2 Patterns and canonical models

In this section, we formalize the definitions outlined in §1. 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 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 $\operatorname {\mathrm {En}}(T)$ . 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 $\operatorname {\mathrm {Int}}(X)$ and $\operatorname {\mathrm {Cl}}(X)$ 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 is a tree map and P is a finite f-invariant set such that $\operatorname {\mathrm {En}}(T)\subset P$ . 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 or simply by , as the smallest closed connected subset of T containing X. When $X=\{x,y\}$ , we will write $[x,y]$ to denote . The notation $(x,y)$ , $(x,y]$ 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 labelled if $\theta (x_i)=x_{i+1}$ for $0\le i<n-1$ 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 $(T',P')$ are equivalent if there exists a bijection which preserves discrete components. The equivalence class of a pointed tree $(T,P)$ will be denoted by $[T,P]$ .

Let $(T,P)$ and $(T',P')$ be equivalent pointed trees, and let and be maps. We will say that $\theta $ and $\theta '$ are equivalent if $\theta '= \phi \circ \theta \circ \phi ^{-1}$ for a bijection that 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 and $\Theta =[f\big \rvert _{_{P}}]$ .

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

$$ \begin{align*} h(\mathcal{P}) := \inf \{h(f) \,\colon (T,P,f)\ \mathrm{is a model exhibiting}\ \mathcal{P} \}. \end{align*} $$

The simplest models exhibiting a given pattern are the monotone ones, defined as follows. Let be a tree map map. Given $a,b\in T$ , we say that $f\big \rvert _{[a,b]}$ is monotone if $f([a,b])$ is either an interval or a point and $f\big \rvert _{[a,b]}$ 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 $f\big \rvert _{[a,b]}$ is monotone for any basic path $\{a,b\}$ . The model $(T,P,f)$ will then be said to be monotone. In such a case, [Reference Alsedà, Guaschi, Los, Mañosas and Mumbrú3, Proposition 4.2] 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).

First, [Reference Alsedà, Guaschi, Los, Mañosas and Mumbrú3, Theorem A] 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. Additionally, [Reference Alsedà, Guaschi, Los, Mañosas and Mumbrú3, Theorem B] states that every pattern has a canonical model. Moreover, given two canonical models $(T,P,f)$ and $(T',P',f')$ of the same pattern, there exists a homeomorphism such that $\phi (P) = P'$ and $f' \circ \phi \big \rvert _{P} = \phi \circ f\big \rvert _{P}$ . Hence, the canonical model of a pattern is essentially unique. Summarizing, we have the following result.

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 $(T',P',f')$ of Figure 1, a representative of the same pattern, cannot be $P'$ -monotone, since in this case, we would have $f'(v) \in f'([x^{\prime }_2,x^{\prime }_6]) \cap f'([x^{\prime }_4,x^{\prime }_5]) = [x^{\prime }_3,x^{\prime }_1] \cap [x^{\prime }_5,x^{\prime }_6] = \emptyset .$

There is a combinatorial procedure to compute the entropy of a pattern $\mathcal {P}$ that 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 $V = \{v_1,v_2,\ldots ,v_k\}$ 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 . 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.

To end this section, we define the patterns that will be shown to have minimum positive entropy. Let $n\in \mathbb {N}$ with $n\ge 3$ . 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 labelled 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 [Reference Štefan31]. In [Reference Alsedà, Juher and Mañosas7], the authors prove that $h(\mathcal {Q}_n)=\log (\unicode{x3bb} _n)$ , where $\unicode{x3bb} _n$ is the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$ . We will use the following properties of the numbers $\unicode{x3bb} _n$ . Statement (a) is proved in [Reference Alsedà, Juher and Mañosas7, Proposition 3.1], while statement (b) is an easy exercise.

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 [Reference Alsedà, Guaschi, Los, Mañosas and Mumbrú3], and another description was proven to be equivalent in [Reference Alsedà, Juher and Mañosas6]. 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, which 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 square root construction [Reference Sharkovskii30] is an early example of a block structure. The notion of extension, first appearing in [Reference Block16], gives rise to some particular cases of block structures. Also, the notion of division, introduced in [Reference Li, Misiurewicz, Pianigiani and Yorke24] for interval periodic orbits and generalized in [Reference Alsedà and Ye11] to study the entropy and the set of periods for tree maps, is a particular case of a 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 labelled 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.

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 $f(c)=a$ .

