Complexity and invariant measure of the period-doubling subshift

TL;DR

This paper provides direct combinatorial proofs for the complexity and invariant measure of the period-doubling subshift, deriving explicit formulas for correlation and recurrence, and analyzing its determinism as the threshold approaches zero.

Contribution

It offers new direct proofs and explicit formulas for key properties of the period-doubling subshift, expanding understanding beyond prior indirect methods.

Findings

Explicit formulas for complexity and invariant measure derived from combinatorial properties.

Correlation integral and recurrence characteristics explicitly calculated.

Determinism converges to 1 as the distance threshold approaches 0.

Abstract

Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and other recurrence characteristics of the period-doubling subshift. As a corollary we obtain that the determinism of this subshift converges to 1 as the distance threshold approaches 0.

Equations

$$\omega ={\omega }_1{\omega }_2\dots =0100010101000100\dots$$

$$\mu ([u])=\frac{2}{3\cdot 2^k}\text{or}\mu ([u])=\frac{1}{3\cdot 2^k},$$

$$p(m)=\{\begin{array}{ll}2& \text{if }k=0;\\ 3\cdot 2^{k-1}+2q& \text{if }k\ge 1\text{ and }q\le 2^{k-1};\\ 4\cdot 2^{k-1}+q& \text{if }k\ge 1\text{ and }q>2^{k-1};\end{array}$$

$${\mathcal{L}}_m(\omega )=\left\{w_i^{(m)}:1\le i\le \frac{3}{2}m\right\},$$

$$r(m)=\{\begin{array}{ll}1& \text{if }k=0;\\ 3\cdot 2^{k-1}-2q& \text{if }k\ge 1\text{ and }q<2^{k-1};\\ 2^k-q& \text{if }k\ge 1\text{ and }q\ge 2^{k-1}.\end{array}$$

$$2^{-m_{\epsilon }}\le \epsilon <2^{-m_{\epsilon }+1}.$$

$$\text{c}(\mu ,\epsilon )=\underset{n\to \infty }{\lim }\text{C}(\omega ,n,\epsilon )=\{\begin{array}{ll}1& \text{ if }{m}_{\epsilon }=0;\\ 5/9& \text{ if }{m}_{\epsilon }=1;\\ (3\cdot 2^{k+1}-4q)/((3\cdot 2^k{)}^2)& \text{ if }{m}_{\epsilon }\ge 2\text{ and }q<2^{k-1};\\ (5\cdot 2^k-2q)/((3\cdot 2^k{)}^2)& \text{ if }{m}_{\epsilon }\ge 2\text{ and }q\ge 2^{k-1};\end{array}$$

$${\text{RR}}_{\ell }(\omega ,\epsilon )=\{\begin{array}{ll}1& \text{ if }{m}_{\epsilon }=0;\\ 5/9& \text{ if }{m}_{\epsilon }=1\text{ and }\ell =1;\\ (3\cdot 2^{k+1}-4q+4\ell -4)/((3\cdot 2^k{)}^2)& \text{ if }{m}_{\epsilon }+\ell \ge 3\text{ and }q<2^{k-1};\\ (5\cdot 2^k-2q+2\ell -2)/((3\cdot 2^k{)}^2)& \text{ if }{m}_{\epsilon }+\ell \ge 3\text{ and }q\ge 2^{k-1};\end{array}$$

$${\text{DET}}_{\ell }(\omega ,\epsilon )=\frac{{\text{RR}}_{\ell }(\omega ,\epsilon )}{{\text{RR}}_1(\omega ,\epsilon )}$$

$$\underset{\epsilon \to 0}{\lim }{\text{DET}}_{\ell }(\omega ,\epsilon )=1.$$

$$\underset{\ell \to \infty }{\lim }{\text{DET}}_{\ell }(\omega ,\epsilon )=0.$$

$$\zeta :A\to A^{\ast },\zeta (0)=01,\zeta (1)=00.$$

$$\zeta (x)=\zeta (x_1)\zeta (x_2)\dots \text{for }x=(x_n{)}_{n\in \mathbb{N}}\in A^{\mathbb{N}}.$$

$$w_i^{(m)}=w_i{w}_{i+1}\dots w_{i+m-1}.$$

$$0^{(m)}={\zeta }^k(0)\text{and}{1}^{(m)}={\zeta }^k(1);$$

$$0^{(2m)}=0^{(m)}{1}^{(m)},1^{(2m)}=0^{(m)}{0}^{(m)}.$$

$$\text{c}(\mu ,\epsilon )=\mu \times \mu \{(x,y)\in X\times X:\rho (x,y)\le \epsilon \}.$$

$$\textup{c}(\mu,\varepsilon)=\sum_{v\in A^{m}}\mu\big{(}[v]\big{)}^{2}.$$

$$\text{C}(x,n,\epsilon )=\frac{1}{n^2}\#\left\{(i,j):0\le i,j<n,\rho ({\sigma }^i(x),{\sigma }^j(x))\le \epsilon \right\}.$$

$${\rho }_{\ell }(\alpha ,\beta )=\underset{0\le k<\ell }{\max }\rho \left({\sigma }^k(\alpha ),{\sigma }^k(\beta )\right).$$

$${\rho }_{\ell }(\alpha ,\beta )=\{\begin{array}{ll}1& \text{ if }{\alpha }_i\ne {\beta }_i\text{ for some }1\le i\le \ell ,\\ 2^{\ell -1}\rho (\alpha ,\beta )& \text{ if }{\alpha }_i={\beta }_i\text{ for all }1\le i\le \ell .\end{array}$$

