Minimal Bounded Speedups of Toeplitz Flows

TL;DR

This paper explores the properties of minimal bounded speedups in Toeplitz flows, showing they can differ from Toeplitz flows and providing criteria to identify when they are Toeplitz.

Contribution

It introduces new techniques to determine whether a minimal bounded speedup of a Toeplitz flow remains Toeplitz and offers conditions for this to occur.

Findings

Minimal bounded speedups may not be Toeplitz flows.

Techniques to identify Toeplitz property in speedups.

Sufficient conditions for speedups to be Toeplitz.

Abstract

We investigate minimal bounded speedups of Toeplitz flows. We demonstrate that the minimal bounded speedup of a Toeplitz flow need not be another Toeplitz flow and describe techniques for determining whether the resulting speedup is Toeplitz. We also provide sufficient conditions to guarantee that the minimal bounded speedup will be a Toeplitz flow.

Equations

$$(x_1,x_2,\dots )+(y_1,y_2,\dots )=(z_1,z_2,\dots )$$

$$T_{\alpha }((x_1,x_2,x_3\dots ))=(x_1,x_2,x_3,\dots )+(1,0,0,\dots ).$$

$$\mathcal{P}=\bigcup _{i=1}^N\{T^j(B_i):0\le j\le h_i-1\}$$

$$\tau (a,\ell )=(a_1,{\ell }_1)(a_2,{\ell }_2)\cdots (a_n,{\ell }_n)$$

$${\tau }_k^l(a,\ell )=(a_1,{\ell }_1)(a_2,{\ell }_2)\cdots (a_n,{\ell }_n)$$

$$p(x)=\{\begin{array}{ll}3& \text{if }x\in A\\ 1& \text{if }x\in \sigma (A)\\ 2& \text{else.}\end{array}$$

$$\psi(\phi(w_{1}))=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCCBCBDBEEEBEB~ABCCDBEE$}}$$

$$\psi(\phi(w_{2}))=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCDBEE~ABCCCBCBDBEEEBEB$}}$$

$$\psi (\varphi (w_3))$$

$$\psi(\phi(w_{4})=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCDBEE\qquad\qquad\qquad\qquad\qquad\qquad\quad$}}$$

$$\psi ({\varphi }^2(w_1))$$

$$\displaystyle=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCCBCBDBEEEBEB~ABCCDBEE$}}$$

$$\psi ({\varphi }^2(w_2))$$

$$\displaystyle=\scalebox{0.86}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCCBCBDBEEEBEB~ABCCDBEE$}}$$

$$\psi ({\varphi }^2(w_3))$$

$$\displaystyle=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCCBCBDBEEEBEB~ABCCDBEE$}}$$

$$\displaystyle\quad\ \scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEEEBEB$}},$$

$$\psi ({\varphi }^2(w_4))$$

$$\displaystyle=\scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCCBCBDBEEEBEB~ABCCDBEE$}}$$

$$\displaystyle\quad\ \scalebox{0.9}{\mbox{$\displaystyle ABCCCBCBDBEE~ABCCDBEEEBEB~ABCCDBEE$}},$$

$$p(x,n)=cn$$

$$p(x)=c+g(x)-g(S(x))\text{ for all x∈X.}$$

$$g(x_1)=c+g(x_0)-p(x_0).$$

$$g(x_n)=nc+g(x_0)-p(x_0,n)\text{ for xn=Sn(x0)}.$$

$$g(S^n(x))=nc+g(x)-p(x,n)=g(x).$$

$$g(x)-g(S(x))=p(x)-c\text{ for all x∈X and g(x)=g(y) for all x,y∈Bik(j),}$$

$$0$$

$$=\sum _{m=0}^{n-1}(g(S^m(x))-g(S^{m+1}(x))$$

$$=\sum _{m=0}^{n-1}(p(S^m(x))-c)=p(x,n)-nc.$$

$$\chi :\{\begin{array}{l}a\to aab\\ b\to abb\end{array}$$

$$p(x,n)=2n\text{ for all first return times under S of x to its cylinder set Bik.}$$

$$p(x,n)=m{h}^k$$

$$n$$

$$=(\#(i,j^{\prime })+\#(i^{\prime },j^{\prime }))o_i^k(j^{\prime })+(\#(i^{\prime },j)+1)o_i^k(j)$$

$$=\#(j^{\prime })o_i^k(j^{\prime })+\#(j)o_i^k(j)=\#(j)h^k$$

Full text

1. Introduction

Given a minimal Cantor system $(X,T)$, a topological speedup of $(X,T)$ is a dynamical system $(X,S)$ where $S$ is a homeomorphism such that $S(x)=T^{p(x)}(x)$ for some function $p:X\to\mathbb{N}$. We call the function $p(x)$ the jump function for the speedup. Throughout this paper we assume that $p(x)$ is continuous (equivalently bounded by [1, Proposition 2.2]) and that the resulting Cantor system $(X,S)$ is minimal. We are interested in investigating which properties of the original Cantor system are preserved under minimal bounded speedups. In particular, this paper focuses on speedups of Toeplitz flows.

There is a rich history of studying speedups of dynamical systems, particularly in the measure-theoretical setting; see, for example, [2, 11, 12]. More recently, there has been a focus on topological speedups. In [4], the authors characterize when one Cantor minimal system is the speedup of another and provide topological analogues to the measure-theoretical results of [2]. In [1], minimal bounded speedups of odometers and substitutions are investigated. It is shown that the minimal bounded speedup of an odometer is a conjugate odometer, whereas the minimal bounded speedup of a primitive substitution is always a different primitive substitution. In [3], the authors use Bratteli diagrams to investigate minimal bounded speedups of primitive substitutions.

Because it is known that minimal bounded speedups of odometers are conjugate odometers, and Toeplitz flows are symbolic, minimal, almost one-to-one extensions of odometers, it was conjectured in [3] that the minimal bounded speedup of a Toeplitz flow is a Toeplitz flow. We demonstrate that this conjecture is false. We investigate minimal bounded speedups of Toeplitz flows and provide sufficient conditions for a speedup to be Toeplitz.

In Section 2 we present background terminology and foundational results about substitution shifts, Toeplitz flows, minimal Cantor systems, and topological speedups of minimal Cantor systems. Section 3 demonstrates the existence of minimal bounded speedups of Toeplitz flows that are not themselves Toeplitz flows. In fact, we show in Theorem 3.5 that no bounded speedup of the Toeplitz flow $(X_{\theta},\sigma)$ generated by the substitution $\theta(a)=ab$ and $\theta(b)=aa$ with orbit number $c=2$ will be a Toeplitz flow.

We demonstrate in Section 4 that a constant speedup $(X,T^{c})$ of a Toeplitz flow $(X,T)$ is a Toeplitz flow if and only if $\gcd(c,p_{k})=1$ for all $k$ where $(p_{k})$ is the period structure of the original Toeplitz flow. Additionally, in this case $(X,T)$ and $(X,T^{c})$ will have the same underlying odometer. The section concludes by presenting a method for determining whether a given speedup of a Toeplitz flow is Toeplitz. Algorithms are provided for both the case where $(X,T)$ is generated by a constant length left proper substitution and the more general Toeplitz flow setting.

The final section of the paper investigates sufficient conditions for the minimal bounded speedup of a Toeplitz flow to be Toeplitz. Namely, if there exists some level of the Kakutani-Rokhlin partition for $(X,T)$ such that every tower has the same orbit permutation and each orbit has the same number of levels in each tower under the jump function $p$, then the speedup $(X,S)$ will be a Toeplitz flow (Theorem 5.1). We demonstrate that this can happen such that $(X,S)$ is conjugate to $(X,T^{c})$ and such that $(X,S)$ is not conjugate to $(X,T^{c})$, as $(X,T^{c})$ need not be minimal. We conjecture that our sufficient conditions in Theorem 5.1 are actually necessary for $(X,S)$ to be Toeplitz because of the strong relationship between Toeplitz flows and odometers.

2. Preliminaries

2.1. Substitution Subshifts

Let $\mathcal{A}$ be a finite alphabet. A substitution over $\mathcal{A}$ is a map $\theta:\mathcal{A}\to\mathcal{A}^{*}$, where $\mathcal{A}^{*}$ denotes the set of nonempty finite words with letters from $\mathcal{A}$. We extend $\theta$ to map from $\mathcal{A}^{*}$ to $\mathcal{A}^{*}$ by concatenation such that $\theta(ab)=\theta(a)\theta(b)$; we then extend to maps from $\mathcal{A}^{\mathbb{N}}$ to $\mathcal{A}^{\mathbb{N}}$ (similarly from $\mathcal{A}^{\mathbb{Z}}$ to $\mathcal{A}^{\mathbb{Z}}$) in the natural way. A substitution is constant length if $|\theta(a)|=|\theta(b)|$ for all $a,b\in\mathcal{A}$. We say a substitution is primitive if there exists some $n\in\mathbb{N}$ such that for every $a\in\mathcal{A}$, $\theta^{n}(a)$ contains every letter of $\mathcal{A}$. A substitution is left proper if there is some $a\in\mathcal{A}$ such that $\theta(b)$ starts with $a$ for every $b\in\mathcal{A}$; if we also have some $c\in\mathcal{A}$ such that $\theta(b)$ ends with $c$ for every $b\in\mathcal{A}$, then we say the substitution is proper.

Without loss of generality (by considering $\theta^{n}$ if necessary), we assume $\theta$ is primitive if and only if there is a letter $a\in\mathcal{A}$ such that $\theta(a)$ begins with $a$, $\lim_{n\to\infty}|\theta^{n}(b)|=\infty$ for all $b\in\mathcal{A}$, and $\lim_{n\to\infty}\theta^{n}(a)$ is a fixed point under the substitution that contains every letter in $\mathcal{A}$. We define the shift space $X_{\theta}\subset\mathcal{A}^{\mathbb{Z}}$ to be the set of all $x\in\mathcal{A}^{\mathbb{Z}}$ such that whenever $i,j\in\mathbb{Z}$ with $i<j$, then $x_{[i,j]}$ is a subword of $\theta^{k}(a)$ for some $k\in\mathbb{N}$ and $a\in\mathcal{A}$. We refer to $(X_{\theta},\sigma)$ as the substitution subshift and emphasize that $\theta$ is primitive if and only if $(X_{\theta},\sigma)$ is minimal.

2.2. Toeplitz Flows

A sequence $x\in\mathcal{A}^{\mathbb{N}}$ (or $\mathcal{A}^{\mathbb{Z}}$) is called a Toeplitz sequence if for every $i\in\mathbb{N}$ there exists a $p_{i}\in\mathbb{N}$ such that $x_{i}=x_{i+kp_{i}}$ for all $k\in\mathbb{N}$ (or $\mathbb{Z}$). The minimal dynamical system given by $(X,\sigma)$ where $X=\overline{\{\sigma^{n}(x):n\geq 0\}}$ is called a Toeplitz flow or Toeplitz shift.

Let $(X,\sigma)$ be a Toeplitz flow with $X\subset\mathcal{A}^{\mathbb{Z}}$. Given $x\in X$, $p\in\mathbb{N}$, and $a\in\mathcal{A}$, we define the following sets.

• •

$\text{Per}_{p}(x,a)=\{k\in\mathbb{Z}\mid x_{k^{\prime}}=a\quad\forall k^{\prime}\equiv k\mod p\}$.

• •

$\text{Per}_{p}(x)=\bigcup_{a\in\mathcal{A}}\text{Per}_{p}(x,a)$.

• •

$\text{Aper}(x)=\mathbb{Z}\setminus\bigcup_{p\in\mathbb{N}}\text{Per}_{p}(x)$.

A sequence $x\in\mathcal{A}^{\mathbb{Z}}$ is Toeplitz if and only if $\text{Aper}(x)=\emptyset$. Additionally, every Toeplitz sequence $x$ is such that for every $n\in\mathbb{N}$, there is a $p\in\mathbb{N}$ such that $\{0,1,2,3,\ldots,n-1\}\subseteq\text{Per}_{p}(x)$.

For any $p\in\mathbb{N}$ and $x\in\mathcal{A}^{\mathbb{Z}}$, the p-skeleton of $x$ is the sequence obtained from $x$ by replacing $x_{i}$ by a ‘hole’ for all $i\notin\text{Per}_{p}(x)$. That is, the p-skeleton of $x$ is the part of $x$ which is periodic with period $p$. We call $p$ an essential period of a sequence $x$ if the $p$-skeleton of $x$ is not periodic with smaller period.