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 non-trivial n-periodic pattern with $n\ge 3$ . For $n>p\ge 2$ , we will say that $\mathcal {P}$  has a p-block structure if there exists a partition $P=P_0\cup P_1\cup \cdots \cup P_{p-1}$ such that $f(P_i)=P_{i+1\bmod p}$ for $i\ge 0$ , and for $i\ne j$ . In this case, p is a strict divisor of n and $|P_i|=n/p$ for $0\le i<p$ . 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 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 $\mathcal {P}$ .

Let $(T,P,f)$ be the canonical model of $\mathcal {P}$ . A p-block structure $P_0\cup P_1\cup \cdots \cup P_{p-1}$ for $\mathcal {P}$ will be said to be separated if for $i\ne j$ . Note that the separability of a block structure for a pattern depends on the particular topology of its canonical model and, as a 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 an 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). In contrast, 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).

Let $\mathcal {P}$ be an n-periodic pattern and let $(T,P,f)$ be the canonical model of $\mathcal {P}$ . Let $P=P_0\cup P_1\cup \cdots \cup P_{p-1}$ be a separated p-block structure for $\mathcal {P}$ . Then, . 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 to a point $x_i$ . Let be the standard projection, which is bijective on and satisfies . Set $Q=\kappa (P)=\{x_0,x_1,\ldots ,x_{p-1}\}$ and define 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 $([S,Q],[\theta ])$ .

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 four trivial blocks. By checking the canonical model $(T,P,f)$ , which is shown in Figure 4 (centre), we see that when $i\ne j$ . 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. (centre) 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 [Reference Alsedà, Guaschi, Los, Mañosas and Mumbrú3, Proposition 8.1]) states.

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.

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 $k\ge 1$ , we will say that $\pi $ splits in k iterates if $f^i(\pi )$ is a basic path of $\mathcal {P}$ for $0\le i<k$ and $f^k(\pi )$ is not a basic path of $\mathcal {P}$ . Equivalently, $f^i(\pi )$ only f-covers $f^{i+1}(\pi )$ for $0\le i<k$ 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 $i\ge 0$ . 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. However, 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 two 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.

4 A mechanism to compare entropies

Another key ingredient to prove Theorem A is a tool, first introduced in [Reference Alsedà, Juher and Mañosas7], 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 $k\ge 2$ 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 $\mathcal {P}$ .

As one may expect from intuition, the entropy of a model decreases when performing an opening, as the following result ([Reference Alsedà, Juher and Mañosas7, Theorem 5.3]) states.

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.

5 Strategy of the proof of Theorem A

In this section, we give a general overview of the proof of Theorem A 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 than 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.

(⋆) $$ \begin{align} \mbox{Every opening of } \mathcal{P} \mbox{ is a zero entropy pattern}. \end{align} $$

Property (⋆) is very restrictive and has a strong consequence: a pattern satisfying property (⋆) is, generically, $\pi $ -reducible. More precisely, we have the following result, that will be proved in §8.

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.

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 (⋆) implies $\pi $ -reducibility: the k-flowers and the triple chain. A k-flower is a pattern consisting on $k\ge 2$ 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 be easily convinced that the flowers and the triple chain are the two sorts of patterns that do not satisfy the property of having at least two inner points and at least three openings. Here, we have, precisely, the step of the proof where we find our candidate $\mathcal {Q}_n$ for minimum entropy, since it has two discrete components attached at a single inner point (a $2$ -flower).

Figure 6

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

The cases of the k-flowers and the triple chain will be tackled in §§9 and 10, respectively. Concerning the k-flowers, the case $k=2$ is especially simple since Theorem A follows directly from a previous result in [Reference Alsedà, Juher and Mañosas8]. However, for $k\ge 3$ , we construct an $n'$ -periodic pattern, where $n'$ 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 $ . As a 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 §5 relies strongly in using property (⋆) 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 §3. It is [Reference Alsedà, Juher and Mañosas6, Proposition 5.6].

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 $\mathcal {S}'$ 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 two trivial blocks, with the associated skeleton $\mathcal {S}'$ being a trivial pattern of two 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 on the topology, next we propose a similar but purely combinatorial simplification mechanism over zero entropy patterns.

Figure 7

(top) A sequence of skeletons. (bottom) The sequence of combinatorial collapses according to Definition 6.4.

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.

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 $r\ge 0$ such that:

1. (a) $\mathcal {P}_r=\mathcal {P}$ ;

2. (b) $\mathcal {P}_0$ is a trivial $p_0$ -periodic pattern;

3. (c) for $1\le i\le r$ , $\mathcal {P}_{i}$ 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 $p_0=3$ , $p_1=2$ , $p_2=3$ .

Figure 8

An example of a zero entropy 18-periodic pattern $\mathcal {P}_2$ and the corresponding sequence of collapses.

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 §8 to prove Theorem D.