$${\text{C}}_{\ell }(x,n,\epsilon )=\frac{1}{n^2}\#\left\{(i,j):0\le i,j<n,{\rho }_{\ell }({\sigma }^i(x),{\sigma }^j(x))\le \epsilon \right\}.$$

$${\text{RR}}_{\ell }=\ell \cdot {\text{C}}_{\ell }-(\ell -1)\cdot {\text{C}}_{\ell +1}\text{ and }{\text{DET}}_{\ell }=\frac{{\text{RR}}_{\ell }}{{\text{RR}}_1},$$

$$(0^{(m)}{)}_i=(1^{(m)}{)}_i\text{for }i<m,(0^{(m)}{)}_m\ne (1^{(m)}{)}_m.$$

$$w_{(i-1)m+1}^{(m)}=\{\begin{array}{ll}{0}^{(m)}& \text{if }{\omega }_i=0,\\ 1^{(m)}& \text{if }{\omega }_i=1.\end{array}$$

$$w_i^{(m)}=\{\begin{array}{ll}{w}_{i-m/2}^{(m)}& \text{if }\frac{3}{2}m<i\le 2m;\\ w_{i-2m}^{(m)}& \text{if }2m<i\le 3m;\\ w_{i-3m}^{(m)}& \text{if }3m<i\le 4m.\end{array}$$

$$\phi (i,j)=\min \{1\le h\le 2^{k+1}:{\omega }_{i+h-1}\ne {\omega }_{j+h-1}\};$$

$$w_i^{(m)}=w_j^{(m)}\text{ if and only if }\phi (i,j)>m.$$

$$\text{if }\phi (i,j)\ge 2\text{then }\phi (i+1,j+1)=\phi (i,j)-1.$$

$$\phi (i,j)=2^{k+1}-(i-1).$$

Full text

1. Introduction

The period-doubling sequence

[TABLE]

can be defined in various ways. First, its $n$-th member is [math] if and only if the largest $k$ such that $k$-th power of $2$ divides $n$, is odd; otherwise it is $1$. Second, $\omega$ is a unique fixed point of the primitive substitution $0\mapsto 01$, $1\mapsto 00$. Third, $\omega$ is the Toeplitz sequence defined by patterns $(0*)$ and $(1*)$; for the general definition of Toeplitz sequences see [12, 8].

The induced subshift, again called period-doubling, is strictly ergodic (i.e. it is minimal and has a unique invariant measure) and has zero topological entropy. Dynamical properties of this subshift were studied already in 50s and 60s, see the book [10] by Gottschallk and Hedlund and the article [12] by Jacobs and Keane; for some recent references see e. g. [5, 1, 4]. In the book [13], period-doubling subshift (called Feigenbaum subshift therein) is mentioned many times as an example with interesting dynamics.

The period doubling sequence is tightly connected with the Thue-Morse sequence, which is a unique fixed point of the primitive substitution $0\mapsto 01,1\mapsto 10$ which starts with [math]. Complexity of this sequence was studied in [3, 6] and the invariant measure was considered in [7].

The period-doubling sequence $\omega$ is a 2-to-1 image of the Thue-Morse sequence [10, Definition 12.51]; every subword $w=w_{1}\ldots w_{n}$ of $\omega$ corresponds to exactly two subwords $u=u_{1}\ldots u_{n+1}$ of the Thue-Morse sequence such that $u_{i}=u_{i+1}$ if and only if $w_{i}=1$. This relation and the results from [3, 6, 7] yield formula (1.1) for the complexity of $\omega$, and a description of the unique invariant period-doubling measure $\mu$; namely for every allowed $m$-word $u$ $(m\geq 1)$ we have

[TABLE]

where $k\geq 0$ is such that $2^{k}\leq m<2^{k+1}$.

These results are well-known, but cannot be easily found in the literature. Since the period-doubling substitution is of constant length, it is possible to study the complexity of it using a general method from [16]; however, it yields a set of non-trivial recurrent formulas and it seems difficult to derive (1.1) from them.

Dekking [7] has described factor frequencies in the Thue-Morse sequence and the Fibonacci sequence. Factor frequencies in generalized Thue-Morse words were studied in [2]. Frid [9] has obtained a precise description of factor frequencies in a wide class of fixed points of substitutions (the so-called circular marked uniform substitutions, for definitions see [9]) including the Thue-Morse sequence, but the period-doubling sequence, being not marked, does not belong to this class.

Here we give a direct proof of formula (1.1) based on the combinatorics of the period-doubling sequence $\omega$, and we derive some other properties of $\omega$. One of them states that if the length $m$ is a power of $2$, then the set of all $m$-words is equal to the set of first $(3/2)m$ subwords of $\omega$.

Theorem 1.1** (Complexity of the period-doubling sequence).**

Let $m\in\mathbb{N}$ be arbitrary. Then the number of $m$-words in the period-doubling sequence is given by

[TABLE]

where $k\geq 0$ and $0\leq q<2^{k}$ are such that $m=2^{k}+q$.

Furthermore, for $m=2^{k}\geq 2$, the set of all $m$-words is

[TABLE]

where $w_{i}^{(m)}=\omega_{i}\ldots\omega_{i+m-1}$.

Further, we can say exactly what is the measure of a given cylinder.

Theorem 1.2**.**

Let $\mu$ be the unique invariant measure of the period-doubling subshift. Let $u$ be an allowed $m$-word $(m\geq 1)$, $k\geq 0$ be such that $2^{k}\leq m<2^{k+1}$ and $i$ be the least integer such that $u=w_{i}^{(m)}$. Then $1\leq i\leq 3\cdot 2^{k}$, and

1. (1)

if $i\leq 2^{k}-q$, or $q<2^{k-1}$ and $2^{k}<i\leq 2^{k}+2^{k-1}-q$, then $\mu\left([u]\right)=2/(3\cdot 2^{k})$; 2. (2)

otherwise $\mu\left([u]\right)=1/(3\cdot 2^{k})$.

