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1 Introduction

We continue the study of combinatorial properties of infinite permutations analogous to those of words. In this approach, infinite permutations are interpreted as equivalence classes of real sequences with distinct elements, such that only the order of elements is taken into account. In other words, an infinite permutation is a linear order in \(\mathbb N\). We consider it as an object close to an infinite word, but instead of symbols, we have transitive relations \(<\) or \(>\) between each pair of elements.

Infinite permutations in the considered sense were introduced in [10]; see also a very similar approach coming from dynamics [6] and summarised in [3]. Since then, they were studied in two main directions: First, a series of results compared properties of infinite permutations with those of infinite words ([4, 10, 11] and others). Secondly, different authors studied permutations directly constructed with the use of general words [7, 13], as well as precise examples: the Thue-Morse word [14, 18], other morphic words [17, 19] or Sturmian words [15].

In the previous paper [5], we introduced the notion of an ergodic permutation, which means that a permutation can be defined by a sequence of numbers from [0,1] such that the frequency of its elements in any interval is equal to the length of the interval. We proved also that the minimal complexity (i.e., the number of subpermutations of length n) of an ergodic permutation is n, and the permutations of minimal complexity are Sturmian permutations in the sense of [15] (and close to the sense of [4]). So, the situation for ergodic permutations is similar to that for words. Note that for the permutations in general, this is not the case: The complexity of an aperiodic permutation can grow slower than any unbounded growing function [10].

In this paper, we focus on permutations generated by words. First of all, we prove that such a permutation is ergodic if and only if its generating word is uniquely ergodic, which explains the choice of the term. Then we generalize the construction of Makarov [14] and give a general method to construct the canonical representative sequence of any permutation generated by a fixed point of a primitive monotone separable morphism. We also discuss why this method cannot be directly extended further, and give some examples.

2 Basic Definitions

We consider finite and infinite words over a finite alphabet \(\varSigma _q = \{0, 1,q-1\}\). A factor of an infinite word is any sequence of its consecutive letters. The factor \(u[i]\cdots u[j]\) of an infinite word \(u=u[0] u[1] \cdots u[n] \cdots \), with \(u[k] \in \varSigma \), is denoted by u[i..j]; prefixes of a finite or an infinite word are as usual defined as starting factors.

The length of a finite word s is denoted by |s|. An infinite word \(u=vww\cdots = vw^{\omega }\) for some non-empty word w is called ultimately (|w|-)periodic; otherwise it is called aperiodic.

When considering words on \(\varSigma _q\), we refer to the order on finite and infinite words meaning lexicographic (partial) order: \(0<1<\ldots <q-1\), and \(u<v\) if \(u[0..i]=v[0..i]\) and \(u[i+1]<v[i+1]\) for some i. For words such that one of them is the prefix of the other the order is not defined.

Now we recall the notion of the uniform frequency of letters and factors in an infinite word. For finite words v and w, we let \(|v|_w\) denote the number of occurrences of w in v. The infinite word u has uniform frequencies of factors if, for every factor w of u, the ratio \(\frac{|u[i..i+n]|_w}{n+1}\) has a limit \(\rho _w(u)\) when \(n\rightarrow \infty \) uniformly in k. For more on uniform frequencies in words we refer to [8].

To define infinite permutations, we will use sequences of real numbers. Analogously to a factor of a word, for a sequence \((a[n])_{n=0}^{\infty }\) of real numbers, any of its finite subsequences \(a[i],a[i+1],\ldots ,a[j]\) is called a factor and is denoted by a[i..j]. We define an equivalence relation \(\sim \) on real infinite sequences with pairwise distinct elements as follows: \((a[n])_{n=0}^{\infty }\sim (b[n])_{n=0}^{\infty }\) if and only if for all i, j the conditions \(a[i]<a[j]\) and \(b[i]<b[j]\) are equivalent. Since we consider only sequences of pairwise distinct real numbers, the same condition can be defined by substituting \((<)\) by \((>)\): \(a[i]>a[j]\) if and only if \(b[i]>b[j]\). An infinite permutation is then defined as an equivalence class of real infinite sequences with pairwise distinct elements. So, an infinite permutation is a linear ordering of the set \(\mathbb N_0=\{0,\ldots ,n,\ldots \}\). We denote it by \(\alpha =(\alpha [n])_{n=0}^{\infty }\), where \(\alpha [i]\) are abstract elements equipped by an order: \(\alpha [i] <\alpha [j]\) if and only if \(a[i]<a[j]\) or, which is the same, \(b[i]<b[j]\) for every representative sequence (a[n]) or (b[n]) of \(\alpha \). So, one of the simplest ways to define an infinite permutation is by a representative, which can be any sequence of pairwise distinct real numbers.