Let $\mathcal {P}$ be an n-periodic pattern and let x be a point of $\mathcal {P}$ of valence $\nu \ge 1$ . 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 labelling 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 labelled 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 $\{0,1,4\}$ .

To understand the following result, it is crucial to keep in mind Remarks 3.3 and 6.5 concerning the labelling 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 $\mathcal {P}$ .

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 $P_i$ . However, 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 $Z'$ containing $Q_j$ . The point x will be called bidirectional if $Z\ne Z'$ .

Proof. Let $\mathcal {P}=([T,P],[f])$ be a zero entropy pattern and let $\mathcal {C}$ be the combinatorial collapse of $\mathcal {P}$ . Let $P_0\cup P_1\cup \cdots \cup P_{p-1}$ and $Q_0\cup Q_1\cup \cdots \cup Q_{q-1}$ 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. To not overload the notation, assume without loss of generality that $x=0$ . By the standing labelling conventions, $0\in P_0$ and the collapse of $P_0$ is the point of $\mathcal {C}$ labelled 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

$$ \begin{align*} X:=\bigcup_{1\le k<p/q} P_{kq}. \end{align*} $$

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

(7.1) $$ \begin{align} P_0\cup X\mbox{ is contained in a single 0-branch } Z \mbox{ of } \mathcal{P}. \end{align} $$

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 $0\le i<p/q$ must intersect a single discrete component D of $\mathcal {P}$ . Since $X\cap C=\emptyset $ , by property (7.1), this is only possible if $C=P_0$ (as claimed), D is contained in the 0-branch Z and D is adjacent to C. See Figure 9(centre). Let $x'$ be the only point in $C\cap D=P_0\cap D$ , whose collapse is the point 0 in $\mathcal {C}$ . Then, the $x'$ -branch containing $P_0$ and the $x'$ -branch containing $Q_0$ are different. Therefore, $x'$ is bidirectional and we are done.

Figure 9

The two cases in the proof of Lemma 7.3. The arrows mark the two different $x'$ -branches implying that $x'$ is bidirectional.

Case 2. $X\cap C\ne \emptyset \mbox { and }X\not \subset C$ . In this case, all blocks $P_{iq}$ intersect C and at least one block, say $P_{jq}$ , has an inner point $x'$ in common with C, whose collapse is the point $jq$ in $\mathcal {C}$ . See Figure 9(right). Then, the $x'$ -branch containing $P_{jq}$ and the $x'$ -branch containing $Q_0$ are different. Therefore, $x'$ is bidirectional and we are done.

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, as a consequence, $X\subset C$ . Since $P_0\subset C$ , we get that

$$ \begin{align*} \tilde{P}_0:=P_0\cup X=\bigcup_{0\le k<p/q} P_{kq}\subset D. \end{align*} $$

Set $\tilde {P}_i:=\bigcup _{k=0}^{(p/q)-1} P_{i+kq}$ for $0\le i<q$ . 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 $\tilde {P}_0\cup \tilde {P}_1\cup \cdots \cup \tilde {P}_{q-1}$ is a trivial block structure for $\mathcal {P}$ , in contradiction with the maximality of the structure $P_0\cup P_1\cup \cdots \cup P_{p-1}$ .

Let x be a point of an n-periodic pattern $\mathcal {P}$ and let $\nu \ge 1$ 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 labelled 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, which coincides with its set of discrete components (petals) when y is labelled 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 labelled 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) so that the central point of the flower reads as 0, the petals are written as $\{0,3,5,6\}$ , $\{0,2\}$ and $\{0,4,1\}$ , which are precisely the x-branches of $\mathcal {P}$ .

The previous remark says that in fact the notation $\mathcal {F}_{x}(\mathcal {P})$ , which 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 labelled as 0.

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) $p_i\ge 2$ for $0\le i\le r$ ;

2. (bs2) $\delta _1=1$ ;

3. (bs3) for any $1\le i\le r$ , 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 $\nu \ge 1$ . 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 $1\le i\le r$ , satisfies properties (bs1)–(bs3). 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.

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 $n=p_0p_1\cdots p_r$ 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 $p_0p_1\cdots p_i$ has been defined, with $d_i:=\max \{\delta _j\}_{j=0}^i$ discrete components labelled 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 $p_0p_1\cdots p_{i+1}$ 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 property (bs3), either $\delta _{i+1}\le d_i$ , 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 labelled 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 $j\ne 0$ 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 labelled 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, or 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 $\delta _{i+1}\ne \delta _i$ for all $0\le i<r$ .