Corollary 1.3**.**

Let $m=2^{k}+q$ with $k\geq 0$ and $0\leq q<2^{k}$. Denote by $r(m)$ the number of $m$-words $u$ such that $\mu([u])=2/(3\cdot 2^{k})$. Then

[TABLE]

Precise knowledge of the invariant measure $\mu$ allows us to derive formulas for correlation integrals (for corresponding definitions see Section 2). For $\varepsilon>0$ define $m_{\varepsilon}\in\mathbb{N}$ as follows: if $\varepsilon\geq 1$ then $m_{\varepsilon}=0$; otherwise $m_{\varepsilon}$ is a unique positive integer such that

[TABLE]

Theorem 1.4**.**

Let $\varepsilon>0$. Then the correlation integral of the unique invariant measure $\mu$ of the period-doubling subshift is

[TABLE]

where $k\geq 0$ and $0\leq q<2^{k}$ are integers such that $m_{\varepsilon}=2^{k}+q$.

For simple inequalities for $\textup{c}(\mu,\varepsilon)$ see Corollary 5.1. Theorem 1.4 together with the results from [11] yield asymptotic values for two of the basic measures of recurrence quantification analysis: recurrence rate (RR) and determinism (DET).

Theorem 1.5** (Recurrence rate of $\omega$).**

Let $\ell\geq 1$ and $\varepsilon>0$. Then the recurrence rate $\textup{RR}_{\ell}(\omega,\varepsilon)$ exists and

[TABLE]

there, for $m_{\varepsilon}+\ell\geq 3$, $k\geq 1$ and $0\leq q<2^{k}$ are unique integers such that $m_{\varepsilon}+\ell-1=2^{k}+q$.

Theorem 1.6** (Determinism of $\omega$).**

Let $\ell\geq 2$ and $\varepsilon>0$. Then $\textup{DET}_{\ell}(\omega,\varepsilon)$ exists,

[TABLE]

and

[TABLE]

Moreover, $\textup{DET}_{\ell}(\omega,\varepsilon)=1$ if and only if one of the following three cases happens:

1. (a)

$\varepsilon\geq 1$; 2. (b)

$2^{k}\leq m_{\varepsilon}<m_{\varepsilon}+\ell-1<2^{k}+2^{k-1}$* for some $k\in\mathbb{N}$;* 3. (c)

$2^{k}+2^{k-1}\leq m_{\varepsilon}<m_{\varepsilon}+\ell-1<2^{k+1}$* for some $k\in\mathbb{N}$.*

Figure 1 illustrates $\textup{RR}_{2}$ and $\textup{DET}_{2}$ of the period-doubling sequence.

Remark 1.7*.*

We trivially have that, for every $\varepsilon<1$,

[TABLE]

Theorems 1.4, 1.5 and 1.6 are stated for embedding dimension $1$. For general embedding dimension, see Subsection 5.1. See also [20] for formulas for other recurrence quantifiers.

This paper is organized as follows. Preliminaries are given in Section 2. Complexity of the period-doubling sequence (Theorem 1.1) is derived in Section 3 as a consequence of some other properties of this sequence. In Section 4 we give the proof of Theorem 1.2. In Section 5 we apply these results to prove Theorems 1.4, 1.5 and 1.6. Moreover, we consider a generalization of our results to arbitrary embedding dimension.

2. Preliminaries

The set of positive integers $\{1,2,\dots\}$ is denoted by $\mathbb{N}$. The set $A=\{0,1\}$ is called an alphabet. Put $A^{*}=\bigcup_{k\geq 0}A^{k}$; $A^{*}$ endowed with concatenation is a monoid. Members of $A^{*}$ are called words. A word of length $m$, or an $m$-word ($m\geq 1$) is any $v=v_{1}\dots v_{m}$ from $A^{m}$ ($m\geq 0$); $v_{i}$ is the $i$-th letter of $v$. The empty word (the unique word of length [math]) is denoted by $\varepsilon$. A subword of $v=v_{1}\dots v_{n}$ starting at the $i$-th letter is any word $v_{i}v_{i+1}\dots v_{n^{\prime}}$ with $i\leq n^{\prime}\leq n$.

The period-doubling substitution $\zeta$ is defined as follows:

[TABLE]

The substitution $\zeta$ induces a morphism (denoted also by $\zeta$) of the monoid $A^{*}$ by putting $\zeta(\varepsilon)=\varepsilon$ and $\zeta(w)=\zeta(w_{1})\zeta(w_{2})\dots\zeta(w_{n})$ for any nonempty word $w=w_{1}w_{2}\dots w_{n}$. Likewise, $\zeta$ induces a map (again denoted by $\zeta$) from $A^{\mathbb{N}}$ to $A^{\mathbb{N}}$ by

[TABLE]

The iterates $\zeta^{k}$ ($k\geq 1$) of $\zeta$ are defined inductively by $\zeta^{1}=\zeta$ and $\zeta^{k}=\zeta\circ\zeta^{k-1}$ for $k\geq 2$.

Period-doubling sequence $\omega=0100\,0101\,0100\,0100\,0100\,\dots$ is the unique fixed point of $\zeta:A^{\mathbb{N}}\to A^{\mathbb{N}}$. Recall that, for every $i\in\mathbb{N}$, $\omega_{i}$ is equal to $k_{i}\operatorname{mod}2$, where $k_{i}$ is the largest integer such that $2^{k_{i}}$ divides $i$. For every integers $m,i\geq 1$, the $m$-word starting at the position $i$ is denoted by $w_{i}^{(m)}$:

[TABLE]

For $m=2^{k}$ ($k\geq 1$) put

[TABLE]

note that both $0^{(m)}$ and $1^{(m)}$ are words of length $m$.

