# Characterization of the tree cycles with minimum positive entropy for any
period

David Juher and Francesc Mañosas and David Rojas  Departament d’Informàtica,
Matemàtica Aplicada i Estadística, Universitat de Girona, c/ Maria Aurèlia
Capmany 61, 17003 Girona, Spain. ORCID 0000-0001-5440-1705
<EMAIL_ADDRESS>(Corresponding author) Departament de Matemàtiques,
Edifici C, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès,
Barcelona, Spain. ORCID 0000-0003-2535-0501<EMAIL_ADDRESS>Departament
d’Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, c/
Maria Aurèlia Capmany 61, 17003 Girona, Spain. ORCID 0000-0001-7247-4705
<EMAIL_ADDRESS>
###### Abstract.

Consider, for any integer $n\geq 3$, the set $\operatorname{Pos}_{n}$ of all
$n$-periodic tree patterns with positive topological entropy and the set
$\operatorname{Irr}_{n}\subset\operatorname{Pos}_{n}$ of all $n$-periodic
irreducible tree patterns. The aim of this paper is to determine the elements
of minimum entropy in the families $\operatorname{Pos}_{n}$,
$\operatorname{Irr}_{n}$ and
$\operatorname{Pos}_{n}\setminus\operatorname{Irr}_{n}$. Let $\lambda_{n}$ be
the unique real root of the polynomial $x^{n}-2x-1$ in $(1,+\infty)$. We
explicitly construct an irreducible $n$-periodic tree pattern
$\mathcal{Q}_{n}$ whose entropy is $\log(\lambda_{n})$. We prove that this
entropy is minimum in $\operatorname{Pos}_{n}$. Since the pattern
$\mathcal{Q}_{n}$ is irreducible, $\mathcal{Q}_{n}$ also minimizes the entropy
in the family $\operatorname{Irr}_{n}$. We also prove that the minimum
positive entropy in the set
$\operatorname{Pos}_{n}\setminus\operatorname{Irr}_{n}$ (which is nonempty
only for composite integers $n\geq 6$) is $\log(\lambda_{n/p})/p$, where $p$
is the least prime factor of $n$.

###### Key words and phrases:

tree maps, periodic patterns, topological entropy

###### 1991 Mathematics Subject Classification:

Primary: 37E15, 37E25

This work has been funded by grants PID2020-118281GB-C31 of Ministerio de
Ciencia e Innovación and 2021 SGR 00113 of Generalitat de Catalunya. D.R. is a
Serra Húnter fellow.

##  1\. Introduction

The field of Combinatorial Dynamics has its roots in the striking
Sharkovskii’s Theorem [31], in the sense that the theory grew up as a
succession of progressive refinements and generalizations of the ideas
contained in the original proof of that result. The core of the theory is the
notion of _combinatorial type_ or _pattern_.

Consider a class $\mathcal{X}$ of topological spaces (closed intervals of the
real line, trees, graphs and compact surfaces are classic examples) and the
family $\mathcal{F}_{\mathcal{X}}$ of all maps $\\{\mbox{$f\colon
X\longrightarrow X$}:X\in\mathcal{X}\\}$ satisfying a given property
(continuous maps, homeomorphisms, etc). Any of such maps gives rise, by
iteration, to a discrete dynamical system. Assume now that we have a map
$f\colon X\longrightarrow X$ in $\mathcal{F}_{\mathcal{X}}$ which is known to
have a periodic orbit $P$. The _pattern of $P$_ is the equivalence class
$\mathcal{P}$ of all maps $g\colon Y\longrightarrow Y$ in
$\mathcal{F}_{\mathcal{X}}$ having an invariant set $Q\subset Y$ that, at a
combinatorial level, behaves like $P$. In this case, we say that every map $g$
in the class _exhibits_ the pattern $\mathcal{P}$. Of course we have to
precise in which sense a periodic orbit _behaves as $P$_. So, we have to
decide which feature of $P$ has to be preserved inside the equivalence class
$\mathcal{P}$. The period of $P$, just a natural number, is a first
possibility (Sharkovskii’s Theorem), but a richer option arises from imposing
that

  1. (a)

the relative positions of the points of $Q$ inside $Y$ are the same as the
relative positions of $P$ inside $X$

  2. (b)

the way these positions are permuted under the action of $g$ coincides with
the way $f$ acts on the points of $P$.

An example is given by the family $\mathcal{F}_{\mathcal{M}}$ of surface
homeomorphisms. The pattern (or _braid type_) of a cycle $P$ of a map $f\colon
M\longrightarrow M$ from $\mathcal{F}_{\mathcal{M}}$, where $M$ is a surface,
is defined by the isotopy class, up to conjugacy, of
$f\bigr{\rvert}_{M\setminus P}$ [19, 27].

When $\mathcal{F}_{\mathcal{X}}$ is the family of continuous maps of closed
intervals, the points of an orbit $P$ of a map in $\mathcal{F}_{\mathcal{X}}$
are totally ordered and the pattern of $P$ can be simply identified with a
cyclic permutation in a natural way. The notion of pattern for interval maps
was formalized and developed in the early 1990s [12, 30].

In the last decades, a growing interest has arisen in extending the notion of
_pattern_ from the interval case to more general one-dimensional spaces such
as graphs [2, 10] or trees [6, 13, 14]. Precisely, in this paper we deal with
patterns of periodic orbits of continuous maps defined on trees (simply
connected graphs).

Let us precise the conditions (a,b) above in our context. If $f\colon
T\longrightarrow T$ is a continuous map of a tree and $P\subset T$ is a
periodic orbit of $f$, the triplet $(T,P,f)$ will be called a _model_. Two
points $x,y$ of $P$ will be said to be _consecutive_ if the unique closed
interval of $T$ having $x,y$ as endpoints contains no other points of $P$. Any
maximal subset of $P$ consisting only of pairwise consecutive points will be
called a _discrete component_. We will say that two models $(T,P,f)$ and
$(T^{\prime},P^{\prime},f^{\prime})$ are equivalent if there is a bijection
$\phi$ from $P$ to $P^{\prime}$ which sends discrete components to discrete
components and conjugates the action of $f$ on $P$ and the action of
$f^{\prime}$ on $P^{\prime}$, i.e.
$f^{\prime}\circ\phi\bigr{\rvert}_{P}=\phi\circ f\bigr{\rvert}_{P}$. In Figure
1 we show two equivalent 6-periodic models with two discrete components. Note
that two points $x_{i},x_{j}$ of $P$ are consecutive in $T$ when the
corresponding points $x^{\prime}_{i},x^{\prime}_{j}$ of $P^{\prime}$ are
consecutive in $T^{\prime}$.

A _pattern_ is an equivalence class of models by the above equivalence
relation. A map $f\colon T\longrightarrow T$ is said to _exhibit a pattern
$\mathcal{P}$_ if $f$ has an invariant set $P$ such that
$(T,P,f)\in\mathcal{P}.$

Figure 1. Set $P=\\{x_{i}\\}_{i=0}^{5}$ and
$P^{\prime}=\\{x^{\prime}_{i}\\}_{i=0}^{5}$. If $f\colon T\longrightarrow T$
and $f^{\prime}\colon T^{\prime}\longrightarrow T^{\prime}$ are continuous
maps such that $f(x_{i})=x_{i+1}$ and
$f^{\prime}(x^{\prime}_{i})=x^{\prime}_{i+1}$ for $0\leq i\leq 5$,
$f(x_{5})=x_{0}$ and $f^{\prime}(x^{\prime}_{5})=x^{\prime}_{0}$, then the
models $(T,P,f)$ and $(T^{\prime},P^{\prime},f^{\prime})$ are equivalent and
belong to the same pattern $[T,P,f]=[T^{\prime},P^{\prime},f^{\prime}]$.

A usual way of measuring the dynamical complexity of a map $f\colon
X\longrightarrow X$ of a compact metric space is in terms of its _topological
entropy_ , a notion first introduced in 1965 [1]. It is a non-negative real
number (or infinity) that measures how the iterates of the map mix the points
of $X$. It will be denoted by $h(f)$. An interval map with positive entropy is
_chaotic_ in the sense of Li and Yorke [26]. The same is true for more general
compact metric spaces [15]. On the other hand, the dynamics of a map with zero
topological entropy is much simpler.

Given a pattern $\mathcal{P}$ in $\mathcal{F}_{\mathcal{X}}$, we would like to
establish, only in terms of the combinatorial data encoded by $\mathcal{P}$, a
lower bound for the dynamical complexity that will be present in any map in
$\mathcal{F}_{\mathcal{X}}$ exhibiting $\mathcal{P}$. In view of what have
been said in the previous paragraph, it is natural to define the _topological
entropy of the pattern $\mathcal{P}$_, denoted from now on by
$h(\mathcal{P})$, as the infimum of the topological entropies of all maps in
$\mathcal{F}_{\mathcal{X}}$ exhibiting $\mathcal{P}$.

Although computing the entropy of a continuous map is difficult in general, in
some cases the computation of the entropy of a pattern $\mathcal{P}$ in
$\mathcal{F}_{\mathcal{X}}$ can be easily performed thanks to the existence of
the so called _canonical models_. A _canonical model_ of a pattern
$\mathcal{P}$ in $\mathcal{F}_{\mathcal{X}}$ is a map
$f\in\mathcal{F}_{\mathcal{X}}$ that exhibits $\mathcal{P}$ and satisfies at
least the following properties:

  1. (1)

$f$ is essentially unique and can be constructed from the combinatorial data
enclosed in $\mathcal{P}$

  2. (2)

$f$ has minimum entropy in the set of all maps exhibiting $\mathcal{P}$

  3. (3)

the dynamics of $f$ can be completely described using algebraic tools that, in
particular, allow us to compute $h(f)$.

From (1–3) it follows that $h(\mathcal{P})$, defined as the infimum of
entropies of maps, is in fact a minimum and can be easily computed as the
entropy of the canonical model of $\mathcal{P}$. The existence of canonical
models for patterns has been proved for continuous maps of closed intervals
(see [9] for a list of references), homeomorphisms of compact surfaces [22,
33] and continuous maps on trees [6].

Now we are ready to explain the aim of this paper. Several natural questions
concerning patterns and entropy arise. Fix $n\in\mathbb{N}$ and consider the
(finite) set of all $n$-periodic tree patterns. An important classification in
this set is given by the zero/positive entropy character of its elements. On
the one hand, the zero entropy tree patterns are well understood and several
equivalent characterizations can be found in the literature [18, 6, 5]. On the
other hand, let $\operatorname{Pos}_{n}$ be the subset of all $n$-periodic
tree patterns with positive entropy. One would like to describe the patterns
with maximal/minimal entropy in $\operatorname{Pos}_{n}$.

Several advances in the description of the entropy-maximal tree patterns have
been reported [4], but the problem is still open. In fact, the maximality
problem is unsolved even in the particular case of interval patterns [20, 21,
24]. Indeed, the maximal-entropy cyclic permutations of order $n$, when $n$
has the form $4k+2$, are still unknown, although [3] tackles this case from a
computational point of view and proposes a conjecture.

In this paper we face the opposite problem: the characterization of the
patterns of minimal entropy in $\operatorname{Pos}_{n}$. For interval maps,
the description of the minimum entropy cycles is known when $n$ is not a power
of two (see [9] for a review). In the setting of tree maps and for any $n\geq
3$, an $n$-periodic tree pattern $\mathcal{Q}_{n}$ was defined in [7] that
conjecturally has minimal entropy in the set $\operatorname{Pos}_{n}$ (the
problem makes no sense when $n=1,2$, since every periodic pattern of period 1
or 2 has entropy zero), and the conjecture was proved to be true when $n$ is a
power of a prime. See the canonical model of $\mathcal{Q}_{n}$ in Figure 2.
The entropy of $\mathcal{Q}_{n}$ turns out to be $\log(\lambda_{n})$, where
$\lambda_{n}$ is the unique real root of the polynomial $x^{n}-2x-1$ in
$(1,+\infty)$.

Figure 2. The canonical model $(T,P,f)$ of the pattern $\mathcal{Q}_{n}$, for
which $P=\\{x_{i}\\}_{i=0}^{n-1}$ is time labeled and $f(y)=y$.

The first main result of this paper states that the conjecture is in fact true
for every $n\geq 3$.

######  Theorem A.

Let $n\geq 3$ be a positive integer. Then, $\mathcal{Q}_{n}$ has minimum
entropy in the set $\operatorname{Pos}_{n}$ of all $n$-periodic patterns with
positive entropy. Moreover,
$h(\mathcal{P})>h(\mathcal{Q}_{n})=\log(\lambda_{n})$ for any
$\mathcal{P}\in\operatorname{Pos}_{n}$ such that
$\mathcal{P}\neq\mathcal{Q}_{n}$, where $\lambda_{n}$ is the unique real root
of the polynomial $x^{n}-2x-1$ in $(1,+\infty)$.

Traditionally, reducibility/irreducibility has been another important
classification for tree patterns. A pattern is _reducible_ when it has a block
structure (see Section 3). Roughly speaking, this means that the points of the
orbit can be partitioned into disjoint subtrees that are permuted under the
action of the map. The notion of reducibility arose early in the study of
interval maps and has been recently extended to the setting of tree patterns
[5]. The irreducible tree patterns are closely related to pseudo-Anosov braid
types of periodic orbits of orientation preserving disk homeomorphisms [23].
As we will see, every irreducible tree pattern has positive entropy. The
dynamic relevance of the patterns from $\operatorname{Irr}_{n}$ motivates the
study of the minimality of the entropy in this subclass of
$\operatorname{Pos}_{n}$. For interval maps, the problem was solved in [29].
Since the minimum entropy pattern $\mathcal{Q}_{n}$ turns out to be
irreducible, Theorem A incidentally proves that $\mathcal{Q}_{n}$ also
minimizes the topological entropy in the subclass $\operatorname{Irr}_{n}$.

######  Corollary B.

Let $n\geq 3$ be a positive integer. Then, $\mathcal{Q}_{n}$ has minimum
entropy in the set $\operatorname{Irr}_{n}$ of all $n$-periodic irreducible
patterns. Moreover, $h(\mathcal{P})>h(\mathcal{Q}_{n})=\log(\lambda_{n})$ for
any $\mathcal{P}\in\operatorname{Irr}_{n}$ such that
$\mathcal{P}\neq\mathcal{Q}_{n}$.

Now, the problem of determining the minimum (positive) entropy in the family
of all reducible patterns arises. It is not difficult to see that
$\operatorname{Pos}_{n}\setminus\operatorname{Irr}_{n}\neq\emptyset$ if and
only if $n$ is not a prime and $n\geq 6$. By Theorem A, the minimum positive
entropy for any reducible pattern is strictly larger than $\log(\lambda_{n})$.
The second main result of this paper gives the minimum entropy in
$\operatorname{Pos}_{n}\setminus\operatorname{Irr}_{n}$. In this case,
however, the minimum entropy pattern is not unique.

######  Theorem C.

Let $n\geq 6$ be a composite number. Then, the minimum positive entropy in the
set of all reducible $n$-periodic patterns is $\log(\lambda_{n/p})/p$, where
$p$ is the smallest prime factor of $n$.

This paper is organized as follows. In Section 2 we introduce formally the
basic notions of pattern, canonical model and path transition matrix, and
recall how to compute the topological entropy of a pattern. In Section 3 we
review some classic notions and results about block structures and
reducibility for tree patterns, that we use in Section 6 to recall the
characterization of zero entropy periodic patterns. A deeper study of the
structure of zero entropy paterns is carried out in Section 7. In Section 4 we
briefly recall a mechanism, first introduced in [7], that allows us to compare
the entropies of two patterns $\mathcal{P}$ and $\mathcal{O}$ when
$\mathcal{O}$ has been obtained by joining together several discrete
components of $\mathcal{P}$. Section 5 is devoted to the task of explaining
the strategy of the proof of Theorem A. As we will see, the proof is by
induction on the period $n$ and relies on a core result, Theorem D, that is
stated in the same section and proved in Section 8 using the results of
Section 7. The use of this result allows us to prove Theorem A for almost all
patterns, with two particular exceptions: the _$k$ -flowers_ (patterns with
$k$ discrete components attached at a unique central point) and the _triple
chain_ , a pattern with three consecutive discrete components. We deal with
these two cases in Sections 9 and 10 respectively. Putting all together, we
prove Theorem A in Section 11. Finally, Section 12 is devoted to the proof of
Corollary B and Theorem C.

##  2\. Patterns and canonical models

In this section we formalize the definitions outlined in the Introduction. We
also recall how to compute the topological entropy of a pattern by using
purely combinatorial tools. Finally we define the pattern that will be proved
to have minimum positive entropy.

A _tree_ is a compact uniquely arcwise connected space which is a point or a
union of a finite number of intervals (by an _interval_ we mean any space
homeomorphic to $[0,1]$). Any continuous map $f\colon T\longrightarrow T$ from
a tree $T$ into itself will be called a _tree map_. A set $X\subset T$ is said
to be _$f$ -invariant_ if $f(X)\subset X$. For each $x\in T$, we define the
_valence_ of $x$ to be the number of connected components of
$T\setminus\\{x\\}$. A point of valence different from 2 will be called a
_vertex_ of $T$ and the set of vertices of $T$ will be denoted by $V(T)$. Each
point of valence 1 will be called an _endpoint_ of $T$. The set of such points
will be denoted by $\operatorname{En}(T)$. Also, the closure of a connected
component of $T\setminus V(T)$ will be called an _edge of $T$_.

Given any subset $X$ of a topological space, we will denote by
$\operatorname{Int}(X)$ and $\operatorname{Cl}(X)$ the interior and the
closure of $X$, respectively. For a finite set $P$ we will denote its
cardinality by $|P|$.

A triplet $(T,P,f)$ will be called a _model_ if $f\colon T\longrightarrow T$
is a tree map and $P$ is a finite $f$-invariant set such that
$\operatorname{En}(T)\subset P$. In particular, if $P$ is a periodic orbit of
$f$ and $|P|=n$ then $(T,P,f)$ will be called an _$n$ -periodic model_. Given
$X\subset T$ we will define the _connected hull_ of $X$, denoted by $\langle
X\rangle_{T}$ or simply by $\langle X\rangle$, as the smallest closed
connected subset of $T$ containing $X$. When $X=\\{x,y\\}$ we will write
$[x,y]$ to denote $\langle X\rangle$. The notations $(x,y)$, $(x,y]$ and
$[x,y)$ will be understood in the natural way.

An $n$-periodic orbit $P=\\{x_{i}\\}_{i=0}^{n-1}$ of a map $\theta$ will be
said to be _time labeled_ if $\theta(x_{i})=x_{i+1}$ for $0\leq i<n-1$ and
$\theta(x_{n-1})=x_{0}$.

