. Let $\mathcal{A}_{1}$, $\mathcal{A}_{2}$,
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ be as in §2.3 and set

$\mathcal{A}_{3}:=\\{m\in\mathbb{N}:\mathcal{S}(A_{R_{m}-1})\cap\mathcal{D}_{0}\not=\emptyset\\}$
and
$\mathcal{B}_{3}:=\\{m\in\mathbb{N}:\mathcal{S}(B_{S_{m}-1})\cap\mathcal{D}_{0}\not=\emptyset\\}$.

First, it is clear that
$\mathbb{N}=\mathcal{A}_{1}\cup\mathcal{A}_{2}=\mathcal{A}_{1}\cup(\mathcal{A}_{2}\setminus\mathcal{A}_{3})\cup\mathcal{A}_{3}$.
We set

$\mathcal{A}_{2}^{(1)}:=\\{m\in\mathcal{A}_{2}:m+1\in\mathcal{A}_{1}\\},$
$\mathcal{A}_{2}^{(2)}:=\\{m\in\mathcal{A}_{2}:P(m)\in\mathcal{B}_{3}\\},$
$\mathcal{A}_{2}^{(3)}:=\\{m\in\mathcal{A}_{2}:P(m)+1\in\mathcal{B}_{2}\\},$
and
$\mathcal{A}_{4}:=\mathcal{A}_{2}\setminus(\mathcal{A}_{3}\cup\bigcup_{j=1}^{3}\mathcal{A}_{2}^{(j)})$,

noting that the map $P$ is defined on $\mathcal{A}_{2}$ by Proposition 2.5
(3).
Then we have
$\mathcal{A}_{2}\setminus\mathcal{A}_{3}\subset\bigcup_{j=1}^{3}\mathcal{A}_{2}^{(j)}\cup\mathcal{A}_{4}$,
which implies that

$\mathbb{N}=\mathcal{A}_{1}\cup\bigcup_{j=1}^{3}\mathcal{A}_{2}^{(j)}\cup\mathcal{A}_{3}\cup\mathcal{A}_{4}.$

Hence we can divide the proof into six cases:

(Case A) $j\in\mathcal{A}_{1}$. (Case B) $j\in\mathcal{A}_{2}^{(1)}$. (Case C)
$j\in\mathcal{A}_{2}^{(2)}$.
(Case D) $j\in\mathcal{A}_{2}^{(3)}$. (Case E) $j\in\mathcal{A}_{3}$. (Case F)
$j\in\mathcal{A}_{4}.$

We note that the proofs of (Case A) and (Case B) are similar to those of (Case
1) and (Case 2) in [13] respectively.

(Case C) $j\in\mathcal{A}_{2}^{(2)}$. Take
$D\in\mathcal{S}(B_{S_{P(j)}-1})\cap\mathcal{D}_{0}$. By the definition of
$n_{1}$, we can find an integer $n\leq n_{1}$ and finite vertices
$C_{0},\ldots,C_{n}\in\mathcal{C}$ such that
$C_{[0,n]}\in\mathcal{L}(\Sigma^{+}_{\mathcal{C}})$, $C_{0}=D$ and
$C_{n}=B_{S_{m_{0}}}$. Now, we set
$\underline{p}:=\Psi((B_{[S_{m_{0}},S_{P(j)})}C_{[0,n)})^{\infty})$. Then
$\underline{p}$ is a periodic point with period $l:=S_{P(j)}-S_{m_{0}}+n$.
Since $A_{R_{j}-1}\rightarrow B_{S_{P(j)}}$, we have
$a_{(R_{j-1},R_{j})}=b_{[0,S_{P(j)})}$, which implies (I).

(Case D) $j\in\mathcal{A}_{2}^{(3)}$. Since $P(j)+1\in\mathcal{B}_{2}$, we can
find $u\leq P(j)$ such that $B_{S_{P(j)+1}-1}\rightarrow B_{u}$. We set
$\underline{p}:=\Psi((B_{(S_{P(j)},S_{P(j)+1})}B_{[u,S_{P(j)}]})^{\infty})$.
Then $\underline{p}$ is a periodic point with period $l:=S_{P(j)+1}-u$. Since
$A_{R_{j}-1}\rightarrow B_{S_{P(j)}}$ and $B_{S_{P(j)+1}-1}\rightarrow B_{u}$,
we have
$a_{(R_{j-1},R_{j})}=b_{[0,S_{P(j)})}=b_{[0,u)}b_{[u,S_{P(j)})}=b_{(S_{P(j)},S_{P(j)+1})}b_{[u,S_{P(j)}]}$.
This implies (I).

(Case E) $j\in\mathcal{A}_{3}$. Take
$D\in\mathcal{S}(A_{R_{j}-1})\cap\mathcal{D}_{0}$. By the definition of
$n_{1}$, we can find an integer $n\leq n_{1}$ and finite vertices
$C_{0},\ldots,C_{n}\in\mathcal{C}$ such that
$C_{[0,n]}\in\mathcal{L}(\Sigma^{+}_{\mathcal{C}})$, $C_{0}=D$ and
$C_{n}=A_{R_{m_{0}}}$. Hence if we set
$\underline{p}:=\Psi((A_{[R_{m_{0}},A_{R_{j}})}C_{[0,n)})^{\infty})$, then (I)
holds.

(Case F) $j\in\mathcal{A}_{4}$. In this case, $j,j+1\in\mathcal{A}_{2}$,
$P(j)+1\in\mathcal{B}_{1}$, $j\not\in\mathcal{A}_{3}$ and
$P(j)\not\in\mathcal{B}_{3}$ hold by the definition of $\mathcal{A}_{4}$. We
prove the following proposition, which plays a fundamental role to prove this
case.

######  Proposition 3.3.

We have

$S_{P(j)}^{\infty}\succ\\{r_{j+i}-1\\}_{i=1}^{\infty}\quad\mbox{or}\quad
R_{j}^{\infty}\succ\\{s_{P(j)+i}-1\\}_{i=1}^{\infty}.$

Here $\succ$ denotes the lexicographical order.

###### Proof..

For the notational simplicity, set $r^{(m)}:=\\{r_{m+i}-1\\}_{i=1}^{\infty}$
and $s^{(m)}:=\\{s_{m+i}-1\\}_{i=1}^{\infty}$ for $m\geq 1$. Showing by
contradiction, we assume

$S_{P(j)}^{\infty}\preceq r^{(j)}\quad\mbox{and}\quad R_{j}^{\infty}\preceq
s^{P(j)}.$ (3.1)

For $q\geq R_{j}+1$ set $\mathcal{E}_{q,1}:=\\{m:R_{j}\leq R_{m}<q\\}$ and
$\mathcal{E}_{q,2}:=\\{m:S_{P(j)}\leq R_{m}<q\\}$. Denote also
$\underline{x}:=f_{j+1}(\underline{a})$ and $\underline{y}:={\rm
adj}(\underline{x})=f_{P(j)+1}(\underline{b})$. Note that
$f_{m}(\underline{a})=x$ implies $m\in\mathcal{A}_{2}$. Similarly
$f_{m}(\underline{b})=\underline{y}$ implies $m\in\mathcal{B}_{1}$. To prove
the proposition, it is sufficient to show the following for all $q\geq
R_{j}+1$:

  * •

For any $m\in\mathcal{E}_{q,1}$, we have

$m\not\in\mathcal{A}_{3},\ f_{m+1}(\underline{a})=\underline{x}\text{ and
}S_{P(j)}^{\infty}\preceq r^{(m)}.$ (3.2)

  * •

For any $m\in\mathcal{E}_{q,2}$, we have

$m\not\in\mathcal{B}_{3},\ f_{m+1}(\underline{b})=\underline{y}\text{ and
}R_{j}^{\infty}\preceq s^{(m)}.$ (3.3)

Indeed, letting $\mathcal{C}^{\prime}:=\\{A_{m}:m\geq
R_{j}\\}\cup\\{B_{m}:m\geq S_{P(j)}\\}$, we have
$\bigcup_{C\in\mathcal{C}^{\prime}}\mathcal{S}(C)\cap(\mathcal{C}\setminus\mathcal{C}^{\prime})=\emptyset$
by (3.2) and (3.3). However, this is a contradiction since $\mathcal{C}$ is
irreducible and $A_{n_{0}}\in\mathcal{C}\setminus\mathcal{C}^{\prime}$.

We prove (3.2) and (3.3) by induction with regard to $q\geq R_{j}+1$. Let
$q=R_{j}+1$. Since $R_{j+1}\geq R_{j}+1$, $j+1\notin\mathcal{E}_{R_{j}+1,1}$.
Similarly, $P(j)+1\notin\mathcal{E}_{R_{j}+1,2}$ because
$S_{P(j)+1}>s_{P(j)+1}\geq R_{j}$. Hence we have
$\mathcal{E}_{R_{j}+1,1}=\\{j\\}$ and $\mathcal{E}_{R_{j}+1,2}=\\{P(j)\\}$.
Hence (3.2) and (3.3) immediately follows by (3.1).

Let $q>R_{j}+1$ and assume the inductive hypothesis for $q-1$. Take
$m\in\mathcal{E}_{q,1}\setminus\mathcal{E}_{q-1,1}$ (if no such $m$ exists,
(3.2) automatically holds for all $m\in\mathcal{E}_{q,1}$). We will show that
(3.2) holds for $m$. Since $m\in\mathcal{E}_{q,1}$, we have $R_{m-1}<R_{m}<q$
and hence $m-1\in\mathcal{E}_{q-1,1}$. Therefore, by the inductive hypothesis,
we have $m-1\notin\mathcal{A}_{3}$, $f_{m}(\underline{a})=\underline{x}$ and
$S_{P(j)}^{\infty}\preceq r^{(m-1)}$.

Figure 2. Sketch of the Markov diagram in Case 1.