Example 2.1

Both sequences \((a[n])=(1,-1/2,1/4,\ldots )\) with \(a[n]=(-1/2)^n\) and (b[n]) with \(b[n]=1000+(-1/3)^n\) are representatives of the same permutation \(\alpha =\alpha [0],\alpha [1],\ldots \) defined by

$$\begin{aligned} \alpha [2n]>\alpha [2n+2]>\alpha [2k+3]>\alpha [2k+1] \end{aligned}$$

for all \(n,k\ge 0\).

A factor \(\alpha [i..j]\) of an infinite permutation \(\alpha \) is a finite sequence \((\alpha [i],\alpha [i+1],\ldots ,\alpha [j])\) of abstract elements equipped by the same order as in \(\alpha \). Note that a factor of an infinite permutation can be naturally interpreted as a finite permutation: for example, if in a representative (a[n]) we have a factor (2.5, 2, 7, 1.6), that is, the 4th element is the smallest, followed by the 2nd, 1st and 3rd, then in the permutation, it will correspond to a factor \(\begin{pmatrix} 1 &{} 2 &{} 3 &{} 4 \\ 3 &{} 2 &{} 4 &{} 1 \end{pmatrix}\), which we will denote simply as (3241). Note that in general, we index the elements of infinite objects (words, sequences or permutations) starting with 0 and the elements of finite objects starting with 1.

A factor of a sequence (permutation) should not be confused with its subsequence \(a[n_0],a[n_1],\ldots \) (subpermutation \(\alpha [n_0],\alpha [n_1],\ldots \)) which is defined as indexed with a growing subsequence \((n_i)\) of indices.

Note, however, that in general, an infinite permutation cannot be defined as a permutation of \(\mathbb N_0\). For instance, the permutation from Example 2.1 has all its elements between the first two ones.

3 Ergodic Permutations

Let \((a[i])_{i=0}^{\infty }\) be a sequence of real numbers from the interval [0,1], representing an infinite permutation, a and p also be real numbers from [0,1]. We say that the probability that an element of (a[i]) is less than a exists and is equal to p if the ratio

$$\begin{aligned} \frac{ \# \{a[j+k]|0\le k <n, a[j+k]<a\}}{n}\end{aligned}$$

has a limit p when \(n \rightarrow \infty \) uniformly in j.

In other words, if we substitute all the elements from (a[i]) which are smaller than a by 1, and those which are bigger by 0, the above condition means that the uniform frequency of the letter 1 exists and equals p. So, in fact the probability to be smaller than a is the uniform frequency of the elements which are less than a.

We note that this is not exactly probability on the classical sense, since we do not have a random sequence. But we are interested in permutations where this “probability” behaves in certain sense like the probability of a random sequence uniformly distributed on [0,1]:

Definition 3.1

A sequence \((a[i])_{i=0}^{\infty }\) of real numbers is canonical if

  • all the numbers are pairwise distinct;

  • for all i we have \(0\le a[i] \le 1\);

  • and for all a, the probability for any element a[i] to be less than a is well-defined and equal to a for all \(a\in [0,1]\).

Remark 3.2

The set \(\{a[i]|i \in \mathbb N\}\) for a canonical sequence (a[i]) is dense on [0,1].

Remark 3.3

In a canonical sequence, the frequency of the elements which fall into any interval \((t_1,t_2)\subseteq [0,1]\) exists and is equal to \(t_2-t_1\).

Remark 3.4

Symmetrically to the condition “the probability to be less than a is a” we can consider the equivalent condition “the probability to be greater than a is \(1-a\)”.

Definition 3.5

An infinite permutation \(\alpha =(\alpha [i])_{i=1}^{\infty }\) is called ergodic if it has a canonical representative.

Example 3.6

For any irrational \(\sigma \) and for any \(\rho \), consider the sequence of fractional parts \(\{\rho +n\sigma \}\). It is uniformly distributed in [0, 1), so, the respective permutation is ergodic. In fact, such a permutation is a Sturmian permutation in the sense of [14]; in [4], the considered class of permutations is wider than that. It is easy to see that Sturmian permutations are directly related to Sturmian words [12, 16].

Proposition 3.7

An ergodic permutation \(\alpha \) has a unique canonical representative.

Proof. Given \(\alpha \), for each i we define

$$\begin{aligned} a[i]=\lim _{n\rightarrow \infty } \frac{\#\{\alpha [k]|0\le k <n, \alpha [k]<\alpha [i]\}}{n} \end{aligned}$$

and see that, first, this limit must exist since \(\alpha \) is ergodic, and secondly, a[i] is the only possible value of an element of a canonical representative of \(\alpha \).    \(\square \)

Note, however, that even if for some infinite permutation all the limits above exist, it does not imply the existence of the canonical representative. Indeed, there is another condition to fulfill: for different i the limits must be different.