Let $T$ be a tree and let $P\subset T$ be a finite subset of $T$. The pair
$(T,P)$ will be called a _pointed tree_. Two points $x,y$ of $P$ will be said
to be _consecutive_ if $(x,y)\cap P=\emptyset$. Any maximal subset of $P$
consisting only of pairwise consecutive points will be called a _discrete
component_ of $(T,P)$. We say that two pointed trees $(T,P)$ and
$(T^{\prime},P^{\prime})$ are _equivalent_ if there exists a bijection
$\phi\colon P\longrightarrow P^{\prime}$ which preserves discrete components.
The equivalence class of a pointed tree $(T,P)$ will be denoted by $[T,P]$.

Let $(T,P)$ and $(T^{\prime},P^{\prime})$ be equivalent pointed trees, and let
$\theta\colon P\longrightarrow P$ and $\theta^{\prime}\colon
P^{\prime}\longrightarrow P^{\prime}$ be maps. We will say that $\theta$ and
$\theta^{\prime}$ are _equivalent_ if
$\theta^{\prime}=\phi\circ\theta\circ\phi^{-1}$ for a bijection $\phi\colon
P\longrightarrow P^{\prime}$ which preserves discrete components. The
equivalence class of $\theta$ by this relation will be denoted by $[\theta]$.
If $[T,P]$ is an equivalence class of pointed trees and $[\theta]$ is an
equivalence class of maps then the pair $([T,P],[\theta])$ will be called a
_pattern_. We will say that a model $(T,P,f)$ _exhibits_ a pattern
$(\mathcal{T},\Theta)$ if $\mathcal{T}=[\langle P\rangle_{T},P]$ and
$\Theta=[f\bigr{\rvert}_{{}_{P}}]$.

Despite the fact that the notion of a discrete component is defined for
pointed trees, by abuse of language we will use the expression _discrete
component of a pattern_ , which will be understood in the natural way since
the number of discrete components and their relative positions are the same
for all models of the pattern.

Recall that the topological entropy of a continuous tree map $f$ is denoted by
$h(f)$. Given a pattern $\mathcal{P}$, the topological entropy of
$\mathcal{P}$ is defined to be

$h(\mathcal{P}):=\inf\\{h(f)\,\colon(T,P,f)\ \text{is a model exhibiting}\
\mathcal{P}\\}.$

The simplest models exhibiting a given pattern are the monotone ones, defined
as follows. Let $f\colon T\longrightarrow T$ be a tree map map. Given $a,b\in
T$ we say that $f\bigr{\rvert}_{[a,b]}$ is _monotone_ if $f([a,b])$ is either
an interval or a point and $f\bigr{\rvert}_{[a,b]}$ is monotone as an interval
map. Let $(T,P,f)$ be a model. A pair $\\{a,b\\}\subset P$ will be called a
_basic path of $(T,P)$_ if it is contained in a single discrete component of
$(T,P)$. We will say that $f$ is _$P$ -monotone_ if $f\bigr{\rvert}_{[a,b]}$
is monotone for any basic path $\\{a,b\\}$. The model $(T,P,f)$ will then be
said to be _monotone_. In such case, Proposition 4.2 of [6] states that the
set $P\cup V(T)$ is $f$-invariant (recall that $V(T)$ stands for the set of
vertices of $T$). Hence, the map $f$ is also $(P\cup V(T))$-monotone. Observe
that the notion of $P$-monotonicity is much more restrictive than the usual
topological notion of a _monotone map_ (full preimages of continua are
continua).

Theorem A of [6] states that every pattern $\mathcal{P}$ has monotone models,
and that for every monotone model $(T,P,f)$ of $\mathcal{P}$,
$h(f)=h(\mathcal{P})$. Moreover, there exists a special class of monotone
models, satisfying several extra properties that we omit here, called
_canonical models_. Theorem B of [6] states that every pattern has a canonical
model. Moreover, given two canonical models $(T,P,f)$ and
$(T^{\prime},P^{\prime},f^{\prime})$ of the same pattern there exists a
homeomorphism $\phi\colon T\longrightarrow T^{\prime}$ such that
$\phi(P)=P^{\prime}$ and $f^{\prime}\circ\phi\bigr{\rvert}_{P}=\phi\circ
f\bigr{\rvert}_{P}$. Hence, the canonical model of a pattern is essentially
unique. Summarizing, we have the following result.

######  Theorem 2.1.

Let $\mathcal{P}$ be a pattern. Then the following statements hold.

  1. (a)

There exists a canonical model of $\mathcal{P}$.

  2. (b)

The canonical model $(T,P,f)$ of $\mathcal{P}$ satisfies
$h(f)=h(\mathcal{P})$.

It is worth noticing that the proof of Theorem 2.1 gives a finite algorithm to
construct the canonical model of any pattern. For instance, the model
$(T,P,f)$ in the right picture of Figure 1 is the canonical model of the
corresponding pattern. The $P$-monotonicity of $f$ determines that $f(a)=b,$
$f(b)=c,$ and $f(c)=c.$ Observe also that the left model
$(T^{\prime},P^{\prime},f^{\prime})$ of Figure 1, a representative of the same
pattern, cannot be $P^{\prime}$-monotone, since in this case we would have
$f^{\prime}(v)\in f^{\prime}([x^{\prime}_{2},x^{\prime}_{6}])\cap
f^{\prime}([x^{\prime}_{4},x^{\prime}_{5}])=[x^{\prime}_{3},x^{\prime}_{1}]\cap[x^{\prime}_{5},x^{\prime}_{6}]=\emptyset.$

There is a combinatorial procedure to compute the entropy of a pattern
$\mathcal{P}$ which does not require the construction of its canonical model.
Indeed, $h(\mathcal{P})$ can be obtained from the transition matrix of a
combinatorial directed graph that can be derived independently of the images
of the vertices in any particular monotone model of the pattern. Let us recall
this procedure.

A _combinatorial directed graph_ is a pair $\mathcal{G}=(V,U)$ where
$V=\\{v_{1},v_{2},\dots,v_{k}\\}$ is a finite set and $U\subset V\times V$.
The elements of $V$ are called the _vertices_ of $\mathcal{G}$ and each
element $(v_{i},v_{j})$ in $U$ is called an _arrow_ (from $v_{i}$ to $v_{j}$)
in $\mathcal{G}$. Such an arrow is usually denoted by $v_{i}\rightarrow
v_{j}$. The notions of _path_ and _loop_ in $\mathcal{G}$ are defined as
usual. The _length_ of a path is defined as the number of arrows in the path.
The _transition matrix_ of $\mathcal{G}$ is a $k\times k$ binary matrix
$(m_{ij})_{i,j=1}^{k}$ such that $m_{ij}=1$ if and only if there is an arrow
from $v_{i}$ to $v_{j}$, and $m_{ij}=0$ otherwise.

Let $\\{\pi_{1},\pi_{2},\ldots,\pi_{k}\\}$ be the set of basic paths of the
pointed tree $(T,P)$. We will say that $\pi_{i}$ _$f$ -covers_ $\pi_{j}$,
denoted by $\pi_{i}\rightarrow\pi_{j}$, whenever $\pi_{j}\subset\langle
f(\pi_{i})\rangle_{T}$. The _$\mathcal{P}$ -path graph_ is the combinatorial
directed graph whose vertices are in one-to-one correspondence with the basic
paths of $(T,P)$, and there is an arrow from the vertex $i$ to the vertex $j$
if and only if $\pi_{i}$ $f$-covers $\pi_{j}$. The associated transition
matrix, denoted by $M_{\mathcal{P}}$, will be called the _path transition
matrix of $\mathcal{P}$_. It can be seen that the definitions of the
$\mathcal{P}$-path graph and the matrix $M_{\mathcal{P}}$ are independent of
the particular choice of the model $(T,P,f)$. Thus, they are well-defined
pattern invariants.

For any square matrix $M$, we will denote its _spectral radius_ by $\rho(M)$.
We recall that it is defined as the maximum of the moduli of the eigenvalues
of $M$.

######  Remark 2.2.

Let $M_{\mathcal{P}}$ be the path transition matrix of a pattern
$\mathcal{P}$. Then (see [6]), the topological entropy of $\mathcal{P}$ can be
computed as $h(\mathcal{P})=\log\max\\{\rho(M_{\mathcal{P}}),1\\}$.

To end this section we define the patterns that will be showed to have minimum
positive entropy. Let $n\in\mathbb{N}$ with $n\geq 3$. Let $\mathcal{Q}_{n}$
be the $n$-periodic pattern $([T,P],[\theta])$ such that
$P=\\{x_{0},x_{1},\ldots,x_{n-1}\\}$ is time labeled and $(T,P)$ has two
discrete components, $\\{x_{n-1},x_{0}\\}$ and
$\\{x_{0},x_{1},\ldots,x_{n-2}\\}$. In Figure 2 we show the canonical model of
$\mathcal{Q}_{n}$. Observe that $\mathcal{Q}_{3}$ is nothing but the
3-periodic Štefan cycle of the interval [32]. In [7] the authors prove that
$h(\mathcal{Q}_{n})=\log(\lambda_{n})$, where $\lambda_{n}$ is the unique real
root of the polynomial $x^{n}-2x-1$ in $(1,+\infty)$. We will use the
following properties of the numbers $\lambda_{n}$. Statement (a) is proved in
Proposition 3.1 of [7], while statement (b) is an easy exercise.

######  Proposition 2.3.

Let $n$ be any positive integer with $n\geq 3$. Then:

  1. (a)

$\lambda_{n+1}<\lambda_{n}$

  2. (b)

$\sqrt[n]{4}>\lambda_{n}$.

##  3\. Block structures, skeletons and $\pi$-reducibility

The zero entropy tree patterns will play a central role in this paper. The
characterization of such patterns was first given in [6], and another
description was proven to be equivalent in [5]. We will use this second
approach, and this section is devoted to recall the necessary notions and
results. The characterization of zero entropy periodic patterns relies on the
notion of _block structure_ , that is classic in the field of Combinatorial
Dynamics. In the literature one can find several kinds of block structures and
related notions for periodic orbits. In the interval case, the Sharkovskii’s
_square root construction_ [31] is an early example of a block structure. The
notion of _extension_ , first appeared in [17], gives rise to some particular
cases of block structures. Also the notion of _division_ , introduced in [25]
for interval periodic orbits and generalized in [11] in order to study the
entropy and the set of periods for tree maps, is a particular case of block
structure.

######  Remark 3.1.

All patterns considered in this paper will be periodic. Given an $n$-periodic
pattern $\mathcal{P}$, by abuse of language we will speak about the _points_
of $\mathcal{P}$, and by default we will consider that such points are time
labeled with the integers $\\{0,1,\ldots,n-1\\}$. Often we will identify a
point in $\mathcal{P}$ with its time label. In agreement with such
conventions, the points of the patterns shown in the pictures will be simply
integers in the range $[0,n-1]$. See for instance Figure 3.

A pattern will be said to be _trivial_ if it has only one discrete component.
It is easy to see that the entropy of any trivial pattern is zero.

Let $\mathcal{P}=([T,P],[f])$ be a nontrivial $n$-periodic pattern with $n\geq
3$. For $n>p\geq 2$, we will say that $\mathcal{P}$ _has a $p$-block
structure_ if there exists a partition $P=P_{0}\cup P_{1}\cup\ldots\cup
P_{p-1}$ such that $f(P_{i})=P_{i+1\bmod p}$ for $i\geq 0$, and $\langle
P_{i}\rangle_{T}\cap P_{j}=\emptyset$ for $i\neq j$. In this case, $p$ is a
strict divisor of $n$ and $|P_{i}|=n/p$ for $0\leq i<p$. The sets $P_{i}$ will
be called _blocks_ , and the blocks will be said to be _trivial_ if each
$P_{i}$ is contained in a single discrete component of $\mathcal{P}$
(equivalently, each pattern $([\langle P_{i}\rangle_{T},P_{i}],[f^{p}])$ is
trivial). Note that $\mathcal{P}$ can have several block structures, but only
one $p$-block structure for any given divisor $p$ of $n$. If $\mathcal{P}$ has
structures of trivial blocks, the one with blocks with maximum cardinality
will be called a _maximal structure_.

From the equivalence relation which defines the class of models belonging to
the pattern $\mathcal{P}$ it easily follows that the notions defined in the
previous paragraph do not depend on the particular model $(T,P,f)$
representing $\mathcal{P}$.

######  Remark 3.2 (Standing convention).

Let $\mathcal{P}$ be an $n$-periodic pattern whose points are time labeled as
$\\{0,1,\ldots,n-1\\}$. When $\mathcal{P}$ has a block structure of $p$ blocks
$P_{0}\cup P_{1}\cup\ldots\cup P_{p-1}$, by convention we will always assume
that the time labels of the blocks have been chosen in such a way that $0\in
P_{0}$.

Let $(T,P,f)$ be the canonical model of $\mathcal{P}$. A $p$-block structure
$P_{0}\cup P_{1}\cup\ldots\cup P_{p-1}$ for $\mathcal{P}$ will be said to be
_separated_ if $\langle P_{i}\rangle_{T}\cap\langle
P_{j}\rangle_{T}=\emptyset$ for $i\neq j$. Note that the separability of a
block structure for a pattern depends on the particular topology of its
canonical model and, in consequence, cannot be determined directly from the
combinatorial data of $\mathcal{P}$ a priori. However, recall that the
canonical model of a pattern $\mathcal{P}$ is unique and can be
algorithmically computed from $\mathcal{P}$. So, this is an intrinsic notion.

In Figure 3 we show an example of a 8-periodic pattern $\mathcal{P}$ admitting
both a 4-block structure given by $P_{0}=\\{0,4\\}$, $P_{1}=\\{1,5\\}$,
$P_{2}=\\{2,6\\}$, $P_{3}=\\{3,7\\}$ and a 2-structure given by
$Q_{0}=\\{0,2,4,6\\}$, $Q_{1}=\\{1,3,5,7\\}$. Note that in both cases the
blocks are trivial, and $Q_{0}\cup Q_{1}$ is a maximal structure by
definition. As it has been said, one can determine these block structures
directly in the combinatorial representation of $\mathcal{P}$, without
checking any particular topology. See Figure 3 (left). On the contrary, to
determine the separability of a block structure one has to construct the
canonical model of $\mathcal{P}$, which is shown in the same figure (right).
Here we see that $Q_{0}\cup Q_{1}$ is separated, while $P_{0}\cup P_{1}\cup
P_{2}\cup P_{3}$ is not (the convex hulls of the blocks $P_{0}$ and $P_{2}$,
which are respectively the intervals $[0,4]$ and $[2,6]$, intersect at the
vertex $a$).

Figure 3. Left: an 8-periodic pattern $\mathcal{P}$ admitting two block
structures with trivial blocks. Right: the canonical model $(T,P,f)$ of
$\mathcal{P}$, for which the images of the vertices are $f(a)=c$, $f(b)=0$ and
$f(c)=a$.

Let $\mathcal{P}$ be an $n$-periodic pattern and let $(T,P,f)$ be the
canonical model of $\mathcal{P}$. Let $P=P_{0}\cup P_{1}\cup\ldots\cup
P_{p-1}$ be a separated $p$-block structure for $\mathcal{P}$. Then,
$f(\langle P_{i}\rangle)=\langle P_{i+1\bmod p}\rangle$. The _skeleton of
$\mathcal{P}$_ (associated to this block structure) is a $p$-periodic pattern
$\mathcal{S}$ defined as follows. Consider the tree $S$ obtained from $T$ by
collapsing each tree $\langle P_{i}\rangle$ to a point $x_{i}$. Let
$\kappa\colon T\longrightarrow S$ be the standard projection, which is
bijective on $T\setminus\cup_{i}\langle P_{i}\rangle$ and satisfies
$\kappa(\langle P_{i}\rangle)=x_{i}$. Set
$Q=\kappa(P)=\\{x_{0},x_{1},\ldots,x_{p-1}\\}$ and define $\theta\colon
Q\longrightarrow Q$ by $\theta(x_{i})=x_{i+1\bmod p}$. Then the _skeleton_
$\mathcal{S}$ of $\mathcal{P}$ is defined to be the $p$-periodic pattern
$([S,Q],[\theta])$.

######  Remark 3.3 (Standing convention).

Let $\mathcal{P}$ be an $n$-periodic pattern whose points are time labeled as
$\\{0,1,\ldots,n-1\\}$. Assume that $\mathcal{P}$ has a separated $p$-block
structure. From the convention established in Remark 3.2, each point of
$\mathcal{P}$ labeled as $i$ belongs to the block $P_{i\bmod{p}}$. From now on
we adopt the convention that the $p$ points of the skeleton have time labels
$\\{0,1,\ldots,p-1\\}$ such that the point $i$ of the skeleton corresponds to
the collapse of the block $P_{i}$.

######  Example 3.4.

Let us see an example of construction of the skeleton. Consider the 8-periodic
pattern $\mathcal{P}$ consisting of two discrete components $\\{0,2,6\\}$,
$\\{0,1,3,4,5,7\\}$ (Figure 4, left). Then, $P_{0}=\\{0,4\\}$,
$P_{1}=\\{1,5\\}$, $P_{2}=\\{2,6\\}$, $P_{3}=\\{3,7\\}$ defines a structure of
4 trivial blocks. By checking the canonical model $(T,P,f)$, which is shown in
Figure 4 (center), we see that $\langle P_{i}\rangle_{T}\cap\langle
P_{j}\rangle_{T}=\emptyset$ when $i\neq j$. Thus, the structure is separated.
The corresponding skeleton is obtained by collapsing the convex hull of each
block to a point, giving the 4-periodic pattern $\mathcal{S}$ shown in Figure
4 (right).

Figure 4. Left: an 8-periodic pattern $\mathcal{P}$ with a separated structure
of 4 trivial blocks. Center: the canonical model $(T,P,f)$ of $\mathcal{P}$,
the convex hulls of the blocks marked with thick lines. Right: the
corresponding skeleton.

The entropies of a pattern $\mathcal{P}$ with a separated structure of trivial
blocks and its associated skeleton coincide, as the following result (a
reformulation of Proposition 8.1 of [6]) states.

######  Proposition 3.5.

Let $\mathcal{P}$ be a pattern with a separated structure of trivial blocks.
Let $\mathcal{S}$ be the corresponding skeleton. Then,
$h(\mathcal{S})=h(\mathcal{P})$.

Going back to Example 3.4, note that the obtained skeleton $\mathcal{S}$ is a
zero entropy interval pattern. Then, $h(\mathcal{P})=0$ by Proposition 3.5.

As a consequence of Proposition 3.5 we have the following result, that will be
used in the proof of the main theorem of this paper.

######  Corollary 3.6.

Let $\mathcal{P}$ an $n$-periodic pattern with a separated structure of $p$
trivial blocks. Let $\mathcal{S}$ be the corresponding skeleton. If
$h(\mathcal{S})\geq\log(\lambda_{p})$, then
$h(\mathcal{P})>\log(\lambda_{n})$.