Definition 2.1**.**

A period structure for an aperiodic Toeplitz sequence $x$ is an increasing sequence $\{p_{i}\}_{i\in\mathbb{N}}$ of positive integers satisfying the following properties.

1. (1)

For all $i\in\mathbb{N}$, $p_{i}$ is an essential period for $x$; 2. (2)

$p_{i}\mid p_{i+1}$; 3. (3)

$\bigcup_{i\geq 1}\text{Per}_{p_{i}}(x)=\mathbb{Z}$.

There are many equivalent characterizations of Toeplitz sequences. We note a few of them here which will be used throughout this paper. The following theorem is stated in terms of one-sided Toeplitz sequences but can easily be generalized to two-sided sequences (see, for example, [9]). In fact, if $x\in\mathcal{A}^{\mathbb{Z}}$ is Toeplitz, then so is $x^{\prime}=x_{0}x_{1}x_{2}\cdots\in\mathcal{A}^{\mathbb{N}}$. We state this one-sided version to aid in quickly identifying our KR-partitions.

Theorem 2.2**.**

[5, Theorem 4.90]** The one-sided sequence $x\in\mathcal{A}^{\mathbb{N}}$ is Toeplitz if and only if there is a sequence of constant length substitutions $\theta_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i-1}$ on finite alphabets $\mathcal{A}_{i}$ with $\mathcal{A}=\mathcal{A}_{0}$ such that $\theta_{i}(a)$ starts with the same symbol for each $a\in\mathcal{A}_{i}$, and $x=\lim_{i\to\infty}\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{i}(a)$, where $a\in\mathcal{A}_{i}$ is arbitrarily chosen.

Observe that by using this characterization for the Toeplitz flow, we obtain a period structure $(p_{k})$ where $p_{k}=|\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{k}|$ for all $k\in\mathbb{N}$. We note that by allowing $\theta_{i}=\theta$ for every $i\in\mathbb{N}$, we may obtain a system which is both a substitution shift and a Toeplitz flow. In this case, the underlying period structure is such that $p_{k}=|\theta|^{k}$.

Corollary 2.3**.**

If $(X_{\theta},\sigma)$ is generated by a constant length, left proper substitution $\theta$, then $(X_{\theta},\sigma)$ is a Toeplitz flow.

We now define the $\alpha$-adic odometer, as there is a close relationship between odometers and Toeplitz flows.

Definition 2.4**.**

Let $\alpha=(\alpha_{1},\alpha_{2},\alpha_{3},\ldots)$ be a sequence of integers with each $\alpha_{i}\geq 2$. Denote by $X_{\alpha}$ the set of all infinite sequences of points $\textbf{x}=(x_{1},x_{2},x_{3},\ldots)$ such that $0\leq x_{i}\leq\alpha_{i}-1$ for each $i\in\mathbb{N}$. Addition on $X_{\alpha}$ is defined by

[TABLE]

where $z_{1}=(x_{1}+y_{1})\mod\alpha_{1}$ and for each $i\geq 2$, $z_{i}=x_{i}+y_{i}+r_{i-1}\mod\alpha_{i}$ where $r_{i-1}=0$ if $x_{i-1}+y_{i-1}+r_{i-2}<\alpha_{i-1}$ and $r_{i-1}=1$ otherwise (we set $r_{0}=0$). We define $T_{\alpha}:X_{\alpha}\to X_{\alpha}$ by

[TABLE]

The dynamical system $(X_{\alpha},T_{\alpha})$ is called the $\alpha$-adic odometer.

Theorem 2.5**.**

[7]** Toeplitz flows can be characterized up to topological conjugacy as topological dynamical systems that are symbolic, minimal, almost one-to-one extensions of odometers. Given the period structure $(p_{k})$ of a Toeplitz point $x\in X$, the underlying odometer is $(X_{\alpha},T_{\alpha})$ where $\displaystyle\alpha=\left(p_{1},\frac{p_{2}}{p_{1}},\frac{p_{3}}{p_{2}},\ldots,\frac{p_{i}}{p_{i-1}},\ldots\right)$.

2.3. Kakutani-Rokhlin Partitions

We now introduce Kakutani-Rokhlin partitions, which will be helpful in the study of topological speedups.

Definition 2.6**.**

Given a minimal Cantor system $(X,T)$, a Kakutani-Rokhlin partition (KR-partition) for $(X,T)$ is a partition

[TABLE]

of $X$ into pairwise disjoint clopen sets that cover $X$. We call the set $B=\bigcup_{i=1}^{N}B_{i}$ the base of the KR-partition and the integers $h_{i}$ the heights of each tower $A_{i}=\{T^{j}(B_{i}):0\leq j\leq h_{i}-1\}$. Further, for all $1\leq i\leq N$ we have $T^{h_{i}}(B_{i})\subset B$.

Every continuous minimal Cantor system $(X,T)$ has a sequence of KR-partitions $\mathcal{P}(k)$ satisfying the following properties [10].

1. (1)

The sequence of bases is nested: $B(k+1)\subset B(k)$. 2. (2)

For all $k\in\mathbb{N}$, $\mathcal{P}(k+1)$ refines $\mathcal{P}(k)$. 3. (3)

$\bigcap_{k\in\mathbb{N}}B(k)$ is a single point $x\in X$. 4. (4)

The partitions $\mathcal{P}(k)$ form a basis for the topology of $X$. 5. (5)

For all $k\in\mathbb{N}$, $i\leq N(k)$ and $i^{\prime}\leq N(k-1)$, there exists $0\leq j\leq h_{i}^{k}$ such that $T^{j}(B_{i}^{k})\subset B_{i^{\prime}}(k-1)$. 6. (6)

$B(k)\subset B_{1}(k-1)$ for all $k\in\mathbb{N}$.

We now explicitly define the KR-partitions $\mathcal{P}(k)$ for Toeplitz flows and substitution systems.

Definition 2.7**.**

Suppose $(X,T)$ is a Toeplitz flow generated by the Toeplitz point $x\in X$. By Theorem 2.2, there is a sequence of alphabets $\{\mathcal{A}_{i}\}$ and a sequence of constant length substitutions $\theta_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i-1}$ that generate $(X,T)$. For every $k\in\mathbb{N}$ and $1\leq i\leq|\mathcal{A}_{k}|$, let $B_{i}^{k}$ denote the set of points $x\in X$ such that $x_{[0,|\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{k}|-1]}=\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{k}(i)$ and set $h_{i}^{k}=|\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{k}|=p_{k}$. The sequence of KR-partitions is defined by the sets $\mathcal{P}(k)=\bigcup_{i=1}^{|\mathcal{A}_{k}|}\{T^{j}B_{i}^{k}:0\leq j\leq p_{k}-1\}$. For ease, we denote each tower by $A_{i}^{k}=\{T^{j}B_{i}^{k}:0\leq j\leq h_{i}^{k}-1\}$.

Example 2.8**.**

Let $\mathcal{A}=\mathcal{A}_{0}=\{0,1\}$, $\mathcal{A}_{1}=\{a,b\}$, and $\mathcal{A}_{i}=\{c,d\}$ for all $i\geq 2$. Define $\theta_{1}(a)=011$, $\theta_{1}(b)=100$, $\theta_{2}(c)=ab$ and $\theta_{2}(d)=aa$, and for all $i\geq 2$, define $\theta_{i}(c)=ccd$, and $\theta_{i}(d)=cdd$. Then $\mathcal{P}(3)$ will consist of $2$ towers, each of height $p_{3}=18$; the first tower will have its base floor given by the set of points $x\in X$ with $x_{[0,17]}=011100011100011011$ and the second tower will have its base floor given by the cylinder set $x_{[0,17]}=011100011011011011$.

We can similarly define the KR-partitions $\mathcal{P}(k)$ when the Toeplitz flow is generated by a substitution system. This is a special case of the Toeplitz definition where $\theta_{i}=\theta$ for all $i\in\mathbb{N}$.

Definition 2.9**.**

Suppose that $(X,T)$ is generated by a left proper, primitive, aperiodic substitution on an alphabet $\mathcal{A}=\{1,2,\ldots,N\}$. For every $k\in\mathbb{N}$ and $1\leq i\leq N$, let $B_{i}^{k}$ denote the set of points $x\in X$ such that $x_{[0,|\theta|^{k}-1]}=\theta^{k}(i)$ and set $h_{i}^{k}=|\theta^{k}(i)|$. Then the sequence of KR-partitions is defined by the sets $\mathcal{P}(k)=\bigcup_{i=1}^{N}\{T^{j}B_{i}^{k}:0\leq j\leq h_{i}^{k}-1\}$. For ease, we denote each individual tower by $A_{i}^{k}=\{T^{j}B_{i}^{k}:0\leq j\leq h_{i}^{k}-1\}$

We note that intuitively, given any minimal Cantor system $(X,T)$ we may think of the KR-partition $\mathcal{P}(k+1)$ as being a collection of towers, where each tower is comprised of a stack of towers from $\mathcal{P}(k)$ and the lowest $\mathcal{P}(k)$-tower in each $\mathcal{P}(k+1)$-tower is the same primary tower from $\mathcal{P}(k)$.

2.4. Speedups of Minimal Cantor Systems

Definition 2.10**.**

Let $(X,T)$ be a minimal Cantor system. A bounded speedup of $(X,T)$ is a homeomorphism of the form $(X,S)$ where $S(x)=T^{p(x)}(x)$ for some bounded function $p:X\to\mathbb{N}$. We refer to the bounded function $p(x)$ as the jump function.

In general, one can explore speedups where $p$ is not assumed to be bounded; however, the function $p$ is known to be continuous if and only if it is bounded [1, Propostion 2.2]. Additionally, the resulting speedup of a minimal Cantor system need not be minimal. If all $S$-orbits are dense in $X$, then we say that $(X,S)$ is a minimal speedup of $(X,T)$. Throughout this paper, we will always assume that $p$ is bounded and that $(X,S)$ is a minimal speedup. Given a minimal bounded speedup $(X,S)$ of $(X,T)$, there is a constant $c\in\mathbb{N}$, called the orbit number, such that every $T$-orbit is the union of exactly $c$ different $S$-orbits [1, Lemma 2.3]. We now discuss some properties of jump functions as they relate to the sequence of KR-partitions for the underlying minimal Cantor system $(X,T)$.

Remark 2.11**.**

Let $(X,T)$ be a minimal Cantor system. Suppose that $p:X\to\mathbb{N}$ defines a bounded topological speedup $(X,S)$. Then the sequence of KR-partitions for $(X,T)$ is such that for all $k$ sufficiently large:

1. (1)

The height $h_{i}^{k}$ of each tower $A_{i}^{k}$ is greater than $\max_{x\in X}p(x)$. 2. (2)

$p$ is constant on each level of $\mathcal{P}(k)$. 3. (3)

$p$ is constant on each set $T^{m}(B(k))$ for $0\leq m\leq\max_{x\in X}p(x)$.

By beginning our sequence of KR-partitions with $k$ large enough, we may assume these properties are satisfied for our entire sequence of KR-partitions.

We may thus create an orbit labeling of the levels in $\mathcal{P}(k)$ that models the speedup. This process demonstrates how every $T$-orbit decomposes into $c$ different $S$-orbits.

Definition 2.12**.**

For all $k$ large enough so that the properties in Remark 2.11 hold and for $c$ equal to the orbit number of the speedup, let $\mathcal{L}_{k}:\mathcal{P}(k)\to\{1,2,\ldots,c\}$ be the orbit labeling function defined as follows. For each $1\leq i\leq N(k)$:

• •

Label the base floor $B_{i}^{k}$ of tower $A_{i}^{k}$ with a $1$.

• •

Label any other floor $F$ in $A_{i}^{k}$ with a $1$ if $F=S^{j}(B_{i}^{k})$ and $\sum_{l=0}^{j-1}pS^{l}(x)<h_{i}^{k}$ for all $x\in B_{i}^{k}$.

• •

Label the lowest unlabeled floor $T^{d_{2}}(B_{i}^{k})$ of $A_{i}^{k}$ with a $2$.

