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\ihead{Errata to arXiv:1804.03322v1}
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\begin{document}

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\textbf{Abelian networks IV. Dynamics of nonhalting networks}

\textit{Swee Hong Chan and Lionel Levine}

\href{https://arxiv.org/abs/1804.03322v1}{arXiv:1804.03322v1}

\textbf{Errata and addenda by Darij Grinberg (version of
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\section*{***}

I will refer to the results appearing in the preprint \textquotedblleft
Abelian networks IV. Dynamics of nonhalting networks\textquotedblright\ by the
numbers under which they appear in this preprint (specifically, in its version
of 10 April 2018, posted on the arXiv as
\href{https://arxiv.org/abs/1804.03322v1}{arXiv:1804.03322v1}).

\section{Errata}

The errata below are relatively haphazard -- I have skipped various proofs in
my readthrough. A lot of them are pedantic suggestions rather than serious errors.

\begin{itemize}
\item \textbf{page 3:} \textquotedblleft increasing functions the
form\textquotedblright\ $\rightarrow$ \textquotedblleft increasing functions
of the form\textquotedblright.

\item \textbf{page 7:} In \textquotedblleft the spectral radius of
P\textquotedblright, the \textquotedblleft P\textquotedblright\ should be in mathmode.

\item \textbf{page 9, \S 2.1:} In the definition of the Grothendieck group,
I'd replace \textquotedblleft$\mathcal{M}\times\mathcal{M}/\sim$%
\textquotedblright\ by \textquotedblleft$\left(  \mathcal{M}\times
\mathcal{M}\right)  /\sim$\textquotedblright; otherwise, it may be misread as
\textquotedblleft$\mathcal{M}\times\left(  \mathcal{M}/\sim\right)
$\textquotedblright.

\item \textbf{page 9, \S 2.1:} Before \textquotedblleft Grothendieck group
satisfies the \textit{universal enveloping property}\textquotedblright, add
\textquotedblleft The\textquotedblright.

\item \textbf{page 11, \S 2.1:} You say that \textquotedblleft$\tau\left(
\mathcal{K}\right)  $ is a finite group if $\mathcal{M}$ is finitely
generated\textquotedblright. I would suggest justifying this claim (it follows
from the Fundamental Theorem of finitely generated abelian groups, but this
should probably be said).

\item \textbf{page 11, \S 2.2:} Your definition of \textquotedblleft
irreducible\textquotedblright\ (for an action of $\mathcal{F}$ on $Y$) differs
from the one given in [BL16b]. Are the two definitions equivalent? A few extra
sentences are probably in order, since you refer to [BL16b] for the proof of
Lemma 2.7.

\item \textbf{page 13, \S 3.1:} You begin this section with \textquotedblleft
Let $G=\left(  V\left(  G\right)  ,E\left(  G\right)  \right)  $ be a directed
graph\textquotedblright. But if you want multiple edges, then your directed
graph cannot just be a pair of two sets; it must also store the sources and
the targets of the edges somewhere. I would encode a directed graph with
multiple edges either as a triple $\left(  V,E,\varphi\right)  $ (where
$\varphi$ is the map sending any edge to the pair consisting of its source and
its target) or as a quadruple $\left(  V,E,s,t\right)  $ (where $s$ is the map
sending each edge to its source, and $t$ is the map sending each edge to its target).

\item \textbf{page 13, \S 3.1:} In the definition of \textquotedblleft
message-passing function\textquotedblright, replace \textquotedblleft for each
edge $e=\left(  v,u\right)  $\textquotedblright\ by \textquotedblleft for each
edge $e$ with source $v$ and target $u$\textquotedblright. This is, again,
because you want to allow multiple edges, so you cannot just identify an edge
with a pair of two vertices.

This problem keeps coming up over and over in your paper, if the parts I have
read so far are representative. I personally prefer to abbreviate
\textquotedblleft edge $e$ with source $v$ and target $u$\textquotedblright%
\ by \textquotedblleft edge $e:v\rightarrow u$\textquotedblright\ for brevity
(more so because it reminds me of the directedness of the graph). I don't know
of a good way to quickly search/replace all the appearances of this, though.

\item \textbf{page 13, \S 3.1:} After \textquotedblleft and sends the message
$T_{e}\left(  q,a\right)  $ to $\mathcal{P}_{u}$\textquotedblright, add
\textquotedblleft for each edge $e$ from $v$ to $u$\textquotedblright.

\item \textbf{page 13, \S 3.1:} In the commutativity conditions, you should
add \textquotedblleft For each edge $e$ with source $v$ and target
$u$\textquotedblright\ at the beginning of condition (ii).