###### Proof.

Since $p$ is a strict divisor of $n$, it is a direct consequence of
Propositions 3.5 and 2.3(a). ∎

The existence of a separated structure of trivial blocks for a pattern
$\mathcal{P}$ has a strong connection with the path transition matrix of
$\mathcal{P}$, via the iterative behaviour of some particular basic paths of
$\mathcal{P}$. Let us explain it. Let $\mathcal{P}$ be a periodic pattern and
let $\pi$ be a basic path of $\mathcal{P}$. Consider any model $(T,P,f)$ of
$\mathcal{P}$. For $k\geq 1$, we will say that $\pi$ _splits in $k$ iterates_
if $f^{i}(\pi)$ is a basic path of $\mathcal{P}$ for $0\leq i<k$ and
$f^{k}(\pi)$ is not a basic path of $\mathcal{P}$. Equivalently, $f^{i}(\pi)$
only $f$-covers $f^{i+1}(\pi)$ for $0\leq i<k$ and $f^{k-1}(\pi)$ $f$-covers
at least two different basic paths. We say that a basic path $\pi$ _never
splits_ if $f^{i}(\pi)$ is a basic path for every $i\geq 0$. In this case, we
will say that $\mathcal{P}$ is _$\pi$ -reducible_. As an example, the path
$\pi=\\{0,4\\}$ for the pattern $\mathcal{P}$ in Figure 4 never splits, so
$\mathcal{P}$ is $\pi$-reducible. On the other hand, let $\sigma$ be the path
$\\{4,7\\}$ on the same pattern. Note that $f(\sigma)=\\{5,0\\}$ is a basic
path, while $f^{2}(\sigma)=\\{6,1\\}$ is not. Then, $\sigma$ splits in 2
iterates and $f^{2}$-covers the two basic paths $\\{6,0\\}$ and $\\{0,1\\}$.

The $\pi$-reducibility of a pattern with respect to a basic path $\pi$ is
equivalent to the existence of a separated structure of trivial blocks, as the
following result states.

######  Proposition 3.7.

Let $\mathcal{P}$ be a periodic pattern. Then, $\mathcal{P}$ is
$\pi$-reducible for a basic path $\pi$ if and only if $\mathcal{P}$ has a
maximal and separated structure of trivial blocks. In this case, $\mathcal{P}$
is $\sigma$-reducible for any basic path $\sigma$ contained in a block.

###### Proof.

The ‘only if’ part of the first statement is Proposition 9.5 of [7], while its
‘if’ part and the second claim easily follow from the definition of a trivial
block structure. ∎

##  4\. A mechanism to compare entropies

Another key ingredient to prove Theorem A is a tool, first introduced in [7],
that allows us to compare the entropies of two patterns $\mathcal{P}$ and
$\mathcal{O}$ when $\mathcal{O}$ has been obtained by joining together several
discrete components of $\mathcal{P}$. For the sake of brevity, here we will
give a somewhat informal (though completely clear) version of this procedure.

Let $(T,P,f)$ be a model of a pattern $\mathcal{P}$. We recall that two
discrete components of $(T,P)$ are either disjoint or intersect at a single
point of $P$. Two discrete components $A,B$ of $(T,P)$ will be said to be
_adjacent at $x\in P$_ (or simply _adjacent_) if $A\cap B=\\{x\\}$. A point
$z\in P$ will be said to be _inner_ if $z$ belongs to $k\geq 2$ discrete
components of $(T,P)$, all being pairwise adjacent at $z$.

Now let $x\in P$ be an inner point and let $A,B$ be two discrete components
adjacent at $x$. If we join together $A$ and $B$ to get a new discrete
component $A\cup B$ and keep intact the remaining components, we get a new
pattern $\mathcal{O}$. We will say that $\mathcal{O}$ is an _opening of
$\mathcal{P}$_ (with respect to the inner point $x$ and the discrete
components $A$ and $B$). As an example, see Figure 5, where $\mathcal{O}$ is
an opening of $\mathcal{P}$ with respect to the inner point 5 and the discrete
components $A=\\{2,5,6\\}$ and $B=\\{0,5\\}$, while $\mathcal{R}$ is an
opening of $\mathcal{P}$ with respect to the inner point 5 and the discrete
components $B$ and $C=\\{1,3,5\\}$.

Figure 5. Two different openings of $\mathcal{P}$.

######  Remark 4.1 (Standing convention).

As it is clear from the examples shown in Figure 5, we are implicitly assuming
that the labeling of the points of an $n$-periodic pattern $\mathcal{P}$ fixes
the labeling of the points of any opening of $\mathcal{P}$.

As one may expect from intuition, the entropy of a model decreases when
performing an opening, as the following result (Theorem 5.3 of [7]) states.

######  Theorem 4.2.

Let $\mathcal{P}$ and $\mathcal{O}$ be $n$-periodic patterns. If $\mathcal{O}$
is an opening of $\mathcal{P}$, then $h(\mathcal{P})\geq h(\mathcal{O})$.

We finish this section stating that the property for a pattern of having a
block structure is preserved by openings. The result is a direct consequence
of the definition of a block structure and the fact that no new inner points
are created after performing an opening.

######  Lemma 4.3.

Let $\mathcal{P}$ be a periodic pattern with a block structure and let
$\mathcal{O}$ be an opening of $\mathcal{P}$. Then, $\mathcal{O}$ has a block
structure.

##  5\. Strategy of the proof of Theorem A

In this section we give a general overview of the proof of Theorem A, in order
to justify the need for the several techniques and results deployed in the
subsequent sections.

We will prove Theorem A by induction on the period $n$. So, assume that we
have an $n$-periodic pattern $\mathcal{P}$ and that the result is true for
every pattern with period less than $n$.

The first step is a simplification process based on the opening mechanism.
Recall (Theorem 4.2) that after performing an opening on $\mathcal{P}$, the
entropy $h$ of the obtained pattern is less or equal to $h(\mathcal{P})$. If
$h$ is still positive, we can perform again an opening and so on, until we get
a pattern with positive entropy such that every new opening leads to entropy
zero. In other words, we can assume that $\mathcal{P}$ satisfies the following
property:

($\star$) $\mbox{Every opening of $\mathcal{P}$ is a zero entropy pattern}.$

Property ($\star$ ‣ 5) is very restrictive and has a strong consequence: a
pattern satisfying ($\star$ ‣ 5) is, _generically_ , $\pi$-reducible. More
precisely, we have the following result, that will be proved in Section 8.

######  Theorem D.

Let $\mathcal{P}$ be an $n$-periodic pattern with positive entropy such that
any opening of $\mathcal{P}$ has entropy zero. Assume that $\mathcal{P}$ has
at least two inner points and at least three openings. Then, $\mathcal{P}$ is
$\pi$-reducible for some basic path $\pi$.

If $\mathcal{P}$ satisfies the hypothesis of Theorem D, then it is
$\pi$-reducible. So, we can consider its skeleton $\mathcal{S}$, with the same
entropy but with a period that strictly divides $n$, and use the induction
hypothesis.

Figure 6. A $k$-flower (left) and a triple chain (right).

The above argument is the core idea of the proof of Theorem A, but we are left
with two special cases for which we cannot assure that property ($\star$ ‣ 5)
implies $\pi$-reducibility: the _$k$ -flowers_ and the _triple chain_. A _$k$
-flower_ is a pattern consisting on $k\geq 2$ discrete components (the
_petals_) attached at a unique inner point. A pattern having three discrete
components and two inner points will be called a _triple chain_. See Figure 6.
The reader will find easy to convince that the flowers and the triple chain
are the two sort of patterns that do not satisfy the property of having at
least two inner points and at least three openings.

The cases of the $k$-flowers and the triple chain will be tackled in Sections
9 and 10 respectively. Concerning the $k$-flowers, the case $k=2$ is specially
simple since Theorem A follows directly from a previous result in [8]. On the
other hand, for $k\geq 3$ we construct an $n^{\prime}$-periodic pattern, where
$n^{\prime}$ is a strict divisor of $n$, whose entropy can be put in relation
with that of $\mathcal{P}$, and then we use the induction hypothesis. Finally,
in the case of the triple chain we compute directly lower bounds of the
entropy by counting coverings in the $\mathcal{P}$-path graph (equivalently,
entries in the path transition matrix).

##  6\. Structure of zero entropy patterns

Although a point is an element of a topological space and a pattern is a
combinatorial object defined as an equivalence class of pointed trees, recall
that by abuse of language we talk about the _points_ of a pattern. The same
translation from topology to combinatorics can be applied to the terms
_valence_ , _inner point_ and _endpoint_. The (combinatorial) _valence_ of a
point $x$ of a pattern $\mathcal{P}$ is defined as the number of discrete
components of $\mathcal{P}$ containing $x$. Recall that an _inner point of
$\mathcal{P}$_ has been defined as a point of combinatorial valence larger
than 1. Otherwise, the point will be called an _endpoint of $\mathcal{P}$_.
Let $x$ be a point of $\mathcal{P}$ of combinatorial valence $\nu$. Obviously,
for any model $(T,P,f)$ of $\mathcal{P}$, the (topological) valence of the
point of $T$ corresponding to $x$ is the same and equals $\nu$. In
consequence, $x$ is an endpoint (respectively, an inner point) of
$\mathcal{P}$ if and only if the point corresponding to $x$ in any model
$(T,P,f)$ is an endpoint (respectively, a point of valence larger than 1) of
the tree $T$. So, in what follows we will drop the words _combinatorial_ and
_topological_ and will use these terms indistinctly in both senses.

The strategy outlined in Section 5 relies strongly in using property ($\star$
‣ 5), that depends on the notion of _zero entropy pattern_. So, we start this
section with the following recursive characterization of zero entropy
patterns, that uses the notions of block structure and skeleton presented in
Section 3. It is Proposition 5.6 of [5].

######  Proposition 6.1.

Let $\mathcal{P}$ be an $n$-periodic pattern. Then, $h(\mathcal{P})=0$ if and
only if either $\mathcal{P}$ is trivial or has a maximal separated structure
of trivial blocks such that the associated skeleton has entropy $0$.

Figure 7. Top: a sequence of skeletons. Bottom: the sequence of combinatorial
collapses according to Definition 6.4.

Obviously we can use Proposition 6.1 recursively, in the sense that the
skeleton $\mathcal{S}$, with entropy zero and a period that strictly divides
that of $\mathcal{P}$, has also a maximal separated structure of trivial
blocks with an associated skeleton $\mathcal{S}^{\prime}$ of entropy zero. We
can thus iterate the process as many times as necessary to finally obtain a
trivial pattern. Consider, for instance, the zero entropy pattern
$\mathcal{P}$ of Example 3.4, whose skeleton $\mathcal{S}$ was shown in Figure
4. This skeleton has a maximal separated structure of 2 trivial blocks, with
the associated skeleton $\mathcal{S}^{\prime}$ being a trivial pattern of 2
points. See the complete sequence of skeletons in Figure 7 (top). Note that
the previous simplification process cannot be carried out without checking the
particular topology of the involved canonical models. Indeed, if we ignore the
topology of the tree $T$ in the canonical model $(T,P,f)$ of $\mathcal{P}$
(that is shown in Figure 4), for the skeleton it is not possible to decide,
only from the combinatorics of $\mathcal{P}$, between the patterns
$\mathcal{S}$ and $\mathcal{C}$ depicted in Figure 7. To overcome this
dependence from the topology, next we propose a similar but purely
combinatorial simplification mechanism over zero entropy patterns.

######  Definition 6.2.

Let $\mathcal{P}=([T,P],[f])$ be a zero entropy $n$-periodic pattern. Let
$P_{0}\cup P_{1}\cup\ldots\cup P_{p-1}$ be the maximal and separated structure
of trivial blocks given by Proposition 6.1. A $p$-periodic pattern
$\mathcal{C}=([S,Q],[g])$ will be called the _combinatorial collapse of
$\mathcal{P}$_ if the following properties are satisfied:

  1. (a)

$g(i)=j$ if and only if $f(P_{i})=P_{j}$

  2. (b)

For any $0\leq i<j\leq p-1$, there is a discrete component of $\mathcal{P}$
intersecting the blocks $P_{i},P_{j}$ if and only if there is a discrete
component of $\mathcal{C}$ containing the points $i,j$.

We will say that the point $i$ of $\mathcal{C}$ is the _collapse_ of the block
$P_{i}$ of $\mathcal{P}$. Property (a) above implies that the standing
convention established in Remark 3.3 about the labeling of the points of a
skeleton translates verbatim to the labeling of the points of a combinatorial
collapse.

Note that, by definition, the combinatorial collapse is unique, since it is
always carried out over the maximal structure of trivial blocks.

As an example, the pattern $\mathcal{C}$ shown in Figure 7 (bottom) is the
combinatorial collapse of $\mathcal{P}$. Note that the skeleton $\mathcal{S}$
does not satisfy property (b) of Definition 6.2: the blocks $P_{0}=\\{0,4\\}$
and $P_{1}=\\{1,5\\}$ intersect a single discrete component in $\mathcal{P}$,
while the corresponding points $0,1$ of $\mathcal{S}$ are contained in
different discrete components.

Notice that, if $\mathcal{P}$ is a zero entropy pattern, then the
combinatorial collapse $\mathcal{C}$ of $\mathcal{P}$ can be obtained from the
skeleton $\mathcal{S}$ of $\mathcal{P}$ simply by performing openings. Then,
Theorem 4.2 assures us that $h(\mathcal{C})=h(\mathcal{S})=0$. Therefore, we
get the following translation of Proposition 6.1 to the context of
combinatorial collapses.

Figure 8. An example of a zero entropy 18-periodic pattern $\mathcal{P}_{2}$
and the corresponding sequence of collapses.

######  Proposition 6.3.

Let $\mathcal{P}$ be a nontrivial periodic pattern with entropy zero. Then,
the combinatorial collapse of $\mathcal{P}$ has entropy zero.

######  Definition 6.4.

As an immediate consequence of Proposition 6.3, a zero entropy $n$-periodic
pattern $\mathcal{P}$ has associated a sequence of patterns
$\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ and a sequence of integers
$\\{p_{i}\\}_{i=0}^{r}$ for some $r\geq 0$ such that:

  1. (a)

$\mathcal{P}_{r}=\mathcal{P}$

  2. (b)

$\mathcal{P}_{0}$ is a trivial $p_{0}$-periodic pattern

  3. (c)

For $1\leq i\leq r$, $\mathcal{P}_{i}$ has a maximal separated structure of
$\prod_{j=0}^{i-1}p_{j}$ trivial blocks of cardinality $p_{i}$ and
$\mathcal{P}_{i-1}$ is the corresponding combinatorial collapse.

The sequence $\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ will be called _the sequence of
collapses of_ $\mathcal{P}$. Notice that $\prod_{j=0}^{r}p_{j}=n$. See Figure
8 for an example with $p_{0}=3$, $p_{1}=2$, $p_{2}=3$.

######  Remark 6.5.

Let $\mathcal{P}$ be a zero entropy $n$-periodic pattern and let
$\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ be the corresponding sequence of collapses.
Consider any particular time labeling $\\{0,1,\ldots,n-1\\}$ of the points of
$\mathcal{P}$. By Remark 3.3, this choice fixes the time labels of all points
in all patterns of the sequence of collapses. Note also that, for any $0\leq
i<r$, the integers labeling the points of $\mathcal{P}_{i}$ persist as labels
of points in $\mathcal{P}_{i+1}$. In particular, if $p_{0}$ is the period of
the trivial pattern $\mathcal{P}_{0}$, then $\\{0,1,\ldots,p_{0}-1\\}$ are the
only integers in the rank $\\{0,1,\ldots,n-1\\}$ that persist as labels of
points in any pattern of the sequence of collapses. See Figure 8 for an
example with $p_{0}=3$.

##  7\. Branching sequences

In this Section we dive deeper into the very particular combinatorial
structure of zero entropy patterns. The obtained results will be used in
Section 8 to prove Theorem D.

Let $\mathcal{P}$ be an $n$-periodic pattern and let $x$ be a point of
$\mathcal{P}$ of valence $\nu\geq 1$. Consider any model $(T,P,f)$ of
$\mathcal{P}$. Then, $T\setminus\\{x\\}$ has $\nu$ connected components
$K_{1},K_{2},\ldots,K_{\nu}$. We want to register how the forward iterates of
the point $x$ are distributed among the connected components of
$T\setminus\\{x\\}$. To this end, consider the integer time labeling of the
points of $\mathcal{P}$ such that $x=0$. Now, $\\{P\cap K_{i}\\}_{i=1}^{\nu}$
can be viewed as a partition of $\\{1,2,\ldots,n-1\\}$. The set $(P\cap
K_{i})\cup\\{0\\}$ of points of $\mathcal{P}$ will be called an _$x$ -branch_.
Note that this notion is independent of the chosen model $(T,P,f)$
representing $\mathcal{P}$. As an example, consider the 7-periodic pattern
$\mathcal{P}$ shown in Figure 5. Let $x$ be the point of valence 3 labeled as
5 in that figure. Shift all labels by $-5$ (mod 7). The discrete components of
$\mathcal{P}$ read now as $\\{0,3,5\\}$, $\\{3,6\\}$, $\\{0,2\\}$ and
$\\{0,1,4\\}$. The $x$-branches of $\mathcal{P}$ are then $\\{0,3,5,6\\}$,
$\\{0,2\\}$ and $\\{0,1,4\\}$.

######  Remark 7.1.

Let $\mathcal{P}$ be a periodic pattern and let $x$ be any point of
$\mathcal{P}$. Observe that any discrete component of $\mathcal{P}$ is
contained in a single $x$-branch. As a direct consequence of this fact, if in
addition $\mathcal{P}$ has entropy zero then any block of the maximal
structure of trivial blocks is contained in a single $x$-branch.

To understand the following result, it is crucial to keep in mind Remarks 3.3
and 6.5 concerning the labeling conventions of points and blocks in zero
entropy patterns. In particular, the labels of all points in the combinatorial
collapse of a pattern $\mathcal{P}$ persist as labels of points in
$\mathcal{P}$.

######  Lemma 7.2.

Let $\mathcal{P}$ be a zero entropy periodic pattern with a maximal separated
structure $P_{0}\cup P_{1}\cup\ldots\cup P_{p-1}$ of trivial blocks. Let
$\mathcal{C}$ be the combinatorial collapse of $\mathcal{P}$. If $0\leq
i,j,k<p$ are three points of $\mathcal{C}$ such that $\\{j,k\\}$ is contained
in a single $i$-branch of $\mathcal{C}$, then $P_{j}\cup P_{k}$ is contained
in a single $i$-branch of $\mathcal{P}$.

###### Proof.

Assume by way of contradiction that $P_{j}$ and $P_{k}$ are respectively
contained in two different $i$-branches of $\mathcal{P}$. Then, for some
$N\geq 2$ there exist $N+1$ different points of $\mathcal{P}$,
$x_{0},x_{1},\ldots,x_{N}$, such that:

  1. (a)

$x_{0}=j$ and $x_{N}=k$

  2. (b)

$x_{n}$ is inner for all $0<n<N$

  3. (c)

$\\{x_{n},x_{n+1}\\}$ is contained in a discrete component of $\mathcal{P}$
for $0\leq n<N$

  4. (d)

$x_{m}=i$ for some $1\leq m<N$

Intuitively, the above ordered sequence of points accounts for all points of
$\mathcal{P}$ successively met in the shortest path going from $j$ to $k$. The
assumption that $i$ separates $j$ from $k$ is imposed by property (d).