(Case 1) $S_{P(j)}=r_{m}-1$ (for the situation of the Markov diagram, see
Figure 2). In this case, we have $S_{P(j)}=r_{m}-1=S_{P(m)}$. This implies
$A_{R_{m}-1}\subset B_{S_{P(j)-1}}$ and
$\\#\mathcal{S}(A_{R_{m}-1})\leq\\#\mathcal{S}(B_{S_{P(j)-1}})=2$. Hence we
have $m\notin\mathcal{A}_{3}$. Moreover,
$f_{m+1}(\underline{a})=\underline{x}$ is followed by
$\mathcal{S}(A_{R_{m}-1})=\\{A_{R_{m}},B_{S_{P(j)}}\\}$ and
$f_{S_{P(j)+1}}(\underline{b})=\underline{y}$. The last condition
$S_{P(j)}^{\infty}\preceq r^{(m)}$ follows from $S_{P(j)}=r_{m}-1$ and
$S_{P(j)}^{\infty}\preceq r^{(m-1)}$.

(Case 2) $S_{P(j)}<r_{m}-1$ (For the situation of the Markov diagram, see
Figure 3). In this case, we have $S_{P(j)}<r_{m}-1=S_{P(m)}<R_{m}=q-1$, which
implies that $P(m)-1,\ P(m)\in\mathcal{E}_{q-1,2}$. Hence by the inductive
hypothesis (3.3) for $q-1$, we have $P(m)-1,P(m)\notin\mathcal{B}_{3}$,
$P(m),P(m)+1\in\mathcal{B}_{1}$ and $R_{j}^{\infty}\preceq s^{(P(m)-1)},\
s^{(P(m))}$. Since $A_{R_{m}-1}\rightarrow B_{S_{P(m)}}$, we have
$\\#\mathcal{S}(A_{R_{m}-1})\leq\\#\mathcal{S}(B_{S_{P(m)-1}})=2$, which
implies $m\notin\mathcal{A}_{3}$. Moreover,
$\mathcal{S}(A_{R_{m}-1})=\\{A_{R_{m}},B_{S_{P(m)}}\\}$ implies

$f_{m+1}(\underline{a})={\rm adj}(f_{S_{P(m)+1}}(\underline{b}))={\rm
adj}(\underline{y})=\underline{x}.$

Now we consider the last condition $S_{P(j)}^{\infty}\preceq r^{(m)}$. Since
$Q(P(m))=s_{P(m)}-1\geq R_{j}$ and $Q(P(m))<s_{P(m)}<S_{P(m)}<R_{m}=q-1,$
(3.2) holds for $Q(P(m))$. If $r^{(Q(P(m)))}=r^{(m)}$, (3.2) yields
$S_{P(j)}^{\infty}\preceq r^{(Q(P(m)))}=r^{(m)}$. Otherwise let
$\ell=\inf\\{i\geq 1:r_{Q(P(m))+i}\neq r_{m+i}\\}$. It is easy to see
$S_{P(j)}^{\infty}\preceq r^{(m)}$ is yield by the following: For
$i=0,1,\ldots,\ell$, we have

$A_{R_{m+i}}\subset A_{R_{Q(P(m))+i}}\ \mbox{and}\
\\#\tau(A_{R_{m+i}-1})=\\#\tau(A_{R_{Q(P(m))+i}-1})=2$ (3.4)

In particular, (3.4) implies $r_{Q(P(m))+\ell+1}<r_{m+\ell+1}$ and
$r^{(Q(P(m))+1)}\preceq r^{(m)}$. We close our proof of (3.2) by showing
(3.4). Since $A_{R_{m}-1}\rightarrow B_{S_{P(m)}}$ and
$B_{S_{P(m)}-1}\rightarrow A_{R_{Q(P(m))}}$, we have

$A_{R_{m}-1}\subset B_{S_{P(m)}-1}\subset A_{R_{Q(P(m))}-1}.$

Moreover, $f_{m+1}(a)=f_{Q(P(m))+1}=\underline{x}$ implies
$A_{R_{m}},A_{R_{Q(P(m))}}\subset[x_{0}]$ and $A_{R_{m}}\subset
A_{R_{Q(P(m))}}$. Since (3.2) holds for $Q(P(m))$, we have
$\\#\mathcal{S}(A_{R_{m}-1})=\\#\mathcal{S}(A_{R_{Q(P(m))}-1})=2$.

Figure 3. Sketch of the Markov diagram in Case 2.

Let $0<i\leq\ell$ and assume (3.4) holds for all $0\leq i^{\prime}<i$. Since
$A_{R_{m}+i-1}\subset A_{R_{Q(P(m))}+i-1}$ and $r_{Q(P(m))+i-1}=r_{m+i-1}$, we
have $A_{R_{m+i}-1}\subset A_{R_{Q(P(m))+i}-1}$. Since
$A_{R_{Q(P(m))}}<S_{P((m)}<R_{m}$ yields

$\displaystyle R_{Q(P(m))+i}$
$\displaystyle=R_{Q(P(m))}+\sum_{i^{\prime}=1}^{i}r_{Q(P(m))+i^{\prime}}$
$\displaystyle=R_{Q(P(m))}+\sum_{i^{\prime}=1}^{i}r_{m+i^{\prime}}$
$\displaystyle<R_{m}+R_{m+i}-R_{m}=R_{m+i},$

we have $\\#\mathcal{S}(A_{R_{m+i}-1})=\\#\mathcal{S}(A_{R_{Q(P(m))+i}-1})=2$.
Combining the definition of $\ell$ and the (3.4) for $\ell$, we have
$r_{Q(P(m))+\ell+1}<r_{m+\ell+1}$.

We can prove (3.3) for all $l\in\mathcal{E}_{q,2}$ in a similar manner, which
proves the proposition. ∎

We continue to the proof of Theorem B. By Proposition 3.3, we can divide (Case
F) into the following three cases:

(Case F1) $R_{j}^{\infty}\succ\\{s_{P(j)+i}-1\\}_{i=1}^{\infty}$.
(Case F2) $S_{P(j)}^{\infty}\succ\\{r_{j+i}-1\\}_{i=1}^{\infty}$ and
$P(j)-1\in\mathcal{B}_{1}$.
(Case F3) $S_{P(j)}^{\infty}\succ\\{r_{j+i}-1\\}_{i=1}^{\infty}$ and
$P(j)-1\in\mathcal{B}_{2}$.

The proofs of (Case F1), (Case F2) and (Case F3) are similar to those of (Case
3), (Case 4) and (Case 5) in [13], respectively. Theorem B is proved.

######  Remark 3.1.

In our proof of Theorem B, we improve the method in Hofbauer and Raith’s work
[13] to apply generalized $(\alpha,\beta)$-transformations. For a piecewise
monotonic map with two monotonic pieces, its critical set $\mathcal{CR}$
consists of two points which are clearly adjacent. Moreover, its Markov
diagram has no vertices at which more than three edges start and a word
represented by a path without branching vertices coincides with the word of
either of the points in $\mathcal{CR}$. On the other hand, for a generalized
$(\alpha,\beta)$-transformation, there may exist a vertex at which more than
three edges start in its Markov diagram. Moreover it is difficult to check
which critical point represent a word defined on a path without branching
vertices. Therefore, in the proof of Proposition 3.3 we need to be careful to
check that every vertex has two or less edges and a pair of adjacent critical
points, $\underline{x}=f_{j}(\underline{a})$ and $\underline{y}={\rm
adj}(\underline{x})=f_{P(j+1)}(\underline{b})$, appears in each inductive
step.

Acknowledgement. The authors would like to thank Hajime Kaneko for suggesting
this problem. The first author was partially supported by JSPS KAKENHI Grant
Number 21K13816 and the second author was partially supported by JSPS KAKENHI
Grant Number 21K03321.

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grading $\tilde{\alpha}_{\Sigma}:\Sigma\to\R$ given by the unique constant
function $\tilde{\alpha}_{\Sigma}\in\\{0,1/2\\}$ (in particular,
$\tilde{\alpha}_{\Sigma}\equiv 0$ when $\dim B$ is even). To equip $\Sigma$
with a Pin structure, note that we have an isomorphism $T\Sigma\cong TB\oplus
T^{*}B\oplus T^{*}B$. Hence a Pin structure on $B$ induces one on $\Sigma$.

Note that in our case we have chosen a Pin structure on $B$ (this is an input
to run Family Floer theory, see [Abo14, Section 3.1]). Equip the Lagrangian
correspondence $\Sigma$ with the natural choice of grading and Pin structure
given by 36.

######  Lemma 37.

There are gradings on $\Gamma_{m\theta}$ and $\Gamma^{n[l]}$ inducing any
given grading on $\Gamma^{n[l]}_{m\theta}$.

######  Proof 4.14.

This follows directly by tracing the construction of a grading on the
geometric composition explained in [WW10, Section 3].

######  Lemma 38.

The Pin structure induced on the fiberwise addition of any two tropical
Lagrangians equipped with the preferred Pin structure is again the preferred
Pin structure.

######  Proof 4.15.

Let us first recall how the geometric composition of Lagrangian
correspondences inherits a Pin structure. Given a Lagrangian correspondence
$\Sigma\subset X^{-}\times Y$ and a Lagrangian $L\subset X$, the geometric
composition is defined by
$\Phi_{\Sigma}(L)=\pi_{Y}(L\times_{\Delta_{X}}\Sigma)\subset Y$, where
$\Delta_{X}\subset X\times X$ is the diagonal (we assume that the intersection
is transverse and that the map $\pi_{Y}$ restricts to an embedding). Let
$Z=L\times_{\Delta_{X}}\Sigma$, which is diffeomorphic to $\Phi_{\Gamma}(L)$
via $\pi_{Y}$. Consider the isomorphism $TZ\oplus T\Delta_{X}\cong TL\oplus
T\Sigma$. It follows that Pin structures on $\Sigma$, $L$ and $\Delta_{X}$
determine one on $Z$ (and hence on $\Phi_{\Sigma}(L)\cong Z$).