4 Ergodic Permutations Generated by Words

Consider an aperiodic infinite word \(u=u[0]\cdots u[n] \cdots \) over \(\varSigma _q\) and, as usual, define its nth shift \(T^nu\) as the word obtained from u by erasing the first n symbols: \(T^n u = u[n] u[{n+1}] \cdots \). We can also interpret a word u as a real number 0.u in the q-ary representation.

If the word u is aperiodic, then in the sequence \((0.T^n u)_{n=0}^\infty \) all the numbers are different and thus this sequence is a representative of a permutation which we denote by \(\alpha _u\). Clearly, \(\alpha _u[i]<\alpha _u[j]\) if and only if \(T^i u\) is lexicographically smaller than \(T^j u\). A permutation which can be constructed like this is called valid; the structure of valid permutations has been studied in [13] (for the binary case) and [7] (in general).

Most of results of this paper were inspired by the following construction.

Example 4.1

The famous Thue-Morse word \(0110100110010110\cdots \) is defined as the fixed point starting with 0 of the morphism \(f_{tm}: 0 \mapsto 01, 1 \mapsto 10\) [1, 2]. The respective Thue-Morse permutation defined by the representative (\(0.01101001\cdots \), \(0.11010011\cdots \), \(0.10100110\cdots \), \(0.01001100\cdots \),\(\ldots \)) can also be defined by the following sequence, denoted by \(a_{tm}\):

$$\begin{aligned}\frac{1}{2},1,\frac{3}{4},\frac{1}{4},\frac{5}{8},\frac{1}{8},\frac{3}{8},\frac{7}{8},\cdots ,\end{aligned}$$

that is the fixed point of the morphism \(\varphi _{tm}: [0,1] \mapsto [0,1]^2\):

$$\begin{aligned} \varphi _{tm}(x)={\left\{ \begin{array}{ll} \frac{x}{2}+\frac{1}{4}, \frac{x}{2}+\frac{3}{4}, \text{ if } 0 \le x \le \frac{1}{2},\\ \frac{x}{2}+\frac{1}{4}, \frac{x}{2}-\frac{1}{4}, \text{ if } \frac{1}{2} < x \le 1. \end{array}\right. } \end{aligned}$$

It will be proved below that the latter sequence is canonical and thus the Thue-Morse permutation is ergodic. This construction and the equivalence of the two definitions was proved by Makarov in 2009 [15]; then the properties of the Thue-Morse permutation were studied by Widmer [18].

When is a valid permutation ergodic? The answer is simple and explains the choice of the term “ergodic”.

Lemma 4.2

A valid permutation \(\alpha _u\) for a recurrent non-periodic word u is ergodic if and only if all the uniform frequencies of factors in u exist and are not equal to 0.

Before proving the lemma, we prove the following proposition about words:

Proposition 4.3

Let u be a recurrent aperiodic word and w and v some of its factors. Then in the orbit of w there can be the lexicographically maximal word from its closure starting with w, or the lexicographically minimal word from its closure starting with v, but not both at a time.

Proof. Suppose the opposite: let \(T^k(u)\) be the maximal element of the orbit closure of u starting with w, and \(T^l(u)\) be the minimal element of the orbit closure of u starting with v. Consider the prefix r of u of length \(\max (k+|u|, l+|v|)\). Since u is recurrent, this prefix appears in it an infinite number of times, and since u is not ultimately periodic, there exists an extension p of r to the right which is right special: pa and pb are factors of u for some symbols \(a \ne b\). Suppose that the prefix of u of the respective length is pa, and pb is a prefix of \(T^n(u)\).

If \(a<b\), then \(u < T^n(u)\) and thus \(T^k(u)< T^{k+n}(u)\), where \(T^{k+n}(u)\) starts with w. A contradiction with the maximality of \(T^k(u)\). If by contrary \(a>b\), then \(u > T^n(u)\) and thus \(T^l(u)> T^{l+n}(u)\), where \(T^{l+n}(u)\) starts with v. A contradiction with the minimality of \(T^l(u)\). The proposition is proved.    \(\square \).

Proof of Lemma 4.2.

Suppose first that the frequency \(\mu (w)\) of each factor w in u exists and is non-zero. We should prove that the corresponding valid permutation is ergodic. For every k we define

$$\begin{aligned}a[k]=\lim _{n \rightarrow \infty } \sum _{\begin{array}{c} |v|=n,\\ v \le w[k]\cdots w[{k+n-1}] \end{array}} \mu (v).\end{aligned}$$

Clearly, such a limit exists and is in [0,1], and by the definition, the probability that another element of the sequence (a[i]) is less than a[k] is equal to a[k].