Consider now, for any point $x_{n}$ of the above sequence, the collapse of the
trivial block of $\mathcal{P}$ containing $x_{n}$. It is a point of
$\mathcal{C}$ that we denote by $y_{n}$. Note that $y_{0}=j$, $y_{m}=i$ and
$y_{N}=k$. Observe also that, for a pair of consecutive points
$x_{n},x_{n+1}$, it may happen that $\\{x_{n},x_{n+1}\\}$ is contained in a
block. In this case, since the blocks are trivial,
$\\{x_{n},x_{n+1},x_{n+2}\\}$ is not contained in a block. Therefore,
$y_{n}=y_{n+1}\neq y_{n+2}$. On the other hand, if $\\{x_{n},x_{n+1}\\}$ is
not contained in a block, by the definition of the combinatorial collapse,
$\\{y_{n},y_{n+1}\\}$ is a binary set contained in a discrete component of
$\mathcal{C}$. This observations lead to the existence of a sequence
$z_{0},z_{1},\ldots,z_{M}$ of $M+1\leq N+1$ points of $\mathcal{C}$ such that

  1. (a’)

$z_{0}=j$ and $z_{M}=k$

  2. (b’)

$z_{n}$ is inner for all $0<n<M$

  3. (c’)

$\\{z_{n},z_{n+1}\\}$ is contained in a discrete component of $\mathcal{C}$
for $0\leq n<M$

  4. (d’)

$x_{m^{\prime}}=i$ for some $1\leq m^{\prime}<M$

By property (d’), $j$ and $k$ belong to different $i$-branches in
$\mathcal{C}$, in contradiction with the hypothesis of the lemma. ∎

Let $\mathcal{P}$ be a zero entropy periodic pattern and let $\mathcal{C}$ be
its combinatorial collapse. Let us call $\\{P_{i}\\}$ and $\\{Q_{i}\\}$ the
blocks of the respective maximal structures of trivial blocks. Let $x$ be a
point of $\mathcal{P}$ and let $P_{i}$ be the block of $\mathcal{P}$
containing $x$. Let us call $y$ the point of $\mathcal{C}$ corresponding to
the collapse of $P_{i}$ and let $Q_{j}$ be the block of $\mathcal{C}$
containing $y$. By Remark 7.1, there exists a unique $x$-branch $Z$ containing
$P_{i}$. On the other hand, Remark 7.1 yields also that $Q_{j}$ is contained
in a single $y$-branch of $\mathcal{C}$. Recall now that the labels of the
points in $\mathcal{C}$ persist as labels of points in $\mathcal{P}$. So, we
can view $Q_{j}$ also as a subset of points of $\mathcal{P}$. Then, by Lemma
7.2, there exists a unique $x$-branch $Z^{\prime}$ containing $Q_{j}$. The
point $x$ will be called _bidirectional_ if $Z\neq Z^{\prime}$.

######  Lemma 7.3.

Any periodic pattern with entropy zero has bidirectional inner points.

###### Proof.

Let $\mathcal{P}=([T,P],[f])$ be a zero entropy pattern and let $\mathcal{C}$
be the combinatorial collapse of $\mathcal{P}$. Let $P_{0}\cup
P_{1}\cup\ldots\cup P_{p-1}$ and $Q_{0}\cup Q_{1}\cup\ldots Q_{q-1}$ be the
maximal separated block structures of $\mathcal{P}$ and $\mathcal{C}$
respectively.

Let $x$ be any inner point of $\mathcal{P}$. Assume that $x$ is not
bidirectional. In order to do not overload the notation, assume without loss
of generality that $x=0$. By the standing labeling conventions, $0\in P_{0}$
and the collapse of $P_{0}$ is the point of $\mathcal{C}$ labeled as 0, that
belongs to the block $Q_{0}=\\{0,q,2q,\ldots,(p/q-1)q\\}$. Since $P_{0}$ is a
trivial block, $P_{0}\subset C$ for a discrete component $C$ of $\mathcal{P}$.
Set

$X:=\bigcup_{1\leq k<p/q}P_{kq}.$

The set $X$ is the expansion of all points in $Q_{0}\setminus\\{0\\}$ to the
corresponding blocks in $\mathcal{P}$. Since we are assuming that 0 is not
bidirectional, Remark 7.1 and Lemma 7.2 imply that

(1) $P_{0}\cup X\mbox{ is contained in a single 0-branch $Z$ of }\mathcal{P}.$

We start by distinguishing two cases.

  * Case 1.

$X\cap C=\emptyset$.

We claim that in this case $C=P_{0}$. Indeed, $Q_{0}$ is contained in a
discrete component of $\mathcal{C}$. By definition of the combinatorial
collapse, all blocks $P_{iq}$ for $0\leq i<p/q$ must intersect a single
discrete component $D$ of $\mathcal{P}$. Since $X\cap C=\emptyset$, by (1)
this is only possible if $C=P_{0}$ (as claimed), $D$ is contained in the
0-brach $Z$ and $D$ is adjacent to $C$. See Figure 9 (center). Let
$x^{\prime}$ be the only point in $C\cap D=P_{0}\cap D$, whose collapse is the
point 0 in $\mathcal{C}$. Then, the $x^{\prime}$-branch containing $P_{0}$ and
the $x^{\prime}$-branch containing $Q_{0}$ are different. Therefore,
$x^{\prime}$ is bidirectional and we are done.

  * Case 2.

$X\cap C\neq\emptyset\mbox{ and }X\not\subset C$.

In this case, all blocks $P_{iq}$ intersect $C$ and at least one block, say
$P_{jq}$, has an inner point $x^{\prime}$ in common with $C$, whose collapse
is the point $jq$ in $\mathcal{C}$. See Figure 9 (right). Then, the
$x^{\prime}$-branch containing $P_{jq}$ and the $x^{\prime}$-branch containing
$Q_{0}$ are different. Therefore, $x^{\prime}$ is bidirectional and we are
done.

Figure 9. The two cases in the proof of Lemma 7.3. The arrows mark the two
different $x^{\prime}$-branches implying that $x^{\prime}$ is bidirectional.

Note that if $\mathcal{P}$ has no bidirectional inner points, then from above
we are not in the hypotheses of cases 1 and 2 and, in consequence, $X\subset
C$. Since $P_{0}\subset C$, we get that

$\tilde{P}_{0}:=P_{0}\cup X=\bigcup_{0\leq k<p/q}P_{kq}\subset D.$

Set $\tilde{P}_{i}:=\bigcup_{k=0}^{(p/q)-1}P_{i+kq}$ for $0\leq i<q$. From
above, if $\mathcal{P}$ has no bidirectional inner points then $\tilde{P}_{i}$
is contained in a single discrete component of $\mathcal{P}$. Moreover, since
$f(\tilde{P}_{i})=\tilde{P}_{i+1}$, it follows that
$\tilde{P}_{0}\cup\tilde{P}_{1}\cup\ldots\cup\tilde{P}_{q-1}$ is a trivial
block structure for $\mathcal{P}$, in contradiction with the maximality of the
structure $P_{0}\cup P_{1}\cup\ldots\cup P_{p-1}$. ∎

Let $x$ be a point of an $n$-periodic pattern $\mathcal{P}$ and let $\nu\geq
1$ be the valence of $x$. It is convenient to fix an indexing of the set of
$x$-branches. Next we define a natural indexing method that will be used by
default from now on. Recall that, arithmetically, an $x$-branch is nothing but
a subset of $\\{0,1,\ldots,n-1\\}$. Moreover, each $x$-branch contains 0 by
definition and the intersection of two different $x$-branches is $\\{0\\}$. We
will index the set of $x$-branches according to the minimum (positive) time
distance from $x$ to a point in the branch. More precisely, for any $x$-branch
$Z$, let $d_{Z}$ be the minimum positive integer in $Z$. From now on, we will
assume that the set $\\{Z_{i}\\}_{i=1}^{\nu}$ of $x$-branches is indexed in
such a way that $d_{Z_{i}}<d_{Z_{j}}$ if and only if $i<j$. As an example,
consider the 7-periodic pattern $\mathcal{P}$ shown in Figure 5. Let $x$ be
the point of valence 3 labeled as 5 in that figure. The $x$-branches of
$\mathcal{P}$ are then $X=\\{0,3,5,6\\}$, $Y=\\{0,2\\}$ and $W=\\{0,1,4\\}$,
with $d_{X}=3$, $d_{Y}=2$ and $d_{W}=1$. So, for this example we would denote
the set of $x$-branches as $\\{Z_{1},Z_{2},Z_{3}\\}$, with $Z_{1}=W$,
$Z_{2}=Y$ and $Z_{3}=X$.

Let $\mathcal{P}$ be an $n$-periodic pattern and let $x$ be an inner point of
$\mathcal{P}$, of valence $\nu>1$. There exists a unique $n$-periodic
$\nu$-flower (a pattern with a unique inner point $y$ and $\nu$ discrete
components) whose set of $y$-branches, that coincides with its set of discrete
components (petals) when $y$ is labeled as 0, coincides with the set of
$x$-branches of $\mathcal{P}$. Such a pattern will be denoted by
$\mathcal{F}_{x}(\mathcal{P})$. Note that $\mathcal{F}_{x}(\mathcal{P})$ is in
some sense the simplest pattern having the set of $x$-branches of
$\mathcal{P}$, and is obtained from $\mathcal{P}$ by performing iteratively
all possible openings that do not consist of joining two discrete components
adjacent at $x$. For an example, consider the 7-periodic pattern $\mathcal{P}$
shown in Figure 5. Let $x$ be the point of valence 3 labeled as 5 in that
figure. In this case, $\mathcal{F}_{x}(\mathcal{P})$ is the 3-flower whose
petals are $\\{5,1,3,4\\}$, $\\{5,0\\}$ and $\\{5,2,6\\}$. After shifting the
labels by $-5$ (mod 7) in order that the central point of the flower reads as
0, the petals are written as $\\{0,3,5,6\\}$, $\\{0,2\\}$ and $\\{0,4,1\\}$,
that are precisely the $x$-branches of $\mathcal{P}$.

######  Remark 7.4.

Let $\mathcal{P},\mathcal{Q}$ be $n$-periodic patterns. For any
$x,y\in\\{0,1,\ldots,n-1\\}$, the set of $x$-branches of $\mathcal{P}$ and the
set of $y$-branches of $\mathcal{Q}$ coincide if and only if
$\mathcal{F}_{x}(\mathcal{P})=\mathcal{F}_{y}(\mathcal{Q})$.

The previous remark says that in fact the notation
$\mathcal{F}_{x}(\mathcal{P})$, that denotes a pattern, could have been
reserved to denote simply the (arithmetic) set of $x$-branches of
$\mathcal{P}$. We have used the construction of the flower just as a trick
that hopefully supports the geometric visualization.

The following result is true for any point of a periodic pattern but, in
pursuit of simplicity, is stated without loss of generality for a point
labeled as 0.

######  Lemma 7.5.

Let $\mathcal{P}$ be a zero entropy periodic pattern and let
$\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ be the associated sequence of collapses. For
any $0\leq i\leq r$, let $P_{0}^{i}$ be the block of the maximal structure of
$\mathcal{P}_{i}$ containing the point $0\in\mathcal{P}_{i}$. Then,
$P_{0}^{i}$ is contained in a single 0-branch of $\mathcal{P}$.

###### Proof.

By Proposition 6.3, $h(\mathcal{P}_{i})=0$. Then, by Remark 7.1, $P_{0}^{i}$
is contained in a single 0-branch of $\mathcal{P}_{i}$. The result follows
then immediately by using iteratively Lemma 7.2. ∎

A sequence $\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ of pairs of integers will be
called a _branching sequence_ if the following conditions hold:

  1. (bs1)

$p_{i}\geq 2$ for $0\leq i\leq r$.

  2. (bs2)

$\delta_{1}=1$.

  3. (bs3)

For any $1\leq i\leq r$, if $\delta_{i}\notin\\{\delta_{j}\\}_{j=0}^{i-1}$
then $\delta_{i}=1+\max\\{\delta_{j}\\}_{j=0}^{i-1}$.

Let $\mathcal{P}$ be a zero entropy $n$-periodic pattern and let
$\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ be the associated sequence of collapses. Let
$x$ be any point of $\mathcal{P}$, with valence $\nu\geq 1$. Relabel the
points of $\mathcal{P}$ in such a way that $x=0$. Now, for any pattern
$\mathcal{P}_{i}$ in the sequence of collapses, Lemma 7.5 tells us that the
block of the maximal structure of $\mathcal{P}_{i}$ containing $0$ is
contained in a single 0-branch $\delta_{i}$ of $\mathcal{P}$. It is easy to
check that the sequence $\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$, where $p_{0}$ is
the period of $\mathcal{P}_{0}$ and $p_{i}$ is the cardinality of the blocks
of the maximal structure in $\mathcal{P}_{i}$ for any $1\leq i\leq r$,
satisfies properties (bs1–3) above. It will be called _the branching sequence
of $\mathcal{P}$ around $x$_. Once the indexing of the $x$-branches is fixed
after the accorded convention, it is uniquely determined by the pattern
$\mathcal{P}$ and the chosen point $x$ of $\mathcal{P}$. See Figure 10 for an
example of construction of the branching sequence. For the pattern
$\mathcal{P}$ shown in that figure, the 0-branches are
$Z_{1}=\\{0,1,2,3,5,6,7,8,9,10,11,13,14,15\\}$ and $Z_{2}=\\{0,4,12\\}$. The
maximal trivial blocks have cardinality 2 in each pattern of the sequence of
collapses. The blocks containing 0 are $\\{0,1\\}$ in $\mathcal{P}_{0}$,
$\\{0,2\\}$ in $\mathcal{P}_{1}$, $\\{0,4\\}$ in $\mathcal{P}_{2}$ and
$\\{0,8\\}$ in $\mathcal{P}$. Seen as sets of points of $\mathcal{P}$, they
are respectively contained in $Z_{1}$, $Z_{1}$, $Z_{2}$ and $Z_{1}$.
Collecting it all, we get that the branching sequence of $\mathcal{P}$ around
0 is $\\{(2,1),(2,1),(2,2),(2,1)\\}$.

Figure 10. A pattern $\mathcal{P}$ whose branching sequence around 0 is
$\\{(2,1),(2,1),(2,2),(2,1)\\}$. The two 0-branches in $\mathcal{P}$ are
denoted with $Z_{1}$ and $Z_{2}$ with the standard indexing convention.

The following observation follows directly from the definitions.

######  Remark 7.6.

Let $\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ be the branching sequence of a zero
entropy pattern around an inner point $x$. Then, $x$ is bidirectional if and
only if $\delta_{r-1}\neq\delta_{r}$.

Now we reverse the process and consider an (abstract) branching sequence
$S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$. Let us see that from such a sequence
we can construct a zero entropy $n$-periodic $\nu$-flower, where
$n=p_{0}p_{1}\cdots p_{r}$ and $\nu=\max\\{\delta_{i}\\}_{i=0}^{r}$. Consider
a $p_{0}$-periodic trivial pattern $\mathcal{P}_{0}$ and let us denote its
unique discrete component by $C^{0}_{\delta_{1}}=C^{0}_{1}$ (property (bs2)).
Assume now that a zero entropy periodic pattern $\mathcal{P}_{i}$ of period
$p_{0}p_{1}\cdots p_{i}$ has been defined, with
$d_{i}:=\max\\{\delta_{j}\\}_{j=0}^{i}$ discrete components labeled as
$\\{C^{i}_{1},C^{i}_{2},\ldots,C^{i}_{d_{i}}\\}$, all adjacent to the point 0.
Now we define a new pattern $\mathcal{P}_{i+1}$ of period $p_{0}p_{1}\cdots
p_{i+1}$ by applying the following procedure. For any point $j$ of
$\mathcal{P}_{i}$, set
$K_{j}:=\\{j+p_{i},j+2p_{i},\ldots,j+(p_{i+1}-1)p_{i}\\}$. Note that, by
(bs3), either $\delta_{i+1}\leq d_{i}$, and in this case we set
$d_{i+1}:=d_{i}$, or $\delta_{i+1}=d_{i}+1$, and in this case we set
$d_{i+1}:=d_{i}+1$. The pattern $\mathcal{P}_{i+1}$ is then defined as a
$d_{i+1}$-flower with inner point 0 and discrete components labeled as
$\\{C^{i+1}_{1},C^{i+1}_{2},\ldots,C^{i+1}_{d_{i}+1}\\}$, in such a way that
$K_{0}\subset C^{i+1}_{\delta_{i+1}}$ and for any point $j\neq 0$ of
$\mathcal{P}_{i}$, $K_{j}\subset C^{i+1}_{k}$ if and only if $j\in C^{i}_{k}$.
By iterating $r$ times this procedure, finally we obtain the prescribed
$\nu$-flower $\mathcal{P}_{r}$, with the inner point conventionally labeled as
0 by construction. Such a flower, algorithmically constructed from the
branching sequence $S$, will be denoted by $\mathcal{F}(S)$. To fit the
intuition into the description of the algorithm, note that the combinatorial
collapse of a zero entropy $k$-flower is either a $(k-1)$-flower when a petal
fully coincides with a block of the maximal structure, and a $k$-flower
otherwise.

######  Example 7.7.

Let $S=\\{(2,1),(3,2),(2,2),(2,3)\\}$. In Figure 11 we have shown the sequence
of patterns leading to $\mathcal{F}(S)$ according to the prescribed algorithm.

Figure 11. The steps of the algorithm to generate the flower $\mathcal{F}(S)$
from the branching sequence $S=\\{(2,1),(3,2),(2,2),(2,3)\\}$.

A branching sequence $S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ will be called
_minimal_ if $\delta_{i+1}\neq\delta_{i}$ for all $0\leq i<r$.

######  Lemma 7.8.

Let $S$ and $R$ be minimal branching sequences such that
$\mathcal{F}(S)=\mathcal{F}(R)$. Then $S=R$, i.e. $S$ and $R$ have the same
length and are identical term by term.