• •

Label any other floor $F$ in $A_{i}^{k}$ with a $2$ if $F=S^{j}(T^{d_{2}}(B_{i}^{k}))$ and $\sum_{l=0}^{j-1}pS^{l}(x)<h_{i}^{k}$ for all $x\in T^{d_{2}}(B_{i}^{k})$.

• •

Label the lowest unlabeled floor in $A_{i}^{k}$ with a $3$ and continue.

• •

Continue in this manner until all floors of $A_{i}^{k}$ are labeled with a value from $\{1,2,\ldots,c\}$.

Note that by assuming the properties in Remark 2.11 hold, we guarantee that any two floors in different columns with the same height below the $\max p(x)$ floor will have the same labeling. Suppose that $F$ is the maximal floor in $A_{i}^{k}$ that is labeled $l$; then for all $x,y\in F$, the label of the floor containing $S(x)$ is the same as the label of the floor containing $S(y)$. Hence, for $k$ large enough so that the properties in Remark 2.11 hold, define $\pi_{i}^{(k)}(l)$ to be the label of the floor containing $S(x)$ for all $x$ in the floor of maximal height in $A_{i}^{k}$ labeled $l$.

Proposition 2.13**.**

If $(X,S)$ is a minimal speedup of $(X,T)$ with jump function $p:X\to\mathbb{N}$, then each $\pi_{i}^{(k)}$ is a permutation of the set $\{1,2,\ldots,c\}$.

We now list some known results about speedups of odometers and substitutions that will be helpful in our study of Toeplitz flows. Recall that every KR-partition for an odometer consists of a single tower.

Lemma 2.14**.**

[1, Lemma 3.7]** Suppose $(X,T)=(X_{\alpha},T_{\alpha})$ is an odometer and $(X_{\alpha},S)$ is a bounded speedup of $(X_{\alpha},T_{\alpha})$. Then $(X_{\alpha},S)$ is minimal if and only if for all $k$ sufficiently large, $\pi^{(k)}$ is a cyclic permutation on $\{1,2,\ldots,c\}$, where $c$ is the orbit number.

The following theorem is a combination of several results from [1].

Theorem 2.15**.**

[1]** If $(X,T)=(X_{\alpha},T_{\alpha})$ is an odometer with $\alpha=(\alpha_{1},\alpha_{2},\ldots)$, then there is a minimal bounded speedup with orbit number $c>1$ if and only if there is some $N\in\mathbb{N}$ such that $\gcd(c,\alpha_{i})=1$ for all $i\geq N$. Further, in this case the minimal speedup will be topologically conjugate to the original odometer.

We note that the Toeplitz flows we are interested in studying are each symbolic shift spaces. It thus follows that any minimal speedup of a Toeplitz flow will necessarily result in a system that is not conjugate to the original system.

Theorem 2.16**.**

[1, Theorem 4.6]** If $(X,T)$ is a minimal subshift on an infinite set $X$ and $(X,S)$ is a minimal bounded speedup of $(X,T)$, then $(X,T)$ and $(X,S)$ are not conjugate.

Let $\theta$ be a constant length, left proper, primitive substitution. Because there exists some $K\geq 1$ such that $\pi_{i}^{(K)}=\pi_{i}^{(jK)}$ for all $1\leq i\leq|\mathcal{A}|$ and all $j\in\mathbb{N}$ ([1]), we may assume that $K$ is chosen large enough to satisfy Remark 2.11. Additionally, because the system generated by $\theta$ is conjugate to the system generated by $\theta^{k}$, without loss of generality we assume that $\theta$ satisfies all of these hypotheses with $k=1$ and $K=1$.

Definition 2.17**.**

Let $\theta$ be a constant length, left proper, primitive substitution on the alphabet $\mathcal{A}$. Set $C=\{1,2,\ldots,c\}$ and consider the alphabet $\mathcal{A}\times C$. Define $\tau:\mathcal{A}\times C\to(\mathcal{A}\times C)^{*}$ by

[TABLE]

where $\theta(a)=a_{1}a_{2}\cdots a_{n}$, $\ell_{1}=\ell$, and $\ell_{k+1}=\pi_{i_{k}}(\ell_{k})$ for each $k\geq 1$.

Lemma 2.18**.**

[1, Lemma 4.15]** Suppose $\theta$ is a (left) proper, primitive, aperiodic substitution on $\mathcal{A}$ and $(X,T)$ is the subshift generated by $\theta$. Suppose $(X,S)$ is a bounded speedup of $(X,T)$ with orbit number $c$. Then $(X,S)$ is minimal if and only if $\tau$ is primitive.

Theorem 2.19**.**

[1, Theorem 4.18]** Suppose $(X,T)$ is a minimal substitution system associated with a (left) proper primitive substitution $\theta$. If $(X,S)$ is a minimal bounded speedup of $(X,T)$, then $(X,S)$ is a minimal substitution system.

We now generalize Definition 2.17 and Lemma 2.18 from the substitution setting to the more general S-adic setting for Toeplitz flows.

Definition 2.20**.**

Let $(\mathcal{A}_{i})_{i\geq 0}$ be a sequence of alphabets and $(\theta_{i})_{i\geq 1}$ be a sequence of constant length, left proper, primitive substitutions such that $\theta_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i-1}^{*}$. Set $C=\{1,2,\ldots,c\}$ and consider the sequence of alphabets $((\mathcal{A}_{i}\times C)_{i\geq 0})$. Define $\tau_{k}^{l}:\mathcal{A}_{k}\times C\to(\mathcal{A}_{l}\times C)^{*}$ by

[TABLE]

where $\theta_{l+1}\circ\theta_{l+2}\circ\cdots\circ\theta_{k}(a)=a_{1}a_{2}\cdots a_{n}$, $\ell_{1}=\ell$, and $\ell_{j+1}=\pi_{i_{j}}(\ell_{j})$ for each $j\geq 1$.

Lemma 2.21**.**

Let $(X,T)$ be the Toeplitz flow generated by the sequences $(\mathcal{A}_{i})$ and $(\theta_{i})$. Suppose $(X,S)$ is a bounded speedup of $(X,T)$ with orbit number $c$. Then $(X,S)$ is minimal if and only if for every $l\in\mathbb{N}$ there is a $k\geq l$ such that $\tau_{k}^{l}(a,j)$ contains every letter from $\mathcal{A}_{l}\times C$ where $a\in\mathcal{A}_{k}$ and $j\in C$ are arbitrarily chosen.

Proof.

Suppose for all $l\in\mathbb{N}$ there exists a $k\geq l$ such that $\tau_{k}^{l}(a,j)$ contains every letter from $\mathcal{A}_{l}\times C$, where $a\in\mathcal{A}_{k}$ and $j\in C$ are arbitrarily chosen. Let $x\in X$ and consider the decomposition of $x$ into $\theta_{1}\circ\theta_{2}\circ\cdots\theta_{k}$-words. Within each $\theta_{1}\circ\theta_{2}\circ\cdots\circ\theta_{k}$-word is every possible $\theta_{1}\circ\theta_{2}\circ\cdots\theta_{l}$-word with every possible label. Thus, the $S$-orbit of $x$ intersects all floors of $\mathcal{P}(l)$. Because this is true for all $l\in\mathbb{N}$, the $S$-orbit of $x$ is dense.

Conversely, suppose that $S$ is minimal and fix $x\in X$. There is an $N$ such that any $S$-orbit block of length $N+1$ intersects every floor of $\mathcal{P}(l)$ with all labels. Set $M=\sup_{x\in X}p(x)$ and let $k>l$ be chosen such that $|\tau_{k}^{l}(a,j)|\geq MN$ for all $(a,j)\in\mathcal{A}_{k}\times C$. Then we may conclude that every letter $(a^{\prime},j^{\prime})\in\mathcal{A}_{l}\times C$ appears in $\tau_{k}^{l}(a,j)$ for each $(a,j)\in\mathcal{A}_{k}\times C$. ∎

3. The Speedup of a Toeplitz Flow Need Not be Toeplitz

In this section we present an example of a Toeplitz flow $(X,T)$ and a jump function $p(x):X\to\mathbb{N}$ such that the speedup $(X,S)=(X,T^{p(x)})$ is minimal and not conjugate to a Toeplitz flow. This demonstrates that the conjecture in [3] that the minimal speedup of a Toeplitz flow is Toeplitz is not true.

Example 3.1**.**

Consider the primitive substitution $\theta$ on the alphabet $\mathcal{A}=\{a,b\}$ given by $\theta(a)=ab$ and $\theta(b)=aa$. Note that the subshift $(X_{\theta},\sigma)$ is a Toeplitz flow. Let $A=[\theta^{3}(a)]=\{x\mid x_{[0,7]}=abaaabab\}$ and define $p:X_{\theta}\to\mathbb{Z}^{+}$ by

[TABLE]

With this given jump function $p$, $(X_{\theta},S)$ is a minimal bounded speedup of $(X_{\theta},\sigma)$ where $S=\sigma^{p(x)}(x)$. In this case, the orbit number is $c=2$.

Remark 3.2**.**

For $(X_{\theta},\sigma)$, the tower $A_{1}^{k}$ is given by $\theta^{k}(a)$ and the tower $A_{2}^{k}$ is given by $\theta^{k}(b)$. For all $k\geq 3$, the permutation on the first tower is given by $\pi_{1}^{(k)}(1)=2$ and $\pi_{1}^{(k)}(2)=1$, whereas the permutation on the second tower is given by $\pi_{2}^{(k)}(1)=1$ and $\pi_{2}^{(k)}(2)=2$. Because the permutation depends only on the tower and not on the level of the KR-partition, we may simply use $\pi_{1}(i)$ and $\pi_{2}(i)$ for all KR partitions $\mathcal{P}(k)$.

Also observe that the height of orbit $j$ in tower $A_{i}^{k}$ is given by $o_{i}^{k}(j)=2^{k-1}$ for all $i,j\in\{1,2\}$ and for all $k\geq 3$. In particular, the height of each orbit in each tower of $\mathcal{P}(3)$ is $4$. To see this more clearly, let $A=aba$, $B=aa$, $C=ba$, $D=b$, and $E=ab$. Then tracing through the first orbit of the tower $A_{1}^{3}$ is equivalent to tracing through $ABCC$. Tracing through the second orbit of the tower $A_{1}^{3}$ is equivalent to tracing through $DBEE$. Similarly, tracing through the first orbit of the tower $A_{2}^{3}$ is equivalent to tracing through the $EBEB$ and tracing through the second orbit of $A_{2}^{3}$ is equivalent to tracing through $CBCB$.

Remark 3.3**.**

In Example 3.1, let $ABCC$ denote the first orbit path of the first tower in $\mathcal{P}(3)$, $DBEE$ denote the second orbit path in the first tower, $EBEB$ denote the first orbit path in the second tower, and $CBCB$ denote the second orbit path in the second tower in $\mathcal{P}(3)$. Note that $|ABCC|=|DBEE|=|EBEB|=|CBCB|=4$. Define $\mathcal{C}=\{A,B,C,D,E\}$ and $\mathcal{B}=\{w_{1},w_{2},w_{3},w_{4}\}$.

Define $\psi:\mathcal{B}\to\mathcal{C}^{*}$ by $\psi(w_{1})=ABCCCBCBDBEE$, $\psi(w_{2})=ABCCDBEEEBEB$, $\psi(w_{3})=ABCCCBCBDBEEEBEB$, and $\psi(w_{4})=ABCCDBEE$.

Define $\phi:\mathcal{B}\to\mathcal{B}^{*}$ by $\phi(w_{1})=w_{1}w_{2}w_{3}w_{4}$, $\phi(w_{2})=w_{1}w_{2}w_{4}w_{3}$, $\phi(w_{3})=w_{1}w_{2}w_{3}w_{4}w_{3}$, and $\phi(w_{4})=w_{1}w_{2}w_{4}$.

Then by using the return word method of Durand [8], the speedup $(X_{\theta},S)$ is topologically conjugate to the shift space defined by applying $\psi$ to the fixed point of the substitution $\phi$ and then taking the closure of the shift on this new point. We denote this space $(\psi(X_{\phi}),\sigma)$.

Theorem 3.4**.**

The speedup $(X_{\theta},S)$ defined in Example 3.1 is not a Toeplitz flow.

Proof.