Any subword of $\omega$ (including the empty one) is called allowed. The language $\mathcal{L}_{\omega}$ of $\omega$ is the set of all allowed words. The set of all allowed $m$-words is denoted by $\mathcal{L}_{\omega}^{m}$. Complexity function of $\omega$ is the map $p=p_{\omega}:\mathbb{N}\to\mathbb{N}$ such that, for every $m\in\mathbb{N}$, $p(m)=\#\mathcal{L}_{\omega}^{m}$ is the number of allowed $m$-words.

Note that for every $m=2^{k}$ ($k\geq 0$) we have $0^{(1)}=0,1^{(1)}=1,$ and

[TABLE]

A measure-theoretical dynamical system is a system $(X,\mathcal{B},\mu,f)$, where $X$ is a set, $\mathcal{B}$ is $\sigma$-algebra over $X$, $\mu$ is a probability measure and $f:X\to X$ is a $\mu$-measurable and $\mu$-invariant transformation, i.e. $f^{-1}(B)\in\mathcal{B}$ and $\mu(f^{-1}(B))=\mu(B)$ for every $B\in\mathcal{B}$. The system $(X,\mathcal{B},\mu,f)$ is ergodic if $\mu(B)=0$ or $\mu(B)=1$ for every $B\in\mathcal{B}$ with $f^{-1}(B)=B$.

A pair $(X,f)$ is called a topological dynamical system if $X$ is a compact metric space and $f:X\to X$ is a continuous map. A dynamical system $(X,f)$ is minimal if there is no proper subset $M\subsetneq X$ which is nonempty, closed and $f$-invariant (a set $M$ is $f$-invariant if $f(M)\subseteq M$). Let $\mathcal{B}_{X}$ denote the system of all Borel subsets of $X$. A probability measure $\mu$ is said to be invariant if $\mu(f^{-1}(A))=\mu(A)$ for every $A\in\mathcal{B}_{X}$; that is, $(X,\mathcal{B}_{X},\mu,f)$ is a measure-theoretical dynamical system. By Krylov-Bogolyubov theorem, for every $(X,f)$ there exists an invariant measure $\mu$. System $(X,f)$ is called uniquely ergodic if such a measure $\mu$ is unique. Moreover, if $(X,f)$ is also minimal, we call it strictly ergodic.

Metric $\rho$ on $\Sigma=A^{\mathbb{N}}$ is defined for every $\alpha,\beta\in\Sigma$ by $\rho(\alpha,\beta)=0$ if $\alpha=\beta$, and $\rho(\alpha,\beta)=2^{-k+1}$ if $\alpha\neq\beta$, where $k=\min\{i:\alpha_{i}\neq\beta_{i}\}$. Note that $(\Sigma,\rho)$ is a compact metric space. For an $m$-word $v$ we define the cylinder $[v]$ by $[v]=\{\alpha\in\Sigma:\alpha_{i}=v_{i}\text{ for }i\leq m\}$. Cylinders form a basis of the topology and $[v]=B(x,\varepsilon)$ for every $x\in[v]$ and $\varepsilon=2^{-\lvert v\rvert}$, where $B(x,\varepsilon)$ denotes the closed ball with the center $x$ and radius $\varepsilon$. A shift is the map $\sigma:\Sigma\to\Sigma$ defined by $\sigma(\alpha_{1}\alpha_{2}\alpha_{3}\ldots)=\alpha_{2}\alpha_{3}\ldots$ For each nonempty closed $\sigma$-invariant subset $Y\subseteq\Sigma$, the restriction of $(\Sigma,\sigma)$ to $Y$ is called a subshift. The closure of the orbit $(\sigma^{n}(\alpha))_{n\geq 0}$ of any $\alpha\in\Sigma$ defines a subshift, as it is always nonempty, closed and $\sigma$-invariant set. Period-doubling subshift is the orbit closure of the period-doubling sequence.

Let $(X,\sigma)$ be a subshift over $A$, $\rho$ be the metric defined above and $\mu$ be a $\sigma$-invariant measure. Correlation integral of $\mu$ is defined for $\varepsilon>0$ as follows:

[TABLE]

If $2^{-m}\leq\varepsilon<2^{-m+1}$ then clearly

[TABLE]

For $x\in X,n\in\mathbb{N}$ and $\varepsilon>0$, correlation sum is defined by

[TABLE]

For uniquely ergodic systems, $\lim_{n}\textup{C}(x,n,\varepsilon)=\textup{c}(\mu,\varepsilon)$ for every but countably many $\varepsilon>0$ and every $x\in X$ [18].

For any $\ell\geq 1$ consider Bowen’s metric

[TABLE]

An easy computation gives that we always have

[TABLE]

We can now define

[TABLE]

Recurrence quantification analysis ([22], see also [14, 21]) gives several complexity measures quantifying structures in recurrence plots, which are useful for visualization of recurrence. Two of them are recurrence rate (RR) and determinism (DET). By [11, Proposition 1], recurrence rate and determinism can be expressed by correlation sums as follows:

[TABLE]

where $\ell$ is the minimal required line length; arguments $x,n,\varepsilon$ are omitted and we consider embedding dimension $1$. For general embedding dimension $d$ see Subsection 5.1.

If the limit of $\textup{C}_{\ell}(x,n,\varepsilon)$ for $n\to\infty$ exists, it is denoted by $\textup{C}_{\ell}(x,\varepsilon)$. Analogously we define $\textup{RR}_{\ell}(x,\varepsilon)$ and $\textup{DET}_{\ell}(x,\varepsilon)$.