###### Proof.

Set $S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ and
$R=\\{(q_{i},\kappa_{i})\\}_{i=0}^{t}$. By Remark 7.4, the hypothesis that
$\mathcal{F}(S)$ and $\mathcal{F}(R)$ are the same pattern can be reworded as
follows: if both flowers are labeled in such a way that the respective inner
points read as 0, then the respective sets of 0-branches coincide. In
particular,

(2) $\prod_{i=0}^{r}p_{i}=\prod_{i=0}^{t}q_{i}.$

First we claim that $(p_{1},\delta_{1})=(q_{1},\kappa_{1})$. Indeed, by
property (bs2), $\delta_{1}=\kappa_{1}=1$. Assume by way of contradiction that
$p_{1}<q_{1}$ (the argument is symmetric when $q_{1}<p_{1}$). Then, from (2)
it follows that $r\geq 2$. Moreover, since $S$ is minimal,
$\delta_{2}\neq\delta_{1}$. Property (bs3) yields then that $\delta_{2}=2$.
So, the algorithm of construction of $\mathcal{F}(S)$ and $\mathcal{F}(R)$
implies that the 0-branch indexed as 1 in $\mathcal{F}(S)$ contains the points
$0,1,2,\ldots,p_{1}-1$ and the point $p_{1}$ is contained in the 0-branch
indexed as 2, while the 0-branch indexed as 1 in $\mathcal{F}(R)$ contains at
least the points $0,1,2,\ldots,p_{1}-1,p_{1}$. In consequence,
$\mathcal{F}(S)$ and $\mathcal{F}(R)$ are not the same pattern, a
contradiction that proves the claim.

Assume now that all terms of $S$ and $R$ are identical up to an index $j\geq
1$ (the previous claim states that this is true when $j=1$). In this case, if
$S$ has length $j$, then (2) implies that $R$ has also length $j$ and we are
done. Assume that $r>j$ (the arguments and conclusions are the same if $t>j$).
Set $k:=\prod_{i=0}^{j}p_{i}=\prod_{i=0}^{j}q_{i}$. From the algorithm of
construction of $\mathcal{F}(S)$ and $\mathcal{F}(R)$, it follows that all
points from 0 to $k-1$ are distributed identically inside the 0-branches of
both flowers. The same arguments used above show then that $t>j$, and that if
we assume $(p_{j+1},\delta_{j+1})\neq(q_{j+1},\kappa_{j+1})$, we reach a
contradiction since the points $k,k+1,k+2,\ldots,kp_{j+1}-1$ will be
distributed in different 0-branches of $\mathcal{F}(S)$ and $\mathcal{F}(R)$.
∎

######  Remark 7.9.

If $\mathcal{P}$ is a zero entropy flower, then the branching sequence of
$\mathcal{P}$ around its unique inner point is minimal. Indeed, if for an
index $i$ we had two consecutive terms $(p_{i},\delta_{i})$,
$(p_{i+1},\delta_{i+1})$ with $\delta_{i}=\delta_{i+1}$, then, in the sequence
$\\{\mathcal{P}_{i}\\}_{i=0}^{r}$ of collapses, the trivial blocks for the
pattern $\mathcal{P}_{i}$ would not be maximal, since there would exist
greater trivial blocks of cardinality $p_{i}p_{i+1}$. For example, let
$\mathcal{P}$ be the rightmost pattern shown in Figure 11, that is in fact the
3-flower constructed from $S=\\{(2,1),(3,2),(2,2),(2,3)\\}$. The sequence of
collapses of $\mathcal{P}$ is _not_ $\\{\mathcal{P}_{i}\\}_{i=0}^{3}$ but
$\\{\mathcal{P}^{\prime}_{i}\\}_{i=0}^{2}$, with
$\mathcal{P}^{\prime}_{0}=\mathcal{P}_{0}$,
$\mathcal{P}^{\prime}_{1}=\mathcal{P}_{2}$ and
$\mathcal{P}^{\prime}_{2}=\mathcal{P}_{3}$. The branching sequence of
$\mathcal{P}$ around 0 is then $S^{\prime}=\\{(2,1),(6,2),(2,3)\\}$, which is
minimal.

Let $S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ be a branching sequence. Assume
that $S$ is not minimal, i.e. for some $0\leq j<r$ we have that
$\delta_{j+1}=\delta_{j}$. Then we can consider a _reduced sequence_
$S^{\prime}=\\{(p^{\prime}_{i},\delta^{\prime}_{i})\\}_{i=0}^{r-1}$ defined as
$(p^{\prime}_{i},\delta^{\prime}_{i})=(p_{i},\delta_{i})$ for $0\leq i<j$,
$(p^{\prime}_{j},\delta^{\prime}_{j})=(p_{j}p_{j+1},\delta_{j})$ and
$(p^{\prime}_{i},\delta^{\prime}_{i})=(p_{i+1},\delta_{i+1})$ for $j<i\leq
r-1$. One can easily check that $S^{\prime}$ satisfies (bs1–3) and is thus a
branching sequence. The following result states that $S$ and $S^{\prime}$
generate the same flower. It follows immediately from the algorithm of
construction of $\mathcal{F}(S)$.

######  Lemma 7.10.

Let $S,S^{\prime}$ be branching sequences such that $S^{\prime}$ has been
reduced from $S$. Then, $\mathcal{F}(S^{\prime})=\mathcal{F}(S)$.

The process of reducing a non-minimal branching sequence
$S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ can be iterated as many times as
necessary in order to finally obtain what we call the _sequence fully reduced
from $S$_, a minimal branching sequence
$\widehat{S}=\\{(\widehat{p}_{i},\widehat{\delta}_{i})\\}_{i=0}^{\widehat{r}}$
satisfying $\prod_{i=0}^{r}p_{i}=\prod_{i=0}^{\widehat{r}}\widehat{p}_{i}$.
One can easily check that it is unique and well defined. As a direct corollary
of Lemma 7.10, we get the following result.

######  Corollary 7.11.

Let $S$ be a branching sequence and let $\widehat{S}$ be the sequence fully
reduced from $S$. Then, $\mathcal{F}(S)=\mathcal{F}(\widehat{S})$.

In this section we have defined two procedures to generate a flower
(equivalently, a set of branches). The first one uses openings to get a flower
$\mathcal{F}_{x}(\mathcal{P})$ given a pattern $\mathcal{P}$ and a point $x$
of $\mathcal{P}$, while the second one constructs a flower $\mathcal{F}(S)$
given an abstract branching sequence $S$. The next lemma, that follows
immediately from the definitions and the labeling conventions of the points
and branches, states that if $S$ is precisely the branching sequence of
$\mathcal{P}$ around $x$, both flowers are the same as patterns.

######  Lemma 7.12.

Let $\mathcal{P}$ be a zero entropy pattern. Let $x$ be a point of
$\mathcal{P}$ and let $S$ be the branching sequence of $\mathcal{P}$ around
$x$. Then, $\mathcal{F}(S)=\mathcal{F}_{x}(\mathcal{P})$.

Now we are ready to use all techniques and results of this section to get the
following proposition and the subsequent corollary, that will be crucial in
the proof of Theorem D.

######  Proposition 7.13.

Let $\mathcal{P}$ be a zero entropy periodic pattern and let $x$ be a point of
$\mathcal{P}$. Let $S$ be the branching sequence of $\mathcal{P}$ around $x$
and let $\widehat{S}$ be the sequence fully reduced from $S$. Then, the
branching sequence of $\mathcal{F}_{x}(\mathcal{P})$ around $x$ is
$\widehat{S}$.

###### Proof.

By Lemma 7.12,

(3) $\mathcal{F}(S)=\mathcal{F}_{x}(\mathcal{P}).$

Let $R$ be the branching sequence of $\mathcal{F}_{x}(\mathcal{P})$ around
$x$. We want to see that $R=\widehat{S}$. Since
$\mathcal{F}_{x}(\mathcal{F}_{x}(\mathcal{P}))=\mathcal{F}_{x}(\mathcal{P})$,
using again Lemma 7.12 yields

(4) $\mathcal{F}(\mathcal{R})=\mathcal{F}_{x}(\mathcal{P}).$

On the other hand, by Corollary 7.11,

(5) $\mathcal{F}(S)=\mathcal{F}(\widehat{S}).$

From (3), (4) and (5) we get then that

(6) $\mathcal{F}(R)=\mathcal{R}(\widehat{S}).$

Since $\widehat{S}$ is minimal by definition of a fully reduced sequence and
$R$ is minimal by Remark 7.9, then (6) and Lemma 7.8 imply that
$R=\widehat{S}$. ∎

######  Corollary 7.14.

Let $\mathcal{P}$ and $\mathcal{Q}$ be two zero entropy $n$-periodic patterns.
Let $x$ and $y$ be inner points of $\mathcal{P}$ and $\mathcal{Q}$
respectively. Let $\widehat{S}$ and $\widehat{R}$ be the fully reduced
sequences of $\mathcal{P}$ and $\mathcal{Q}$ around $x$ and $y$ respectively.
If $\mathcal{F}_{x}(\mathcal{P})=\mathcal{F}_{y}(\mathcal{Q})$ then
$\widehat{S}=\widehat{R}$, i.e. both sequences have the same length and are
identical term by term.

##  8\. Proof of Theorem D

Recall that the hypothesis of Theorem D is that we have an $n$-periodic
pattern $\mathcal{P}$ with at least two inner points and at least three
openings. Moreover, $h(\mathcal{P})>0$ and any opening has entropy zero. Under
these conditions, we have to prove that $\mathcal{P}$ is $\pi$-reducible for
some basic path $\pi$. The name $\mathcal{O}$ that we will use to denote zero
entropy patterns in this section stands for _opening_ , in the spirit of
Theorem D.

The following is a simple remark about how the set of $x$-branches, where $x$
is a point of a pattern $\mathcal{P}$, can change after performing an opening
of $\mathcal{P}$.

######  Remark 8.1.

Let $x$ be a point of a pattern $\mathcal{P}$ and let $\mathcal{O}$ be an
opening of $\mathcal{P}$. If $\mathcal{O}$ has been obtained by joining two
discrete components not adjacent at $x$ (equivalently, the valence of $x$ in
$\mathcal{O}$ equals the valence of $x$ in $\mathcal{P}$), then
$\mathcal{F}_{x}(\mathcal{O})=\mathcal{F}_{x}(\mathcal{P})$. As an example,
consider the pattern $\mathcal{P}$ and the opening $\mathcal{O}$ shown in
Figure 5. Take $x=1$. In this case,
$\mathcal{F}_{x}(\mathcal{O})=\mathcal{F}_{x}(\mathcal{P})$ is a 2-flower
whose petals can be labeled as $\\{0,3\\}$ and $\\{0,1,2,4,5,6\\}$. On the
other hand, if $\mathcal{O}$ has been obtained by joining two discrete
components adjacent at $x$ (equivalently, the valence of $x$ in $\mathcal{O}$
is one less than the valence of $x$ in $\mathcal{P}$), then
$\mathcal{F}_{x}(\mathcal{O})$ is an opening of
$\mathcal{F}_{x}(\mathcal{P})$. As an example, take $x=5$ in the previous
example. Here $\mathcal{F}_{x}(\mathcal{P})$ is a 3-flower whose petals can be
labeled as $\\{0,3,5,6\\}$, $\\{0,2\\}$ and $\\{0,1,4\\}$, while
$\mathcal{F}_{x}(\mathcal{O})$ is a 2-flower whose petals can be labeled as
$\\{0,3,5,6\\}$ and $\\{0,1,2,4\\}$, i.e. an opening of
$\mathcal{F}_{x}(\mathcal{P})$.

Recall that the integer labels of the points of a pattern $\mathcal{P}$ are by
default preserved when performing an opening of $\mathcal{P}$. So, in the
following statement we use the same letter $x$ to refer indistinctly to a
point of a pattern and to the corresponding point of an opening.

######  Lemma 8.2.

Let $\mathcal{P}$ be an $n$-periodic pattern with positive entropy such that
any opening of $\mathcal{P}$ has entropy zero. Assume that $\mathcal{P}$ has
at least two inner points and at least three openings. Then, there exist a
point $x$ of $\mathcal{P}$ and two different openings $\mathcal{O}$ and
$\mathcal{R}$ of $\mathcal{P}$ such that:

  1. (a)

$x$ is a bidirectional inner point in $\mathcal{O}$.

  2. (b)

$x$ is an inner point in $\mathcal{R}$.

  3. (c)

One of the following statements holds:

    1. (c1)

$\mathcal{F}_{x}(\mathcal{O})=\mathcal{F}_{x}(\mathcal{R})$

    2. (c2)

$\mathcal{F}_{x}(\mathcal{O})$ is an opening of
$\mathcal{F}_{x}(\mathcal{R})$.

###### Proof.

To prove the result we consider two cases.

  * Case 1.

$\mathcal{P}$ has exactly two inner points.

In this case, the hypothesis imply that at least one inner point has valence
larger than 2 and that $\mathcal{P}$ has at least four different openings. Let
us consider for instance that $\mathcal{P}$ has one inner $\alpha$ with
valence 2 and one inner $\beta$ with valence 3. The proof can be trivially
extended to any other case. In this situation, $\mathcal{P}$ has four discrete
components, which we label by $C_{0}$, $C_{1}$, $C_{2}$ and $C_{3}$. See
Figure 12 for a representation of $\mathcal{P}$ and the three openings that we
will use below.

According to the notation in Figure 12 we consider $\mathcal{O}$ to be the
opening of $\mathcal{P}$ corresponding to the union $C_{0}\cup C_{2}$. The
pattern $\mathcal{O}$ is a triple chain with two inner points $\alpha$ and
$\beta$. Let $x$ be a bidirectional inner point of $\mathcal{O}$, that exists
by Proposition 7.3. Then, (a) holds. Consider now a relabeling of the points
of $\mathcal{P}$ (and, in consequence, of $\mathcal{O}$) such that $x=0$. We
have now two possibilities.

Figure 12. Three possible openings for case 1 in the proof of Lemma 8.2.

If $0=\alpha$ then we take $\mathcal{R}$ as the opening $\mathcal{O}^{\prime}$
corresponding to the union $C_{0}\cup C_{3}$. So, (b) is satisfied. Moreover,
neither $\mathcal{O}$ nor $\mathcal{R}$ have been formed by joining together
discrete components adjacent to 0. It follows that the valence of 0 in
$\mathcal{P}$, $\mathcal{O}$ and $\mathcal{R}$ is the same and (c1) follows
from Remark 8.1.

If $0=\beta$ then we take $\mathcal{R}$ as the opening
$\mathcal{O}^{\prime\prime}$ corresponding to the union $C_{0}\cup C_{1}$. So,
(b) is satisfied. Moreover, $\mathcal{R}$ has only one inner point, $\beta=0$.
In this case the valence of $0$ in $\mathcal{R}$ equals the valence of $0$ in
$\mathcal{P}$ and it is one larger than in $\mathcal{O}$. Thus, (c2) follows
from Remark 8.1.

  * Case 2.

$\mathcal{P}$ has at least three inner points.

Let $\mathcal{O}$ be an arbitrary opening of $\mathcal{P}$. Let $x$ be a
bidirectional inner point of $\mathcal{O}$, that exists by Proposition 7.3.
Then, (a) holds. Consider now a relabeling of the points of $\mathcal{P}$
(and, in consequence, of $\mathcal{O}$) such that $x=0$. Let $\alpha\neq 0$
and $\beta\neq 0$ be two different inner points of $\mathcal{P}$.

If $\mathcal{O}$ has been obtained by joining two discrete components adjacent
to $\alpha$, then we choose $\mathcal{R}$ as any opening obtained by joining
two discrete components adjacent to $\beta$. In this case, the valence of $0$
is the same in the three patterns $\mathcal{P}$, $\mathcal{O}$ and
$\mathcal{R}$ (See Figure 13). Thus, (b) holds and, by Remark 8.1, (c1) is
also satisfied.

Figure 13. Illustration of case 2 (first subcase) in the proof of Lemma 8.2.
Figure 14. Illustration of case 2 (second subcase) in the proof of Lemma 8.2.

Finally, if $\mathcal{O}$ has been formed by joining two discrete components
adjacent to $0$, then we choose $\mathcal{R}$ as an opening obtained by
joining two discrete components adjacent to $\beta$. In this case, the valence
of $0$ in $\mathcal{P}$ and $\mathcal{R}$ is the same and one larger than the
valence of $0$ in $\mathcal{O}$. In particular, the valence of $0$ in
$\mathcal{P}$ and $\mathcal{R}$ is larger than two. Thus, (b) holds and, by
Remark 8.1, (c2) is satisfied (see Figure 14). ∎

To prove Theorem D, we will use branching sequences in the two situations (c1)
and (c2) given by Lemma 8.2(c). To deal with (c2), we need to relate the
branching sequences of both a flower $\mathcal{F}$ and an opening
$\mathcal{F}^{\prime}$ of $\mathcal{F}$.

######  Lemma 8.3.

Let $\mathcal{F}$ be a zero entropy periodic $\nu$-flower and let
$R=\\{(q_{i},\kappa_{i})\\}_{i=0}^{t}$ be the branching sequence of
$\mathcal{F}$ around its unique inner point $x$. Let $\mathcal{F}^{\prime}$ be
an opening of $\mathcal{F}$ obtained by joining two discrete components
corresponding to two $x$-branches labeled as $j_{1},j_{2}$, with $1\leq
j_{1}<j_{2}\leq\nu$. Set
$R^{\prime}:=\\{(q_{i},\kappa^{\prime}_{i})\\}_{i=0}^{t}$, with
$\kappa^{\prime}_{i}$ defined as

$\kappa^{\prime}_{i}=\left\\{\begin{array}[]{lcl}\kappa_{i}&\mbox{if}&\kappa_{i}<j_{2}\\\
j_{1}&\mbox{if}&\kappa_{i}=j_{2}\\\
\kappa_{i}-1&\mbox{if}&\kappa_{i}>j_{2}\end{array}\right.$

Then, $R^{\prime}$ is a branching sequence and the sequence fully reduced from
$R^{\prime}$ is the branching sequence of $\mathcal{F}^{\prime}$ around $x$.

###### Proof.

It is easy to check directly from the definition of $R^{\prime}$ that
properties (bs1–3) satisfied by $R$ are inherited by $R^{\prime}$. Thus,
$R^{\prime}$ is a branching sequence. By checking the steps of the algorithm
of construction of the flower $\mathcal{F}(R^{\prime})$, one easily gets that
$\mathcal{F}(R^{\prime})=\mathcal{F}^{\prime}$.