Suppose that $(X_{\theta},S)$ is a Toeplitz flow. Then without loss of generality, there exists some word $y\in X_{\theta}$ that is Toeplitz under $S$. Without loss of generality, we may assume that the decomposition of $y$ into $\mathcal{C}$-words begins with $y_{[0,8]}=ABCC$. It follows that the decomposition of $y$ into these $\mathcal{C}$-words is a Toeplitz sequence. Hence there exists some $t\in\mathbb{N}$ such that $A$ appears in the $\mathcal{C}$-decomposition of $y$ in every $t$ positions. Further, because $y$ is necessarily minimal and the substitution $\phi$ is recognizable, $\psi(\phi^{k}(w_{2}))$ appears in this decomposition of $y$ for every $k\in\mathbb{N}$. Hence, $A$ must appear in $\psi(\phi^{k}(w_{2}))$ exactly every $t$ positions for every $k\in\mathbb{N}$.

Observe the following $\psi(\phi)$ words (formatted with spaces to ease identifying the locations of $A$).

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By inspection, it necessarily follows that $t\geq 24$. Additionally, we can look at the $\psi(\phi^{2})$ words.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By applying the same logic as above, we see that necessarily $t\geq 96$. Suppose that for some $k\geq 1$, $A$ appears every $t$ positions in $\psi(\phi^{k}(w_{2}))$ for $t=6\cdot 4^{k}$. We now outline the following facts about $\psi\circ\phi^{k}$ and $\psi\circ\phi^{k+1}$.

1. (1)

Facts about the lengths of the $\psi(\phi^{k}(w))$ words:

• •

$|\psi(\phi^{k}(w_{1}))|=6\cdot 4^{k}+6\cdot 4^{k}$.

• •

$|\psi(\phi^{k}(w_{2}))|=6\cdot 4^{k}+6\cdot 4^{k}$

• •

$|\psi(\phi^{k}(w_{3}))|=6\cdot 4^{k}+6\cdot 4^{k}+4\cdot 4^{k}$

• •

$|\psi(\phi^{k}(w_{4}))|=6\cdot 4^{k}+2\cdot 4^{k}$ 2. (2)

Facts about concatenation of words in $\psi(\phi^{k+1}(w_{2}))$:

• •

$\psi(\phi^{k+1}(w_{2}))=\psi(\phi^{k}(w_{1})\phi^{k}(w_{2})\phi^{k}(w_{4})\phi^{k}(w_{3}))$

• •

$|\psi(\phi^{k+1}(w_{2}))|=|\psi(\phi^{k}(w_{1})\phi^{k}(w_{2})\phi^{k}(w_{4})\phi^{k}(w_{3}))|=6\cdot 4^{k}+6\cdot 4^{k}+6\cdot 4^{k}+6\cdot 4^{k}+6\cdot 4^{k}+2\cdot 4^{k}+6\cdot 4^{k}+6\cdot 4^{k}+4\cdot 4^{k}$

• •

The alignment of $\psi(\phi^{k}(w_{4})\phi^{k}(w_{3}))$ is such that $A$ does not appear at all multiples of $6\cdot 4^{k}$. It does hold that $A$ appears at all multiples of $6\cdot 4^{k+1}$ in $\psi(\phi^{k}(w_{4})\phi^{k}(w_{3}))$. Observe that $A$ will also appear at all multiples of $6\cdot 4^{k+1}$ in $\psi(\phi^{k}(w_{1}w_{2}))$.

• •

We thus conclude that $A$ appears at all multiples of $6\cdot 4^{k+1}$ in $\psi(\phi^{k+1}(w_{2}))$. Therefore, it follows that if $A$ appears at every $t$ positions in the $\mathcal{C}$-decomposition of $y$, then $t\geq 6\cdot 4^{k+1}$.

For each $k\in\mathbb{N}$, it follows that in order for $A$ to appear within $\psi(\phi^{k}(w_{2}))$ at every $t$ positions, then $t\geq 6\cdot 4^{k}$. Since $t$ is assumed to be a finite number that works for every $k\in\mathbb{N}$, we obtain a contradiction. It thus follows that the decomposition of $y$ into $\mathcal{C}$-words is not a Toeplitz sequence, and hence $y$ will not be a Toeplitz sequence under $S$. We conclude that no point in $X_{\theta}$ will be Toeplitz under $S$, and therefore $(X_{\theta},S)$ is not a Toeplitz flow. ∎

We now show that no minimal speedup of the Toeplitz flow $(X_{\theta},\sigma)$, where $\theta$ is defined as in Example 3.1, with orbit number $c=2$ is a Toeplitz flow.

Theorem 3.5**.**

Any minimal speedup of the Toeplitz flow $(X_{\theta},\sigma)$ generated by the substitution $\theta(a)=ab$ and $\theta(b)=aa$ with orbit number $c=2$ will not be a Toeplitz flow.

Proof.

Let $p(x)$ be a jump function such that $(X_{\theta},S)$ is a minimal bounded speedup of $(X_{\theta},\sigma)$ with orbit number $c=2$. Choose a KR-partition $\mathcal{P}(k)$ with $k$ chosen large enough such that the properties in Remark 2.11 hold. Consider the permutations $\pi_{a}^{(k)}$ and $\pi_{b}^{(k)}$ induced by the speedup. Observe that if $\pi_{a}^{(k)}=\pi_{b}^{(k)}=\text{id}$, then the associated speedup would not be minimal. Similarly, if $\pi_{a}^{(k)}(1)=\pi_{b}^{(k)}(1)=2$, the associated speedup will not be minimal as $\pi_{a}^{(k+1)}=\pi_{b}^{(k+1)}=\text{id}$. Thus, one of tower permutations must be the identity, and other tower permutation must be the cyclical permutation. Observe that in either case, for all $k\geq K$ (there exists $K\in\mathbb{N}$), it follows that $\pi_{a}^{(k)}(1)=2$ and $\pi_{b}^{(k)}=\text{id}$. One can check via Lemma 2.18 that the associated speedup will be minimal.

Let $o_{a}^{k}(i)$ denote the number of levels of the $\theta^{k}(a)$ tower in the KR-partition $\mathcal{P}(k)$ that are labeled with $i$ for each $i\in\{1,2\}$; similarly define $o_{b}^{k}(i)$. Observe that $o_{a}^{k}(1)+o_{a}^{k}(2)=|\theta^{k}|$ and $o_{b}^{k}(1)+o_{b}^{k}(2)=|\theta^{k}|$. Further note that the substitution $\phi:\mathcal{B}\to\mathcal{B}^{\ast}$ defined as in Remark 3.3 is the same as the substitution defined by checking returns to $(a,1)$ in the substitution defined by $\tau$ in Definition 2.17. Here $\mathcal{B}=\{w_{1}=(a,1)(b,2)(a,2),w_{2}=(a,1)(a,2)(b,1),w_{3}=(a,1)(b,2)(a,2)(b,1),w_{4}=(a,1)(a,2)\}$, $\phi(w_{1})=w_{1}w_{2}w_{3}w_{4}$, $\phi(w_{2})=w_{1}w_{2}w_{4}w_{3}$, $\phi(w_{3})=w_{1}w_{2}w_{3}w_{4}w_{3}$, and $\phi(w_{4})=w_{1}w_{2}w_{4}$. Observe that when we consider the orbit heights of each orbit path, the length of $w_{1}$ is $o_{a}^{k}(1)+o_{b}^{k}(2)+o_{a}^{k}(2)=|\theta|^{k}+o_{b}^{k}(2)$, the length of $w_{2}$ is $o_{a}^{k}(1)+o_{a}^{k}(2)+o_{b}^{k}(1)=|\theta|^{k}+o_{b}^{k}(1)$, the length of $w_{3}$ is $o_{a}^{k}(1)+o_{b}^{k}(2)+o_{a}^{k}(2)+o_{b}^{k}(1)=2|\theta|^{k}$, and the length of $w_{4}$ is $o_{a}^{k}(1)+o_{a}^{k}(2)=|\theta|^{k}$.

We now follow the same logic as in the previous theorem to demonstrate that $(X_{\theta},S)$ is not Toeplitz. Suppose that $(X_{\theta},S)$ is Toeplitz. Then there is some sequence $x\in X_{\theta}$ that is Toeplitz under $S$. We can decompose $x$ uniquely into its $w_{i}$ representation. As $x$ is Toeplitz, there exists some $l\in\mathbb{N}$ such that if the decomposition of $x$ into $w_{i}$ words begins with $w_{1}$, then the decomposition of $S^{l\cdot n}(x)$ begins with $w_{1}$ for all $n\in\mathbb{N}$. By our substitution, $|\phi(w_{1})|=|\phi(w_{2})|=6|\theta|^{k}$, $|\phi(w_{3})|=8|\theta|^{k}$,and $|\phi(w_{4})|=4|\theta|^{k}$. Hence, any return between our desired occurrences of $w_{1}$ in the $w_{i}$-decomposition of $x$ gives $l\geq 8|\theta|^{k}$. By looking at $\phi^{2}(w_{1})=w_{1}w_{2}w_{3}w_{4}w_{1}w_{2}w_{4}w_{3}w_{1}w_{2}w_{3}w_{4}w_{3}w_{1}w_{2}w_{4}$, we obtain that $l\geq 12|\theta|^{k}$. By looking at $\phi^{3}(w_{1})$, we obtain that $l\geq 24|\theta|^{k}$. In general, we will never be able to find a single value $l$ such that our $w_{i}$-decomposition of $x$ returns to $w_{1}$ every $l$ steps. Therefore $(X_{\theta},S)$ is not a Toeplitz flow. ∎

This result leads to the following natural question.

Question 3.6**.**

Suppose $(X_{\theta},\sigma)$ is an arbitrary Toeplitz flow generated by the constant length substitution $\theta$. Is there any minimal speedup $(X_{\theta},S)$ of $(X_{\theta},\sigma)$ with orbit number $c=|\theta|$ such that $(X_{\theta},S)$ is Toeplitz? What if the orbit number $c$ is such that $\gcd(c,|\theta|)\neq 1$?

4. Proving that a Speedup is (Not) Toeplitz

Theorem 4.1**.**

Suppose that $(X,T)$ is a Toeplitz flow with underlying period structure $(p_{k})$. Then $(X,T^{c})$ is a Toeplitz flow if and only if $\gcd(c,p_{k})=1$ for all $k$.

Proof.

Consider the sequence of KR-partitions $\mathcal{P}(k)$ for $(X,T)$ as defined in Definition 2.7. Recall that this sequence of KR-partitions can be built from the $S$-adic structure for an arbitrary fixed Toeplitz sequence $x\in X$. Fix a level $N$ such that all of the towers in the KR-partition $\mathcal{P}(k)$ have height $p_{k}>c$ for every $k\geq N$. Look at the permutations of the orbits on each tower. By construction, each tower has the same permutation of orbits; thus set $\pi_{i}^{(k)}=\pi$ for each $i$ (here we understand that $\pi$ depends on the level $\mathcal{P}(k)$, but we only work with one fixed $k$ at a time).

First suppose that $\gcd(c,p_{k})=1$ for all $k$. In this case $\pi$ is a cyclical permutation on $c$ elements, so $\pi^{c}=\text{id}$. If one starts on the first floor of any tower (i.e., the first orbit level), after tracing through exactly $c$ towers, they will return to the base floor of some tower. By construction, the base floor of every tower in $\mathcal{P}(k)$ is contained in the same base floor of the same primary tower in $\mathcal{P}(k)$. Hence, we have that for our fixed Toeplitz sequence $x$, $T^{c\cdot p_{k}\cdot n}(x)=(T^{c})^{p_{k}\cdot n}(x)$ is in the same cylinder set as $x$ for all $n\in\mathbb{N}$. Since we may follow this argument for every $k\geq N$, it follows that $x$ is a Toeplitz point under the map $T^{c}$.

It remains to show that $T^{c}$ is minimal. Fix $k$ such that $p_{k}>c$. Since $(X,T)$ is minimal, there is some $t>k$ such that the primary tower of $\mathcal{P}(t-1)$, whose base floor contains the base floor of every tower in $\mathcal{P}(t)$, is a stack that contains every tower from $\mathcal{P}(k)$. Whenever one starts in the base floor of any tower in $\mathcal{P}(t)$, one will thus trace through every orbit of the primary tower from $\mathcal{P}(t-1)$ that contains every tower from $\mathcal{P}(k)$. That is, one will trace through every possible floor of every tower from the KR-partition $\mathcal{P}(k)$. Since $k$ was arbitrarily chosen, we see that $T^{c}$ is minimal. Hence, it follows that if $\gcd(c,p_{k})=1$, then $(X,T^{c})$ is a Toeplitz flow.