\item \textbf{page 14, \S 3.1:} In the first equality on this page, you seem
to use the notation $\operatorname*{Out}\left(  v\right)  $ for the set of all
edges with source $v$. Please say so.

\item \textbf{page 14, \S 3.1:} Also, in the first equality on this page, you
want $v$ to be the vertex of $G$ satisfying $a\in A_{v}$.

\item \textbf{page 14, \S 3.1:} In \textquotedblleft Lemma 3.1(ii) implies
that the function\textquotedblright, replace \textquotedblleft
function\textquotedblright\ by \textquotedblleft functions\textquotedblright.

\item \textbf{page 14, \S 3.2:} Do you really mean \textquotedblleft the
letters in $\mathcal{N}$\textquotedblright, or perhaps \textquotedblleft the
letters in $w$\textquotedblright?

\item \textbf{page 14, \S 3.2:} On the second-to-last line of this page,
\textquotedblleft for all $i\in\left\{  1,\ldots,l\right\}  $%
\textquotedblright\ $\rightarrow$ \textquotedblleft for all $i\in\left\{
1,\ldots,\ell\right\}  $\textquotedblright\ (%
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ell, not just l).

\item \textbf{page 15, Definition 3.2:} Explain, perhaps, what
\textquotedblleft sends\textquotedblright\ means (namely, $w$
\textquotedblleft sends\textquotedblright\ $\mathbf{x}.\mathbf{q}$ to $\pi
_{w}\left(  \mathbf{x}.\mathbf{q}\right)  $)\textbf{.}

\item \textbf{page 15, \S 3.2:} \textquotedblleft The \textit{support} of a
vector $\mathbf{z}$\textquotedblright\ $\rightarrow$ \textquotedblleft The
\textit{support} of a vector $\mathbf{u}$\textquotedblright.

\item \textbf{page 15, Lemma 3.3:} In part (ii), replace \textquotedblleft%
$\operatorname*{supp}\left(  \mathbf{w}\right)  $\textquotedblright\ by
\textquotedblleft$\operatorname*{supp}\left(  \left\vert w\right\vert \right)
$\textquotedblright.

\item \textbf{page 15, \S 3.3:} Your definition of a finite network seems to
entail that the state space $Q$ can be infinite, but only the $Q_{v}$ and
$A_{v}$ for each given $v\in V$ are required to be finite (whereas $V$ may be
infinite). But I don't think this is what you want, because you next claim
that $M$ is finite. I don't know why you want to define finiteness for a
single processor to begin with; do you ever use it? Why not just define a
network $\mathcal{N}$ to be finite if its (total) state space $Q$ and its
(total) alphabet $A$ are finite?

\item \textbf{page 15, proof of Lemma 3.4:} In \textquotedblleft Since
$\mathbf{n}\geq1$\textquotedblright, the \textquotedblleft$1$%
\textquotedblright\ should be boldfaced.

\item \textbf{page 15, \S 3.3:} I don't understand the definition of
\textquotedblleft idempotent vector\textquotedblright. Seeing that you're only
using this word one time in the whole paper, maybe ditch it altogether?

\item \textbf{page 16, \S 3.4:} In \textquotedblleft\textit{negative part}
$\mathbf{z}^{-}$ of $\mathbf{x}$\textquotedblright, replace \textquotedblleft%
$\mathbf{x}$\textquotedblright\ by \textquotedblleft$\mathbf{z}$%
\textquotedblright.

\item \textbf{page 16, \S 3.4:} After \textquotedblleft such that
$\mathbf{z}=\mathbf{z}^{+}-\mathbf{z}^{-}$\textquotedblright, add
\textquotedblleft and $\operatorname*{supp}\left(  \mathbf{z}^{+}\right)
\cap\operatorname*{supp}\left(  \mathbf{z}^{-}\right)  =\varnothing
$\textquotedblright.

\item \textbf{page 16, \S 3.4:} When defining \textquotedblleft locally
irreducible\textquotedblright, you probably also want to require that
$Q\neq\varnothing$. Or, at least, you should require it in Definition 3.8,
since otherwise $P\left(  \mathbf{k}\right)  $ is not defined (you need a
$\mathbf{q}\in\operatorname*{Loc}\left(  \mathcal{N}\right)  $ to ensure that
$P_{\mathbf{q}}\left(  \mathbf{k}\right)  $ makes sense).

\item \textbf{page 17, \S 3.5:} You may want to point out what $\lambda\left(
P\right)  $ means when the matrix $P$ is empty. (This happens, e.g., when
$A=\varnothing$ in your network.)