Let $\widehat{R^{\prime}}$ be the sequence fully reduced from $R^{\prime}$. By
Corollary 7.11,
$\mathcal{F}^{\prime}=\mathcal{F}(R^{\prime})=\mathcal{F}(\widehat{R^{\prime}})$.
Let $B$ be the branching sequence of $\mathcal{F}^{\prime}$ around its unique
inner point $x$. We want to see that $B=\widehat{R^{\prime}}$. Since
$\mathcal{F}_{x}(\mathcal{F}^{\prime})=\mathcal{F}^{\prime}$, Lemma 7.12
yields $\mathcal{F}(B)=\mathcal{F}^{\prime}$. Therefore,
$\mathcal{F}(B)=\mathcal{F}(\widehat{R^{\prime}})$. Since
$\widehat{R^{\prime}}$ is minimal by definition of a fully reduced sequence
and $B$ is minimal by Remark 7.9, the previous equality and Lemma 7.8 imply
$B=\widehat{R^{\prime}}$. ∎

To illustrate Lemma 8.3, let $\mathcal{F}$ be the 4-flower shown in Figure 15.
The discrete components (equivalently, the 0-branches) of $\mathcal{F}$ are
$Z_{1}=\\{0,1,3,5,7,9,11,13,15\\}$, $Z_{2}=\\{0,2,6,10,14\\}$,
$Z_{3}=\\{0,4,12\\}$, $Z_{4}=\\{0,8\\}$. One can check that the branching
sequence of $\mathcal{F}$ around 0 is $R=\\{(2,1),(2,2),(2,3),(2,4)\\}$. Now
let $\mathcal{F}^{\prime}$ be the opening obtained by joining the discrete
components $Z_{1}$ and $Z_{3}$. The 0-branches of $\mathcal{F}^{\prime}$,
indexed according to the standing convention, are then $Y_{1}=Z_{1}\cup
Z_{3}$, $Y_{2}=Z_{2}$, $Y_{3}=Z_{4}$. The sequence $R^{\prime}$ defined in the
statement of Lemma 8.3 is $R^{\prime}=\\{(2,1),(2,2),(2,1),(2,3)\\}$, that is
minimal. According to Lemma 8.3, it is the branching sequence of
$\mathcal{F}^{\prime}$ around 0. As another example, let
$\mathcal{F}^{\prime}$ be the opening of $\mathcal{F}$ obtained by joining the
discrete components $Z_{2}$ and $Z_{3}$. In this case, the 0-branches of
$\mathcal{F}^{\prime}$ are $Y_{1}=Z_{1}$, $Y_{2}=Z_{2}\cup Z_{3}$ and
$Y_{3}=Z_{4}$. The sequence $R^{\prime}$ defined in the statement of Lemma 8.3
reads as $R^{\prime}=\\{(2,1),(2,2),(2,2),(2,3)\\}$ and its fully reduced
sequence $\\{(2,1),(4,2),(2,3)\\}$ is the branching sequence of
$\mathcal{F}^{\prime}$ around 0.

Figure 15. A 16-periodic 4-flower with entropy zero.

Now we are in position of proving Theorem D.

###### Proof of Theorem D.

Let $\mathcal{O}$ and $\mathcal{R}$ be the two openings of $\mathcal{P}$ given
by Lemma 8.2, let $S=\\{(p_{i},\delta_{i})\\}_{i=0}^{r}$ and
$R=\\{(q_{i},\kappa_{i})\\}_{i=0}^{t}$ be the corresponding branching
sequences around $x$, and let
$\widehat{S}=\\{(\widehat{p}_{i},\widehat{\delta}_{i})\\}_{i=0}^{\widehat{r}}$
and
$\widehat{R}=\\{(\widehat{q}_{i},\widehat{\kappa}_{i})\\}_{i=0}^{\widehat{t}}$
be the sequences fully reduced, respectively, from $S$ and $R$. From the
definition of a reduced sequence,

(7) $\widehat{q}_{\widehat{t}}=q_{t-j}q_{t-j+1}\cdots q_{t-1}q_{t}\mbox{ for
some }j\geq 0.$

On the other hand, since $x$ is bidirectional in $\mathcal{O}$, then, by
Remark 7.6, $\delta_{r-1}\neq\delta_{r}$. Therefore, using again the
definition of a reduced sequence we get

(8)
$(\widehat{p}_{\widehat{r}},\widehat{\delta}_{\widehat{r}})=(p_{r},\delta_{r}).$

We claim that $q_{t}$ divides $p_{r}$. To prove this claim we will consider
the two cases produced by Lemma 8.2(c).

Assume first that Lemma 8.2(c1) holds. Then, by Corollary 7.14, $\widehat{S}$
and $\widehat{R}$ are identical term by term. In particular,
$\widehat{q}_{\widehat{t}}=\widehat{p}_{\widehat{r}}$, which is equal to
$p_{r}$ by (8). Thus, (7) implies that $q_{t}$ divides $p_{r}$, as claimed.

Assume now that Lemma 8.2(c2) holds. From Proposition 7.13 we have that
$\widehat{S}$ is the branching sequence of the flower
$\mathcal{F}^{\prime}:=\mathcal{F}_{x}(\mathcal{O})$ and $\widehat{R}$ is the
branching sequence of the flower $\mathcal{F}:=\mathcal{F}_{x}(\mathcal{R})$.
Since $\mathcal{F}^{\prime}$ is an opening of $\mathcal{F}$, Lemma 8.3 tells
us that
$\widehat{S}=\\{(\widehat{p}_{i},\widehat{\delta}_{i})\\}_{i=0}^{\widehat{r}}$
has been obtained from
$\widehat{R}=\\{(\widehat{q}_{i},\widehat{\kappa}_{i})\\}_{i=0}^{\widehat{t}}$
in two steps. First, we consider a sequence
$\widehat{R}^{\prime}=\\{(\widehat{q}_{i},\widehat{\kappa}^{\prime}_{i})\\}_{i=0}^{\widehat{t}}$
and then fully reduce it to obtain
$\\{(\widehat{p}_{i},\widehat{\delta}_{i})\\}_{i=0}^{\widehat{r}}$. Again the
definition of a reduction implies that

$\widehat{p}_{\widehat{r}}=\widehat{q}_{\widehat{t}-\ell}\widehat{q}_{\widehat{t}-\ell+1}\cdots\widehat{q}_{\widehat{t}-1}\widehat{q}_{\widehat{t}}\mbox{
for some }\ell\geq 0.$

The previous equality and (7) imply that $q_{t}$ divides $p_{r}$ also in this
case. In consequence, the claim is proved.

To end up we claim that the divisibility of $p_{r}$ by $q_{t}$ implies the
$\pi$-reducibility of $\mathcal{P}$. We recall that $p_{r}$ and $q_{t}$ are
the cardinalities of the trivial blocks in the respective maximal structures
of $\mathcal{O}$ and $\mathcal{R}$ given by Proposition 6.3. Relabel if
necessary the points of $\mathcal{P}$ in such a way that $x=0$. The inner
point $0$ belongs to the block of $\mathcal{O}$

$O_{0}=\\{0,\tfrac{n}{p_{r}},\tfrac{2n}{p_{r}},\dots,\tfrac{(p_{r}-1)n}{p_{r}}\\}.$

By Proposition 3.7, $\mathcal{O}$ is $\pi$-reducible for any basic path $\pi$
contained in $O_{0}$. On the other hand, the inner point $0$ belongs to the
block of $\mathcal{R}$

$R_{0}=\\{0,\tfrac{n}{q_{t}},\tfrac{2n}{q_{t}},\dots,\tfrac{(q_{t}-1)n}{q_{t}}\\}.$

Again, $\mathcal{R}$ is $\pi$-reducible for any basic path $\pi$ contained in
$R_{0}$. Since $q_{t}$ divides $p_{r}$, the point $\frac{n}{q_{t}}$ belongs to
$O_{0}\cap R_{0}$. Take $\pi:=\\{0,\frac{n}{q_{t}}\\}$. Note that $\pi$ is a
basic path in $\mathcal{P}$, $\mathcal{Q}$ and $\mathcal{R}$. Moreover, $\pi$
never splits in both $\mathcal{O}$ and $\mathcal{R}$. Since all inner points
of $\mathcal{P}$ are inner points either in $\mathcal{O}$ or in $\mathcal{R}$,
it follows that $\pi$ never splits in $\mathcal{P}$. ∎

##  9\. $k$-Flowers

Following the sketch of the proof of Theorem A outlined in Section 5, we have
to deal now with the special case of patterns with only one inner point and
$k\geq 2$ discrete components. When $k=2$, the following result (Theorem 5.2
of [8]) does the job.

######  Theorem 9.1.

Let $\mathcal{P}$ be an $n$-periodic pattern with two discrete components. If
$h(\mathcal{P})>0$, then $h(\mathcal{P})\geq\log(\lambda_{n})$.

For $k\geq 3$, and in the spirit of the proof by induction outlined in Section
5, we need to relate our pattern of period $n$ with another pattern with
period less than $n$ and positive entropy. So, let $\mathcal{P}=([T,P],[f])$
be an $n$-periodic pattern. A pattern $\mathcal{P}^{\prime}$ will be said to
be _subordinated to_ $\mathcal{P}$ if for some divisor $n>p>1$ of $n$ there is
an $(n/p)$-periodic orbit $P^{\prime}\subset P$ of $f^{p}$ such that
$\mathcal{P}^{\prime}=([\langle
P^{\prime}\rangle_{T},P^{\prime}],[f^{p}\bigr{\rvert}_{P^{\prime}}])$.
Clearly, this definition is independent of the particular model $(T,P,f)$
representing $\mathcal{P}$.

The following result is Lemma 9.1 of [7]. It allows us to estimate the entropy
of a pattern from the entropy of a subordinated.

######  Lemma 9.2.

Let $\mathcal{P}$ be an $n$-periodic pattern. Let $\mathcal{P}^{\prime}$ be an
$n^{\prime}$-periodic pattern subordinated to $\mathcal{P}$. If
$h(\mathcal{P}^{\prime})\geq\log(\lambda_{n^{\prime}})$ then
$h(\mathcal{P})>\log(\lambda_{n})$.

A discrete component of a pattern will be said to be _extremal_ if it contains
only one inner point. As an example, the discrete components $A$, $B$ and $D$
are extremal for the pattern $\mathcal{P}$ shown in Figure 5.

Let $(T,P,f)$ be a model of a periodic pattern $\mathcal{P}$. Let $C$ be a
discrete component of $(T,P)$. We will say that a point $x\in C$ _escapes from
$C$_ if $f(x)$ does not belong to the connected component of
$T\setminus\\{x\\}$ that intersects $\operatorname{Int}(\langle C\rangle)$.
Any discrete component $C$ of $(T,P)$ without points escaping from it will be
called a _scrambled component_ of $\mathcal{P}$. Clearly, this notion does not
depend on the particular chosen model of $\mathcal{P}$. So, it makes sense to
say that the pattern $\mathcal{P}$ _has a scrambled component_. As an example,
the point 7 escapes from $\\{1,7,13\\}$ in the 18-periodic pattern
$\mathcal{P}_{2}$ shown in Figure 8, while does not scape from
$C:=\\{0,3,5,7,9,11,15,17\\}$. In fact, no point in $C$ escapes from $C$. So,
$C$ is a scrambled component for $\mathcal{P}_{2}$. It is easy to see that
every periodic pattern has scrambled components (Lemma 4.2 of [7]).

######  Theorem 9.3.

Let $\mathcal{P}$ be an $n$-periodic pattern with positive entropy and at
least three discrete components. Assume that any opening of $\mathcal{P}$ has
entropy zero. If $\mathcal{P}$ has an extremal scrambled component, then
$\mathcal{P}$ has subordinated patterns with positive entropy.

###### Proof.

Let $(T,P,f)$ be a model of $\mathcal{P}$ and let $C$ be the extremal
scrambled component of $\mathcal{P}$. Then, there is only one inner point $x$
in $C$, and $f(x)\in C$ by definition of a scrambled component. Consider a
sequence of openings that joins together all discrete components different
from $C$ into a single discrete component $D$, leading to a pattern
$\mathcal{P}^{\prime}$ with two discrete components, $C$ and $D$. Since
$h(\mathcal{P^{\prime}})=0$ by hypothesis, $\mathcal{P^{\prime}}$ has a
_division_ [7] with respect to $C$. In consequence, there exists $p\geq 2$, a
divisor of $n$, such that $f^{i}(D)\subset C$ for $1\leq i<p$ and
$f^{p}(D)=D$. In other words, $\cup_{i=0}^{p-1}P_{i}$ is a $p$-block structure
for $\mathcal{P}$, where $P_{i}:=f^{i}(D)$. Note that the blocks
$P_{1},P_{2},\ldots P_{p-1}$ are contained in $C$ and are, thus, trivial.
Consider the pattern $\mathcal{Q}:=([\langle
P_{0}\rangle_{T},P_{0}],[f^{p}\bigr{\rvert}_{P_{0}}])$. Then, $\mathcal{Q}$ is
subordinated to $\mathcal{P}$. Moreover, its entropy is positive, for
otherwise the fact that all blocks but one are trivial would easily imply that
$h(\mathcal{P})=0$. ∎

When a pattern has only one inner point $x$, the discrete component containing
the image of $x$ is clearly scrambled and extremal. So, we have the next
result as an immediate consequence of Theorem 9.3.

######  Corollary 9.4.

Let $\mathcal{P}$ be a positive entropy $k$-flower, with $k\geq 3$. Assume
that any opening of $\mathcal{P}$ has entropy zero. Then, $\mathcal{P}$ has
subordinated patterns with positive entropy.

##  10\. Triple chains

The final stage in the proof of Theorem A outlined in Section 5 leaves us with
the special case of a pattern with exactly two inner points and three discrete
components, a _triple chain_. In order to find lower bounds for the entropy of
a triple chain $\mathcal{P}$, it is unavoidable to count coverings in the
$\mathcal{P}$-path graph (equivalently, entries in the path transition
matrix). This section is devoted to this task. In our context, it is assumed
that $\mathcal{P}$ is $\pi$-irreducible and any of the two possible openings
of $\mathcal{P}$ has entropy zero (property ($\star$ ‣ 5)). Note that any
opening of a triple chain has two discrete components. So, to obtain a lower
bound of the entropy of $\mathcal{P}$ we will proceed in two steps. First, we
will study the coverings in the path graph of zero entropy patterns with two
discrete components. This is the aim of Lemmas 10.3 and 10.6. Finally, we will
study how the previous coverings, present in the two possible openings of the
triple chain $\mathcal{P}$, imply the existence of a number of coverings in
the $\mathcal{P}$-path graph (Lemma 10.8) that forces enough entropy for our
purposes.

The results mentioned in the previous scheme are extremely technical. Readers
are cautioned to follow the arguments using examples, as the ones shown in the
figures.

A basic path $\pi$ for a pattern with a separated structure of trivial blocks
will be said to be _in-block_ if it is contained in a block. Otherwise, it
will be said to be _inter-block_. As an example, $\\{1,13\\}$ is an in-block
basic path of $\mathcal{P}_{2}$ in Figure 8, while $\\{0,15\\}$ is inter-
block. The second statement of Proposition 3.7 says that an in-block path
never _splits_ (as defined in page 3.6). On the other hand, next result states
that inter-block basic paths do always split.

######  Lemma 10.1.

Let $\mathcal{P}$ an $n$-periodic pattern with a separated structure of
trivial blocks. Then any inter-block basic path of $\mathcal{P}$ splits before
$n$ iterates.

###### Proof.

The proof strongly relies on the construction of the maximal separated
structure of trivial blocks in Proposition 9.5 of [7] and its uniqueness (see
Section 3). The construction shows that if $\mathcal{P}$ is $\sigma$-reducible
for a basic path $\sigma$, then each trivial block is obtained as the set of
endpoints of a connected component of $\cup_{i\geq 0}\langle
f^{i}(\sigma)\rangle$. In particular, $\sigma$ is contained in one block and
is thus an in-block basic path. The uniqueness of the maximal structure of
trivial blocks implies that the same is true for any basic path
$\sigma^{\prime}\neq\sigma$ such that $\mathcal{P}$ is
$\sigma^{\prime}$-reducible. Now we note that if $\pi$ does not split in $n$
iterates, then $\mathcal{P}$ is $\pi$-reducible. Then, by the previous
discussion, $\pi$ has to be in-block, a contradiction. ∎

Let $\mathcal{P}$ be a non-trivial $n$-periodic pattern with entropy zero. By
Proposition 6.3, $\mathcal{P}$ has a maximal structure of trivial blocks and
the corresponding combinatorial collapse $\mathcal{C}$ has entropy zero. Let
$x$ be a point of $\mathcal{P}$. The point of $\mathcal{C}$ corresponding to
the collapse of the block containing $x$ will be denoted by $\overline{x}$,
and this will be a standing notation throughout this section. In fact, if $x$
is contained in the block $P_{i}$, then $\overline{x}$ is precisely the point
of $\mathcal{C}$ labeled as $i$. Let $\pi=\\{x,y\\}$ be an inter-block basic
path of $\mathcal{P}$. Then, $\overline{x}\neq\overline{y}$. The binary set
$\\{\overline{x},\overline{y}\\}$ will be denoted by $\overline{\pi}$. Note
that, by property (b) of Definition 6.2, $\overline{\pi}$ is a basic path in
$\mathcal{C}$. As an example, consider the pattern $\mathcal{P}_{2}$ shown in
Figure 8. The basic paths $\pi_{1}=\\{11,8\\}$ and $\pi_{2}=\\{0,7\\}$ are
inter-block. In this case, $\overline{\pi}_{1}=\\{5,2\\}$ and
$\overline{\pi}_{2}=\\{0,1\\}$ are (respectively, in-block and inter-block)
basic paths of the combinatorial collapse $\mathcal{P}_{1}$.

The notation $\mathcal{O}$ for patterns of entropy zero used in the statements
of this section suggests, as in Section 8, the term _opening_.

######  Lemma 10.2.

Let $\mathcal{O}$ be a zero entropy periodic pattern and let $\mathcal{C}$ be
the combinatorial collapse of $\mathcal{O}$. Let $\pi$ be an inter-block basic
path of $\mathcal{O}$. If $\overline{\pi}$ splits in $\ell$ iterates on
$\mathcal{C}$, then $\pi$ splits in at most $\ell$ iterates on $\mathcal{O}$.

###### Proof.

Consider any model $(T,P,f)$ of $\mathcal{O}$ and assume that $f^{i}(\pi)$ is
a basic path for $0\leq i<\ell$. From the definition of a block structure it
follows that, for all $0\leq i<\ell$, the basic path $f^{i}(\pi)$ is inter-
block in $\mathcal{O}$. Set $\\{a,b\\}:=f^{\ell}(\pi)$. By hypothesis,
$\overline{f^{i}(\pi)}$ is a basic path in $\mathcal{C}$ for $0\leq i<\ell$,
while $\overline{a}$ and $\overline{b}$ are separated by at least one inner
point in $\mathcal{C}$. We have to see that $a$ and $b$ are also separated in
$\mathcal{O}$. Assume, by way of contradiction, that there exists a discrete
component $D$ of $\mathcal{O}$ containing $\\{a,b\\}$. In particular, the
trivial blocks $K_{a}$ and $K_{b}$ in $\mathcal{O}$ whose collapse gives
respectively the points $\overline{a}$ and $\overline{b}$ of $\mathcal{C}$
satisfy $K_{a}\cap D\neq\emptyset$ and $K_{b}\cap D\neq\emptyset$. By
definition of the combinatorial collapse, this implies that there exists a
single discrete component of $\mathcal{C}$ containing
$\\{\overline{a},\overline{b}\\}$, a contradiction. ∎

Given a pattern $\mathcal{P}=([T,P],[f])$ and two basic paths $\pi$ and
$\sigma$ of $\mathcal{P}$, we will say that $\pi$ is a _strict pre-image of_
$\sigma$ if there exists $j\geq 1$ such that $f^{i}(\pi)$ is a basic path for
$0\leq i\leq j$ and $f^{j}(\pi)=\sigma$. Note that, in this case, $f^{i}(\pi)$
are also strict pre-images of $\sigma$ for $1\leq i<j$.