It remains to prove that \(a[k]\ne a[l]\) for \(k \ne l\), that is, that the sequence (a[n]) is indeed a representative of a permutation.

Suppose the opposite: \(a[k]=a[l]\) for \(k \ne l\). Let \(m\ge 0\) be the first position such that \(w[{k+m}]\ne w[{l+m}]\): say, \(w[{k+m}]<w[{l+m}]\). The only possibility for a[l] and a[k] to be equal is that \(T^k(w)=w[k] w[{k+1}]\cdots \) is the maximal word in the orbit closure of w starting with \(w[k]\cdots w[{k+m}]\), and \(T^l(w)=w[l] w[{l+1}]\cdots \) is the minimal word in the orbit closure of w starting with \(w[l]\cdots w[{l+m}]\). Due to Proposition 4.3, this is a contradiction. So, the values a[k] are indeed all different, and thus the permutation is well-defined. Together with the condition on the probabilities we proved above, we get that the corresponding valid permutation is ergodic.

The proof of the converse is split into two parts. First we prove that for a valid ergodic permutation the frequencies of factors in the corresponding word must exist, then we prove that they are non-zero.

So, first we suppose that the frequencies of (some) factors of w do not exist. We are going to prove that the permutation is not ergodic, that is, that the canonical representative sequence (a[n]) is not well-defined.

Let us take the shortest and lexicographically minimal factor w whose frequency does not exist and consider the subsequence \((a[{n_i}])\) of the sequence (a[n]) corresponding to suffixes starting with w. The upper limit of \((a[{n_i}])\) should be equal to the sum of frequencies of the words of length |w| less than or equal to w, but since the frequency of w is the only one of them that does not exist, this limit also does not exist. So, the sequence (a[n]) is not well-defined and hence the corresponding valid permutation is not ergodic.

The remaining case is that of zero frequencies: Suppose that w is the shortest and lexicographically minimal factor whose frequency is zero, and consider again the subsequence \((a[{n_i}])\) of the sequence (a[n]) corresponding to suffixes starting with w. The subsequence \((a[{n_i}])\) is infinite since u is recurrent, but all its elements must be equal: Their value is the sum of frequencies of words of length |w| lexicographically less than w. So, the sequence (a[n]) does not correctly define a permutation, and hence in the case of zero frequencies the corresponding valid permutation is not ergodic.   \(\square \)

We have seen above in Example 3.6 how the canonical representatives of permutations corresponding to Sturmian words are built.

Example 4.4

Let us continue the Thue-Morse example started above and prove that the representative \(a_{tm}\) is canonical. We should prove that the probability for any element a[j] to be less than a is well-defined and equal to a. Let us prove by induction on k that the probability for an element to be in any binary rational interval \((d/2^{k},(d+1)/2^k]\), where \(0\le d < 2^k\), is exactly \(1/2^k\). Indeed, by the construction, the intervals (0,1/2] and (1 / 2, 1] correspond to the zeros and ones in the original Thue-Morse word whose frequencies are 1/2. The morphic image of any of these intervals is, consecutively, two intervals: \((0,1/2] \mapsto (1/4,2/4],(3/4,4/4]\), and \((1/2,1] \mapsto (2/4,3/4],(0,1/4]\). So, in both cases, the intervals are of the form \((d/2^2,(d+1)/2^2]\), \(d=0,\ldots ,3\). Each of them is twice rarer than its pre-image; the four intervals cover (0, 1] and do not intersect, so, the probability for a point a[i] to be in each of them is 1/4. But exactly the same argument works for any of these four intervals: its image is two intervals which are twice smaller and twice rarer than the pre-image interval. No other points appear in that shorter interval since each mapping corresponding to a position in the morphism is linear, and their ranges do not intersect. So, the probability for a point to be in an interval \((d/2^3,(d+1)/2^3]\) is 1/8, and so on. By induction, it is true for any binary rational interval and thus for all interval subsets of (0, 1]: the frequency of elements in this interval is equal to its length. This proves that \(a_{tm}\) is indeed the canonical representative of the Thue-Morse permutation.

Remark 4.5

This example shows that the natural way of constructing the canonical representative of a valid permutation has little in common with frequencies of factors in the underlying word. The frequencies of symbols look important, but, for example, the frequency of 00 in the Thue-Morse word is 1/6, whereas all the elements of the canonical representative are binary rationals.

Remark 4.6

In Lemma 4.2, we assumed that the word is recurrent. Indeed, if a word is not recurrent, the permutation can be ergodic. As an example, consider the word

$$\begin{aligned}01221211221121221\cdots ,\end{aligned}$$

that is, 0 followed by the Thue-Morse word on the alphabet \(\{1,2\}\). The respective permutation is still ergodic with the canonical representative \(0,a_{tm}=0,1/2,1,3/4,1/4,\ldots \).