\item \textbf{page 17, Lemma 3.10:} I assume you require eigenvectors to be
$\neq0$, or otherwise part (i) wouldn't be as interesting as you may want it
to be. Best to say so.

\item \textbf{page 19, \S 3.6:} Somewhere here it helps to say that $G$
denotes the underlying digraph of the network; $V$ its set of vertices; and
$E$ its set of edges.

\item \textbf{page 19, \S 3.6:} I suggest adding \textquotedblleft
Eulerian\textquotedblright\ after \textquotedblleft into a\textquotedblright%
\ in \textquotedblleft Any undirected graph can be changed into a directed
graph\textquotedblright.

\item \textbf{page 20, \S 3.6:} In \textquotedblleft For $n\geq3$, the
\textit{bidirected cycle }$C_{n}$ ($n\geq3$)\textquotedblright, remove the
\textquotedblleft($n\geq3$)\textquotedblright\ part (you already said it half
a sentence ago).

\item \textbf{page 20, Example 3.11:} What is a cyclic list? (I assume it is a
family indexed by $\mathbb{Z}_{m}$ for some $m>0$. Please define it.) Same for
the concept of a \textquotedblleft cyclic total order\textquotedblright\ later
(in Example 3.14). The two things mean the same, right?

\item \textbf{page 20, Example 3.11:} \textquotedblleft by a \textit{rotor
configurations}\textquotedblright\ $\rightarrow$ \textquotedblleft by a
\textit{rotor configuration}\textquotedblright.

\item \textbf{page 20, Example 3.11:} Not just in this example, but also in
others, I'd recommend adding a few sentences about what $\operatorname*{Loc}%
\left(  \mathcal{N}\right)  $, $e$ and $K$ are. It is not hard to figure out
(at least in Examples 3.11--3.19, the transition functions $T_{a}$ are
bijective, so that the total transition functions $t_{a}$ are bijective as
well, and thus $M$ is a sub-unital submonoid of $\operatorname*{End}\left(
Q\right)  $, so that $e=1$, and therefore $\operatorname*{Loc}\left(
\mathcal{N}\right)  =Q$; finding $K$ is a bit trickier), but these things are
among the trickiest to understand (being non-issues in classical chip-firing
theory), so some handholding would help readers a lot here.

\item \textbf{page 20, Example 3.12:} In \textquotedblleft$Q_{v}:=\left\{
0,1\ldots,\operatorname*{outdeg}\left(  v\right)  -1\right\}  $%
\textquotedblright, add a comma after \textquotedblleft$0,1$\textquotedblright.

\item \textbf{page 21, Remark:} \textquotedblleft has a subtle
difference\textquotedblright\ $\rightarrow$ \textquotedblleft have a subtle
difference\textquotedblright.

\item \textbf{page 21, Example 3.13:} You don't want to fix a cyclic list
\textquotedblleft of the target vertices of $v$\textquotedblright. You want to
fix a cyclic list of the outgoing edges $e_{j}^{v}$ of $v$, and then define
$u_{j}^{v}$ as the target of $e_{j}^{v}$. Otherwise, it is not clear how to
order parallel edges.

\item \textbf{page 22, Example 3.14:} In \textquotedblleft is identical to the
sinkless height-arrow networks\textquotedblright, replace \textquotedblleft
networks\textquotedblright\ by \textquotedblleft network\textquotedblright.

\item \textbf{page 22, Example 3.14:} \textquotedblleft is equal the
matrix\textquotedblright\ $\rightarrow$ \textquotedblleft is equal to the
matrix\textquotedblright.

\item \textbf{page 22, Example 3.14:} I suspect that the claim
\textquotedblleft$\lambda\left(  P\right)  <1$\textquotedblright\ (and, hence,
subcriticality) requires $P$ to be strongly connected.

\item \textbf{page 23, Example 3.15:} \textquotedblleft with positive diagonal
entries\textquotedblright\ $\rightarrow$ \textquotedblleft with positive
integer diagonal entries\textquotedblright, I guess?

\item \textbf{page 23, Example 3.15:} \textquotedblleft For each $v\in V$
of\textquotedblright\ $\rightarrow$ \textquotedblleft For each vertex $v\in V$
of\textquotedblright.

\item \textbf{page 23, Example 3.15:} In \textquotedblleft$Q_{v}:=\left\{
0,1\ldots,d_{v}-1\right\}  $\textquotedblright, add a comma after
\textquotedblleft$0,1$\textquotedblright.