In the present case, $X=\mathcal{K}$ comes equipped with a polarisation
$\mathcal{P}$ given by the tangent space to the fibers, giving an isomorphism
$T\mathcal{K}\cong T\mathcal{P}\oplus T\mathcal{P}$. Since fibers have a
preferred Pin structure induced from a choice of Pin structure on the base,
$\mathcal{K}$ (and thus $\Delta_{\mathcal{K}}$) have a natural Pin structure.
One can then see that, with the choice of Pin structure for $\Sigma$ explained
above, the fiberwise addition of two tropical Lagrangians equipped with the
preferred Pin structure yields another Lagrangian with the preferred Pin
structure (the reason being that the inclusion $TZ\cong TK\hookrightarrow
TL\oplus T\Sigma\cong(TK)^{\oplus 3}\oplus(T^{*}K)^{\oplus 2}$ can be
homotoped to the map that identifies $TZ$ with the first $TK$ factor on the
right, hence the Pin structure induced on $Z$ is that of $K$).

We now turn to the second issue, namely the admissibility of the surgery
cobordisms. The main observation is that all the Lagrangians we are
considering are product-type when pulled back to the cover $T^{4}=T^{2}\times
T^{2}\to\mathcal{K}$. Furthermore, the surgery cobordisms
$V\subset\mathcal{K}\times\C$ involved in Equation 17 pull back under
$p:T^{4}\times\C\to\mathcal{K}\times\C$ to either

(36) $p^{-1}(V)=S^{1}\times W\subset T^{2}\times T^{2}\times\C\quad\text{ or
}\quad p^{-1}(V)=W\times S^{1}\subset T^{2}\times\C\times T^{2},$

where $W\subset T^{2}\times\C$ is a surgery cobordism in $T^{2}$ and
$S^{1}\subset T^{2}$ is the zero section.

######  Lemma 39.

The cobordism $V$ admits a grading.

######  Proof 4.16.

Note that the phase map of $p^{-1}(V)$ is the product of the phase maps of
$S^{1}$ and $W$. The former is constant and the latter is null-homotopic by
[Hau15, Lemma 5.2]; thus $p^{-1}(V)$ admits a grading. Since furthermore
$\alpha_{p^{-1}(V)}=\alpha_{V}\circ p|_{p^{-1}(V)}$ and
$p|_{p^{-1}(V)}:p^{-1}(V)\to V$ induces an injection on $H^{1}(-;\Z)$, the map
$\alpha_{V}$ is also null-homotopic. Thus $V$ admits a grading and, given a
grading on $\Gamma_{m\theta}$ (resp. $\Gamma^{n[l]}$), Equation 17 holds for
$L_{f_{m\theta}}$ (resp. $L_{f^{nl}}$) and $\Gamma_{0}$ equipped with the
induced grading from $V$. Note that Lagrangian cobordism classes are invariant
under a grading change of $2k$, whereas changing the grading by $2k+1$
modifies their class in the cobordism group by a factor of $-1$ (this can be
seen using a product-type cobordism $\gamma\times L$, where $\gamma\subset\C$
is a C-shaped curve).

######  Lemma 40.

The cobordism $V$ admits a unique Pin structure restricting to the preferred
one on its two section ends.

######  Proof 4.17.

Let us analyze the first case in Equation 36 (the second is similar but
simpler). The topology of $W$ is that of a pair of pants. In particular, $TW$
is trivial and it admits a canonical Pin structure, which in turn induces one
on $[0,1]\times W$. Let $t$ be a coordinate on $[0,1]$ and $(p,z)$ be
coordinates on $W\subset T^{2}\times\C$. Then giving a Pin structure on

$V\simeq\frac{[0,1]\times W}{(t,p,z)\sim(t+1,-p,z)}$

amounts to choosing a lift to $Pin_{3}$ of the linearisation of the gluing
map. One can see that in a suitable trivialisation of $TW$, the linearization
of the gluing map takes the form
$(\partial_{u},\partial_{v},\partial_{t})\mapsto(\partial_{u},-\partial_{v},\partial_{t})$,
which is a reflection along the hyperplane $\partial_{v}=0$. Recalling that
lifts of reflections $R\in O_{n}$ to $Pin_{n}$ are identified with co-
orientations of the fixed hyperplane, the two choices of Pin structure above
can be identified with the vectors $\pm\partial_{v}$. Given such a choice, we
equip the ends of the cobordism with the restricted Pin structure. Note that
the induced Pin structure is the same on all three ends.

Lastly, recall that Pin structures on $V$ are a torsor over
$H^{1}(V;\Z_{2})\cong\Z_{2}^{2}\oplus\Z_{2}$. The restriction map
$H^{1}(V;\Z_{2})\to H^{1}(\partial
V;\Z_{2})\cong\Z_{2}^{2}\oplus\Z_{2}^{2}\oplus\Z_{2}^{2}$ is given by

(37) $(a_{1},a_{2},b)\mapsto(a_{1},b)\oplus(a_{2},b)\oplus(a_{1}+a_{2},b)$

where we have decomposed $\partial V=\sqcup_{i=1}^{3}Z_{i}$ as a disjoint
union of its three boundary Klein bottles. Since the canonical Pin structure
on $W\times[0,1]$ together with a choice $\pm\partial_{t}$ induces the same
Pin structure on the three ends, it follows from this and Equation 37 that
there exists a unique Pin structure on $V$ restricting to the preferred one on
$\Gamma^{n[l]}$ and $\Gamma_{0}$.

######  Lemma 41.

Given a local system
$\eta=\eta_{\Z}\otimes\eta_{\Z_{2}}:H_{1}(\Gamma^{n[l]}_{m\theta})\to G$ on
$\Gamma^{n[l]}_{m\theta}$, one can put local systems on $L_{f_{m\theta}}$ and
$\Gamma_{0}$ and the trivial local system on $L_{f^{nl}}$ such that 24 holds.

######  Proof 4.18.

Let us first recall how the construction in [Sub10] interacts with local
systems. Let $\Sigma\subset X^{-}\times Y$ be a Lagrangian correspondence and
consider a Lagrangian $L\subset X$ equipped with a local system $\eta$. Then,
assuming $\pi_{Y}:L_{1}\times_{X}\Sigma\to Y$ is an embedding, one equips the
Lagrangian $\Phi_{\Sigma}(L)=\pi_{Y}(L_{1}\times_{X}\Sigma)$ with the local
system given by the composition

$\pi_{1}(\Phi_{\Sigma}(L))\xrightarrow{(\pi_{Y}^{-1})_{\\#}}\pi_{1}(L_{1}\times_{X}\Sigma)\xrightarrow{(p_{1}\circ
i)_{\\#}}\pi_{1}(L_{1})\xrightarrow{\eta}G$

where $i:L_{1}\times_{X}\Sigma\hookrightarrow L_{1}\times\Sigma$ is the
inclusion and $p_{1}:L_{1}\times\Sigma\to L_{1}$ is the projection.

The above construction shows that, given any local system $\eta$ on
$\Gamma^{n[l]}_{m\theta}$, one has

$(\Gamma^{n[l]}_{m\theta},\eta)=(\Gamma_{m\theta},\eta)\otimes(\Gamma^{n[l]},\mathbf{1}),$

Here, in the right-hand side we denote still by $\eta$ the local systems on
$\Gamma_{m\theta}$ induced by the canonical diffeomorphism between
$\Gamma^{n[l]}_{m\theta}$ and $\Gamma_{m\theta}$, and we denote by
$\mathbf{1}$ the trivial local system.

We first study the potential restrictions on local systems for the existence
of a cobordism $V$ between $\Gamma_{m\theta}$, $\Gamma_{0}$ and
$L_{f_{m\theta}}$ as in Equation 17. One can use a Mayer-Vietoris
decomposition of the cobordism $V$ between $\Gamma_{m\theta}$, $\Gamma_{0}$
and $L_{f_{m\theta}}$ to show that

$H_{1}(Z)\cong\frac{H_{1}(\Gamma_{m\theta})\oplus H_{1}(\Gamma_{0})}{v\sim
w}\oplus\Z^{m-1}$

where $v,w\in\Z_{2}$ are the generators of the torsion factor of
$H_{1}(\Gamma_{m\theta})$ and $H_{1}(\Gamma_{0})$ respectively. The last
factor $\Z^{m-1}$ can be identified with cycles living in the conormal
direction of $m-1$ of the Lagrangians in the surgery
$L_{f_{m\theta}}\cong\sqcup_{i=1}^{m}N^{*}S^{1}/N^{*}_{\Z}S^{1}$. Hence, given
the local system
$\eta=\eta_{\Z}\otimes\eta_{\Z_{2}}:H_{1}(\Gamma_{m\theta})\to G$, Equation 17
can be upgraded to

$(\Gamma_{m\theta},\eta_{\Z}\otimes\eta_{\Z_{2}})\sim(\Gamma_{0},h\otimes\eta_{\Z_{2}})+(L_{f_{m\theta}},\oplus_{i=1}^{m}(\nu_{i}\otimes\eta_{\Z_{2}})),$

where $h:H_{1}(\Gamma_{0})\to G$ is a local system that factors through
$H_{1}(\Gamma_{0})/\Z_{2}$ and $\nu_{i}$ are local systems on the $i$-th copy
of $L_{f_{m\theta}}$ such that $\eta_{\Z}(v)=h(v)\prod_{i}\nu_{i}(v_{i})$.
Here, $v_{i}$ are the generators of the fiber direction of $L_{f_{m\theta}}$
and $v$ is the generator of $\Z\subset\Z\oplus\Z_{2}\cong
H_{1}(\Gamma_{m\theta})\cong H_{1}(\Gamma_{0})$.