The following result computes the number of iterations necessary for an inter-
block basic path to split in a zero entropy pattern with two discrete
components. At this point we recover the notation introduced in Section 2 and
write $\\{a,b\\}\rightarrow\\{c,d\\}$ to indicate that the basic path
$\\{a,b\\}$ $f$-covers the basic path $\\{c,d\\}$.

######  Proposition 10.3.

Let $\mathcal{O}=([T,P],[f])$ be a zero entropy $n$-periodic pattern with two
discrete components and a maximal structure of trivial blocks of cardinality
$q$. Let $\mathcal{C}$ be the corresponding combinatorial collapse. Assume
that $\mathcal{O}$ is labeled in such a way that 0 is the unique inner point.
Let $\pi$ be an inter-block basic path $\pi$ of $\mathcal{O}$. Then,

  1. (i)

either splits in at most $\frac{n}{q}$ iterates,

  2. (ii)

or it is an strict pre-image of a basic path $\sigma=\\{0,a+\frac{q-1}{q}n\\}$
with $0<a<\frac{n}{q}$. In this case, $\overline{\sigma}$ is in-block in
$\mathcal{C}$ and $\pi$ splits in at most $\frac{2n}{q}$ iterates.

If in addition $\overline{\pi}$ is in-block in $\mathcal{C}$, then the
following statements hold:

  1. (a)

If $\pi=\\{0,a\\}$ with $0<a<\frac{n}{q}$, then $\pi$ splits in
$\frac{n}{q}-a$ iterates.

  2. (b)

If $\pi=\\{0,a+\tfrac{q-1}{q}n\\}$ with $0<a<\frac{n}{q}$, then $\pi$ splits
in $\frac{n}{q}$ iterates.

###### Proof.

Let $\\{\mathcal{O}_{i}\\}_{i=0}^{s}$ be the sequence of collapses of
$\mathcal{O}$ according to Remark 6.4 and let $q_{i}$ be the cardinality of
the blocks of $\mathcal{O}_{i}$. Since $\mathcal{O}$ is not trivial, $s\geq
1$. The proof follows in two steps. First, we prove the result for a sequence
of collapses of length $s=1$. Then we tackle the general case $s>1$ using the
case $s=1$ on a particular subordinated pattern of $\mathcal{O}$.

Assume first that $s=1$ and let $q_{0}=n/q_{1}$ be the period of the
combinatorial collapse $\mathcal{C}=\mathcal{O}_{0}$, which is a trivial
pattern. The pattern $\mathcal{O}=\mathcal{O}_{1}$ is formed by
$q_{0}=n/q_{1}$ trivial blocks of $q_{1}$ points. Let us denote by $P_{i}$,
$i=0,\dots,q_{0}-1$, the trivial blocks of the pattern $\mathcal{O}$ according
to the standing convention in Remark 3.2. Notice that the block $P_{0}$,
formed by the multiples of $q_{0}$ (mod $n$), is one of the two discrete
components of $\mathcal{O}$.

Let $\pi$ be an inter-block basic path of $\mathcal{O}$. Since $\mathcal{C}$
is trivial, $\overline{\pi}$ is in-block in $\mathcal{C}$. The labeling of
$\mathcal{O}$ is fixed by the unique inner point. So, we can write the points
in $P_{i}$ as $i+\ell q_{0}$ with $0\leq\ell\leq q_{1}-1$. The point
$i+(q_{1}-1)q_{0}$ will be called the last point in $P_{i}$. The inter-block
basic path $\pi$ connects a point of the block $P_{i}$ with a point of the
block $P_{j}$. Along the proof we consider that the blocks are ordered in such
a way that $0\leq i<j\leq q_{0}-1$. We distinguish three types of inter-block
basic paths of $\mathcal{O}$ depending on the points that are connected.

Figure 16. The three types of paths in Proposition 10.3.

Type I. $\pi$ connects any point of $P_{i}$ with one of $P_{j}$ that is not
the last one, $0\leq i<j\leq q_{0}-1$. In this situation, we can write
$\pi=\\{i+\ell q_{0},j+rq_{0}\\}$ with $0\leq\ell\leq q_{1}-1$ and $0\leq
r\leq q_{1}-2$.

Type II. $\pi$ connects a point of the block $P_{i}$ that is not the last one
with the last point of $P_{j}$, $0\leq i<j\leq q_{0}-1$. In this case,
$\pi=\\{i+\ell q_{0},j+(q_{1}-1)q_{0}\\}$ with $0\leq\ell\leq q_{1}-2$.

Type III. $\pi$ connects the last points of the blocks $P_{i}$ and $P_{j}$,
$1\leq i<j\leq q_{0}-1$. In this latter case,
$\pi=\\{i+(q_{1}-1)q_{0},j+(q_{1}-1)q_{0}\\}$.

Since $\pi$ is an inter-block basic path, if $i=0$ for Type I and II then
$\ell=0$. That is, only the point $0\in P_{0}$ can be connected to a point of
a different trivial block. For this reason, $i\geq 1$ in Type III.

Notice that an inter-block basic path $\pi=\\{a,b\\}$ splits in $k$ iterates
if $k$ is the smallest integer such that either $a+k$ or $b+k$ is a multiple
of $q_{0}$ different from $0$ (mod $n$). Indeed, since $\pi$ is inter-block,
$a+k$ and $b+k$ cannot be both multiple of $q_{0}$. Otherwise, $f^{k}(\pi)$ is
a basic path joining two points of $P_{0}$ and is, therefore, in-block. Since
$0$ is the only inner point and $P_{0}$ is a whole discrete component, the
previous condition implies that $a+k$ and $b+k$ are on different discrete
components. On account of the previous, now we compute the iterates that an
inter-block basic path of each type requires to split.

If $\pi$ is of Type I, then it splits in $q_{0}-j$ iterates:

$\pi\rightarrow\overset{q_{0}-j)}{\cdots}\rightarrow\\{i+q_{0}-j+\ell
q_{0},0\\}\cup\\{0,(r+1)q_{0}\\}.$

Indeed, since $0\leq i<j\leq q_{0}-1$ then $j+rq_{0}$ reaches the point
$(r+1)q_{0}$ in $q_{0}-j$ iterates, whereas $i+\ell q_{0}$ needs
$q_{0}-i>q_{0}-j$ iterates to reach a multiple of $q_{0}$. Since $0\leq r\leq
q_{1}-2$ then $(r+1)q_{0}\neq 0$ (mod $n$), so the splitting occurs.

If $\pi$ is of Type II, then it splits in $q_{0}-i$ iterates:

$\pi\rightarrow\overset{q_{0}-i)}{\cdots}\rightarrow\\{(\ell+1)q_{0},0\\}\cup\\{0,j-i\\}.$

Indeed, in this case, although $i<j$ and $j+(q_{1}-1)q_{0}$ reaches a multiple
of $q_{0}$ in $q_{0}-j$ iterates, the multiple is $q_{1}q_{0}=n=0$ (mod $n$).
Therefore, there is no splitting in $q_{0}-j$ iterates. On the other hand, in
$q_{0}-i$ iterates the splitting occurs.

Finally, if $\pi$ is of Type III, then it splits in $2q_{0}-j$ iterates.
Indeed, in $q_{0}-j$ iterates:

$\pi\rightarrow\overset{q_{0}-j)}{\cdots}\rightarrow\\{i+q_{0}-j+(q_{1}-1)q_{0},0\\}$

The basic path $\\{i+q_{0}-j+(q_{1}-1)q_{0},0\\}$ is of Type II with $i=0$.
So, it splits in $q_{0}$ iterates. Summing up, $\pi$ splits in $2q_{0}-j$
iterates.

The previous discussion proves the result for $s=1$. Indeed, every inter-block
basic path splits in at most $q_{0}=\frac{n}{q_{1}}$ iterates with the
exception of the strict pre-images of $\\{0,a+(q_{1}-1)q_{0}\\}$ with
$0<a=i+q_{0}-j<q_{0}$, which split in $2q_{0}-j<\frac{2n}{q_{1}}$ iterates.
Moreover, the case (a) corresponds to a Type I basic path by taking
$i=\ell=0$, $j=a$ and $r=0$, so $\pi$ splits in $q_{0}-a=n/q_{1}-a$ iterates.
Taking $i=\ell=0$ and $j=a$ on a Type II basic path,
$\pi=\\{0,a+(q_{1}-1)q_{0}\\}$ splits in $q_{0}=n/q_{1}$ iterates, proving
(b). In Figure 16 we show examples of each type for $n=16$ and $q_{0}=4$.

Figure 17. Notation in the proof of Proposition 10.3.

Assume now that the sequence of collapses of $\mathcal{O}$ has length $s>1$.
Set $\mathcal{C}:=\mathcal{O}_{s-1}$ and $\mathcal{O}:=\mathcal{O}_{s}$.
Moreover, each pattern $\mathcal{O}_{i}$ for $1\leq i\leq s$ has a unique
inner point, labeled as $0$ according to Remark 3.3. Let $q_{0}$ be the period
of $\mathcal{O}_{0}$ and, for $1\leq i\leq n$, let $q_{i}$ be the cardinality
of the blocks of $\mathcal{O}_{i}$. Then, $n=\prod_{i=0}^{s}q_{i}$. According
to the notation in the statement, $q_{s}=q$. The pattern $\mathcal{O}$ has a
maximal structure of $n/q$ trivial separated blocks of $q$ points, and the
pattern $\mathcal{C}$ has a maximal structure of $n/(q_{s-1}q)$ trivial blocks
of cardinality $q_{s-1}$. Let us denote by $S_{i}$ and $P_{i}$ the blocks of
the patterns $\mathcal{C}$ and $\mathcal{O}$, respectively. Since 0 is the
unique inner point in $\mathcal{O}$, from Lemma 7.3 it follows that 0 is
bidirectional. Then, the trivial block $P_{0}$ of $\mathcal{O}$ and the
trivial block $S_{0}$ of $\mathcal{C}$ are contained in different 0-branches.
See Figure 17 for an example with $s=2$, $q_{0}=3$, $q_{1}=4$, $q=q_{2}=2$,
$n=24$.

Let $\pi$ be an inter-block basic path of $\mathcal{O}$. If $\overline{\pi}$
is inter-block in $\mathcal{C}$, by Lemma 10.1 $\overline{\pi}$ splits before
$n/q$ iterates. By Lemma 10.2, this property is inherited by $\pi$, which
splits in at most $n/q$ iterates, as desired.

Let us assume now that $\pi$ is an inter-block basic path of $\mathcal{O}$
such that $\overline{\pi}$ is in-block in $\mathcal{C}$. That is,
$\overline{\pi}$ is contained in $S_{\eta}$ for some
$\eta\in\\{0,\dots,\frac{n}{q_{s-1}q}-1\\}$. Note that all points in a block
of $\mathcal{C}$ differ by a multiple of $n/(q_{s-1}q)$, while all points in a
block of $\mathcal{O}$ differ by a multiple of $n/q$. It follows that
$\overline{\pi}$ has the form

$\overline{\pi}=\\{\overline{a},\overline{b}\\}=\\{\eta+i\tfrac{n}{q_{s-1}q},\eta+j\tfrac{n}{q_{s-1}q}\\}$

with $0\leq\eta\leq\frac{n}{q_{s-1}q}-1$ and $0\leq i<j\leq q_{s-1}-1$, while
$\pi$ has the form

$\pi=\\{a,b\\}=\\{\overline{a}+\ell\tfrac{n}{q},\overline{b}+r\tfrac{n}{q}\\}$

with $0\leq\ell,r\leq q-1$. For the sake of intuition, note that $\eta$ labels
the block $S_{\eta}$ of $\mathcal{C}$ containing $\overline{\pi}$, while the
blocks of $\mathcal{O}$ containing $a$ and $b$ are, respectively,
$P_{\overline{a}}$ and $P_{\overline{b}}$. Going back to the example shown in
Figure 17, if we take $\pi=\\{16,22\\}$, then $\overline{\pi}=\\{4,10\\}$,
$\eta=1$, $i=1$, $j=3$, $\overline{a}=4$, $\overline{b}=10$, $\ell=r=1$.

Let us study the iterates $f^{i}(\pi)$. Since 0 is the unique inner point in
$\mathcal{O}$, a pair $\\{x,y\\}$ of points of $\mathcal{O}$ is not a basic
path if and only if 0 separates $x$ and $y$. Since $\overline{\pi}$ is in-
block in $\mathcal{C}$, it never splits by Proposition 3.7. Moreover, $0\in
P_{0}$. It follows that a basic path $\pi$ may split in $k$ iterates only if
$\overline{f^{k}(\pi)}\subset S_{0}$. Let $\pi_{0}$ be the first iterate of
$\pi$ such that $\overline{\pi}_{0}\subset S_{0}$. Then,

$\pi_{0}=\\{\tfrac{n}{q_{s-1}q}(i+\ell
q_{s-1}),\tfrac{n}{q_{s-1}q}(j+rq_{s-1})\\}$

for some $0\leq i<j\leq q_{s-1}-1$ and $0\leq\ell,r\leq q-1$. Let us look at
the worst-case scenario by assuming that $\pi_{0}$ is a basic path. That is to
say, we have the sequence of non-splitting coverings

$\pi\rightarrow f(\pi)\rightarrow
f^{2}(\pi)\rightarrow\overset{\frac{n}{q_{s-1}q}-\eta)}{\cdots}\rightarrow\pi_{0}.$

In order to bound the number of iterates required by $\pi$ to split, we study
$\pi_{0}$. Let us consider the subordinated pattern
$\mathcal{O}^{\prime}:=([\langle
P_{0}\rangle_{T},P_{0}],[f^{\frac{n}{q_{s-1}q}}])$. Note that
$\mathcal{O}^{\prime}$ has two discrete components, entropy zero and a maximal
structure of $q_{s-1}$ trivial blocks given by

$P_{0}\cup P_{\frac{n}{q_{s-1}q}}\cup\ldots\cup
P_{\frac{(q_{s-1}-1)n}{q_{s-1}q}}.$

Moreover, the corresponding combinatorial collapse $\mathcal{C}^{\prime}$ is a
trivial pattern of period $q_{s-1}$. In other words, the sequence of collapses
of $\mathcal{O}^{\prime}$ reduces to
$\\{\mathcal{C}^{\prime},\mathcal{O}^{\prime}\\}$ and thus we can apply the
discussion about types of basic paths and coverings used in the case $s=1$.
Let us take the labeling of $\mathcal{O}^{\prime}$ such that the only inner
point reads as 0. See Figure 18 for a picture of the patterns
$\mathcal{C}^{\prime}$ and $\mathcal{O}^{\prime}$ corresponding to the example
shown in Figure 17.

Figure 18. The subordinated pattern $\mathcal{O}^{\prime}$ and its collapse
$\mathcal{C}^{\prime}$ for the example shown in Figure 17.

Notice that there is a correspondence between the basic path $\pi_{0}$ in
$\mathcal{O}$ and the basic path $\\{i+\ell q_{s-1},j+rq_{s-1}\\}$ in
$\mathcal{O}^{\prime}$. Since $\pi_{0}$ may only split when
$\overline{\pi}_{0}$ returns to $S_{0}$, it suffices to study the number of
iterates required by $\\{i+\ell q_{s-1},j+rq_{s-1}\\}$ to split in
$\mathcal{O}^{\prime}$ and then multiply the length of the sequence of paths
by $\frac{n}{q_{s-1}q}$. As it was stated in the discussion of the case $s=1$,
we have three situations depending on the type of path.

If $\\{i+\ell q_{s-1},j+rq_{s-1}\\}$ in $\mathcal{O}^{\prime}$ is of Type I
then $\pi_{0}$ splits in
$\frac{n}{q_{s-1}q}(q_{s-1}-j)=\frac{n}{q}-\frac{j}{q_{s-1}q}n$ iterates in
$\mathcal{O}$. Taking $i=\ell=r=0$ and $a=\frac{j}{q_{s-1}q}n$, this proves
(a).

If $\\{i+\ell q_{s-1},j+rq_{s-1}\\}$ in $\mathcal{O}^{\prime}$ is of Type II
then $r=q-1$ and $\pi_{0}$ splits in
$\frac{n}{q_{s-1}q}(q_{s-1}-i)=\frac{n}{q}-\frac{i}{q_{s-1}q}n$ iterates in
$\mathcal{O}$. Taking $i=\ell=0$ and $a=\frac{j}{q_{s-1}q}n$, this proves (b).

Lastly, if $\\{i+\ell q_{s-1},j+rq_{s-1}\\}$ in $\mathcal{O}^{\prime}$ is of
Type III then $\ell=r=q-1$ and $\pi_{0}$ splits in
$\frac{n}{q_{s-1}q}(2q_{s-1}-j)=\frac{2n}{q}-\frac{j}{q_{s-1}q}n$ iterates in
$\mathcal{O}$ and it is an strict pre-image of the basic path
$\\{0,\frac{n}{q_{s-1}q}(i+q_{s-1}-j+(q-1)q_{s-1})\\}$.

The previous holds for $\pi_{0}$. In order to bound the iterates required by
$\pi$ to split we add $\frac{n}{q_{s-1}q}-\eta$ to the previous. So, depending
on the types before, for $1\leq\eta\leq\frac{n}{q_{s-1}q}-1$, either

  * •

$\pi$ splits in $\frac{n}{q}-\frac{j-1}{q_{s-1}q}n-\eta$ iterates, or

  * •

$\pi$ splits in $\frac{n}{q}-\frac{i-1}{q_{s-1}q}n-\eta$ iterates, or

  * •

$\pi$ splits in $\frac{2n}{q}-\frac{j-1}{q_{s-1}q}n-\eta$ iterates and it is
an strict pre-image of

$\\{0,\tfrac{n}{q_{s-1}q}(i+q_{s-1}-j+(q-1)q_{s-1})\\}.$

This proves that every inter-block basic path of $\mathcal{O}$ splits after at
most $2n/q$ iterates. Moreover, the only inter-block basic paths splitting in
more than $n/q$ iterates are strict pre-images of some
$\\{0,a+\frac{q-1}{q}n\\}$, with
$0<a=\frac{n}{q_{s-1}q}(i+q_{s-1}-j)<\frac{n}{q}$, proving the result. ∎

The previous result states that almost every inter-block basic path of a zero
entropy pattern with two discrete components splits in at most $n/q$ iterates
with the exception of those considered in (ii). The following results are
concerned with the bound for the latter case. The first result states that the
“time reverse” of a zero entropy pattern $\mathcal{O}$ with two discrete
components coincides with $\mathcal{O}$. Figure 19 shows an example that
illustrates this remarkable property, that is not true for general zero
entropy patterns. It is possible to prove it using sequences of collapses and
Proposition 6.1, but we use a result from [8] to get a considerably shorter
proof.