3. Complexity of the period-doubling sequence

3.1. Length $m=2^{k}$

In this section, we prove Theorem 1.1 in the special case when the length $m$ is a power of $2$. We start with two lemmas. The first one follows by induction using (2.2) and the second one is a direct consequence of $\zeta^{k}(\omega)=\omega$.

Lemma 3.1**.**

For any $m=2^{k}$ ($k\geq 0$), the $m$-words $0^{(m)},1^{(m)}$ differ exactly at the $m$-th letter:

[TABLE]

Moreover, if $k$ is even then $(0^{(m)})_{m}=0$ and $(1^{(m)})_{m}=1$, and if $k$ is odd then $(0^{(m)})_{m}=1$ and $(1^{(m)})_{m}=0$.

Lemma 3.2**.**

Let $m=2^{k}$ ($k\geq 0$). Then the period-doubling sequence $\omega$ can be written in the form $\omega=(\omega_{1})^{(m)}(\omega_{2})^{(m)}\dots$. That is, for every $i\in\mathbb{N}$,

[TABLE]

Lemma 3.3**.**

For the period-doubling sequence $\omega$, $p(1)=2$ and $p(2)=3$. Moreover, the allowed $1$-words are $w_{1}^{(1)}=0$ and $w_{2}^{(1)}=1$, and the allowed $2$-words are $w_{1}^{(2)}=01$, $w_{2}^{(2)}=10$, and $w_{3}^{(2)}=00$.

Proof.

We only need to prove that the word $11$ is not allowed. But this immediately follows from the fact that $\omega_{2i-1}=0$ for every $i$. ∎

Lemma 3.4**.**

Let $m=2^{k}$ ($k\geq 1$). Then the words $w_{i}^{(m)}$ ($1\leq i\leq\frac{3}{2}m$) are pairwise distinct.

Proof.

We start by showing that, for $\frac{3}{2}m<i\leq 4m$,

[TABLE]

To see this, realize that $\omega=0^{(m)}1^{(m)}0^{(m)}0^{(m)}\,0^{(m)}\ldots$ by Lemma 3.2. Hence, by Lemma 3.1, $w_{i}^{(m)}=w_{i-2m}^{(m)}$ for $2m<i\leq 3m$ and $w_{i}^{(m)}=w_{i-3m}^{(m)}$ for $3m<i\leq 4m$. Furthermore, $\omega=0^{(n)}1^{(n)}0^{(n)}0^{(n)}\,0^{(n)}1^{(n)}\ldots$, where $n=m/2$. So analogously, $w_{i}^{(m)}=w_{i-n}^{(m)}$ for $\frac{3}{2}m<i\leq 2m$.

We now proceed by induction on $k$. For $k=1$, the claim follows from Lemma 3.3. Assume now that the claim is valid for some $k\geq 1$; we are going to show that it is valid for $k+1$. Put $m=2^{k}$. Since $w_{i}^{(2m)}=w_{i}^{(m)}w_{i+m}^{(m)}$, (3.1) and the induction hypothesis yield that the words $w_{i}^{(2m)}$ for $1\leq i\leq 3m$ are pairwise distinct. ∎

Lemma 3.5**.**

Let $m=2^{k}$ ($k\geq 1$) and $v$ be any allowed $m$-word. Then exactly one of the following is true:

1. (1)

$v$* is a subword of $0^{(m)}1^{(m)}$ starting at the $i$-th letter with $i\leq m$;* 2. (2)

$v$* is a subword of $1^{(m)}0^{(m)}$ starting at the $i$-th letter with $i\leq m/2$.*

Proof.

We start by showing that at least one of (1), (2) is true. If $v\in\{0^{(m)},1^{(m)}\}$, we are done. Otherwise, by Lemma 3.2, $v$ is a subword of $0^{(m)}0^{(m)}$ or $0^{(m)}1^{(m)}$ or $1^{(m)}0^{(m)}$, starting at an index $j\leq m$. By Lemma 3.1, $v$ is a subword of $0^{(m)}1^{(m)}$ or $1^{(m)}0^{(m)}$. In the former case we have (1). In the latter case, we have (2) since $1^{(m)}0^{(m)}=0^{(n)}0^{(n)}0^{(n)}1^{(n)}$ by (2.2), where $n=m/2$.

Moreover, $\omega$ starts with $0^{(m)}1^{(m)}0^{(m)}$, so $v=w_{i}^{(m)}$ for some $1\leq i\leq\frac{3}{2}m$. By Lemma 3.4 the words $w_{i}^{(m)}$ ($1\leq i\leq\frac{3}{2}m$) are pairwise distinct, so only one of (1) and (2) is true. ∎

Proposition 3.6**.**

Let $m=2^{k}$ ($k\geq 1$). Then $p(m)=\frac{3}{2}m$ and $\mathcal{L}_{\omega}^{m}=\{w_{i}^{(m)}:1\leq i\leq\frac{3}{2}m\}$.

Proof.

Lemma 3.5 gives $p(m)\leq\frac{3}{2}m$. On the other hand, $p(m)\geq\frac{3}{2}m$ by Lemma 3.4. The description of $\mathcal{L}_{\omega}^{m}$ now follows from Lemma 3.4. ∎

Remark 3.7*.*

For $m=2^{k}\,(k\geq 1)$ we also have $\mathcal{L}_{\omega}^{m}=\{w_{i}^{(m)}:\frac{3}{2}m<i\leq 3m\}$; this follows from (3.1).