On the other hand, there is clearly a cobordism relation

$(\Gamma^{n[l]},\mathbf{1})\sim(\Gamma_{0},\mathbf{1})+(L_{f^{n[l]}},\mathbf{1}).$

Choosing $h=\mathbf{1}$ and $\nu_{i}\in G$ satisfying the relation
$\eta_{\Z}=\prod_{i}\nu_{i}$, we get cobordism relations compatible with local
systems

$\displaystyle\begin{split}(\Gamma_{m\theta}^{n[l]},\eta)&=(\Gamma_{m\theta},\eta_{\Z}\otimes\eta_{\Z_{2}})\otimes(\Gamma^{n[l]},\mathbf{1})\\\
&\sim((L_{f_{m\theta}},\oplus_{i=1}^{m}(\nu_{i}\otimes\eta_{\Z_{2}}))+(\Gamma_{0},\eta_{\Z_{2}}))\otimes((L_{f^{n[l]}},\mathbf{1})+(\Gamma_{0},\mathbf{1}))\\\
&=(L_{f_{m\theta}}\otimes
L_{f^{n[l]}},\oplus_{ij}\nu_{i})+(L_{f_{m\theta}},\oplus_{i}(\nu_{i}\otimes\eta_{\Z_{2}}))+(L_{f^{n[l]}},\mathbf{1})+(\Sigma_{0},\eta_{\Z_{2}}).\end{split}$

######  Lemma 42.

The pull back to $T^{4}\times\C$ of the cobordism $V$ is tautologically
unobstructed.

######  Proof 4.19.

Recall from Equation 36 that, up to reordering of the factors, $p^{-1}(V)$ is
of the form

$p^{-1}(V)=S^{1}\times W\subset T^{2}\times(T^{2}\times\C)$

for $S^{1}\subset T^{2}$ a non-contractible curve and $W$ a surgery cobordism
between Lagrangians circles in $T^{2}$ (in fact, it is always the case for us
that one of the Lagrangians is the $0$-section and the other is the graph of a
tropical function). It was proved in [Hau15, Lemma 5.2] that $W$ has Maslov
index 0, thus it bounds no non-constant $J$-holomorphic disks for a generic
choice of almost complex structure $J$ on $T^{2}\times\C$. On the other hand,
the projection to the first factor of any disk must be null-homotopic for
topological reasons; furthermore, choosing the almost complex structure on
$T^{2}\times(T^{2}\times\C)$ to be of product type $j_{T^{2}}\oplus
j_{T^{2}\times\C}$ the projection of the disk is a null-homotopic pseudo-
holomorphic disk, hence constant. Thus $p^{-1}(V)$ does not bound any non-
constant pseudo-holomorphic disks for a complex structure of the form
$j_{T^{2}}\oplus J$.

The almost complex structure for which $W$ does not bound any holomorphic disk
need not be invariant under the covering group. Hence, we cannot say that $V$
is tautologically unobstructed (the almost complex structure does not descend
to $\mathcal{K}$). This is the reason why we need to work with cobordisms with
the weaker property that their pullback is tautologically unobstructed.
Nonetheless, we argue in Appendix C that such cobordisms still induce iterated
cone decompositions between their ends (see 48).

Summarizing, we have proved the following Proposition:

######  Proposition 43.

Consider a Lagrangian $G$-brane supported on $\Gamma_{m\theta}$ equipped with
the preferred Pin structure and an arbitrary grading and local system. There
exists a Lagrangian cobordism $V\subset\mathcal{K}\times\C$ between
$\Gamma_{m\theta},\Gamma_{0}$ and $L_{f_{m\theta}}$ such that:

  1. 1.

$V$ admits a $G$-brane structure restricting to the given $G$-brane structure
on $\Gamma_{m\theta}$;

  2. 2.

the restriction of the Pin structure of $V$ to $\Gamma_{0}$ and
$L_{f_{m\theta}}$ is the preferred one;

  3. 3.

the pullback of $V$ to $T^{4}\times\C$ is tautologically unobstructed.

A similar statement holds for the Lagrangians $\Gamma^{n[l]},\Gamma_{0}$ and
$L_{f^{nl}}$.

##  Appendix A Homology with local coefficients

Let $\mathcal{K}$ be a symplectic bielliptic surface and $\pi:\mathcal{K}\to
K$ its Lagrangian torus fibration over a tropical Klein bottle. Denote by
$\det_{\Z}K=\wedge^{2}T^{*}_{\Z}K$ the (integral) orientation bundle of $K$
and let $\xi:=\pi^{*}\det_{\Z}K$ be the local system on $\mathcal{K}$ obtained
by pullback. In this Appendix we compute the second singular homology
$H_{2}(\mathcal{K};\xi)$ of $\mathcal{K}$ with coefficients in the local
system $\xi$.

$>$$<$$>>$$>>$$V_{1}$$>$$<$$>>$$>>$$V_{2}$$>$$<$$>>$$>>$$V_{1}\cap V_{2}$
Figure 3: Decomposition of the Klein bottle to run Mayer-Vietoris. Note that
both $V_{1}$ and $V_{2}$ retract to a circle, whereas $V_{1}\cap V_{2}$
retracts to the disjoint union of two circles

######  Proposition 44.

The second homology of $\mathcal{K}$ with coefficients in the local system
$\xi$ is given by

(38) $H_{2}(\mathcal{K};\xi)\cong\Z^{4}\oplus\Z_{2}.$

Furthermore, every class has a Lagrangian representative.

######  Proof A.1.

We will use a Mayer-Vietoris decomposition $\mathcal{K}=U_{1}\cup U_{2}$
coming from a decomposition $K=V_{1}\cup V_{2}$ of the base (see Figure 3),
i.e. $U_{i}=\pi^{-1}(V_{i})$. Note the restricted fibrations $U_{i}\to V_{i}$
and $U_{1}\cap U_{2}\to V_{1}\cap V_{2}$ are trivial. Furthermore, since the
local system $\xi$ is trivial on the fiber direction (it’s pulled back from
the base) and the homology of the fiber is free, the Künneth isomorphism for
homology with local coefficients says131313Although Equations (39)-(40) seem
to forget the local system, we will see shortly that the maps appearing in the
Mayer-Vietoris long exact sequence are different from those computing integral
homology.

(39)
$\displaystyle\begin{split}H_{j}(U_{i};\xi|_{U_{i}})&\cong\bigoplus_{k=0}^{j}H_{k}(V_{i};\det_{\Z}K|_{V_{i}})\otimes
H_{j-k}(T^{2};\Z)\\\ &\cong\bigoplus_{k=0}^{j}H_{k}(S^{1};\Z)\otimes
H_{j-k}(T^{2};\Z)\end{split}$ (40) $H_{j}(U_{1}\cap U_{2};\xi|_{U_{1}\cap
U_{2}})\cong\left(\bigoplus_{k=0}^{j}H_{k}(S^{1};\Z)\otimes
H_{j-k}(T^{2};\Z)\right)^{\oplus 2}.$

To simplify notation, from now on we will omit the restriction notation and
simply write $H_{k}(U;\xi)\equiv H_{k}(U;\xi|_{U})$ for any
$U\subset\mathcal{K}$. We will also write $H_{k}(U)\equiv H_{k}(U;\Z)$
whenever we are taking integral homology.

The Mayer-Vietoris long exact sequence splits to give

(41) $0\to\frac{H_{2}(U_{1};\xi)\oplus H_{2}(U_{2};\xi)}{\im(i^{2}_{1}\oplus
i^{2}_{2})_{*}}\to H_{2}(\mathcal{K};\xi)\to\ker(i^{1}_{1}\oplus
i^{1}_{2})_{*}\to 0$

where the maps $(i^{j}_{1}\oplus i^{j}_{2})_{*}:H_{j}(U_{1}\cap U_{2};\xi)\to
H_{j}(U_{1};\xi)\oplus H_{j}(U_{2};\xi)$ are induced by the inclusions
$i_{k}:U_{1}\cap U_{2}\hookrightarrow U_{k}$. One can check that
$\ker(i^{1}_{1}\oplus i^{1}_{2})_{*}\cong\Z^{2}$, with one factor generated by
the difference of the two primitive elements of $H_{1}(V_{1}\cap
V_{2};\Z)\otimes H_{0}(T^{2})$ and the other by the difference of two
primitive elements of $H_{0}(V_{1}\cap V_{2};\Z)\otimes\langle
dy\rangle\subset H_{0}(V_{1}\cap V_{2};\Z)\otimes H_{1}(T^{2})$ (recall there
is a canonical identification $H_{1}(F_{p};\Z)\simeq T^{*}_{p,\Z}K$).
Analyzing the Mayer-Vietoris boundary map, one sees the short exact sequence
(41) is split by the map $\ker(i^{1}_{1}\oplus i^{1}_{2})_{*}\to
H_{2}(\mathcal{K};\xi)$ sending these two generators to the $0$-section (which
is a cycle with $\xi$-coefficients) and the conormal lift of the tropical
curve $\\{y=1/2\\}$. We now compute the quotient in the left of Equation 41:

  * •

Using (39) we get $H_{2}(U_{1};\xi)\cong\Z\oplus\Z^{2}\cong H_{2}(U_{2};\xi)$,
the first factor generated by a fiber and the last two by the conormal and
cotangent lift of the tropical circle $\\{x=1/2\\}\subset K$.

  * •

To compute $\im(i^{2}_{1}\oplus i^{2}_{2})_{*}$, note that $H_{2}(U_{1}\cap
U_{2};\xi)\cong(\Z\oplus\Z^{2})^{\oplus 2}$ by Equation 40. Denoting by
$a_{1},b_{1},c_{1},a_{2},b_{2},c_{2}$ the generators of $H_{2}(U_{1}\cap
U_{2};\xi)\cong(\Z\oplus\Z^{2})^{\oplus 2}$, the map
$(i^{2}_{1},i^{2}_{2})_{*}:H_{2}(U_{1}\cap U_{2};\xi)\to
H_{2}(U_{1};\xi)\oplus H_{2}(U_{2};\xi)$ is given by

$(a_{1},b_{1},c_{1},a_{2},b_{2},c_{2})\mapsto(a_{1}+a_{2},b_{1}-b_{2},c_{1}+c_{2})\oplus(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2}).$

Hence

$\im(i^{2}_{1}\oplus
i^{2}_{2})_{*}=\langle(u,v-w,z)\oplus(u,v+w,z),\,u,v,w,z\in\Z\rangle\subset
H_{2}(U_{1};\xi)\oplus H_{2}(U_{2};\xi).$

  * •

Putting everything together, we get

$\frac{H_{2}(U_{1};\xi)\oplus H_{2}(U_{2};\xi)}{\im(i^{2}_{1}\oplus
i^{2}_{2})_{*}}\cong\Z\oplus\Z_{2}\oplus\Z$

generated by a fiber and the cotangent and conormal lift to the curve
$\\{x=1/2\\}$ respectively.