Conversely, suppose that $\gcd(c,p_{k})=n>1$ for some $k\in\mathbb{N}$. By labeling the orbits of each tower, one sees $c$ different orbit labels. However, the permutation $\pi$ on these orbit labels is such that $\pi$ decomposes into $n$ distinct cyclic permutations. As a result, the speedup $(X,T^{c})$ will not be minimal, since one cannot switch between the $n$ distinct collections of orbit labels. Thus, if $\gcd(c,p_{k})\neq 1$, it follows that $(X,T^{c})$ is not a Toeplitz flow. ∎

Example 4.2**.**

Consider the primitive substitution $\theta$ on the alphabet $\mathcal{A}=\{a,b\}$ given by $\theta(a)=ab$ and $\theta(b)=aa$. The subshift $(X_{\theta},T)$ is a Toeplitz flow with period structure $(p_{k})=(2^{k})$ for all $k\geq 1$. Consider the constant speedup $(X_{\theta},T^{c})$ where $c=3$. By Theorem 4.1, $(X_{\theta},T^{3})$ is a Toeplitz flow. One can verify that $(X_{\theta},T^{3})$ is conjugate to the substitution shift generated by $\gamma:\{A,B,C,D,E\}\to\{A,B,C,D,E\}^{\ast}$ given by $\gamma(A)=ABAC$, $\gamma(B)=ABDE$, $\gamma(C)=AEDC$, $\gamma(D)=ABDC$, and $\gamma(E)=AEDE$. Observe that this substitution is once again constant length and left proper. One associated period structure for $(X_{\theta},T^{3})$ is $(q_{k})=(4^{k})$ for every $k\in\mathbb{N}$. In this case, we note that the underlying odometer for both $(X_{\theta},T)$ and $(X_{\theta},T^{3})$ is the dyadic odometer; in fact $(q_{k})$ is a period structure for $(X_{\theta},T)$, even though $(p_{k})$ is not a period structure for $(X_{\theta},T^{3})$.

We want to emphasize that the previous example generated a minimal speedup which could also be represented as a left proper, constant length substitution. In general, a minimal speedup of a constant length substitution shift need not be conjugate to another constant length substitution. Every example we have constructed where the minimal speedup of a Toeplitz flow generated by a constant left proper substitution is again a Toeplitz flow has the property that the resulting speedup can be represented as a constant length and left proper substitution. This leads us to ask if this is always the case.

Question 4.3**.**

If we begin with a Toeplitz flow generated from a constant length substitution $\theta$, and the speedup $(X,S)$ is also a Toeplitz flow, must there be a constant length representation of the substitution generating $(X,S)$?

Theorem 4.4**.**

Suppose $(X,T)$ is a Toeplitz flow. If $(X,T^{c})$ is a Toeplitz flow, then $(X,T^{c})$ has the same underlying odometer as $(X,T)$.

Proof.

Let $(X,T)$ be a Toeplitz flow and suppose that $(X,T)$ has period structure $(p_{k})$. Suppose further that $\gcd(c,p_{k})=1$ for all $k\in\mathbb{N}$. First note that for every $k\in\mathbb{N}$ with $p_{k}>c$, if one starts at the base floor of any tower in the KR-partition $\mathcal{P}(k)$ and jumps $c$ levels at a time, then one will return to a base floor of some tower after exactly $p_{k}$ jumps. Look at the KR-partition $\mathcal{P}(k+1)$. Observe that there is a primary tower in $\mathcal{P}(k)$ such that the base of every tower in $\mathcal{P}(k+1)$ is contained in the base of the primary tower from $\mathcal{P}(k)$. Thus, if one starts in any base level of any tower of the KR-partition $\mathcal{P}(k+1)$, one is at the base level of the primary tower for $\mathcal{P}(k)$. Hence, one returns to the cylinder set defined by the primary tower of $\mathcal{P}(k)$ exactly every $p_{k+1}$ jumps. Hence, there exists some $N\in\mathbb{N}$ (here $p_{N}>c$) such that we may choose $p_{N+1}$ as the first essential period for our period structure.

Observe that the primary tower from $\mathcal{P}(N+1)$ contains every tower from $\mathcal{P}(N+2)$. Now the entire cylinder set associated with the base floor of the primary tower from $\mathcal{P}(N+1)$ will be entered into every $p_{N+2}$ jumps. Further, note that we do not return to this cylinder set every $p_{N+1}$ jumps, as our space $X$ is not periodic. Hence $p_{N+2}$ is an essential period.

By induction, we have that $p_{N+k}$ is an essential period for every $k\in\mathbb{N}$, $p_{N+k}$ divides $p_{N+k+1}$ for every $k\in\mathbb{N}$, and because the Toeplitz point $x\in X$ generating the KR-partition for $(X,T)$ will be Toeplitz under $T^{c}$ (because every cylinder set of $x$ will return to itself for some $p_{N+k}$), it follows that $(p_{N+k})$ is a period structure for $(X,T^{c})$. Additionally, as $(p_{N+k})\subseteq(p_{k})$, it follows that $(p_{N+k})$ is a period structure for $(X,T)$. Because $(X,T)$ and $(X,T^{c})$ can be generated by the same period structure, it follows that they have the same underlying odometers. ∎

We now generalize the technique used in the Proof of Theorem 3.4 into an algorithm to determine whether the speedup of a Toeplitz flow is Toeplitz. We present the technique in two versions: the constant length substitutive Toeplitz case and the general (non-substitutive) Toeplitz case.

Algorithm to determine whether a given speedup of a substitutive Toeplitz flow is Toeplitz:

1. (1)

Given the constant length and left proper substitution $\theta$ and the jump function $p(x)$, find the tower permutations $\pi_{i}^{(k)}$ for the speedup. Observe that we may choose $k$ large enough and consider the substitution $\theta^{k}$ so that without loss of generality, every KR-partition $\mathcal{P}(k)$ is such that the $i$th tower at every level has the same permutation and Remark 2.11 is satisfied. 2. (2)

Define the substitution $\tau:\mathcal{A}\times C\to(\mathcal{A}\times C)^{\ast}$ as in Definition 2.17. Note that this substitution can be rewritten as an equivalent substitution $\phi:\mathcal{B}\to\mathcal{B}^{*}$ by taking the fixed point of the substitution $\tau$ generated by $(a,1)$ and using Durand’s method (see [8]) to create a left proper substitution by looking at returns to $(a,1)$. 3. (3)

Define a substitution $\psi$ that encodes the individual jumps within each tower orbit into a new alphabet $\mathcal{C}$; this substitution extends to the $\phi$-words. 4. (4)

Take a bi-infinite fixed sequence $x$ under $\phi$ and apply $\psi$. Then the shift space generated by $\psi(x)$ is conjugate to the speedup $(X,S)$. Note that $x$ can be found by extending the labeling rule defined in Definition 2.17 to the bi-infinite fixed point of $\theta$ where we assume $\ell_{0}=1$, $\ell_{k}=\pi_{i_{k}}(\ell_{k-1})$ if $k\geq 1$ and $\ell_{j}=\pi_{i_{j}}^{-1}(\ell_{j+1})$ if $j\leq-1$. 5. (5)

Observe if $\psi\circ\phi$ has a constant length and left proper representation, then immediately $(X,S)$ is Toeplitz. 6. (6)

Suppose that some point $y\in X$ is Toeplitz under $S$. There is a unique decomposition of $y$ into $\psi(\phi^{k})$-words for every $k\in\mathbb{N}$. We call this decomposition $y^{\prime}$. Since the [math]th coordinate of $y^{\prime}$ appears every $p$ positions (there exists some $p\in\mathbb{N}$, since $y^{\prime}$ is Toeplitz under the shift map), it follows that this symbol must appear every $p$ positions in every $\psi(\phi^{k})$ word for every $k\in\mathbb{N}$. 7. (7)

Thus, if one can show that no such $p$ can possibly exist (through checking larger and larger $k$ values), we obtain a contradiction and $(X,S)$ is not Toeplitz. 8. (8)

If it follows that there is a $p\in\mathbb{N}$ such that the [math]th coordinate of $y^{\prime}$ appears every $p$ positions in $y^{\prime}$, then we set $p=p_{1}$ and we suppose there is some $p_{2}\in\mathbb{N}$ (necessarily a multiple of $p_{1}$) such that the word $y^{\prime}_{[0,p_{1}-1]}$ appears every $p_{2}$ positions in every $\psi(\phi^{k})$ word for every $k\in\mathbb{N}$ and continue recursively. If we can find some infinite sequence $(p_{k})$ of integers, this will form our period structure for $y^{\prime}$ under the shift map and we will have verified that $y$ is Toeplitz under $S$. If at any stage we cannot find a valid $p_{k}$, then the speedup will not be Toeplitz.

Algorithm to determine whether a given speedup of a general Toeplitz flow is Toeplitz:

1. (1)

Given the sequence of alphabets $(\mathcal{A}_{i})$ and the sequence of constant length left proper substitutions $(\theta_{i}:\mathcal{A}_{i}\to\mathcal{A}_{i-1})$ that generate the Toeplitz flow $(X,T)$ and the jump function $p(x)$ for the speedup, find the tower permutations $\pi_{i}^{(k)}$ for the speedup for each KR-partition $\mathcal{P}(k)$. 2. (2)

Define a substitution $\tau_{i+1}:\mathcal{A}_{i+1}\times I\to(\mathcal{A}_{i}\times I)^{\ast}$ by $\tau_{i+1}(A,j)=(b_{1},j_{1})(b_{2},j_{2})\cdots(b_{n},j_{n})$ where $\theta_{i+1}(A)=b_{1},b_{2}\ldots b_{n}$, $j_{1}=j$, and for all $i\geq 2$, $j_{i}=\pi_{b_{i-1}}^{(i)}(j_{i-1})$. 3. (3)

Define a substitution $\psi_{l}$ that encodes the individual jumps within each tower orbit $(a,j)\in\mathcal{A}_{l}\times C$ into a new alphabet $\mathcal{C}$. 4. (4)

Take a bi-infinite sequence $x\in X$ and decode it into into $\mathcal{A}_{l}\times C$-words and apply $\psi_{l}$; denote this new sequence as $x^{\prime}$ and note that the shift space generated by $x^{\prime}$ is conjugate to the speedup $(X,S)$. Note that if we choose $x\in X$ to be a Toeplitz sequence (under $T$), then the decomposition of $x$ into $\mathcal{A}_{l}\times C$ words can be found by extending the labeling rule defined in Definition 2.20 to $x$ where we assume $l_{0}=1$, $l_{j}=\pi_{i_{j}}(l_{j-1})$ if $j\geq 1$ and $l_{j}=\pi_{i_{j}}^{-1}(l_{j+1})$ if $j\leq-1$. Hence we may apply $\psi_{l}$ to this bi-infinite sequence and generate the same shift space conjugate to $(X,S)$. 5. (5)

Suppose that some point $y\in X$ is Toeplitz under $S$. There is a unique decomposition of $y$ into $\psi_{l}(\mathcal{A}_{l}\times C)$-words for every $l\in\mathbb{N}$. We call this decomposition $y^{\prime}$ and recall that the shift map on $y^{\prime}$ acts in the same way that $S$ acts on $y$. Since the [math]th coordinate of $y^{\prime}$ appears every $p$ positions (there exists $p\in\mathbb{N}$ since $y^{\prime}$ is Toeplitz under the shift map), it follows that this symbol must appear every $p$ positions in every $\psi_{l}(\mathcal{A}_{l}\times C)$-word for every $l\in\mathbb{N}$. 6. (6)

Thus, if one can show that no such $p$ can possibly exist (through checking larger and larger $l$ values), we obtain a contradiction and $(X,S)$ is not Toeplitz. 7. (7)

If it follows that there is a $p\in\mathbb{N}$ such that the [math]th coordinate of $y^{\prime}$ appears every $p$ positions in $y^{\prime}$, then we set $p=p_{1}$ and we suppose there is some $p_{2}\in\mathbb{N}$ (necessarily a multiple of $p_{1}$) such that the word $y^{\prime}_{[0,p_{1}-1]}$ appears every $p_{2}$ positions in every $\psi_{l}(\mathcal{A}_{l}\times C)$ word for every $l\in\mathbb{N}$ and continue recursively. If we can find some infinite sequence $(p_{k})$ of integers, this will form our period structure for $y^{\prime}$ under the shift map and we will have verified that $y$ is Toeplitz under $S$. If at any stage we cannot find a valid $p_{k}$, then the speedup will not be Toeplitz.