3.2. General length $m$

Lemma 3.8**.**

Let $m=2^{k}+q$, where $k\geq 1$ and $1\leq q<2^{k}$. Let $1\leq i<j\leq 3\cdot 2^{k}$. Then $w_{i}^{(m)}=w_{j}^{(m)}$ if and only if exactly one of the following conditions holds:

1. (1)

$1\leq i\leq 2^{k}-q$* and $j=i+2^{k+1}$;* 2. (2)

$q<2^{k-1}$, $2^{k}+1\leq i\leq 3\cdot 2^{k-1}-q$, and $j=i+2^{k-1}$.

Consequently, for every $1\leq i<3\cdot 2^{k}$ there is at most one $j$ such that $i<j\leq 3\cdot 2^{k}$ and $w_{i}^{(m)}=w_{j}^{(m)}$.

Proof.

For $1\leq i<j\leq 3\cdot 2^{k}$ put

[TABLE]

it is well-defined by Lemma 3.4 applied to the length $2^{k+1}$. Note that

[TABLE]

It is clear that

[TABLE]

Fix $1\leq i<j\leq 3\cdot 2^{k}$ and assume that $w_{i}^{(m)}=w_{j}^{(m)}$; we are going to show that either (1) or (2) is true. Since $m\geq 2^{k}$ we have that $w_{i}^{(2^{k})}=w_{j}^{(2^{k})}$ and, by (3.1), exactly one of the following is true:

1. (a)

$1\leq i\leq 2^{k}$ and $j=i+2^{k+1}$; 2. (b)

$2^{k}<i\leq 3\cdot 2^{k-1}$, and $j=i+2^{k-1}$.

Assume that (a) is true. Since $\varphi(1,2^{k+1}+1)=2^{k+1}$ and $j=i+2^{k+1}\leq 3\cdot 2^{k}$, (3.3) implies

[TABLE]

Since $w_{i}^{(m)}=w_{j}^{(m)}$ by assumption, (3.2) implies $2^{k+1}-(i-1)>m$, that is $i\leq 2^{k}-q$. So we have (1).

If (b) is true then, by Lemma 3.2, $w_{1+2^{k}}^{(2^{k+1})}=0^{(n)}0^{(n)}0^{(n)}1^{(n)}$ and $w_{1+2^{k}+2^{k-1}}^{(2^{k+1})}=0^{(n)}0^{(n)}1^{(n)}0^{(n)}$ for $n=2^{k-1}$. From Lemma 3.1 it follows that $\varphi(1+2^{k},1+2^{k}+2^{k-1})=3\cdot 2^{k-1}$. Since $2^{k}<i\leq 3\cdot 2^{k-1}$ and $j=i+2^{k-1}$, (3.3) yields

[TABLE]

(notice that $0<p\leq 2^{k-1}$). By the assumption $w_{i}^{(m)}=w_{j}^{(m)}$ and so, by (3.2), $\varphi(i,j)>m=2^{k}+q$. Now (3.5) gives $3\cdot 2^{k-1}-q\geq i$, so we have (2).

Now assume that one of the conditions (1), (2) holds. If (1) holds we have $w_{i}^{(m)}=w_{j}^{(m)}$, since (3.4) implies $\varphi(i,j)>m$. Similarly, if (2) is true then $\varphi(i,j)>m$ by (3.5), so again $w_{i}^{(m)}=w_{j}^{(m)}$.

∎

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

It is clear from Lemma 3.3 that (1.1) is true for $k=0$, so we may assume that $k>0$. Let $n=2^{k+1}$. By Proposition 3.6,

[TABLE]

If $q\geq 2^{k-1}$ then only (1) from Lemma 3.8 occurs, consequently, $p(m)=p(n)-(2^{k}-q)=4\cdot 2^{k-1}+q$. Otherwise, both (1) and (2) from Lemma 3.8 occur and so $p(m)=p(n)-(2^{k}-q)-(2^{k-1}-q)=3\cdot 2^{k-1}+2q$. ∎

From Theorem 1.1 we immediately have that

[TABLE]

and

[TABLE]

4. Invariant measure of the period-doubling subshift

Let $(X,\sigma)$ be the period-doubling subshift; i.e. $X$ is the orbit closure of $\omega$ and $\sigma:X\to X$ is the left shift. By [15] (see also [19, Proposition 5.2 and Theorem 5.6]), $(X,\sigma)$ is strictly ergodic.

Denote the unique invariant measure of $(X,\sigma)$ by $\mu$. By [17],

[TABLE]

for every $v\in\mathcal{L}^{m}$. In this section we prove Theorem 1.2 which gives an explicit formula for measures of cylinders $[v]$. We follow [19, Sections 5.3-5.4]. Fix an integer $m\in\mathbb{N}$ and recall that $\mathcal{L}^{m}$ is the set of all $m$-words in $\omega$. Define a substitution $\zeta^{(m)}$ over alphabeth $\mathcal{L}^{m}$ as follows: for $u\in\mathcal{L}^{m}$, write $\zeta(u)=y_{1}y_{2}\ldots y_{2m}$, and define $\zeta^{(m)}(u)=(y_{1}\ldots y_{m})(y_{2}\ldots y_{m+1}).$ Let $M^{m}$ be the composition matrix of $\zeta^{(m)}$, that is $M^{m}$ is a $p(m)\times p(m)$ non-negative matrix such that, for $u,v\in\mathcal{L}^{m}$, $(M^{m})_{uv}$ is the number of occurencies of $v$ in $\zeta^{(m)}(u)$. Trivially every member of $M^{m}$ belongs to $\{0,1,2\}$.

By [19, Corollary 5.2], the Perron-Frobenius eigenvalue of $M^{m}$ is $\lambda=2$. Furthermore, if $d^{m}=(d_{u}^{m})_{u\in\mathcal{L}^{m}}$ is the unique normalized eigenvector of $M^{m}$ corresponding to $\lambda$, then $\mu([u])=d_{u}^{m}$ by [19, Corollary 5.4], see also [9, Proposition 1].