Reordering the terms, we have computed

$H_{2}(\mathcal{K};\xi)\cong\Z^{4}\oplus\Z_{2},$

with generators a fiber, the $0$-section, the conormal lifts to the curves
$\\{x=1/2\\}$ and $\\{y=1/2\\}$ and the cotangent lift of the curve
$\\{x=1/2\\}$.

For the last part of the Theorem, note all but the last homology class have
been given Lagrangian representatives. For the last one, the $3$-chain
obtained by taking the $y$-cotangent direction to the $2$-chain given by the
bottom-half of $K$ has boundary
$T^{*}\\{x=0\\}-N^{*}\\{y=1/2\\}+N^{*}\\{y=0\\}$, showing that the symplectic
torus $T^{*}\\{x=0\\}$ representing the $\Z_{2}$-factor has the difference of
Lagrangians $N^{*}\\{y=1/2\\}-N^{*}\\{y=0\\}$ as a representative.

Note there are important differences between $H_{2}(\mathcal{K};\xi)$ and
$H_{2}(\mathcal{K};\Z)$. Most notably for our purposes, the zero-section is
not a cycle with integer coefficients, whereas it is with $\xi$-coefficients
and furthermore it defines a non-zero homology class. A similar phenomenon
happens for the conormal lift of the tropical circles $\\{y=0\\}$ and
$\\{y=1/2\\}$: they are topologically Klein bottles and do not define cycles
with integer coefficients, but they do with $\xi$-coefficients. Also, the
Lagrangian (conormal) lift of the $y$-axis is $2$-torsion in
$H_{2}(\mathcal{K};\Z)$, but it has infinite order in
$H_{2}(\mathcal{K};\xi)$.

##  Appendix B Integral Chern character

In this appendix we show that for a bielliptic surface $Y$ the Chern character
$ch:K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}^{*}(Y)_{\Q}$ can be lifted
to an integral isomorphism. That is, there exists a well-defined map
$\widetilde{ch}:K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}^{*}(Y)$ and it
is an isomorphism. Similar results appear in [Huy16, Corollary 1.5], where
Huybrechts shows that for a K3 surface the Chern character
$ch:K_{0}(K3)\to\operatorname{\mathrm{CH}}^{*}(K3)_{\Q}$ factors through the
natural map
$\operatorname{\mathrm{CH}}^{*}(K3)\to\operatorname{\mathrm{CH}}^{*}(K3)_{\Q}$
in a canonical way, and that the corresponding map
$K_{0}(K3)\to\operatorname{\mathrm{CH}}^{*}(K3)$ is an isomorphism. A crucial
property for his proof to work is that $\operatorname{\mathrm{CH}}_{0}(K3)$ is
torsion-free. That clearly does not hold in our case, where
$\operatorname{\mathrm{CH}}_{0}(Y)=\Z\oplus\operatorname{\mathrm{Alb}}(Y)$,
the latter factor being an elliptic curve.

We first state a preliminary result, whose proof is a simple computation:

######  Lemma 45.

Let $V=\oplus_{i}V_{i}$ be a graded $\Z$-module and let $W$ be another
$\Z$-module. Assume we have a commutative map $V\otimes V\to W$ and maps
$H_{i}:V_{i}\to W$ such that

(42) $H_{i}(\alpha+\beta)=H_{i}(\alpha)+H_{i}(\beta)+\alpha\beta.$

Then the map $H:V\to W$ defined by

$H(\oplus_{i}v_{i}):=\sum_{i}H_{i}(v_{i})+\sum_{i<j}v_{i}v_{j}$

satisfies Equation 42.

######  Proposition 46.

Let $Y$ be an algebraic bielliptic surface. There is an isomorphism of abelian
groups

$ch:K_{0}(D^{b}Coh(Y))\to CH^{*}(Y).$

######  Proof B.1.

Recall from Section 4.8 that
$\operatorname{\mathrm{CH}}^{1}(Y)=\Z^{2}\oplus\Z_{2}^{2}\oplus\Pic^{0}(Y)$.
Write $Y=(E_{1}\times E_{2})/\Z_{2}$ as in 17 and let $\pi_{i}:Y\to
E_{i}/\Z_{2}$, where $E_{1}/\Z_{2}$ is again an elliptic curve and
$E_{2}/G\cong\mathbb{P}^{1}$. Then, following [Ser90, Ber+19], we can give the
following explicit generators for $\operatorname{\mathrm{CH}}^{1}(Y)$. The
first two $\Z$-factors are generated by a fiber of $\pi_{1}$ and a reduced
multiple fiber of $\pi_{2}$; call these $D_{1}$ and $D_{2}$. The second two
$\Z_{2}$-factors are generated by $F_{1}-F_{2}$ and $F_{1}-F_{4}$, where
$F_{i}$ are the four reduced multiple fibers of $\pi_{2}$ and we take
$F_{1}=D_{2}$; call these two generators $D_{3}$ and $D_{4}$. Lastly,
$\Pic^{0}(Y)$ is an elliptic curve, whose points correspond to the divisor
classes $D_{p}:=D_{1}-\pi_{1}^{-1}(p)$ for $p\in
E_{1}/\Z_{2}\cong\Pic^{0}(Y)$; we will write $D_{5}^{p}$ for these generators.
Since the normal bundles to both $D_{1}$ and $D^{p}_{5}$ are trivial, it
follows that
$D_{1}^{2}=(D^{p}_{5})^{2}=0\in\operatorname{\mathrm{CH}}_{0}(Y)$.
Furthermore, we have

$D_{3}^{2}=(F_{1}-F_{2})^{2}=F_{1}^{2}+F_{2}^{2}=0$

since $F_{i}^{2}=F_{j}^{2}$ for all $i,j$ and these classes are necessarily
$2$-torsion. The same argument shows that $D_{4}^{2}=0$.

Now write
$\operatorname{\mathrm{CH}}^{1}(Y)=\Z^{2}\oplus\Z_{2}^{2}\oplus\Pic^{0}(Y)=\oplus_{i}V_{i}$,
where $V_{i}$ denotes the $i$-th summand of
$\operatorname{\mathrm{CH}}^{1}(Y)$ as in 45. Then the previous discussion
shows that the maps $H_{i}\equiv 0$ for $i\neq 2$ satisfy Equation 42.
Furthermore, define $H_{2}(kD_{2})=k^{2}P$, where
$P\in\operatorname{\mathrm{CH}}^{2}(Y)$ is chosen such that $2P=D_{2}^{2}$
(that such $P$ exists follows from $D_{2}^{2}$ being a $2$-torsion element in
$\operatorname{\mathrm{CH}}_{0}(Y)$, hence it lives in the divisible subgroup
$\operatorname{\mathrm{Alb}}(Y)\subset\operatorname{\mathrm{CH}}_{0}(Y)$).
Then

$H_{2}(mD_{2}+nD_{2})=(m+n)^{2}P=H_{2}(mD_{2})+H_{2}(nD_{2})+2mnP$

with $2mnP=mnD_{2}^{2}$, so $H_{2}$ also satisfies Equation 42. Let
$H:\operatorname{\mathrm{CH}}^{1}(Y)\to\operatorname{\mathrm{CH}}^{2}(Y)$ be
the map defined in 45. We define a modified Chern character
$\tilde{ch}:K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}^{*}(Y)$ via

$\tilde{ch}(\mathcal{F})=rk(\mathcal{F})+c_{1}(\mathcal{F})+(H\circ
c_{1}-c_{2})(\mathcal{F}).$

Note that $\tilde{ch}$ is a group homomorphism: for
$\mathcal{F}_{1},\mathcal{F}_{2}\in K_{0}(D^{b}Coh(Y))$ we have

$\displaystyle\tilde{ch}_{2}(\mathcal{F}_{1}\oplus\mathcal{F}_{2})$
$\displaystyle=H(c_{1}(\mathcal{F}_{1}\oplus\mathcal{F}_{2}))-c_{2}(\mathcal{F}_{1}\oplus\mathcal{F}_{2})$
$\displaystyle=H(c_{1}(\mathcal{F}_{1})+c_{1}(\mathcal{F}_{2}))-(c_{2}(\mathcal{F}_{1})+c_{2}(\mathcal{F}_{2})+c_{1}(\mathcal{F}_{1})c_{1}(\mathcal{F}_{2}))$
$\displaystyle=H(c_{1}(\mathcal{F}_{1}))+H(c_{1}(\mathcal{F}_{2}))+c_{1}(\mathcal{F}_{1})c_{1}(\mathcal{F}_{2})-(c_{2}(\mathcal{F}_{1})+c_{2}(\mathcal{F}_{2})+c_{1}(\mathcal{F}_{1})c_{1}(\mathcal{F}_{2}))$
$\displaystyle=\tilde{ch}_{2}(\mathcal{F}_{1})+\tilde{ch}_{2}(\mathcal{F}_{2}).$