5. Sufficient Conditions for Obtaining Toeplitz Flows

In this section we provide sufficient conditions to guarantee that a minimal bounded speedup of a Toeplitz flow will result in another Toeplitz flow. Recall the following definition: for each $k\in\mathbb{N}$ large enough such that Remark 2.11 holds and for each $j\in\{1,2,\ldots,c\}$, we let $o_{i}^{k}(j)$ denote the height of orbit $j$ in tower $A_{i}^{k}$; that is, $o_{i}^{k}(j)$ is precisely the number of floors in tower $A_{i}^{k}$ that are labeled $j$.

Theorem 5.1**.**

Suppose that $(X,T)$ is a Toeplitz flow and $(X,S)$ is a minimal speedup of $(X,T)$. If there exists some $k\in\mathbb{N}$ such that for the KR-partition we have that every tower has the same orbit permutation and each orbit has the same ‘height’ in each tower (i.e, for every $1\leq j\leq c$, the total number of levels labeled with orbit number $j$ in each tower is the same), then $(X,S)$ will be a Toeplitz flow.

Proof.

First observe that if every tower has the same permutation for some fixed $k\in\mathbb{N}$, then we may denote $\pi_{i}^{(k)}=\pi$ for simplicity (when $k$ is understood). Also note that if $(X,S)$ is minimal, then $\pi$ must be a cyclic permutation on $\{1,2,\ldots,c\}$. If $\pi$ were not a cyclic permutation, then there would be some subcycle of length $\ell<c$ such that $\pi^{\ell}(1)=\pi\circ\cdots\circ\pi(1)=1$. In this case we would have points in $X$ that do not enter into every orbit labeling, which contradicts the minimality of $(X,S)$. We thus have that $\pi^{c}(j)=j$ for all $j\in\{1,2,\ldots,c\}$. By the construction of the KR-partitions, the orbit labeling, and the fact that $(X,T)$ is a Toeplitz flow, it will follow that for every $k^{\prime}\geq k$, each tower in $\mathcal{P}(k^{\prime})$ will have the same permutation.

Now suppose for some $k\in\mathbb{N}$ we have that for every pair of towers $A_{i}^{k}$ and $A_{l}^{k}$ and every orbit label $j\in\{1,2,\ldots,c\}$ it holds that $o_{i}^{k}(j)=o_{l}^{k}(j)$. Because $(X,T)$ is Toeplitz, $\mathcal{P}(k+1)$ has precisely $N(k+1)$ towers each formed by stacking $p_{k+1}/p_{k}$ towers from level $\mathcal{P}(k)$ on top of each other (where $(p_{k})$ is the period structure for $(X,T)$), and each tower in $\mathcal{P}(k)$ has the same permutation and the same orbit labeling, it follows that each tower in $\mathcal{P}(k+1)$ will also have the same orbit labeling. That is, for all $k^{\prime}>k$, for any two towers $A_{i}^{k^{\prime}}$ and $A_{l}^{k^{\prime}}$ and any orbit label $j\in\{1,2,\ldots,c\}$ it holds that $o_{i}^{k^{\prime}}(j)=o_{l}^{k^{\prime}}(j)$.

Fix $k\in\mathbb{N}$ and let $x\in B(k)$, the base level of the KR-partition $\mathcal{P}(k)$. Because every tower in $\mathcal{P}(k)$ has the same cyclic permutation and the same number of floors labeled $j$ for all $j\in\{1,2,\ldots,c\}$, we have that $S^{o_{i}^{k}(1)+o_{i}^{k}(2)+\cdots+o_{i}^{k}(c)}(x)\in B(k)$. Let $m_{k}=o_{i}^{k}(1)+o_{i}^{k}(2)+\cdots+o_{i}^{k}(c)$. This implies that $S^{m_{k}\cdot n}(x)\in B(k)$ for all $x\in B(k)$ and all $n\in\mathbb{N}$.

Let $x=\bigcap_{k\in\mathbb{N}}B(k)$. Fix $\varepsilon>0$; there exists some $K_{\varepsilon}\in\mathbb{N}$ such that for all $k\geq K_{\varepsilon}$ if $x,y\in B(k)$, then $d(x,y)<\varepsilon$. For each $k$ large enough, we thus have that $d(S^{m_{k}\cdot n}(x),x)<\varepsilon$. Hence $x$ is a Toeplitz point under $S$. Because $(X,S)$ is a minimal system that can be generated by a Toeplitz point, $(X,S)$ is thus a Toeplitz flow. ∎

Question 5.2**.**

If for all $\mathcal{P}(k)$, the towers have different orbit permutations, is it possible for the minimal speedup $(X,S)$ to be a Toeplitz flow? What if the original flow is generated by a constant length substitution?

Question 5.3**.**

If for all $\mathcal{P}(k)$, the towers have different orbit heights, is it possible for the minimal speedup $(X,S)$ to be a Toeplitz flow? What if the original flow is generated by a constant length substitution?

We now investigate co-boundaries for Toeplitz speedups. Given a topological dynamical system $(X,T)$, the continuous function $g:X\to\mathbb{R}$ is a $T$-coboundary if there exists a continuous function $f:X\to\mathbb{R}$ such that $g=f-f(T)$. In [1] it was shown that if $c$ is the orbit number for the speedup $(X,S)$ with jump function $p(x)$, then $p(x)-c$ is a $T$-coboundary. Additionally, if $p(x)-c$ is an $S$-coboundary and $T^{c}$ is a minimal action on $X$, then $(X,S)$ is conjugate to $(X,T^{c})$; we note that in general, $p(x)-c$ need not be an $S$-coboundary.

In the following proofs we use $p(x,n)=\sum_{i=0}^{n-1}pS^{i}(x)$ to denote the unique natural number satisfying $S^{n}(x)=T^{p(x,n)}(x)$.

Proposition 5.4**.**

Let $(X,S)$ be the minimal bounded speedup of a minimal system $(X,T)$ with jump function $p(x)$ and orbit number $c$. There exists a level $\mathcal{P}(k)$ of the Kakutani-Rokhlin partition of $(X,T)$ such that

[TABLE]

for all $n\in\mathbb{N}$ such that $n$ is a return time of $x$ to its cylinder set of level $k$ under $S$, if and only if we can define a function $g\in C(X,\mathbb{Z})$ such that

[TABLE]

In other words, if and only if $p(x)-c$ is an $S$-coboundary.

Proof.

Take $k$ such that $p(x)$ is constant on every cylinder set in $\mathcal{P}(k)$. As $S$ is minimal, for any KR-partition $\mathcal{P}(k)$ there exists $k^{\prime}>k$ such that any $T$-tower of $\mathcal{P}(k^{\prime})$ contains all towers of $\mathcal{P}(k)$ with every possible orbit labeling (see Lemma 2.21). We construct the function $g$ along the S-orbit of a base point of the partition. For $x_{0}\in B_{1}^{k^{\prime}}$ we set $g(x_{0})=0$ and then for $x_{1}=S(x_{0})$ we have

[TABLE]

By induction we can define $g$ for any $x$ in the S-orbit of $x_{0}$,

[TABLE]

This construction gives every cylinder set $B^{k}_{i}(j)$ in $\mathcal{P}(k)$ the $g$-value of $x_{l}$ such that $x_{l}\in B^{k}_{i}(j)$; here $B^{k}_{i}(j)$ indicates the $j$-th floor of the $A_{i}^{k}$-tower in $\mathcal{P}(k)$. Thus for $p(x)-c$ to be an $S$-coboundary, any return to a previously visited cylinder set has to give the same value of $g$. Hence, for any $x$ in any base $B_{i}^{k}$ of the KR-partition $\mathcal{P}(k)$ and $n$ such that $S^{n}(x)$ returns to that base,

[TABLE]

Therefore the function $g$ is constant on every cylinder set of $\mathcal{P}(k)$.

On the other hand, if there exists a continuous function $g:X\mapsto\mathbb{Z}$ and a level $k$ such that

[TABLE]

then for any return time $n$ of $x$ to $B_{i}^{k}(j)$ under $S$

[TABLE]

∎

Example 5.5**.**

Consider the length $3$-Toeplitz shift on two symbols $\{a,b\}$ generated by the following substitution

[TABLE]

and take any jump function with orbit number $c=2$ such that

• •

the speedup $(X,S)$ is minimal,

• •

the orbit permutations are $\pi^{(k)}_{a}=\pi^{(k)}_{b}=(12)$ and

• •

$o_{a}^{k}(1)=o_{b}^{k}(1)$* (and therefore also $o_{a}^{k}(2)=o_{b}^{k}(2)$).*

Then $(X,S)$ is conjugate to $(X,T^{2})$ and is therefore Toeplitz.

We may choose $k$ such that the KR-partition $\mathcal{P}(k)$ has a constant jump function on every floor and such that every tower in the KR-partition $\mathcal{P}(k+2)$ contains every tower from $\mathcal{P}(k)$. To apply Proposition 5.4, we need to show that

[TABLE]

The jump function $p(x,n)$ counts the number of T-steps between $x\in B_{i}^{k}$ and its return to the same cylinder set. Then

[TABLE]

where $m$ is the number of T-towers traversed between those consecutive returns to $B_{i}^{k}$ and $h^{k}$ is the height of the towers in $\mathcal{P}(k)$. By the permutation $\pi^{(k)}$, every return under $S$ to the base passes through the same number of towers with orbit label $j$ and towers with orbit label $j^{\prime}$; thus $m=\#(j)+\#(j^{\prime})=2\#(j)$, where $\#(j)$ denotes the number of towers traversed with orbit label $j$. For $x\in A_{i}^{k}$ on floor $d$ with orbit label $j$, we can count the number of $S$-steps until its return to the same floor

[TABLE]

where $\#(i,j)$ is the number of occurrences of towers $A_{i}^{k}$ with orbit label $j$ between $x$ and its return under $S$. Then it follows that

[TABLE]

We now demonstrate that the only minimal speedups of the Toeplitz flow generated by $\chi$ with orbit number $c=2$ which are also Toeplitz are conjugate to $(X,T^{2})$ and must arise as described in the above example.

Example 5.6**.**

Consider the Toeplitz shift $(X,T)$ on two symbols $\{a,b\}$ generated by

[TABLE]

Every minimal speedup with orbit number $c=2$ that is a Toeplitz flow is conjugate to $(X,T^{2})$.

Proof.

Case I: Let $S$ be a minimal speedup with orbit number $c=2$. If $\pi_{a}^{(k)}=\pi_{b}^{(k)}=(12)$ for some $k\in\mathbb{N}$, then note that $\pi_{a}^{(l)}=\pi_{b}^{(l)}=(12)$ for all $l\geq k$. Consider the substitution $\tau:\left(\{a,b\}\times\{1,2\}\right)\to\left(\{a,b\}\times\{1,2\}\right)^{\ast}$ as defined in Definition 2.17 that models the speedup in terms of the towers and orbit paths traced.

[TABLE]

Then $\tau$ is a primitive substitution that is conjugate to the substitution $\phi:\{A,B,C,D\}\to\{A,B,C,D\}^{\ast}$ by

[TABLE]

Let $o_{1}(a)$ denote the number of jumps made by $S$ in $(a,1)$, $o_{2}(a)$ denote the number of jumps made by $S$ in $(a,2)$, and similarly for $o_{1}(b)$ and $o_{2}(b)$. Note that $o_{1}(a)+o_{2}(a)=3^{k}$ and $o_{1}(b)+o_{2}(b)=3^{k}$ for some fixed $k\in\mathbb{N}$.