Lemma 4.1**.**

Let $m=2^{k}\,(k\geq 1)$. Then $d^{m}=\frac{2}{3m}(1,1,\ldots,1)$. Consequently, $\mu\left([v]\right)=\frac{2}{3m}$ for every allowed $m$-word $v$.

Proof.

It is enough to show that every row sum of $M_{m}$ is equal to $2$. For $m=2$ it is easy. So assume that $m\geq 4$. By (3.1) we have

[TABLE]

Hence, for $1\leq j\leq m$, the word $w_{j}^{(m)}$ occurs in $\zeta^{(m)}(w_{i}^{(m)})$ for $i=\lceil\frac{j}{2}\rceil$ and $i=\lceil\frac{j}{2}\rceil+m$. Further, for $m<j\leq\frac{3}{2}m$, the word $w_{j}^{(m)}$ occurs in $\zeta^{(m)}(w_{i}^{(m)})$ for $i=\lceil\frac{j}{2}\rceil$ and $i=\lceil\frac{j}{2}\rceil+\frac{m}{4}$. The proof is complete.

∎

Proof of Theorem 1.2.

Theorem 1.2 holds for $q=0$ by the previous lemma, so let $q\geq 1$. Put $n=2^{k+1}$. If (1) is true then, by Lemma 3.8, there is exactly one index $j$ such that $i<j\leq 3\cdot 2^{k}$ and $w_{i}^{(m)}=w_{j}^{(m)}$; in this case $[v]=[w_{i}^{(n)}]\sqcup[w_{j}^{(n)}]$. Otherwise, $[v]=[w_{i}^{(n)}]$. Now the theorem follows from Lemma 4.1. ∎

5. Correlation integral and RQA measures

Proof of Theorem 1.4 .

By [18], modified to uniquely ergodic systems, $\lim\textup{C}(\omega,n,\varepsilon)=\textup{c}(\mu,\varepsilon)$ provided $\textup{c}(\mu,\varepsilon)$ is continuous at $\varepsilon$. Since the metric $\rho$ attains only values from $2^{-\mathbb{N}_{0}}\cup\{0\}$, $\textup{C}(\omega,n,\varepsilon)$ and $\textup{c}(\mu,\varepsilon)$ are constant on $\varepsilon\in\left[2^{-m},2^{-m+1}\right)$ for every $m$. This easily implies $\lim_{n}\textup{C}(\omega,n,\varepsilon)=\textup{c}(\mu,\varepsilon)$ for every $\varepsilon$. Since

[TABLE]

Theorem 1.2 and Corollary 1.3 yield the desired result. ∎

Corollary 5.1**.**

Let $0<\varepsilon<\frac{1}{2}$ and $m_{\varepsilon}$ be defined as in (1.2). Then

[TABLE]

Moreover, if $m_{\varepsilon}\in\{2^{k},2^{k}+2^{k-1},k\geq 1\}$ then $\textup{c}(\mu,\varepsilon)=\frac{2}{3m_{\varepsilon}}$, and if $m_{\varepsilon}\in\{2^{k}+2^{k-2},k\geq 1\}$ then $\textup{c}(\mu,\varepsilon)=\frac{25}{36m_{\varepsilon}}$.

Proof.

Write $m_{\varepsilon}=2^{k}+q$ with $k\geq 1$ and $0\leq q<2^{k}$. Let $x=\frac{q}{m_{\varepsilon}}\in\left[0,\frac{1}{2}\right)$. Using Theorem 1.4 and substituting $2^{k}=m_{\varepsilon}-q$ into $m_{\varepsilon}\textup{c}(\mu,\varepsilon)$ we get

[TABLE]

Using elementary calculus we obtain that $\frac{2}{3}\leq m_{\varepsilon}\textup{c}(\mu,\varepsilon)\leq\frac{25}{36}\,$ if $0\leq q<2^{k-1}$ and $\frac{2}{3}\leq m_{\varepsilon}\textup{c}(\mu,\varepsilon)\leq\frac{49}{72}<\frac{25}{36}\,$ if $2^{k-1}\leq q<2^{k}$. Moreover, minimum is attained at the points $x=0$ and $x=\frac{1}{3}$, corresponding to $q=0$ and $q=2^{k-1}$, and maximum is attained at the point $x=\frac{1}{5}$ corresponding to $q=2^{k-2}$. ∎

Proof of Theorem 1.5.

If $\varepsilon\geq 1$ then, by (2.5) and Theorem 1.4, $\textup{RR}_{\ell}(\omega,n,\varepsilon)=1$ for every $n$, hence $\textup{RR}_{\ell}(\omega,\varepsilon)=1$. So assume that $0<\varepsilon<1$. By (2.3), for every $x,y\in X$ we have $\rho_{\ell}(x,y)\leq\varepsilon$ if and only if $\rho(x,y)\leq 2^{-\ell+1}\varepsilon$. So

[TABLE]

Thus, by (2.5) and Theorem 1.4,

[TABLE]

Notice that $m_{2^{-\ell}\varepsilon}=m_{\varepsilon}+\ell$ and $m_{2^{-\ell+1}\varepsilon}=m_{\varepsilon}+\ell-1$, since $\varepsilon<1$ and $\ell\geq 1$. Put $m=m_{\varepsilon}+\ell-1$. If $m=1$ (i.e., $m_{\varepsilon}=1$ and $\ell=1$), then $\textup{RR}_{\ell}(\omega,\varepsilon)=5/9$ by (5.1) and Theorem 1.4. So we may assume that $m\geq 2$ (i.e. $m_{\varepsilon}+\ell\geq 3$) and hence we may write $m=2^{k}+q$ with $k\geq 1$ and $0\leq q<2^{k}$.