Figure 19. A pattern $\mathcal{O}$ and its time reverse $\mathcal{Q}$ as
defined in Lemma 10.4.

######  Lemma 10.4.

Let $(T,P,f)$ be the canonical model of an $n$-periodic pattern $\mathcal{O}$
with entropy zero and two discrete components. Let $P=\\{x_{i}\\}_{i=0}^{n-1}$
be time labeled. Consider the relabeling of $P$ given by $y_{i}:=x_{n-i\bmod
n}$ and the map $g\colon P\longrightarrow P$ defined by $g(y_{i}):=y_{i+1\bmod
n}$ for $0\leq i<n$. Then, $([T,P],[g])=\mathcal{O}$.

###### Proof.

Assume without loss of generality that $x_{0}$ is the unique inner point of
$\mathcal{O}$. From the definitions we get that $P$ is an $n$-periodic orbit
of $g$, time labeled as $P=\\{y_{i}\\}_{i=0}^{p-1}$. Thus, $([T,P],[g])$ is an
$n$-periodic pattern $\mathcal{Q}$. By definition, $y_{0}=x_{0}$, so that
$y_{0}$ is the only inner point of $\mathcal{Q}$. To see that
$\mathcal{O}=\mathcal{Q}$ we have to show that both patterns have the same
discrete components.

For any tree map $F\colon S\longrightarrow S$, an ordered set $(a,b,c)$ of
three points of $S$ is called a _forward triplet of $F$_ if $b\in(a,c)$,
$f(a)=b$, $f(b)=c$, and $\\{a,b,c\\}$ is contained in a periodic orbit of $F$.
By Theorem 1.1 of [8], $F$ has positive entropy if and only if there exists
$k\geq 1$ such that $F^{k}$ has a forward triplet. Thus, since
$h(f)=h(\mathcal{O})=0$, $f$ cannot have forward triplets. It easily follows
that both $x_{i}$ and $x_{n-i}$ belong to the same discrete component of
$\mathcal{O}$ for all $1\leq i<n$. But
$\\{x_{i},x_{n-i}\\}=\\{y_{n-i},y_{i}\\}$, implying that both $\mathcal{O}$
and $\mathcal{Q}$ have exactly the same discrete components. ∎

######  Lemma 10.5.

The basic path $\sigma=\\{0,a+\frac{q-1}{q}n\\}$ with $0<a<\frac{n}{q}$ in
(ii) of Proposition 10.3 has at most $a-1$ strict pre-images. Moreover, a
basic path $\pi=\\{0,y\\}$ cannot be an strict pre-image of $\sigma$.

###### Proof.

By Lemma 10.4, the pattern $\mathcal{O}$ coincides with its time reverse. In
particular, the basic path $\\{0,a+\frac{q-1}{q}n\\}$ has as many pre-images
as basic paths are covered by $\\{0,\frac{n}{q}-a\\}$ before splitting. The
basic path $\sigma$ is inter-block and $\overline{\sigma}$ is in-block in the
corresponding combinatorial collapse $\mathcal{C}$ of $\mathcal{O}$.
Therefore, the same is true for the basic path $\\{0,\frac{n}{q}-a\\}$. Since
$0<\frac{n}{q}-a<\frac{n}{q}$, by Proposition 10.3 (a), the basic path
$\\{0,\frac{n}{q}-a\\}$ splits in
$\frac{n}{q}-\bigl{(}\frac{n}{q}-a\bigr{)}=a$ iterates. This proves the first
assertion of the lemma.

The second assertion, using the time reverse property, is equivalent to show
that the basic path $\\{0,n-y\\}$ is not covered by $\\{0,\frac{n}{q}-a\\}$
before splitting. That is, before $a$ iterates. This is clear, since neither
$0$ nor $\frac{n}{q}-a$ map on $0$ before $a$ iterates. ∎

Now we can use Lemma 10.5 together with Proposition 10.3 to find the desired
coverings.

######  Lemma 10.6.

Let $\mathcal{O}$ be a zero entropy $n$-periodic pattern with two discrete
components and a maximal structure of trivial blocks of cardinality $q$. If
$q\geq 3$ then any inter-block basic path of $\mathcal{O}$ covers at least
four basic paths in $n$ iterates.

###### Proof.

Let us label $\mathcal{O}$ in such a way that $0$ is the unique inner point
and let $\pi$ be an inter-block basic path of $\mathcal{O}$. By Proposition
10.3,

  1. (i)

either $\pi$ splits in at most $\frac{n}{q}$ iterates,

  2. (ii)

or $\pi$ is a strict pre-image of a basic path $\\{0,a+\frac{q-1}{q}n\\}$ with
$0<a<\frac{n}{q}$ which splits in $\frac{n}{q}$ iterates.

In the case (i), $\pi$ covers two basic paths $\\{0,y\\}$ and $\\{0,z\\}$
before $\frac{n}{q}$ iterates. Notice that both $\\{0,y\\}$ and $\\{0,z\\}$
cannot be in-block basic paths of $\mathcal{O}$. Otherwise, since $0$ is inner
of $\mathcal{O}$ and $y$ and $z$ are contained in different discrete
components, the trivial block that contains $0$ would contain points of two
different discrete components, a contradiction. Therefore, we can assume
$\\{0,y\\}$ to be an inter-block basic path of $\mathcal{O}$. Moreover, by
Lemma 10.5, an inter-block basic path of the form $\\{0,y\\}$ cannot be an
strict pre-image of a basic path of the form $\\{0,a+\frac{q-1}{q}n\\}$ with
$0<a<\frac{n}{q}$. Therefore, again by Proposition 10.3, $\\{0,y\\}$ splits in
at most $\frac{n}{q}$ iterates, covering two basic paths $\\{0,y_{1}\\}$ and
$\\{0,y_{2}\\}$. Again one of them is inter-block of $\mathcal{O}$ and splits
in at most $\frac{n}{q}$ iterates. Therefore, $\pi$ covers at least four basic
paths in $\frac{3n}{q}\leq n$ iterates. This proves the result in the case
(i).

In the case (ii), by Lemma 10.5, $\pi$ covers $\\{0,a+\frac{q-1}{q}n\\}$ in at
most $a-1$ iterates and $\\{0,a+\frac{q-1}{q}n\\}$ covers $\\{0,a\\}$ and
$\\{0,\frac{n}{q}\\}$ in $\frac{n}{q}$ iterates. By Proposition 10.3(a), the
basic path $\\{0,a\\}$ splits and covers two basic paths $\\{0,u\\}$ and
$\\{0,v\\}$ in $\frac{n}{q}-a$ iterates and, since one must be inter-block in
$\mathcal{O}$, again splits in at most $\frac{n}{q}$ iterates as shown before.
Therefore, $\pi$ covers at least four basic paths in
$a-1+\frac{n}{q}+\frac{n}{q}-a+\frac{n}{q}=\frac{3n}{q}-1<n$ iterates, proving
the result in the case (ii). ∎

######  Remark 10.7.

Let $\mathcal{P}$ be a pattern and let $\mathcal{O}$ be an opening of
$\mathcal{P}$. Let $\pi$ be a basic path of $\mathcal{P}$. Then $\pi$ is also
a basic path of $\mathcal{O}$. Moreover, if $\pi$ covers $k$ basic paths in
$\ell$ iterates in $\mathcal{O}$ then $\pi$ covers at least $k$ basic paths in
$\ell$ iterates in $\mathcal{P}$.

By collecting all previous results, finally we get the desired lower bound for
coverings in a triple chain.

Figure 20. An illustration of the proof of Proposition 10.8. Some loops of the
$\mathcal{P}$-path graph obtained in the proof are shown. The underlined basic
paths are in-block in $\mathcal{O}_{2}$.

######  Proposition 10.8.

Let $\mathcal{P}$ be an $n$-periodic $\pi$-irreducible triple chain. Assume
that the two possible openings $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ of
$\mathcal{P}$ have entropy zero. Then, any basic path $\pi$ of $\mathcal{P}$
covers at least four basic paths in $n$ iterates.

###### Proof.

By Remark 10.7 a basic path $\pi$ of $\mathcal{P}$ is also a basic path of
both $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. We claim that $\pi$ is inter-
block for some $\mathcal{O}_{i}$. Indeed, if $\pi$ is in-block in both
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, then $\pi$ does not split through any
of the two inner points of $\mathcal{P}$. Consequently, $\pi$ never splits in
$\mathcal{P}$ and so $\mathcal{P}$ is $\pi$-reducible, a contradiction.

The patterns $\mathcal{O}_{i}$, $i=1,2$, have zero entropy. So, by Proposition
6.1, each of them has a maximal structure of trivial blocks of cardinality
$q_{i}\geq 2$.

Let us first prove the result when $q_{i}\geq 3$. As stated before, $\pi$ is
inter-block for some of the openings, let us say $\mathcal{O}_{1}$ without
loss of generality. Since $q_{1}\geq 3$, by Lemma 10.6, $\pi$ covers at least
four basic paths in $n$ iterates in $\mathcal{O}_{1}$. By Remark 10.7, this
property is inherited in $\mathcal{P}$, so $\pi$ covers at least four basic
paths in $n$ iterates in $\mathcal{P}$. This proves the result in the first
situation.

Now assume that $q_{1}=q_{2}=2$. In this case, the basic path
$\\{0,\frac{n}{2}\\}$ is in-block in both $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$. By the discussion at the beginning of the proof, this
produces contradiction with the $\pi$-irreducibility of $\mathcal{P}$.

We are left with the case $q_{1}=2$ and $q_{2}\geq 3$. Again, $\pi$ is inter-
block in $\mathcal{O}_{1}$ or $\mathcal{O}_{2}$. If $\pi$ is inter-block in
$\mathcal{O}_{2}$, the result follows as in the first case since $q_{2}\geq
3$. So, we can assume that $\pi$ is in-block in $\mathcal{O}_{2}$ and, in
consequence, inter-block in $\mathcal{O}_{1}$.

Let us relabel $\mathcal{P}$ and, accordingly, the openings $\mathcal{O}_{i}$,
in such a way that the inner point of $\mathcal{O}_{1}$ is $0$. We denote by
$j$ the inner point of $\mathcal{O}_{2}$. The basic path $\pi$ is in-block in
$\mathcal{O}_{2}$, so in $\mathcal{P}$ the first splitting is through the
inner $0$. Since $\pi$ is inter-block in $\mathcal{O}_{1}$, by Proposition
10.3, one of the following situations occurs in $\mathcal{O}_{1}$:

  1. (i)

either $\pi$ splits in at most $\frac{n}{2}$ iterates,

  2. (ii)

or $\pi$ is a strict pre-image of a basic path $\\{0,a+\frac{n}{2}\\}$ with
$0<a<\frac{n}{2}$, which splits in $\frac{n}{2}$ iterates.

In both cases $\pi$ covers two basic paths in $\mathcal{O}_{1}$ after the
first splitting. Since $\mathcal{P}$ is a triple chain, at least one of such
paths is also a basic path in $\mathcal{P}$. For the sake of brevity, we will
focus on the worst scenario which corresponds to assuming that the two basic
paths covered in $\mathcal{O}_{1}$ are also basic paths in $\mathcal{P}$. The
reader may easily check that if this is not the case, then a third basic path
is covered in $\mathcal{P}$ during the first splitting, and the upper bounds
obtained below are valid for the basic path shared between $\mathcal{O}_{1}$
and $\mathcal{P}$.

Consider the case (i). Since $0$ is the inner point of $\mathcal{O}_{1}$, then
$\pi$ covers in $\mathcal{O}_{1}$ two basic paths $\\{0,y\\}$ and $\\{0,z\\}$
in at most $\frac{n}{2}$ iterates. As noticed above, we are assuming that both
$\\{0,y\\}$ and $\\{0,z\\}$ are basic paths in $\mathcal{P}$. Clearly, $y\neq
z$ and so we can assume also $z\neq\frac{n}{2}$. Consequently, $\\{0,z\\}$ is
an inter-block in $\mathcal{O}_{1}$. Moreover, by Lemma 10.5, $\\{0,z\\}$ is
not an strict pre-image of a basic path of the form $\\{0,a+\frac{n}{2}\\}$.
Then, by Proposition 10.3, $\\{0,z\\}$ splits in at most $\frac{n}{2}$
iterates covering two basic paths $\\{0,z_{1}\\}$ and $\\{0,z_{2}\\}$. Since
$\\{0,z\\}$ is a basic path in $\mathcal{P}$, then $\\{0,z\\}$ covers at least
two basic paths in $\frac{n}{2}$ iterates in $\mathcal{P}$. Now we have two
cases depending on the value of $y$. If $y\neq\frac{n}{2}$ the same argument
applies for $\\{0,y\\}$ and, summing up, $\pi$ covers at least four basic
paths in $n$ iterates in $\mathcal{P}$, proving the result in this case. The
following diagram illustrates the coverings in this first situation inside
case (i).

If $y=\frac{n}{2}$ then $\\{0,\frac{n}{2}\\}$ is in-block in
$\mathcal{O}_{1}$. Since $\\{0,\frac{n}{2}\\}$ is a basic path in
$\mathcal{P}$, it is also a basic path in $\mathcal{O}_{2}$. Moreover, it must
be inter-block. By Proposition 10.3, either $\\{0,\frac{n}{2}\\}$ covers two
basic paths before $\frac{n}{q_{2}}$ iterates or it is a strict pre-image of a
basic path $\\{j,j+b+\frac{(q_{2}-1)n}{q_{2}}\\}$, where $0<b<\frac{n}{q_{2}}$
and $j$ is the inner point of $\mathcal{O}_{2}$. The second alternative,
however, cannot be satisfied. Indeed, the time distance between the two points
of an iterate of a basic path is conserved while there is no splitting. If
$\\{0,\frac{n}{2}\\}$ is a strict pre-image of
$\\{j,j+b+\frac{(q_{2}-1)n}{q_{2}}\\}$, then the distance should be conserved,
but $b+\frac{(q_{2}-1)n}{q_{2}}\geq\frac{n}{2}$. Therefore,
$\\{0,\frac{n}{2}\\}$ covers two basic paths in $\mathcal{O}_{2}$ in at most
$\frac{n}{q_{2}}$ iterates. Since $q_{2}\geq 3$ then, summing up, $\pi$ covers
at least four basic paths in $n$ iterates in $\mathcal{P}$, proving the result
for the case (i). The following diagram illustrates the coverings in this
second situation inside case (i).

The basic paths $\\{0,8\\}$ and $\\{3,7\\}$ in Figure 20 are examples of
maximal length of case (i). The basic path $\\{0,8\\}$ splits in
$\frac{n}{2}=6$ iterates and covers $\\{0,y\\}=\\{0,6\\}$ and
$\\{0,z\\}=\\{0,2\\}$. The path $\\{0,6\\}$ is of the form
$\\{0,\frac{n}{2}\\}$, so it is in-block in $\mathcal{O}_{1}$ and inter-block
in $\mathcal{O}_{2}$. It splits in $3<\frac{n}{2}=6$ iterates. The path
$\\{0,2\\}$ is inter-block in both $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ and
splits in $2<\frac{n}{2}=6$ iterates. A similar phenomenon occurs for
$\\{3,7\\}$.

Let us now consider the case (ii). By Lemma 10.4 the basic path
$\\{0,a+\frac{n}{2}\\}$ has, at most, $a-1$ strict pre-images. Thus, $\pi$
covers $\\{0,a+\frac{n}{2}\\}$ in at most $a-1$ iterates. Since $\pi$ is in-
block in $\mathcal{O}_{2}$, $\\{0,a+\frac{n}{2}\\}$ must also be an in-block
path of $\mathcal{O}_{2}$. Hence, $a+\frac{n}{2}=k\frac{n}{q_{2}}$ for some
$1\leq k\leq q_{2}-1$. Moreover, $\\{0,a+\frac{n}{2}\\}$ splits in
$\frac{n}{2}$ iterates and covers the basic paths $\\{0,a\\}$ and
$\\{0,\frac{n}{2}\\}$. Recall that we are assuming that both $\\{0,a\\}$ and
$\\{0,\frac{n}{2}\\}$ are basic paths in $\mathcal{P}$ and so in
$\mathcal{O}_{2}$. The basic path $\\{0,\frac{n}{2}\\}$ is inter-block in
$\mathcal{O}_{2}$ and, as proved in case (i), covers two basic paths before
$\frac{n}{q_{2}}$ iterates. On the other hand, $\\{0,a\\}$ is inter-block for
$\mathcal{O}_{1}$ and, since $a<\frac{n}{2}$, it covers two basic paths in
$\frac{n}{2}-a$ iterates by Proposition 10.3(a). Summing up, $\pi$ covers two
basic paths in $a-1+\frac{n}{2}+\frac{n}{q_{2}}=(k+1)\frac{n}{q_{2}}-1\leq
n-1$ iterates through $\\{0,\frac{n}{2}\\}$ and two basic paths in
$a-1+\frac{n}{2}+\frac{n}{2}-a=n-1$ iterates through $\\{0,a\\}$, which proves
that $\pi$ covers at least four basic paths in $n$ iterates. The following
diagram illustrates the coverings in case (ii).

The basic path $\\{11,7\\}$ is the only one satisfying case (ii) in Figure 20.
Here $\\{0,a+\frac{n}{2}\\}=\\{0,8\\}$ with $a=2$. Indeed, $\\{0,8\\}$ has at
most $a-1=1$ pre-images and $\\{0,8\\}$ splits exactly in $\frac{n}{2}=6$
iterates covering $\\{0,a\\}=\\{0,2\\}$ and $\\{0,\frac{n}{2}\\}=\\{0,6\\}$. ∎

Let $A=(a_{ij})$ be an $n\times n$ nonnegative matrix. Recall that $\rho(A)$
stands for the spectral radius of $A$. For $1\leq i\leq n$, let
$r_{i}(A)=\sum_{j=1}^{n}a_{ij}$ be the $i$-th row sum of $A$. The following
result is well-known [28].

######  Theorem 10.9.

If $A$ is a nonnegative matrix then

$\min_{1\leq i\leq n}r_{i}(A)\leq\rho(A)\leq\max_{1\leq i\leq n}r_{i}(A).$

######  Corollary 10.10.

Let $\mathcal{P}$ be an $n$-periodic and $\pi$-irreducible triple chain.
Assume that the two possible openings of $\mathcal{P}$ have entropy zero.
Then, $h(\mathcal{P})>\log(\sqrt[n]{4})$.

###### Proof.

By Remark 2.2, $h(\mathcal{P})=\log\max\\{\rho(M),1\\}$, where $M$ is the path
transition matrix of $\mathcal{P}$. By Proposition 10.8, any basic path of
$\mathcal{P}$ covers at least four basic paths in $n$ iterates. In particular,
the sum of the elements on each row of $M^{n}$ is $r_{i}(M^{n})\geq 4$. By
Theorem 10.9, $4\leq\rho(M^{n})\leq\rho(M)^{n}$. In consequence,
$\rho(M)\geq\sqrt[n]{4}$ and the result follows. ∎

##  11\. Proof of Theorem A