Note also that this property depends on the order of symbols. For example, the permutation associated with the word

$$\begin{aligned}20110100110010110\cdots =2 u_{tm}\end{aligned}$$

is not ergodic since \(a_tm[0]\) can be equal only to 1. On the other hand, it is well known that the first shift of the Thue-Morse word is the lexicographically largest element in its shift orbit closure. So, \(a_tm[1]\) must also be equal to 1.

4.1 Morphisms on Words and Intervals

In this subsection, we generalize the above construction for the Thue-Morse word to a class of fixed points of morphisms: for any word from that class, we construct a morphism similar to the Thue-Morse interval morphism \(\varphi _{tm}\) defined in Example 4.1.

Let \(\varphi : \{0,\ldots ,q-1\}^*\mapsto \{0,\ldots ,q-1\}^*\) be a morphism and \(u=\varphi (u)\) be its aperiodic infinite fixed point starting with a letter a if it exists. In what follows we give a construction of the canonical representative \(a_u\) of the permutation \(\alpha _u\) provided that the morphism \(\varphi \) is primitive, monotone and separable. We will now define what these properties mean.

Recall that the matrix A of a morphism \(\varphi \) is a \(q\times q\)-matrix whose element \(a_{ij}\) is equal to the number of occurrences of i in \(\varphi (j)\). A matrix A and a morphism \(\varphi \) are called primitive if in some power \(A^n\) of A all the entries are positive, i.e., for every \(b \in \{0,\ldots ,q-1\}\) all the symbols of \(\{0,\ldots ,q-1\}\) appear in \(\varphi ^n(b)\) for some n. A classical Perron-Frobenius theorem says that a primitive matrix has a dominant positive Perron-Frobenius eigenvalue \(\theta \) such that \(\theta >|\lambda |\) for any other eigenvalue \(\lambda \) of A. It is also well-known that a fixed point of a primitive morphism is uniquely ergodic, and that the vector \(\mu =(\mu (0),\ldots ,\mu (q-1))^t\) of frequencies of symbols is the normalized Perron-Frobenius eigenvector of A:

$$\begin{aligned}A\mu =\theta \mu .\end{aligned}$$

We say that a morphism \(\varphi \) is monotone on an infinite word u if for any \(n,m>0\) we have \(T^n(u)<T^m(u)\) if and only if \(\varphi (T^n(u))<\varphi (T^m(u))\); here \(<\) denotes the lexicographic order. A morphism is called monotone if it is monotone on all infinite words, or, equivalently, if for any infinite words u and v we have \(u<v\) if and only if \(\varphi (u)<\varphi (v)\).

Example 4.7

The Thue-Morse morphism \(\varphi _{tm}\) is monotone since \(01=f_{tm}(0)<f_{tm}(1)=10\).

Example 4.8

The Fibonacci morphism \(\varphi _f: 0 \rightarrow 01, 1 \rightarrow 0\) is not monotone since \(01=\varphi _f(0)>\varphi _f(10)=001\), whereas \(0<10\). At the same time, \(\varphi _f^2: 0 \rightarrow 010, 1 \rightarrow 01\) is monotone since for all \(x,y\in \{0,1\}\) we have \(\varphi ^2_f(0x)=0100x'<0101y'=\varphi _f(1y)\), where \(x',y'\in \{0,1\}^*\). So, to use our construction to the Fibonacci word \(u_f=01001010\cdots \) which is the fixed point of \(\varphi _f\), we should consider \(u_f\) as the fixed point of \(\varphi _f^2\).

Example 4.9

As an example of a morphism which does not become monotone even when we consider its powers, consider \(g: 0 \rightarrow 02, 1 \rightarrow 01, 2 \rightarrow 21\). It can be easily seen that \(g^n(0)>g^n(1)\) for all \(n \ge 1\).

The last condition we require from our morphism is to be separable. To define this property, consider the fixed point u as the infinite catenation of morphic images of its letters and say that the type \(\tau (n)\) of a position n is the pair (a, p) such that \(u[n]=\varphi (a)[p]\) in this “correct” decomposition into images of letters. So, there are \(\sum _{a=0}^{q-1} |\varphi (a)|\) different types of positions in u. Also note that we index the elements of u starting with 0 and the elements of finite words \(\varphi (a)\) starting from 1, so that, for example, \(\tau (0)=(u[0],1)\).

We say that a fixed point u of a morphism \(\varphi \) is separable if for every n, m such that \(\tau (n)\ne \tau (m)\) the relation between \(T^n(u)\) and \(T^m(u)\) is uniquely defined by the pair \(\tau (n), \tau (m)\). For a separable morphism \(\varphi \) we write \(\tau (n)\preceq \tau (m)\) if and only if \(T^n(u) \le T^m(u)\).