After establishing the existence of a well-defined integral Chern character
$\tilde{ch}:K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}^{*}(Y)$, the fact
that it is an isomorphism is a minor adaptation of the argument in [Huy16,
Corollary 1.5]. Namely, we have $\tilde{ch}(\mathcal{O})=1$ so
$\operatorname{\mathrm{CH}}_{2}(Y)\subset\im(\tilde{ch})$. Since
$\tilde{ch}(\mathcal{O}_{p})=c_{2}(\mathcal{O}_{p})=p$, it follows that also
$\operatorname{\mathrm{CH}}_{0}(Y)\subset\im(\tilde{ch})$. Lastly,
$\operatorname{\mathrm{CH}}_{1}(Y)\subset\im(\tilde{ch})$ because
$\tilde{ch}(\mathcal{O}(D))=1+D+H(D)$ and both $1$ and $H(D)$ are in
$\im(\tilde{ch})$. Hence $\tilde{ch}$ is surjective. For injectivity, note
that for any smooth surface $S$ we have isomorphisms [Ful13, Example 15.3.6]

$\displaystyle rk:F_{0}K_{0}(S)/F_{1}K_{0}(S)$ $\displaystyle\to\Z$
$\displaystyle c_{1}:F_{1}K_{0}(S)/F_{2}K_{0}(S)$ $\displaystyle\to\Pic(S)$
$\displaystyle c_{2}:F_{2}K_{0}(S)$
$\displaystyle\to\operatorname{\mathrm{CH}}_{0}(S),$

where $F_{i}K_{0}(S)\subset K_{0}(S)$ is the filtration given by sheaves whose
support has dimension at most $2-i$. Suppose now that
$\tilde{ch}(\mathcal{F})=0$. Since $\tilde{ch}(\mathcal{F})=0$ implies
$0=rk(\mathcal{F})=c_{1}(\mathcal{F})$, we get $\mathcal{F}\in
F_{2}K_{0}(D^{b}Coh(Y))$. Furthermore
$0=H(c_{1}(\mathcal{F}))-c_{2}(\mathcal{F})=-c_{2}(\mathcal{F})$, so using
that $c_{2}:F_{2}K_{0}(D^{b}Coh(Y))\to\operatorname{\mathrm{CH}}_{0}$ is an
isomorphism we get $\mathcal{F}=0$.

Unlike for K3 surfaces, our proof does not produce a canonical lift of the
Chern character: we have made a choice of the ‘half class’
$P\in\operatorname{\mathrm{CH}}_{0}(Y)_{hom}$ such that $2P=D_{2}^{2}$.

In Section 4.8 we argue that
$\operatorname{\mathrm{CH}}^{*}(Y)=\Z^{4}\oplus\Z_{2}^{2}\oplus E^{2}$ and
give explicit generators: we denote these by $Z_{i},i=1,\dots,6$ and
$Z_{p},Z^{\prime}_{p^{\prime}},(p,p^{\prime})\in E^{2}$. More explicitly,
$Z_{1}=Y,Z_{2}$ is some reference point $p_{0}\in Y$, the divisors
$Z_{3},\dots,Z_{6},Z_{p}$ are the divisors denoted by
$D_{1},\dots,D_{4},D_{5}^{p}$ in the proof of 46, and
$Z^{\prime}_{p^{\prime}}$ is represented by
$p_{0}-p^{\prime}\in\operatorname{\mathrm{CH}}_{0}(Y)_{hom}$. We consider
classes
$[\mathcal{O}_{Z_{i}}],[\mathcal{O}_{Z_{p}}],[\mathcal{O}_{Z^{\prime}_{p^{\prime}}}]\in
K_{0}(D^{b}Coh(Y))$ represented by the structure sheaves of such generators of
$\operatorname{\mathrm{CH}}^{*}(Y)$. The following result will be useful to
prove 35:

######  Corollary 47.

There is a well-defined group homomorphism

$\displaystyle h:\Z^{4}\oplus\Z_{2}^{2}\oplus E^{2}$ $\displaystyle\to
K_{0}(D^{b}Coh(Y))$ $\displaystyle(n_{1},\dots,n_{6},p,p^{\prime})$
$\displaystyle\mapsto\sum_{i}n_{i}[\mathcal{O}_{Z_{i}}]+[\mathcal{O}_{Z_{p}}]+[\mathcal{O}_{Z^{\prime}_{p^{\prime}}}]$

and it is an isomorphism.

######  Proof B.2.

To see it is well-defined we must check that the relations

$\displaystyle 2[\mathcal{O}_{Z_{i}}]$ $\displaystyle=0,\,i=5,6$
$\displaystyle[\mathcal{O}_{Z_{p_{1}+p_{2}}}]$
$\displaystyle=[\mathcal{O}_{Z_{p_{1}}}]+[\mathcal{O}_{Z_{p_{2}}}],\,p_{j}\in
E$ $\displaystyle[\mathcal{O}_{Z^{\prime}_{p^{\prime}_{1}+p^{\prime}_{2}}}]$
$\displaystyle=[\mathcal{O}_{Z^{\prime}_{p^{\prime}_{1}}}]+[\mathcal{O}_{Z^{\prime}_{p^{\prime}_{2}}}],\,p^{\prime}_{j}\in
E$

hold in $K_{0}(D^{b}Coh(Y))$. It is equivalent to show that their images under
$\tilde{ch}$ satisfy such relations. For the first two, note that
$\widetilde{ch}([\mathcal{O}_{Z_{i}}])=Z_{i}$ for $i=5,6$, and that the
classes $Z_{i}$ are $2$-torsion in $\operatorname{\mathrm{CH}}^{*}(Y)$. For
the last two, we have $\widetilde{ch}([\mathcal{O}_{Z_{p}}])=Z_{p}$ for any
$p\in E$ (and similarly for $Z_{p^{\prime}}$), so the relations also hold.

To see it is an isomorphism, note that
$\widetilde{ch}([\mathcal{O}_{Z_{i}}])=Z_{i}$ for all $i\neq 4$ and
$\widetilde{ch}([\mathcal{O}_{Z_{4}}])=Z_{4}+P$, hence the composition
$\widetilde{ch}\circ h$ is given by

$\displaystyle\Z^{4}\oplus\Z_{2}^{2}\oplus E^{2}$
$\displaystyle\to\operatorname{\mathrm{CH}}^{*}(Y)$
$\displaystyle(n_{1},\dots,n_{6},p,p^{\prime})$
$\displaystyle\mapsto\sum_{i}n_{i}Z_{i}+n_{4}P+Z_{p}+Z^{\prime}_{p^{\prime}}.$

This map is clearly an isomorphism, hence so is $h$.

##  Appendix C Fukaya categories from multivalued perturbations

In this appendix we sketch a definition of a Fukaya category
$\mathcal{F}uk(X;Y)$ of a symplectic manifold together with a finite covering
$\pi:Y\to X$ using multivalued perturbations. We then apply this construction
to $Y=T^{4}\times\C$ and $X=\mathcal{K}\times\C$ and argue that quasi exact
cobordisms in $\mathcal{F}uk(\mathcal{K}\times\C;T^{4}\times\C)$ between
weakly exact Lagrangians still give iterated cone decomposition results as in
[BCS21], as claimed in Section 4.8.

Let $X$ be a symplectic manifold and $\pi:Y\to X$ a finite covering with
covering group $H$. Given a collection of Lagrangians $\mathcal{L}(X)$, we
define two Fukaya categories:

  * •

$\mathcal{F}uk(X)=\mathcal{F}uk(X;\mathcal{L}(X))$ is the Fukaya category
whose objects are pairs $(L,J_{L})$ for $L\in\mathcal{L}(X)$ and $J_{L}$ an
almost complex structure such that $L$ bounds no non-constant
$J_{L}$-holomorphic disk. Morphism spaces and the $A_{\infty}$-structure are
defined as usual [Fuk+09, Sei08].

  * •

$\mathcal{F}uk(X;Y)=\mathcal{F}uk(X;\mathcal{L}(X),Y)$ is a Fukaya category
whose objects are pairs $(L,\tilde{J}_{L})$ for $L\in\mathcal{L}(X)$ and such
that $\pi^{-1}(L)$ bounds no $\tilde{J}_{L}$ holomorphic disks (note that now
$\tilde{J}_{L}$ is an almost complex structure on $Y$, not on $X$). Morphism
spaces in $\mathcal{F}uk(X;Y)$ are defined to be the same as in
$\mathcal{F}uk(X)$. To define the $A_{\infty}$-operations, let
$(L_{0},\tilde{J}_{0}),\dots,(L_{k},\tilde{J}_{k})$ be objects of
$\mathcal{F}uk(X;Y)$ and let $D_{k}$ be a disk with $k+1$ boundary punctures
$p_{0},\dots,p_{k}\in\partial D$. Given an element
$\mathtt{h}=(h_{0},\dots,h_{k})\in H^{k+1}$, we define an
$\mathtt{h}$-perturbation datum to be a map
$J_{g}:D_{k}\to\mathcal{J}=\\{\omega-\text{compatible a.c.s on }Y\\}$ whose
restriction to the $i$-th boundary component of $D_{k}$ agrees with
$(h_{i})^{*}\tilde{J}_{i}$ (here we denote by $h_{i}:Y\to Y$ the covering
automorphism induced by $h_{i}$). A universal choice of perturbation datum is
a choice of $\mathtt{h}$-perturbation datum for each $\mathtt{h}\in H^{k+1}$
and each fiber of the universal curve over the moduli-space of disks with
$k+1$ punctures; these should be regular and compatible in the usual sense
(where we think of the elements $h_{i}$ as labeling the $i$-th boundary
component of $D_{k}$). Given $y$ the data of $y_{0},\dots,y_{k-1}$ Hamiltonian
chords from $L_{i}$ to $L_{i+1}$ and $y_{k}$ a Hamiltonian chord from $L_{0}$
to $L_{k}$ and $\mathtt{h}\in H^{k+1}$, we define $\mathcal{M}(\mathtt{h};y)$
to be the moduli-space of disks $u:D^{2}\setminus\\{p_{1},\dots,p_{k}\\}\to Y$
that are $J_{g}$-holomorphic for the given choice of $\mathtt{h}$-perturbation
datum and asymptotic to some lift of the $y_{i}$ at each boundary puncture
$p_{i}$. Then we have $A_{\infty}$-operations

(43)
$\displaystyle\begin{split}\mu_{k}:\hom((L_{k-1},\tilde{J}_{k-1}),(L_{k},\tilde{J}_{k}))\otimes\dots\otimes\hom&((L_{0},\tilde{J}_{0}),(L_{1},\tilde{J}_{1}))\to\hom((L_{0},\tilde{J}_{0}),(L_{k},\tilde{J}_{k}))\\\
\langle\mu_{k}(y_{0}\otimes\dots\otimes
y_{k-1}),y_{k}\rangle&=\sum_{\mathtt{h}\in
H^{k+1}}\frac{\\#\mathcal{M}(\mathtt{h};y)}{|H|^{k+2}}\end{split}$

The proof that these operations satisfy the $A_{\infty}$-relations is the
usual one.