We denote by $|A|=o_{1}(a)+o_{2}(a)=3^{k}$ the number of jumps made by $S$ in $A=(a,1)(a,2)$. We similarly define $|B|=o_{1}(b)+o_{2}(a)$, $|C|=o_{1}(a)+o_{2}(b)$, and $|D|=3^{k}$.

Observe that if $(X,S)$ is Toeplitz, then there is some point $x\in X$ that is Toeplitz under $S$. Further, there is a unique decomposition of $x$ into its $\phi$-words. Using the minimality of $x$ and the definition of Toeplitz, there exist $l,j\in\mathbb{N}$ such that $S^{ln+j}(x)$ has a $\phi$-decomposition that begins with $A$ for all $n\in\mathbb{N}$.

[TABLE]

Because $\phi^{2}(C)$ contains only one $A$, it follows that $l\geq|\phi^{2}(C)|=8\cdot 3^{k}+o_{1}(a)+o_{2}(b)$. Similarly, we can conclude that $l\geq|\phi^{2}(D)|=9\cdot 3^{k}$. Further observe that $|\phi^{2}(A)|=9\cdot 3^{k}$ and $|\phi^{2}(B)|=8\cdot 3^{k}+o_{1}(b)+o_{2}(a)$. Note that for all $m\in\mathbb{N}$, $\phi^{m}(A)$ and $\phi^{m}(C)$ differ in only one position and $\phi^{m}(C)$ and $\phi^{m}(D)$ differ in only one position.

[TABLE]

Thus we return to $A$ every $l=9\cdot 3^{k}$ jumps if and only if $|\phi^{2}(A)|=|\phi^{2}(B)|=|\phi^{2}(C)|=|\phi^{2}(D)|=9\cdot 3^{k}$, which occurs if and only if $|A|=|C|$ and $o_{1}(a)=o_{1}(b)$. Otherwise, $l\geq\max\{|\phi^{2}(B)\phi^{2}(D)|,|\phi^{2}(C)\phi^{2}(B)|,|\phi^{2}(A)\phi^{2}(B)|\}=\max\{17\cdot 3^{k}+o_{1}(b)+o_{2}(b),18\cdot 3^{k}\}.$ If $l=18\cdot 3^{k}$, we again deduce $o_{1}(a)=o_{1}(b)$; otherwise $l\geq\max\{|\phi^{3}(A)|,|\phi^{3}(B)|,|\phi^{3}(C)|,|\phi^{3}(D)|\}$. Inductively, we obtain that $l$ is a finite value if and only if $o_{1}(a)=o_{1}(b)$.

We thus conclude that if $(X,S)$ is a Toeplitz flow with orbit number $c=2$ and $\pi_{a}^{(k)}=\pi_{b}^{(k)}=(12)$, then $(X,S)$ is topologically conjugate to $(X,T^{2})$.

Case II: If $c=2$ and $\pi_{a}^{(k)}=\pi_{b}^{(k)}=\text{id}$, then $(X,S)$ is not minimal.

Case III: Suppose we have that $\pi_{a}^{(1)}=(12)$ and $\pi_{b}^{(1)}=\text{id}$. Then $\pi_{a}^{(2l+1)}=(12)$ and $\pi_{b}^{(2l+1)}=\text{id}$ for all $l\in\mathbb{N}$. We again consider our substitution $\tau$ as defined in Definition 2.17, which is given by

[TABLE]

We obtain the conjugate substitution

[TABLE]

As in Case I, we get $|A|=3^{k}+o_{1}(b)$, $|B|=3^{k}+2o_{2}(b)$, $|C|=3^{k}+o_{2}(b)$, $|D|=2\cdot 3^{k}+o_{1}(b)$, and $|E|=3\cdot 3^{k}$ for some $k\in\mathbb{N}$. Additionally, we have $|\phi(A)|=13\cdot 3^{k}+o_{1}(b)$, $|\phi(B)|=17\cdot 3^{k}+2o_{2}(b)$, $|\phi(C)|=13\cdot 3^{k}+o_{2}(b)$, $|\phi(D)|=22\cdot 3^{k}+o_{1}(b)$, and $|\phi(E)|=27\cdot 3^{k}$.

If there is a point $x\in X$ that is Toeplitz under $S$, then there exist $j,l\in\mathbb{N}$ such that $S^{ln+j}(x)$ has a $\phi$-decomposition that begins with $AAB$ for all $n\in\mathbb{N}$. However, we claim that there are no values for $o_{1}(b)$ and $o_{2}(b)$ such that this is possible. Hence, it is not possible for $(X,S)$ to be Toeplitz when $c=2$ with $\pi_{a}^{(1)}=(12)$ and $\pi_{b}^{(1)}=\text{id}$.

Case IV: Suppose we have $\pi_{a}^{(1)}=\text{id}$ and $\pi_{b}^{(1)}=(12)$. Then $\pi_{a}^{(2l+1)}=\text{id}$ and $\pi_{b}^{(2l+1)}=(12)$ for all $l\in\mathbb{N}$. We again consider the substitution $\tau$ as in Definition 2.17, given by

[TABLE]

Similar to the above cases, we obtain the conjugate substitution

[TABLE]

where

[TABLE]

By counting the number of jumps $S$ makes in each letter, we see that $|A|=o_{1}(a)$, $|B|=2\cdot 3^{k}+o_{2}(a)$, $|C|=3^{k}+o_{1}(a)$, $|D|=4\cdot 3^{k}+3o_{2}(a)$, and $|E|=3\cdot 3^{k}+2o_{2}(a)$ for some $k\in\mathbb{N}$. We further have $|\phi(A)|=4\cdot 3^{k}+o_{1}(a)$, $|\phi(B)|=22\cdot 3^{k}+o_{2}(a)$, $|\phi(C)|=13\cdot 3^{k}+o_{1}(a)$, $|\phi(D)|=48\cdot 3^{k}+3o_{2}(a)$, and $|\phi(E)|=35\cdot 3^{k}+2o_{2}(a)$.

If there is a point $x\in X$ that is Toeplitz under $S$, then there exist $j,l\in\mathbb{N}$ such that $S^{ln+j}(x)$ has a $\phi$-decomposition that begins with $ABC$ for all $n\in\mathbb{N}$. Similar to Case III, due to the formulas for the number of $S$-jumps in each letter, there are no possible values for $o_{1}(a)$ and $o_{2}(a)$ such that any such $l\in\mathbb{N}$ can exist. We thus conclude that it is not possible for $(X,S)$ to be Toeplitz when $c=2$, $\pi_{a}^{(1)}=\text{id}$, and $\pi_{b}^{(1)}=(12)$.

By considering all cases, we conclude that the only minimal speedups $(X,S)$ of $(X,T)$ (where (X,T) is generated by $\chi$) with orbit number $c=2$ that are Toeplitz occur precisely when $\pi_{a}=\pi_{b}=(12)$ and $o_{1}(a)=o_{1}(b)$. Therefore, every Toeplitz speedup of $(X,T)$ with $c=2$ is conjugate to $(X,T^{2})$. ∎

To further analyze Toeplitz flows under constant speedups, we first give an algorithm on how to find the substitution of a constant speedup and then use the previous example of the length $3$-Toeplitz system to look for factors of constantly sped-up systems. Let $(X,T)$ be a Toeplitz subshift generated by a proper, primitive, constant length substitution $\theta$. To find the substitution of the constant speedup $(X,T^{c})$ with $\gcd(|\theta|,c)=1$ we do the following:

• •

Define the alphabet $\mathcal{B}$ of the sped-up substitution as the set of all words of length $c$ in the language of $\theta$.

• •

For every such word $w$ of length $c$ we recode the mapping $\theta(w)$ into the new alphabet $\mathcal{B}$. This recoding will be the substitution generating the subshift $(X,T^{c})$. It (or one of its powers) will be a proper, primitive, and constant length substitution, as it inherits these properties from $\theta$.

Example 5.7**.**

Let $(X_{\chi},T)$ be a Toeplitz subshift generated by $\chi:a\mapsto aab,\ b\mapsto abb$. The substitution $\tau$ of the constant 2-speedup will be on the alphabet $\mathcal{B}=\{A,B,C,D\}$ where we sort the 2-words lexicographically. Then $\tau$ is defined in the following way.

[TABLE]

We see that $\tau^{2}$ is a proper, primitive substitution of constant length $3^{2}$.

[TABLE]

To find a factor map $\mathcal{F}$ from $(X_{\tau},T^{2})$ to $(X_{\chi},T)$ we use techniques from [6]. First we scale both substitutions to the same length and analyze which positions of the substitution are not coincidences. For $\tau$ there are two such positions in $w_{0}w_{1}\cdots w_{8}$: at position $2$ there is a choice between $B$ and $D$ and at position $6$ there is a choice between between $A$ and $B$. For the original $\chi$ we see that the only difference is at position $4$

[TABLE]

For a factor map $\mathcal{F}$ to exist, we need to map non-singular fibers of $X_{\tau}$ to all non-singular fibers of $X_{\chi}$. All other non-singular fibers must collapse to singular fibers. First we shift $\chi^{2}$ by 7 such that the difference is at the same position as a difference in $\tau^{2}$ and compare:

[TABLE]

We define the map $\mathcal{F}$ such that

[TABLE]

the non-singular fiber of the pair $\{B,D\}$ collapses to a singular fiber, and the pair $\{A,B\}$ stays non-singular. The choice of which fiber collapses was arbitrary, as there exists another factor map where $\{A,B\}$ collapses by shifting $\chi^{2}$ by 2. Constant speedups with an even orbit number have no non-singular fiber at the middle position, thus these cases always need to shift the substitution to get overlapping differences to the original system $X_{\chi}$.

Factor maps from other constant speed ups of the same Toeplitz system onto the original $X_{\chi}$ can be found in a similar way. We now take a few moments to comment on several examples to highlight the connection between specific constant speedups of the same Toeplitz flow.

Example 5.8**.**

Let $(X_{\chi},T)$ be a Toeplitz subshift generated by $\chi:a\mapsto aab,\ b\mapsto abb$.

1. (1)

Letting $c=5$, there are 10 words of length 5 in the language of $X_{\chi}$. The proper, primitive substitution of the speedup $(X_{\chi},T)$ is of length 27.

[TABLE]

To collapse all but the middle fiber we map $\{A,B,E,F,H\}$ to $a$ and $\{C,D,G,I,J\}$ to $b$. Hence we see that the original Toeplitz flow is a factor of the Toeplitz flow $(X_{\chi},T^{5})$. 2. (2)

If we take $c=7$, then there are 14 words of length 7 in the language of $X_{\chi}$. The seven non-singular fibers are at positions 1 with $\{L,M\}$, at 5 with $\{H,I\}$, at 9 with $\{A,B\}$, at 13 with $\{C,E\}$, at 17 with $\{K,I\}$, at 21 with $\{B,G\}$, and at 25 with $\{D,N\}$. Thus, the factor maps $\{A,B,C,F,G,L,M\}$ to $a$ and $\{D,E,H,I,J,K,N\}$ to $b$. Again, the original Toeplitz flow is a factor of the Toeplitz flow $(X_{\chi},T^{7})$. 3. (3)

For the constant speedup by $c=4$, we take the alphabet $\{0,1,\ldots,7\}$ of size 8. We find 4 non-singular fibers at positions 1 with $\{2,7\}$, at 3 with $\{0,3\}$, at 5 with $\{4,5\}$ and at 7 with $\{1,2\}$.

To map this onto the constant $2$-speedup (see Example 5.7), we shift the $2$-speedup once to have the differences at the same positions.

[TABLE]

Thus, we see that mapping $\{5\}$ to A, $\{4,6,7\}$ to B, $\{0,3\}$ to C and $\{1,2\}$ to D gives us a factor map from the constant $4$-speedup to the constant $2$-speedup. That is, $(X_{\chi},T^{2})$ is a factor of $(X_{\chi},T^{4})$. 4. (4)

For the constant speedup by $c=8$ or $c=10$, a similar procedure shows that the system sped-up by any divisor of $c$ is a factor. The constant $8$-speedup has the constant $4$-speedup as a factor and therefore also the constant $2$-speedup and the original system as factors. The constant $10$-speedup has the constant $2$-, the constant $5$-speedup, and the original system as factors.

We believe that the relationships highlighted in the previous example hold for any constant length left proper substitution $\theta$, any $c>1$ with $\gcd(|\theta|,c)=1$, and any factor $l$ of $c$.