Now we consider four cases: $q<2^{k-1}-1$, $q=2^{k-1}-1$, $2^{k-1}\leq q<2^{k}-1$, and $q=2^{k}-1$. In the first and third cases we have $m_{2^{-\ell}\varepsilon}=m+1=2^{k}+(q+1)$ with $q+1<2^{k-1}$ and $2^{k-1}\leq q<2^{k}$, respectively. So (5.1) and Theorem 1.4 give the formulas for $\textup{RR}_{\ell}(\omega,\varepsilon)$.

In the second case ($q=2^{k-1}-1$) we can write $m_{2^{-\ell}\varepsilon}=2^{k}+2^{k-1}$ and in the fourth case ($q=2^{k}-1$) we can write $m_{2^{-\ell}\varepsilon}=2^{k+1}+0$; as above, (5.1) and Theorem 1.4 yield the formula for $\textup{RR}_{\ell}(\omega,\varepsilon)$.

∎

Proof of Theorem 1.6.

From (2.5) and the definition of determinism, we have

[TABLE]

Using (5.1) and the fact that $\textup{RR}_{1}(\omega,\varepsilon)=\textup{c}(\mu,\varepsilon)>0$, we obtain

[TABLE]

It is clear that $\textup{DET}_{\ell}(\omega,\varepsilon)=1$ for $\varepsilon\geq 1$, so assume that $\varepsilon<1$. Let $m_{\varepsilon}=2^{k}+q$ and $m_{\varepsilon}+\ell-1=2^{k^{\prime}}+q^{\prime}$, where $k,k^{\prime}\geq 0,\,0\leq q<2^{k}$ and $0\leq q^{\prime}<2^{k^{\prime}}$. We now compute $\textup{DET}_{\ell}(\omega,\varepsilon)$ using Theorems 1.4, 1.5 and (5.2). We distinguish three cases.

(a) Let $\varepsilon\in(0,1)$ be such that $k^{\prime}=k$; then $q^{\prime}=q+\ell-1$; we write $\varepsilon\in E_{a}$. If $0\leq q<q^{\prime}<2^{k-1}$ or $2^{k-1}\leq q<q^{\prime}<2^{k}$, we immediately have $\textup{DET}_{\ell}(\omega,\varepsilon)=1.$ Otherwise $0\leq q<2^{k-1}\leq q^{\prime}<2^{k}$ and

[TABLE]

Here $2^{k-1}-\ell+1\leq q<2^{k-1}$ and so $q2^{-k}\to 1/2$ for $\varepsilon\to 0$. Thus we have

[TABLE]

(b) Let $\varepsilon\in(0,1)$ be such that $k^{\prime}=k+1$; we write $\varepsilon\in E_{b}$. Then $q^{\prime}=q+\ell-1-2^{k}$, and so

[TABLE]

Clearly

[TABLE]

(c) If $\varepsilon\in(0,1)\backslash(E_{a}\cup E_{b})$, then $k^{\prime}\geq k+2$ and we again have $\textup{DET}_{\ell}(\omega,\varepsilon)<1$. Since this can happen only for large enough $\varepsilon$, this case does not affect the limit $\lim\limits_{\varepsilon\to 0}\textup{DET}_{\ell}(\omega,\varepsilon)$. (In fact, if $\varepsilon<\min(2^{-(\ell-2)},1)\,$ then $m_{\varepsilon}\geq\ell-1$, and so $2^{k^{\prime}}+q^{\prime}=m_{\varepsilon}+\ell-1\leq 2m_{\varepsilon}=2(2^{k}+q)$. From this we immediately have $k^{\prime}\leq k+1$.)

Thus we have proved that $\textup{DET}_{\ell}(\omega,\varepsilon)=1$ if and only if one of (a)–(c) happens (otherwise $\textup{DET}_{\ell}(\omega,\varepsilon)<1$) and that $\lim\limits_{\varepsilon\to 0}\textup{DET}_{\ell}(\omega,\varepsilon)=1$.

∎

5.1. General embedding dimension

Up to now we considered recurrence characteristics without embedding. The results can be easily generalized to arbitrary embedding dimension $d\geq 1$.

If $x$ is a sequence over $A=\{0,1\}$, then the embedded sequence $x^{d}$ is a sequence over $A^{d}=\{0,1\}^{d}$ defined by

[TABLE]

A metric $\rho^{d}$ in the embedding space $(A^{d})^{\mathbb{N}}$ is defined as in Section 2; that is,

[TABLE]

If $k>1$ then trivially

[TABLE]

So for correlation sums $\textup{C}_{\ell}^{d}$, defined by (2.4) with $\rho_{\ell}$ replaced by $\rho_{\ell}^{d}$, it holds that

[TABLE]

for every $x\in A^{\mathbb{N}},\varepsilon\in(0,1)$ and $n\in\mathbb{N}$. This together with Theorem 1.4 yield an explicit formula for (embedded) correlation integrals $\textup{c}_{\ell}^{d}(\omega^{d},\varepsilon)$ for the period-doubling sequence $\omega$. To obtain formulas for $\textup{RR}_{\ell}^{d}(\omega^{d},\varepsilon)$ and $\textup{DET}_{\ell}^{d}(\omega^{d},\varepsilon)$ it suffices to use (2.5):

[TABLE]

Acknowledgements

The author gratefully acknowledges the many helpful suggestions of Vladimír Špitalský. This work was supported by the Slovak Research and Development Agency under the contract No. APVV-15-0439, and by VEGA grant 1/0768/15.