Example 4.10

The Thue-Morse word is separable since for \(\tau (n)=(0,1)\) and \(\tau (m)=(1,2)\) we always have \(T^n(u_{tm})>T^m(u_{tm})\), i.e., all zeros which are first symbols of \(f_{tm}(0)=01\) give greater words than zeros which are second symbols of \(f_{tm}(1)=10\). Symmetrically, all ones which are first symbols of \(f_{tm}(1)=10\) give smaller words than ones which are second symbols of \(f_{tm}(0)=01\), that is, for \(\tau (n)=(1,1)\) and \(\tau (m)=(0,2)\) we always have \(T^n(u_{tm})<T^m(u_{tm})\).

Example 4.11

The fixed point

$$\begin{aligned}u=001001011001001011001011011\cdots \end{aligned}$$

of the morphism \(0 \rightarrow 001, 1 \rightarrow 011\) is inseparable. Indeed, compare the following shifts: \(T^2(u)=1001011001\cdots \), \(T^5(u)=1011\cdots \) and \(T^{17}(u)=1001011011\cdots \). We see that \(T^2(u)<T^{17}(u)<T^5(u)\). At the same time, \(\tau (2)=\tau (5)=(0,3)\), and \(\tau (17)=(1,3)\).

Note that the class of primitive monotone separable morphisms includes in particular all morphisms considered by Valyuzhenich [17] who gave a formula for the permutation complexity of respective fixed points.

Similarly to morphisms on words, we define a morphism on sequences of numbers from an interval [a, b] as a mapping \(\varphi : [a,b]^*\mapsto [a,b]^*\). A fixed point of the morphism \(\varphi \) is defined as an infinite sequence \(a[0], a[1], \dots \) of numbers from [a, b], such that \(\varphi (a[0], a[1], \dots ) = a[0], a[1], \dots \). Clearly, if a morphism \(\varphi \) has a fixed point, then there exists a number \(c\in [a,b]\) such that \(\varphi (c) = c, c[1], \dots , c[k]\) for some \(k\ge 1\) and \(c[i] \in [a,b]\) for \(i=1, \dots k\). Clearly, a fixed point of a morphism on sequences of numbers defines an infinite permutation (more precisely, its representative) if and only if all the elements of the sequence are distinct. The example of morphism defining an infinite permutation is given by the Thue-Morse permutation described in Example 4.1.

The rest of the section is organized as follows: First we provide the construction of a morphic ergodic permutation, then we give some examples, and finally we prove the correctness of the construction.

The construction of ergodic permutation corresponding to a separable fixed point of a monotone primitive morphism.

Now let us consider a separable fixed point u of a monotone primitive morphism \(\varphi \) over the alphabet \(\{0,\ldots ,q-1\}\), and construct the canonical representative \(a_u\) of the premutation \(\alpha _u\) generated by it. To do it, we first look if u contains lexicographically minimal or maximal elements of the orbit with a given prefix. Note that due to Proposition 4.3, it cannot contain both of them. So, if u does not contain lexicographically maximal elements, we consider all the intervals to be half-open \([\cdot )\); in the opposite case, we can consider them to be half-open \((\cdot ]\), like in the Thue-Morse case. Without loss of generality, in what follows we write the intervals \([\cdot )\), but the case of \((\cdot ]\) is symmetric.

So, let \(\mu =(\mu _0,\ldots ,\mu _{q-1})\) be the vector of frequencies of symbols in u. Take the intervals \(I_0=[0,\mu _0)\), \(I_1=[\mu _0,\mu _0+\mu _1)\), \(\ldots \), \(I_{q-1}=[1-\mu _{q-1},1)\). An element e of \(a_u\) is in \(I_b\) if for another element of \(a_u\) the probability to be less than e is greater than the sum of frequences of letters less than b, and the probability to be greater than e is greater than the sum of frequences of letters greater than b. In other words, e is in \(I_b\) if and only if the respective symbol of u is b.

Now let us take all the \(k=\sum _{a=0}^{q-1} |\varphi (a)|\) types of positions in u and denote them according to the order \(\preceq \):

$$\begin{aligned}\tau _1\prec \tau _2 \prec \cdots \prec \tau _k,\end{aligned}$$

with \(\tau _i=(a_i,p_i)\).

For each \(\tau _i\) the frequency \(l_i=\mu _{a_i}/\theta \), where \(\theta \) is the Perron-Frobenius eigenvalue of \(\varphi \), is the frequency of symbols of type \(\tau _i\) in u. Indeed, the \(\varphi \)-images of \(a_i\) are \(\varTheta \) times rarer in u than \(a_i\), and \(\tau _i\) corresponds just to a position in such an image. Denote

$$\begin{aligned}J_1=[0,l_1), J_2=[l_1,l_1+l_2),\ldots , J_k=[1-l_k,1);\end{aligned}$$

so that in general, \(J_i=[\sum _{m=1}^{i-1}l_m,\sum _{m=1}^{i}l_m)\). We will also denote \(J_i=J{a_i,p_i}\).