From the above expression one sees that $|H|$ should be invertible in the
coefficient ring used to define the Fukaya category.

Note that if $\tilde{J}_{i}$ is invariant under $H$ then there is an action of
$H$ on $\mathcal{M}(\mathtt{h};y)$ via $h\cdot u=h\circ u$ (which is still
holomorphic for the original choice of $\mathtt{h}$-perturbation datum due to
the invariance of $\tilde{J}_{i}$). In the particular case where
$\tilde{J}_{i}=\pi^{*}J_{i}$ is an almost complex structure pulled back by
$\pi$, we get an isomorphism
$\mathcal{M}(\mathtt{h};y)/H\cong\mathcal{M}_{X}(y)$ for $\mathcal{M}_{X}(y)$
the moduli-space of disks used in the definition of $\mathcal{F}uk(X)$. Hence
$\\#\mathcal{M}_{X}(y)=\frac{\mathcal{M}(\mathtt{h};y)}{|H|}$, so it follows
from Equation 43 that we have an embedding

(44)
$\displaystyle\begin{split}\iota:\mathcal{F}uk(X)&\hookrightarrow\mathcal{F}uk(X;Y)\\\
(L,J_{L})&\mapsto(L,\pi^{*}J_{L}).\end{split}$

More generally, we have an action of $H$ on $\cup_{\mathtt{h}\in
H^{k+1}}\mathcal{M}(\mathtt{h};y)$ that, for any $h\in H$, sends a disk
$u\in\mathcal{M}(\mathtt{h};y)$ to the disc $h\circ
u\in\mathcal{M}(h^{\prime}\cdot\mathtt{h};y)$, where
$h\cdot\mathtt{h}:=(h\cdot h_{0},\dots,h\cdot h_{k})$. We think of the
quotient moduli-space

$\frac{\bigcup_{\mathtt{h}\in
H^{k+1}}\mathcal{M}(\mathtt{h};y)}{H}\cong\bigcup_{\mathtt{h}\in
H^{k+1}/H}\mathcal{M}(\mathtt{h};y)$

as a moduli space of multi-perturbed holomorphic disks in $X$, in the sense of
[Sal99, Section 5]. An $H$-orbit of a disk $u$ in $Y$ corresponds to the
projected disk $\pi\circ u$ inside $X$.

Now we apply the above construction to $\pi:Y\times\C\to X\times\C$. We
restrict the objects $\mathcal{L}(X\times\C)$ to be Lagrangian cobordisms in
$X\times\C$ equipped with all the extra data that we would like the
corresponding cobordism group to have (for us, they come equipped with
$G$-brane structures). This gives a Fukaya category
$\mathcal{F}uk_{cob}(X\times\C;Y\times\C)$ whose objects are by definition
Lagrangian cobordisms in $X\times\C$ (carrying a $G$-brane structure) that are
tautologically unobstructed when pulled back under $\pi$. The machinery of
[BC14, BCS21] shows that these objects induce iterated cone decompositions in
$\mathcal{F}uk(X;Y)$. Using that the functor $\iota$ in Equation 44 is an
embedding, it follows that cobordisms in $\mathcal{F}uk(X\times\C;Y\times\C)$
with ends $\iota(L_{0}),\dots,\iota(L_{k})$ induce cone decompositions between
the $L_{i}$ in $\mathcal{F}uk(X)$. In particular, we have the following:

######  Proposition 48.

Let $\pi:Y\to X$ be a finite covering, with both $Y$ and $X$ symplectic
manifolds and $\pi^{*}\omega_{X}=\omega_{Y}$. Let
$\operatorname{\mathrm{Cob}}(X)$ be a cobordism group whose generators are
weakly exact Lagrangians in $X$, and whose relations come from cobordisms
$V\subset X\times\C$ such that $(\pi\times\id_{\C})^{-1}(V)\subset Y\times\C$
is tautologically unobstructed. Then there is a well-defined map

$\operatorname{\mathrm{Cob}}(X)\to K_{0}(\mathcal{F}uk(X)).$

Specializing to the case $X=\mathcal{K}$ and $Y=T^{4}$ gives the result
claimed in Section 4.8.

##  Appendix D Proof of 2

In this appendix we prove 2 by using the arguments in [SS20, Section 7] and
applying some elementary linear algebra. It is worth pointing out that, while
[SS20, Lemma 7.9] should be thought of as the symplectic analog of Mumford’s
theorem [Mum69], our extension is the symplectic analog of a more general
theorem due to Roitman [Roi71, Theorem 5]. In fact, the key result from linear
algebra that we use was already encountered and proved by Roitman in his
original paper.

Let $L\subset X$ be a Lagrangian submanifold. Weinstein’s tubular neighborhood
theorem gives an identification between deformations of $L$ and closed
$1$-forms on $L$ that are close to $0$. Since exact $1$-forms define
Hamiltonian isotopic Lagrangians, we obtain an identification between nearby
Lagrangians to $L$ modulo Hamiltonian isotopy and a neighborhood of $0$ in the
first deRham cohomology group $H^{1}_{dR}(L)\cong H^{1}(L;\R)$. More
precisely, recall that given a Lagrangian isotopy
$\mathcal{L}=\\{L_{t}\\}_{t\in[0,1]}$, there is an associated flux map
$Flux\,\mathcal{L}\in H^{1}(L;\R)\cong\Hom(\pi_{1}(L),\R)$ defined via

$\displaystyle\begin{split}Flux\,\mathcal{L}:\pi_{1}(L)&\to\R\\\
[\gamma]&\mapsto\int_{\Gamma}\omega\end{split}$

where $\Gamma:=\cup_{t}\gamma_{t}$ is the cylinder obtained by parallel
transporting $\gamma=\gamma_{0}$ along the isotopy $\mathcal{L}$ (see [Sol13,
Section 6] for more details). Since the flux is invariant under homotopies of
$\mathcal{L}$ with fixed end-points and it vanishes if (and only if for nearby
Lagrangians) $L_{0}$ and $L_{1}$ are Hamiltonian isotopic,141414Similar
statements for the flux of a path of symplectomorphisms can be found in [MS17,
Section 10.2], and were adapted to this setting in [Sol13]. the mapping

$L^{\prime}\mapsto Flux\,\mathcal{L}_{L^{\prime}}$

sending a Lagrangian $L^{\prime}$ close to $L$ to the flux of an isotopy
$\mathcal{L}_{L^{\prime}}=\\{L_{t}\\}_{t\in[0,1]}$ with $L_{0}=L$ and
$L_{1}=L^{\prime}$ provides an explicit indentification between Lagrangians
close to $L$ modulo Hamiltonian isotopy and a neighborhood of the origin
$\mathcal{U}_{L}\subset H^{1}(L;\R)$ (note that for sufficiently close
Lagrangians, the isotopy $\mathcal{L}_{L^{\prime}}$ is well-defined up to
homotopy with fixed end-points).

######  Definition 49.

Given a class $\alpha\in\mathcal{U}_{L}$, we denote by $L(\alpha)$ the
corresponding deformation of $L$.

Note the Lagrangian $L(\alpha)$ can be explicitly represented (up to
Hamiltonian isotopy) as the graph of any $1$-form representing the cohomology
class $\alpha$.

Let $\mathbb{L}:=(L_{1},\dots L_{k})$ be a tuple of Lagrangians and let
$\mathbb{U}_{\mathbb{L}}:=\prod_{i}\mathcal{U}_{L_{i}}$. It follows from the
invariance of the Lagrangian cobordism class under Hamiltonian isotopy that
there is a well-defined map

$\displaystyle\begin{split}f_{\mathbb{L}}:\mathbb{U}_{\mathbb{L}}&\to\operatorname{\mathrm{Cob}}(X)\\\
\alpha&\mapsto\sum_{i}L(\alpha_{i}).\end{split}$

One can easily incorporate local systems into the picture so that if
$\mathbb{L}=(L_{1},\dots L_{k})$ is a tuple of Lagrangians as above and
$\mathbb{U}_{\mathbb{L}}:=\prod_{i}(\mathcal{U}_{L_{i}}\times
H^{1}(L_{i};G))$, there is a well-defined map

$\displaystyle\begin{split}f_{\mathbb{L}}:\mathcal{U}_{\mathbb{L}}&\to\operatorname{\mathrm{Cob}}(X)\\\
(\alpha_{i},\eta_{i})&\mapsto\sum_{i}(L(\alpha_{i}),\eta_{i}).\end{split}$

Here, the notation $(L(\alpha_{i}),\eta_{i})$ means the Lagrangian
$L(\alpha_{i})$ equipped with the local system induced from
$\eta_{i}:\pi_{1}(L_{i})\to G$ via the isotopy. Writing
$\dim\mathbb{U}_{\mathbb{L}}:=\dim(\prod_{i}\mathcal{U}_{L_{i}})$, we present
the following notion of dimensionality of Lagrangian cobordism groups:

######  Definition 50.

We say the Lagrangian cobordism group $\operatorname{\mathrm{Cob}}(X)$ is
$d$-dimensional if $d$ is the minimum $N\in\N$ such that there exists a
sequence $(\mathbb{L}_{i},\mathbb{U}_{\mathbb{L}_{i}})_{i\in\N}$ of Lagrangian
tuples $\mathbb{L}_{i}=(L^{(i)}_{1},\dots,L^{(i)}_{n_{i}})$ and (not-
necessarily open) subsets
$\mathbb{U}_{\mathbb{L}_{i}}\subset\prod_{j}(H^{1}(L^{(i)}_{j};\R)\times
H^{1}(L^{(i)}_{j};G))$ containing $0$ and of dimension at most $N$ such that
$\cup_{i}\im(f_{\mathbb{L}_{i}})=\operatorname{\mathrm{Cob}}(X)$.