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 labelled in such a way that the respective inner points read as 0, then the respective sets of 0-branches coincide. In particular,

(7.2) $$ \begin{align} \prod_{i=0}^r p_i=\prod_{i=0}^t q_i. \end{align} $$

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 equation (7.2), it follows that $r\ge 2$ . Moreover, since S is minimal, $\delta _2\ne \delta _1$ . 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 $0,1,2,\ldots ,p_1-1,p_1$ . As a consequence, $\mathcal {F}(S)$ and $\mathcal {F}(R)$ are not the same pattern, which is a contradiction that proves the claim.

Assume now that all terms of S and R are identical up to an index $j\ge 1$ (the previous claim states that this is true when $j=1$ ). In this case, if S has length j, then equation (7.2) 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 $(p_{j+1},\delta _{j+1})\ne (q_{j+1},\kappa _{j+1})$ , 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 $\mathcal {F}(R)$ .

Let $S=\{(p_i,\delta _i)\}_{i=0}^r$ be a branching sequence. Assume that S is not minimal, that is, for some $0\le j<r$ , we have that $\delta _{j+1}=\delta _j$ . Then, we can consider a reduced sequence $S'=\{(p^{\prime }_i,\delta ^{\prime }_i)\}_{i=0}^{r-1}$ defined as $(p^{\prime }_i,\delta ^{\prime }_i)=(p_i,\delta _i)$ for $0\le i<j$ , $(p^{\prime }_j,\delta ^{\prime }_j)=(p_jp_{j+1},\delta _j)$ and $(p^{\prime }_i,\delta ^{\prime }_i)=(p_{i+1},\delta _{i+1})$ for $j<i\le r-1$ . One can easily check that $S'$ satisfies properties (bs1)–(bs3) and is thus a branching sequence. The following result states that S and $S'$ generate the same flower. It follows immediately from the algorithm of construction of $\mathcal {F}(S)$ .

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 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.

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, which follows immediately from the definitions and the labelling 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.

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

Proof. By Lemma 7.12,

(7.3) $$ \begin{align} \mathcal{F}(S)=\mathcal{F}_x(\mathcal{P}). \end{align} $$

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

(7.4) $$ \begin{align} \mathcal{F}(\mathcal{R})=\mathcal{F}_x(\mathcal{P}). \end{align} $$

However, by Corollary 7.11,

(7.5) $$ \begin{align} \mathcal{F}(S)=\mathcal{F}(\widehat{S}). \end{align} $$

From equations (7.3), (7.4) and (7.5), we get then that

(7.6) $$ \begin{align} \mathcal{F}(R)=\mathcal{R}(\widehat{S}). \end{align} $$

Since $\widehat {S}$ is minimal by definition of a fully reduced sequence and R is minimal by Remark 7.9, then equation (7.6) and Lemma 7.8 imply that $R=\widehat {S}$ .

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 $\mathcal {P}$ .

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.

Proof. To prove the result, we consider two cases.

Case 1. $\mathcal {P}$ has exactly two inner points. In this case, the hypothesis implies 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.

Figure 12

Three possible openings for Case 1 in the proof of Lemma 8.2.

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, statement (a) holds. Consider now a relabelling of the points of $\mathcal {P}$ (and, as a consequence, of $\mathcal {O}$ ) such that $x=0$ . We have now two possibilities.

If $0=\alpha $ , then we take $\mathcal {R}$ as the opening $\mathcal {O}'$ corresponding to the union $C_0\cup C_3$ . So, statement (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 statement (c1) follows from Remark 8.1.

If $0=\beta $ , then we take $\mathcal {R}$ as the opening $\mathcal {O}"$ corresponding to the union $C_0\cup C_1$ . So, statement (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, statement (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, statement (a) holds. Consider now a relabelling of the points of $\mathcal {P}$ (and, as a consequence, of $\mathcal {O}$ ) such that $x=0$ . Let $\alpha \ne 0$ and $\beta \ne 0$ 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, statement (b) holds and, by Remark 8.1, statement (c1) is also satisfied.

Figure 13

Illustration of Case 2 (first 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, statement (b) holds and, by Remark 8.1, statement (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 situation (c2), we need to relate the branching sequences of both a flower $\mathcal {F}$ and an opening $\mathcal {F}'$ of $\mathcal {F}$ .

Figure 14

Illustration of Case 2 (second subcase) in the proof of Lemma 8.2.

Figure 15

A 16-periodic 4-flower with entropy zero.