The interval \(J_i\) is the range of elements of \(a_u\) corresponding to the symbols of type \(\tau _i\) in u. Note that all symbols of the same type are equal, and on the other hand, each symbol is of some type. For example, we have a collection of possible positions of 0 in images of letters, that is, a collection of types corresponding to 0, and all these types are less than any other type corresponding to any other symbol. So, the union of elements \(J_i\) corresponding to 0 is exactly \(I_0\), and the same argument can be repeated for any greater symbol. In particular, each \(J_i\) is a subinterval of some \(I_{a}\).

Now we define the morphism \(\psi : [0,1]^* \mapsto [0,1]^*\) as follows: For \(x \in I_a\) we have

$$\begin{aligned}\psi (x) = \psi _{a,1}(x),\ldots \psi _{a,|\varphi (a)|}.\end{aligned}$$

Here \(\psi _{a,p}\) is a linear mapping \(\psi _{a,p}:I_a \mapsto J_{a,p}\): If \(I_a=[x_1,x_2)\) and \(J_{a,p}=[y_1,y_2)\), then

$$\begin{aligned} \psi _{a,p}(x)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1. \end{aligned}$$
(1)

Now we can define the starting point, that is, the value of \(a_1\). Suppose that the first symbol of u is b; then \(\varphi (b)\) starts with b, which means that \(J_{b,1}\subset I_b\), and the mapping \(\psi _{b,1}\) has a fixed point x: \(\psi _{b,1}(x)=x\). We take \(a_1\) to be this fixed point: \(a_1=x\). Note that if \(a_1\) is the upper end of \(J_{b,1}\), then we should take all the intervals to be \((\cdot ]\); if it is the lower end, the intervals are \([\cdot )\); if it is in the middle of the interval, the ends are never attained. The situation when \(a_1\) is an end of \(J_{b,1}\) corresponds to the situation when there are the least or the greatest infinite words starting from some prefix in the orbit of u; as we have seen in Proposition 4.3, only one of these situations can appear at a time. In particular, in this situation, u is the least (or greatest) element of its orbit starting with b.

This construction may look bulky, but in fact, it is just a natural generalization of that for the Thue-Morse word. Indeed, in the Thue-Morse word, \(\mu _0=\mu _1=1/2\), \(\theta =2\), and the order of types is given in Example 4.10. So, \(I_0=[0,1/2]\), \(I_1=[1/2,1]\), \(J_{0,1}=[1/4,1/2]\), \(J_{0,2}=[3/4,1]\), \(J_{1,1}=[1/2,3/4]\), \(J_{1,2}=[0,1/4]\). Here the intervals are written as closed since at this stage we do not yet know whether we must take them \([\cdot )\) or \(( \cdot ]\). However, it becomes clear as soon as we consider the mapping \(\psi _{0,1}\) which is the linear order-preserving mapping \(I_0 \mapsto J_{0,1}\). Its fixed point is 1/2, that is, the upper end of both intervals. Thus, the intervals must be chosen as \((\cdot ]\). The mappings \(\psi _{a,p}\) are explicitly written down in Example 4.1.

To give another example, consider the square of the Fibonacci morphism mentioned in Example 4.8.

Example 4.12

Consider the Fibonacci word as the fixed point of the square of the Fibonacci morphism: \(\varphi ^2_{f}: 0 \rightarrow 010, 1 \rightarrow 01\). This morphism is clearly primitive; also, it is monotone as we have seen in Example 4.8, and separable: we can check that \((0,3)\preceq (0,1) \preceq (1,1) \preceq (0,2) \preceq (1,2)\). In particular, this means that zeros which are first symbols of \(\varphi ^2_f\) are in the middle among other zeros. So, in what follows we can consider open intervals since their ends are never attained.

The Perron-Frobenius eigenvalue is \(\theta =(3+\sqrt{5})/2\), the frequencies of symbols are \(\mu _0=(\sqrt{5}-1)/2\) and \(\mu _1=(3-\sqrt{5})/2\). So, we have

$$\begin{aligned}I_0=\left( 0,\frac{\sqrt{5}-1}{2}\right) ,I_1=\left( \frac{\sqrt{5}-1}{2},1 \right) \!\!,\end{aligned}$$

and divide their lengths by \(\theta \) to get the lengths of intervals corresponding to symbols from their images:

$$\begin{aligned}|J_{0,1}|=|J_{0,2}|=|J_{0,3}|=\frac{\mu _0}{\theta }=\sqrt{5}-2, \; |J_{1,1}|=|J_{1,2}|=\frac{\mu _1}{\theta }=\frac{7-3\sqrt{5}}{2}.\end{aligned}$$

The order of intervals is shown at Fig. 1.