The above notion of dimensionality of cobordism groups is similar, but not
equal, to that in [SS20]—even without the presence of local systems. The key
difference is that we allow arbitrary sets $\mathcal{U}_{L_{i}}$ containing
$0$, whereas Sheridan-Smith require them to be open. For instance, for
$X=T^{2}$ it is known by Haug [Hau15] (see also our Section 3.2) that the
Lagrangian cobordism group of a $2$-torus is
$\operatorname{\mathrm{Cob}}(T^{2})\cong\Z^{2}\oplus S^{1}$. This group is
$1$-dimensional under 50, but it is not finite-dimensional if one requires the
subsets $\mathbb{U}_{\mathbb{L}_{i}}$ to be open neighborhoods of $0$ and
their dimensions to be bounded.

Sheridan-Smith prove an infinite-dimensionality result for cobordism groups as
follows. First, they prove:

######  Lemma 51.

([SS20, Lemma 7.4]) Given $\mathbb{L}$ and $\mathbb{L}^{\prime}$ finite tuples
of Lagrangians, the subset

$\mathcal{Z}_{\mathbb{L},\mathbb{L}^{\prime}}=\\{(\alpha,\alpha^{\prime})\in\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}|f_{\mathbb{L}}(\alpha)=f_{\mathbb{L}^{\prime}}(\alpha^{\prime})\\}$

is a countable union of (open subsets of) affine subspaces of
$\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}$.

Consider now the $n$-form $\Omega\in\wedge^{n}T^{*}\mathbb{U}_{\mathbb{L}}$
defined by

(45)
$\Omega(\alpha^{1},\dots,\alpha^{n})=\sum_{i}\int_{L_{i}}\alpha^{1}_{i}\cup\dots\cup\alpha^{n}_{i}$

for
$\alpha^{j}=(\alpha^{j}_{1},\dots,\alpha^{j}_{k})\in\mathcal{U}_{L_{1}}\times\dots\times\mathcal{U}_{L_{k}}\simeq
T\mathbb{U}_{\mathbb{L}}$. It can be naturally extended to an $n$-form on
$\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}$ via the
identification
$\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}\simeq\mathbb{U}_{\mathbb{L}\cup\mathbb{L}^{\prime}}$.

######  Lemma 52.

([SS20, Lemma 7.5]) The $n$-form $\Omega$ vanishes on
$\mathcal{Z}_{\mathbb{L},\mathbb{L}^{\prime}}$.

The key property is that $\Omega$ is non-degenerate for $n=2$. Hence we have a
symplectic form on
$\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}$ and an
isotropic subspace
$\mathcal{Z}_{\mathbb{L},\mathbb{L}^{\prime}}\subset\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}}$,
which gives a bound
$\dim\mathcal{Z}_{\mathbb{L},\mathbb{L}^{\prime}}\leq\frac{1}{2}\dim(\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}})$.
Using this, Sheridan-Smith state the infinite-dimensionality of
$\operatorname{\mathrm{Cob}}(X)$:

######  Lemma 53.

([SS20, Lemma 7.9]) Suppose $X$ is a symplectic four manifold containing a
Lagrangian of genus at least one. Then, for any countable family
$\\{\mathbb{L}_{i}\\}$ of finite tuples
$\mathbb{L}_{i}=(L^{i}_{1},\dots,L^{i}_{n_{i}})$ such that
$\sup_{i}\dim\mathbb{U}_{\mathbb{L}_{i}}<\infty$, the cobordism group
$\operatorname{\mathrm{Cob}}(X)$ is not covered by the images of the maps
$f_{\mathbb{L}_{i}}$.

The dimension bound
$\dim\mathcal{Z}_{\mathbb{L}\mathbb{L}^{\prime}}\leq\frac{1}{2}\dim(\mathbb{U}_{\mathbb{L}}\times\mathbb{U}_{\mathbb{L}^{\prime}})$
is the key to make their proof work. This bound comes from having a symplectic
form, which only works in the case $n=2$. For arbitrary $n$, recall from
Equation 45 that we have an $n$-form $\Omega$ of the form
$\Omega=\sum_{i}pr_{i}^{*}\omega_{i}$, where
$pr_{i}:\mathbb{U}_{\mathbb{L}}\to\mathcal{U}_{L_{i}}$ is the projection and
$\omega_{i}:H^{1}(L_{i};\R)^{\otimes n}\to\R$ is given by cup-product and
integration. It turns out that in a situation like this one still has a useful
bound:

######  Lemma 54.

([Roi71, Lemma 9]) Let $V=\oplus_{j=1}^{m}V_{j}$ be a graded vector space and
$0\neq\omega_{j}\in\wedge^{q}V_{j}^{*}$ non-zero $q$-forms on $V_{j}$, $q\geq
2$. Denote by $pr_{j}:V\to V_{j}$ the natural projection. If $W\subset V$ is
an isotropic subspace for the $q$-form $\Omega=\sum_{j}pr^{*}_{j}\omega_{j}$,
then $\dim W\leq\dim V-m$.

With this result, the proof of [SS20, Lemma 7.9] translates almost directly:

######  Proof D.1 (Proof of 2).

Let $N$ be such that $\dim\mathbb{U}_{\mathbb{L}_{i}}<N$ for all $i$, and
consider the family $\mathbb{L}^{(N)}=(T,\dots,T)$ consisting of $N$ copies of
the Lagrangian torus $T$. We choose $\mathcal{U}_{T}$ to be an open
neighborhood of $0$ and as before write
$\mathbb{U}_{\mathbb{L}^{(N)}}=\prod_{k=1}^{N}\mathcal{U}_{T}$. We will prove
that the images of the maps $f_{\mathbb{L}_{i}}$ can’t cover
$\im(f_{\mathbb{L}^{(N)}})\subset\operatorname{\mathrm{Cob}}(X)$, so that in
particular they can’t cover $\operatorname{\mathrm{Cob}}(X)$. Note that the
images of the maps $f_{\mathbb{L}_{i}}$ cover $\im(f_{\mathbb{L}^{(N)}})$ if
and only if the projections

$\mathcal{Z}_{\mathbb{L}_{i},\mathbb{L}^{(N)}}\subset\mathbb{U}_{\mathbb{L}_{i}}\times\mathbb{U}_{\mathbb{L}^{(N)}}\to\mathbb{U}_{\mathbb{L}^{(N)}}$

cover all of $\mathbb{U}_{\mathbb{L}^{(N)}}$. We will show that
$\dim\mathcal{Z}_{\mathbb{L}_{i},\mathbb{L}^{(N)}}<\dim\mathbb{U}_{\mathbb{L}^{(N)}}$
for all $i$, and since countably many sets of strictly lower dimension cannot
surject onto $\mathbb{U}_{\mathbb{L}^{(N)}}$ the result will follow.

Consider the decomposition
$\mathbb{U}_{\mathbb{L}_{i}}\oplus\mathbb{U}_{\mathbb{L}^{(N)}}=\oplus_{j}V_{j}$
given by $V_{1}=\mathbb{U}_{\mathbb{L}_{i}}\oplus\mathcal{U}_{T}$ and
$V_{j}=\mathcal{U}_{T}$ for $j=2,\dots,N$. Given a Lagrangian $L$ we denote by
$\omega_{L}\in\wedge^{n}H^{1}(L;\R)^{*}$ the $n$-form on $H^{1}(L;\R)$ given
by cup-product and integration; they naturally restrict to $n$-forms on
$\mathcal{U}_{L}\subset H^{1}(L;\R)$. We then define $n$-forms
$\omega_{j}\in\wedge^{n}V_{j}^{*}$ by
$\omega_{1}=\sum_{k}\omega_{L_{k}^{(i)}}+\omega_{T}$ and
$\omega_{j}=\omega_{T}$ for $j=2,\dots,N$. Note that $\omega_{j}\neq 0$ for
all $j$ since $T$ is a torus and thus $\cup:H^{1}(T;\R)^{\otimes n}\to
H^{n}(T;\R)$ is a surjection. The form

$\Omega=\sum_{j=1}^{N}pr_{j}^{*}\omega_{j},\quad
pr_{j}:V=\mathbb{U}_{\mathbb{L}_{i}}\oplus\mathbb{U}_{\mathbb{L}^{(N)}}=\oplus_{j}V_{j}\to
V_{j}$

vanishes on $\mathcal{Z}_{\mathbb{L}_{i},\mathbb{L}^{(N)}}$ by 52, hence we
have:

$\displaystyle\dim\mathcal{Z}_{\mathbb{L}_{i},\mathbb{L}^{(N)}}$
$\displaystyle\leq\dim(\mathbb{U}_{\mathbb{L}_{i}}\times\mathbb{U}_{\mathbb{L}^{(N)}})-N\quad$
(54) $\displaystyle<N+\dim\mathbb{U}_{\mathbb{L}^{(N)}}-N\quad$
$\displaystyle(\dim\mathbb{U}_{\mathbb{L}_{i}}<N)$
$\displaystyle=\dim\mathbb{U}_{\mathbb{L}^{(N)}}.$

In particular the projection
$\cup_{i}\mathcal{Z}_{\mathbb{L}_{i},\mathbb{L}^{(N)}}\subset\cup_{i}\mathbb{U}_{\mathbb{L}_{i}}\times\mathbb{U}_{\mathbb{L}^{(N)}}\to\mathbb{U}_{\mathbb{L}^{(N)}}$
cannot be surjective.

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