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 $\mathcal {F}'$ be an opening of $\mathcal {F}$ obtained by joining two discrete components corresponding to two x-branches labelled as $j_1,j_2$ , with $1\le j_1<j_2\le \nu $ . Set $R':=\{(q_i,\kappa ^{\prime }_i)\}_{i=0}^t$ , with $\kappa ^{\prime }_i$ defined as

$$ \begin{align*} \kappa^{\prime}_i=\begin{cases} \kappa_i & \mbox{if } \kappa_i<j_2, \\ j_1 & \mbox{if } \kappa_i=j_2, \\ \kappa_i-1 & \mbox{if } \kappa_i>j_2. \end{cases} \end{align*} $$

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

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 $\mathcal {F}'$ be the opening obtained by joining the discrete components $Z_1$ and $Z_3$ . The 0-branches of $\mathcal {F}'$ , 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 $R'$ defined in the statement of Lemma 8.3 is $R'=\{(2,1),(2,2),(2,1),(2,3)\}$ , which is minimal. According to Lemma 8.3, it is the branching sequence of $\mathcal {F}'$ around 0. As another example, let $\mathcal {F}'$ be the opening of $\mathcal {F}$ obtained by joining the discrete components $Z_2$ and $Z_3$ . In this case, the 0-branches of $\mathcal {F}'$ are $Y_1=Z_1$ , $Y_2=Z_2\cup Z_3$ and $Y_3=Z_4$ . The sequence $R'$ defined in the statement of Lemma 8.3 reads as ${R'=\{(2,1),(2,2),(2,2),(2,3)\}}$ and its fully reduced sequence $\{(2,1),(4,2),(2,3)\}$ is the branching sequence of $\mathcal {F}'$ around 0.

Now, we are in a position to prove 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,

(8.1) $$ \begin{align} \widehat{q}_{\widehat{t}}=q_{t-j}q_{t-j+1}\cdots q_{t-1}q_{t}\quad\mbox{for some }j\ge0. \end{align} $$

However, 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

(8.2) $$ \begin{align} (\widehat{p}_{\widehat{r}},\widehat{\delta}_{\widehat{r}})=(p_r,\delta_r). \end{align} $$

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 equation (8.2). Thus, equation (8.1) 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 $\mathcal {F}':=\mathcal {F}_x(\mathcal {O})$ and $\widehat {R}$ is the branching sequence of the flower $\mathcal {F}:=\mathcal {F}_x(\mathcal {R})$ . Since $\mathcal {F}'$ 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 $\widehat {R}'=\{(\widehat {q}_i,\widehat {\kappa }^{\prime }_i)\}_{i=0}^{\widehat {t}}$ 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

$$ \begin{align*} \widehat{p}_{\widehat{r}}=\widehat{q}_{\widehat{t}-\ell}\widehat{q}_{\widehat{t}-\ell+1}\cdots\widehat{q}_{\widehat{t}-1}\widehat{q}_{\widehat{t}}\quad\mbox{for some }\ell\ge0. \end{align*} $$

The previous equality and equation (8.1) imply that $q_t$ divides $p_r$ also in this case. As a 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 $\mathcal {O}$ ,

$$ \begin{align*} O_0=\bigg\{0,\frac{n}{p_r},\frac{2n}{p_r},\ldots,\frac{(p_r-1)n}{p_r}\bigg\}. \end{align*} $$

By Proposition 3.7, $\mathcal {O}$ is $\pi $ -reducible for any basic path $\pi $ contained in $O_0$ . However, the inner point $0$ belongs to the block of $\mathcal {R}$ ,

$$ \begin{align*} R_0=\bigg\{0,\frac{n}{q_t},\frac{2n}{q_t},\ldots,\frac{(q_t-1)n}{q_t}\bigg\}. \end{align*} $$

Again, $\mathcal {R}$ is $\pi $ -reducible for any basic path $\pi $ contained in $R_0$ . Since $q_t$ divides $p_r$ , the point ${n}/{q_t}$ belongs to $O_0\cap R_0$ . Take $\pi :=\{0,{n}/{q_t}\}$ . 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 §5, we have to deal now with the special case of patterns with only one inner point and $k\ge 2$ discrete components. When $k=2$ , the following result ([Reference Alsedà, Juher and Mañosas8, Theorem 5.2]) does the job.

For $k\ge 3$ , and in the spirit of the proof by induction outlined in §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 $\mathcal {P}'$ 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 $P'\subset P$ of $f^p$ such that . Clearly, this definition is independent of the particular model $(T,P,f)$ representing $\mathcal {P}$ .

The following result is [Reference Alsedà, Juher and Mañosas7, Lemma 9.1]. It allows us to estimate the entropy of a pattern from the entropy of a subordinate.

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 . 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 it does not escape 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 (as in [Reference Alsedà, Juher and Mañosas7, Lemma 4.2]).