Fig. 1.
figure 1

Intervals for the Fibonacci permutation morphism

Full size image

Now the morphism \(\psi \) can be completely defined:

$$\begin{aligned}\psi (x)={\left\{ \begin{array}{ll} \psi _{0,1}(x),\psi _{0,2}(x),\psi _{0,3}(x) \text{ for } x \in I_0,\\ \psi _{1,1}(x), \psi _{1,2}(x) \text{ for } x \in I_1. \end{array}\right. } \end{aligned}$$

Here the mappings \(\psi _{a,p}: I_a \mapsto J_{a,p}\) are defined according to (1). In particular, \(\psi _{0,1}: (0,(\sqrt{5}-1)/2) \mapsto (\sqrt{5}-2,2(\sqrt{5}-2))\) has the fixed point \(x=\psi _{0,1}(x)=(3-\sqrt{5})/2\). This is the starting point \(a_1\) of the fixed point a of \(\psi \).

We remark that we could prove directly that the sequence a constructed above is exactly the canonical representative of the permutation associated with the Fibonacci word, using the fact that Fibonacci word belongs to the family of Sturmian words. However, we do not provide the proof for this example, since we now give a more general proof of the correctness of the general construction: the fixed point of the morphism \(\psi \) described above is indeed the canonical representative of our permutation.

Proof of correctness of the construction of the morphism \(\psi \) .

First we show that the fixed point of \(\psi \) is a representative of our permutation. Indeed, if \(T^n(u)<T^m(u)\), and n and m are of different types, then, since the morphism is separable and by the construction, a[n] and a[m] are in different intervals \(J_{a,p}\), and \(a[n]<a[m]\). Now suppose that n and m are of the same type (a, p), that is, the nth (mth) symbol of u is the symbol number p of the image \(\varphi (a)\), where a is the symbol number \(n'\) (\(m'\)) of u, i.e., \(u[n']=a\), \(u[n]=\varphi (a)[p]\), and applying the morphism \(\varphi \) to u sends \(u[n']\) to \(u[n-p+1 .. n-p+|\varphi (a)|]\). Then, since the morphism is monotone, \(T^n(u)<T^m(u)\) if and only of \(T^{n'}(u)<T^{m'}(u)\). Exactly the same condition is true for the relation \(a_[n]<a_{m}\) if and only if \(a_{n'}<a_{m'}\), since the mapping \(\psi _{a,p}\) preserves the order. Now we can apply the same arguments to \(m'\) and \(n'\) instead of m and n, and so on. So, by the induction on the maximal power of \(\varphi \) involved, we also get that \(T^n(u)<T^m(u)\) if and only if \(a_[n]<a_[m]\). So, the sequence a is indeed a representative of the permutation generated by u.

It remains to prove that this representative is canonical. As above for the Thue-Morse word, it is done inductively on the intervals

$$\begin{aligned}\psi _{b_k,p_k}(\psi _{b_{k-1},p_{k-1}}( \ldots \psi _{b_1,p_1}(I_{b_1})\ldots )).\end{aligned}$$

We prove that the probability for an element of a to be in this interval is equal to its length. For the intervals \(I_b\), it is true by the construction as well as for their images. To make an induction step, we observe that the image of an interval under each \(\psi _{b,p}\) is \(\theta \) times smaller than the initial interval and corresponds to the situation which is \(\theta \) times rarer. So, we have a partition of (0, 1) to arbitrary small intervals for which the length is equal to the frequency of occurrences. This is sufficient to make sure that in fact, this is true for all intervals.    \(\square \)

Remark 4.13

In Example 4.12, we constructed a morphism for the Fibonacci permutation. However, it is not unique, and even not unique among piecewise linear morphisms. For example, the canonical representative b of each Sturmian permutation \(\beta (\sigma ,\rho )\) defined by \(\beta _n=\{\sigma n + \rho \}\) for \(n \ge 0\) is the fixed point of the following morphism \([0,1]^* \mapsto [0,1]^*\): \(x \rightarrow \{2x-\rho \}, \{2x-\rho +\sigma \}.\) Indeed, this is exactly a morphism which sends \(\{\sigma n + \rho \}\) to \(\{\sigma (2n) + \rho \}\), \(\{\sigma (2 n+1) + \rho \}\). It is clearly piecewise linear as well as the function \(\{\cdot \}\). Also, the same idea can be generalized to a k-uniform morphism for any \(k \ge 2\).

Remark 4.14

We remark that the considerations used in the proof of the correctness of the construction are closely related to so-called Dumont-Thomas numeration systems [9].