\item \textbf{page 24, Example 3.16:} In the definition of the message-passing
function, replace \textquotedblleft$(\operatorname{mod}\mathbb{Z}%
_{\operatorname*{outdeg}\left(  v\right)  })$\textquotedblright\ by
\textquotedblleft$(\operatorname{mod}\operatorname*{outdeg}\left(  v\right)
)$\textquotedblright\ (or by \textquotedblleft in $\mathbb{Z}%
_{\operatorname*{outdeg}\left(  v\right)  }$\textquotedblright).

\item \textbf{page 24, Example 3.16:} \textquotedblleft the states of this
network can\textquotedblright\ $\rightarrow$ \textquotedblleft a state of this
network can\textquotedblright.

\item \textbf{page 25, Example 3.17:} In \textquotedblleft$Q_{v}:=\left\{
0,1\ldots,t_{v}-1\right\}  $\textquotedblright, add a comma after
\textquotedblleft$0,1$\textquotedblright.

\item \textbf{page 25, Example 3.18:} In \textquotedblleft where $M:=\left(
m_{i,j}\right)  _{0\leq i,j\leq\ell}$\textquotedblright, replace
\textquotedblleft$\ell$\textquotedblright\ by \textquotedblleft$k$%
\textquotedblright.

\item \textbf{page 26, Example 3.19:} \textquotedblleft For each $v\in V$
of\textquotedblright\ $\rightarrow$ \textquotedblleft For each vertex $v\in V$
of\textquotedblright.

\item \textbf{page 26, Example 3.19:} \textquotedblleft for all $i\in
\mathbb{Z}_{m}$\textquotedblright\ $\rightarrow$ \textquotedblleft for all
$i\in\mathbb{Z}_{m_{v}}$\textquotedblright.

\item \textbf{page 27, and several times further:} \textquotedblleft
occurences\textquotedblright\ $\rightarrow$ \textquotedblleft
occurrences\textquotedblright.

\item \textbf{page 32, Lemma 4.11:} The first comma on the second line of this
lemma should be a period.

\item \textbf{page 33, proof of Lemma 4.13:} \textquotedblleft$u=a_{j+1}\cdots
a_{k}$\textquotedblright\ $\rightarrow$ \textquotedblleft$u=a_{j+1}\cdots
a_{j+k}$\textquotedblright.

\item \textbf{page 34, proof of Proposition 4.9:} Shouldn't \textquotedblleft
The conclusion of the lemma\textquotedblright\ be \textquotedblleft The
conclusion of the proposition\textquotedblright?

\item \textbf{page 35, proof of Lemma 4.14:} \textquotedblleft The
Perron-Frobenius theorem (Lemma 3.10(iii)) then implies that $\lambda\left(
P\right)  =1$\textquotedblright\ is overkill: You can just as well argue that
$P\left\vert w\right\vert =\left\vert w\right\vert $ reveals that $\left\vert
w\right\vert $ is an eigenvector of $P$ for eigenvalue $1$, and thus
$\lambda\left(  P\right)  \geq1$, which is equally sufficient for your aims.

\item \textbf{page 44, proof of Lemma 5.3, (i) implies (ii):}
\textquotedblleft$\overset{w^{\prime}w^{\prime\prime\prime}}{\longrightarrow}%
$\textquotedblright\ $\rightarrow$ \textquotedblleft$\overset{w^{\prime
}w^{\prime\prime}}{\longrightarrow}$\textquotedblright.

\item \textbf{page 51, Example 5.15:} Why do you say that \textquotedblleft
the capacity of a sinkless rotor network is equal to zero\textquotedblright%
\ and not $-\infty$ ? Is the $\mathbb{Z}^{A}$ in Definition 5.14 meant to be
an $\mathbb{N}^{A}$ ?

\item \textbf{page 52, proof of Lemma 5.16:} Does this really prove that the
capacity is well-defined? From what I see, it only shows that the maximum
defining the capacity exists as a supremum; why does it exist as a maximum?

\item \textbf{page 52, Definition 5.17:} \textquotedblleft
finitely\textquotedblright\ $\rightarrow$ \textquotedblleft
finite\textquotedblright?

\item \textbf{page 82, \S 8.3:} In the definition of the weight function,
replace \textquotedblleft$\operatorname{mod}\mathbb{Z}_{n}$\textquotedblright%
\ by \textquotedblleft in $\mathbb{Z}_{n}$\textquotedblright.

\item \textbf{page 82, \S 8.3:} What do you mean by \textquotedblleft
level\textquotedblright\ here? I thought all levels in the sinkless rotor
network would be $0$; do you mean \textquotedblleft level in the sandpile
sense\textquotedblright? (Perhaps better to define this independently.)
\end{itemize}


\end{document}