Conjecture 5.9**.**

Given any constant length, left proper substitution $\theta$ and $c>1$ such that $\gcd(c,|\theta|)=1$, if $l$ is a factor of $c$ with $1\leq l<c$, then the Toeplitz flow $(X_{\theta},T^{l})$ is a factor of $(X_{\theta},T^{c})$.

We now return our attention to $S$-coboundaries for minimal speedups of Toeplitz flows.

Theorem 5.10**.**

Suppose that $(X,T)$ is a Toeplitz flow and $(X,S)$ is a minimal speedup of $(X,T)$. If there exists some $k\in\mathbb{N}$ such that for the KR-partition $\mathcal{P}(k)$ we have that every tower has the same cyclic orbit permutation and each orbit has the same ‘height’ in each tower (i.e., for every $1\leq j\leq c$, $o_{i}^{k}(j)=o_{i^{\prime}}^{k}(j)$ for all towers $i,i^{\prime}$), then there exists an $S$-coboundary for $p(x)-c$.

Proof.

Let $k$ be such that the jump function $p(x)$ is constant on every cylinder set of $\mathcal{P}(k)$. For an arbitrary $x\in B_{i}^{k}$, take a return time $n$ such that $S^{n}(x)=T^{p(x,n)}(x)\in B_{i}^{k}$. Therefore we know $h^{k}=p_{k}$ divides $p(x,n)$ and by the cyclic permutation $\pi^{(k)}$, it takes a multiple of $c$ steps to return to the same label. Thus $p(x,n)=cmh^{k}$ where $m$ is the number of $T$-towers with label $j$ between $x$ and $T^{p(x,n)}(x)$. To compute the number of steps under $S$, we take sums over all labels $j$ and towers $i$ between $x$ and $S^{n}(x)$, recall that $\#(i,j)$ is the number of occurrences of tower $A_{i}^{k}$ with label $j$,

[TABLE]

with $o_{i}^{k}(j)=o_{i^{\prime}}^{k}(j)$ for all towers $i,i^{\prime}$ and by the permutation every label has the same number of occurrences $\#(j)=\#(j^{\prime})$ in the $cm$ towers between. Then $p(x,n)=cmh^{k}=cn$ and by Proposition 5.4 there exists an $S$-coboundary for $p(x)-c$. ∎

Corollary 5.11**.**

Suppose that $(X,T)$ is a Toeplitz flow and $(X,S)$ is a minimal speedup of $(X,T)$ such that $\gcd(c,p_{k})=1$ for all $k$ and the assumptions of the previous theorem hold. Then $(X,S)$ is conjugate to $(X,T^{c})$.

Proof.

Under the condition $\gcd(c,p_{k})=1$ for all $k$ we know that $(X,T^{c})$ is a Toeplitz flow, see Theorem 4.1. As $(X,S)$ has an $S$-coboundary for $p(x)-c$, Proposition 2.6 of [1] proves the conjugacy. ∎

Conjecture 5.12**.**

If $(X,T)$ is a Toeplitz flow and $(X,S)$ is a minimal speedup of $(X,T)$ that is also a Toeplitz flow, then $p(x)-c$ is an $S$-coboundary. Further, if $\gcd(c,p_{k})=1$ for all $k$, then $(X,S)$ is topologically conjugate to $(X,T^{c})$.

We now demonstrate that there exist minimal speedups $(X,S)$ of Toeplitz flows $(X,T)$ with orbit number $c$ such that $(X,S)$ is Toeplitz and not conjugate to $(X,T^{c})$. By construction, the speedup $(X,S)$ will have an $S$-coboundary for $p(x)-c$ even though $(c,p_{k})>1$ for every $k$.

Example 5.13**.**

Let $\mathcal{A}=\mathcal{A}_{0}=\{0,1\}$ and for each $i\in\mathbb{N}$, define $\mathcal{A}_{i}=\{a,b\}$. Define $\theta_{1}:\mathcal{A}_{1}\to\mathcal{A}$ by $\theta_{1}(a)=1001$ and $\theta_{1}(b)=1010$. For all $i\geq 2$, define $\theta_{i}(a)=aab$ and $\theta_{i}(b)=abb$. Then the S-adic shift generated by this sequence of substitutions generates a Toeplitz flow $(X,T)$ with period structure $(p_{k})=(4\cdot 3^{k})$.

Let $A=[\theta_{1}(\theta_{2}(a))]=\{x\mid x_{[0,11]}=1001~1001~1010\}$ and $B=[\theta_{1}(\theta_{2}(b))]=\{x\mid x_{[0,11]}=1001~1010~1010\}$ and define $p:X\to\mathbb{Z}^{+}$ by

[TABLE]

With this given jump function $p$, $(X,S)$ is a minimal bounded speedup of $(X,T)$ where $S(x)=T^{p(x)}(x)$. In this case, the orbit number is $c=2$. It is easy to check that for all $k\in\mathbb{N}$, the orbit permutations are $\pi_{a}^{(k)}=\pi_{b}^{(k)}=(12)$ and the orbit heights are such that $o_{a}^{k}(1)=o_{a}^{k}(2)=o_{b}^{k}(1)=o_{b}^{k}(2)$. By Theorem 5.1, it follows that $(X,S)$ is a Toeplitz flow.

We can represent $(X,S)$ as an S-adic shift generated by the sequence of alphabets $\{\mathcal{B}_{i}\}_{i\geq 0}$ and substitutions $(\phi_{i}\}_{i\geq 1}$ given by $\mathcal{B}_{0}=\{0,1,2,\ldots,6\}$, $\mathcal{B}_{i}=\{a,b,c,d\}$ for all $i\geq 1$, $\phi_{1}(a)=012304230156$, $\phi_{1}(b)=015604230156$, $\phi_{1}(c)=012304230456$, $\phi_{1}(d)=015604230456$, and for all $i\geq 2$ define $\phi_{i}(a)=abccbcabd$, $\phi_{i}(b)=abdcbcabd$, $\phi_{i}(c)=abccbccbd$, and $\phi_{i}(d)=abdcbccbd$.

Observe that $(X,S)$ has period structure $(q_{k})=(12\cdot 9^{k})$. We conclude that $(X,S)$ and $(X,T)$ have the same underlying odometer. Further note that although $p(x)-c$ is an $S$-coboundary by Theorem 5.10, $(X,S)$ is not topologically conjugate to $(X,T^{c})=(X,T^{2})$, since $(X,T^{2})$ is not minimal.

Recall that there exists a speedup of the odometer $(X_{\alpha},T)$ with $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ that has orbit number $c$ if and only if there is some $N\in\mathbb{N}$ such that $\gcd(c,\alpha_{i})=1$ for all $i\geq N$. We now demonstrate how one can generalize Theorem 2.15 to the Toeplitz flow setting.

Theorem 5.14**.**

If $(X,T)$ is a Toeplitz flow with period structure $(p_{k})$ and $c\geq 1$ is an integer such that for some $N\in\mathbb{N}$, $\gcd\left(c,\frac{p_{k+1}}{p_{k}}\right)=1$ for all $k\geq N$, then there is a minimal bounded speedup $(X,S)$ of $(X,T)$ with orbit number $c$ such that $(X,S)$ is a Toeplitz flow with the same underlying odometer as $(X,T)$. Further, $(X,S)$ and $(X,T)$ are not conjugate.

Proof.

Let $(X,T)$ be a Toeplitz flow with period structure $(p_{k})$ and underlying odometer given by $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ where $\alpha_{1}=p_{1}$, and $\alpha_{i}=\frac{p_{i}}{p_{i-1}}$ for all $i\geq 2$. Suppose there exist $c,N\in\mathbb{N}$ such that for all $i\geq N$, $\gcd(c,\alpha_{i})=1$. By [1, Theorem 3.8], there exists some speedup of the odometer $(X_{\alpha},T_{\alpha})$ with orbit number $c$; further, by [1, Lemma 3.7], there exists $K$ such that for all $k\geq K$ $\pi^{(k)}$ is a cyclic permutation on $\{1,2,\ldots,c\}$ for $(X_{\alpha},T_{\alpha})$. Let $M=\max(N,K)$. Define the jump function $p(x)$ on the towers of the KR-partition for $(X,T)$ at level $M$ so that each tower has the same orbit labellings as the single tower at level $M$ for the KR-partition for $(X_{\alpha},T_{\alpha})$ with orbit number $c$. Therefore, for $(X,T)$ with speedup $p(x)$, every tower has the same orbit labeling permutation $\pi^{(M)}$ which is a cyclic permutation on $\{1,2,\ldots,c\}$; additionally, the heights for each orbit agree in every tower.

We first show that $(X,T^{p(x)})$ is minimal. Note that for every level $k$ in the sequence of KR-partitions for $(X,T)$, every tower can be interpreted as a stack of towers from the $(k-1)$st level such that the base tower is the same tower from the $(k-1)$st level, which we will call the primary tower at level $k-1$. Without loss of generality, suppose that every tower at level $k$ contains every tower from level $k-1$ (we may always choose a subsequence of our KR-partitions so this is true). Since $\pi^{(k)}$ is a permutation of $\{1,2,\ldots,c\}$ for all $k\geq M$ and every tower has the same permutation, if one starts on the base floor of any tower at level $k\geq M$, one will traverse through every orbit labelling of every tower from level $k-1$ after traversing through $c$ towers. As this is true for all $k\geq M$, we conclude that our speedup will be minimal.

Lastly, because every tower in the KR-partition for $(X,T)$ with $k$ large enough has every tower with the same orbit permutation and every orbit has the same height in each tower, by Theorem 5.1, we conclude that $(X,S)=(X,T^{p(x)})$ is a Toeplitz flow. By Theorem 2.16, we conclude that $(X,S)$ and $(X,T)$ are not conjugate. ∎

Remark 5.15**.**

Observe that it immediately follows from Theorem 5.14 that every Toeplitz flow $(X,T)$ with underlying odometer $(X_{\alpha},T_{\alpha})$ has a minimal speedup $(X,S)$ which is a Toeplitz flow if there exists some prime $p$ such that $p$ only divides finitely many $\alpha_{i}$. Thus, given a Toeplitz flow $(X,T)$ with underlying odometer given by $\alpha$, where $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ does not have every natural number as a divisor, it will follow that there is some minimal speedup that generates a Toeplitz flow with the same underlying odometer. In particular, a Toeplitz flow with underlying odometer given by $\alpha=(2,3,5,7,\ldots)$ will have a minimal speedup which is a Toeplitz flow for every orbit number $c\in\mathbb{N}$.

Question 5.16**.**

Is it possible to speedup a Toeplitz flow in such a way to obtain a Toeplitz flow with a different underlying odometer?

We conclude with two conjectures about speedups of Toeplitz flows.

Conjecture 5.17**.**

If $\gcd\left(c,\frac{p_{k}}{p_{k-1}}\right)>1$ for infinitely many $k$, then $(X,T)$ does not have a minimal speedup with orbit number $c$ that is a Toeplitz flow. Hence, if $\theta$ is a constant length left proper substitution, then the Toeplitz flow $(X_{\theta},T)$ does not have a minimal speedup with orbit number $c=|\theta|$ that is also a Toeplitz flow.

Conjecture 5.18**.**

Suppose $(X,T)$ is a Toeplitz flow with period structure $(p_{k})$ and suppose $(X,S)$ is a bounded speedup of $(X,T)$ with orbit number $c$. Then $(X,S)$ is a Toeplitz flow if and only if for all sufficiently large $k$, $\pi_{i}^{(k)}$ is the same cyclic permutation on $\{1,2,\ldots,c\}$ for each tower of the KR-partition $\mathcal{P}(k)$ and the orbit heights are the same in each tower, (i.e., $o_{i}^{k}(m)=o_{j}^{k}(m)$ for each $m\in\{1,2,\ldots,c\}$ and all $i,j$).

Acknowledgements. SR would like to thank the OeAD Marietta Blau-Scholarship for funding her research stay at Furman University. LA gratefully acknowledges financial support for this project from the Henry Keith and Ellen Hard Townes endowed professorship at Furman University and from the Fulbright U.S. Scholar Program. Additionally, the authors would like to thank Sarah Frick for helpful conversations in the early stages of this project and Reem Yassawi for her expertise in finding factor maps of constant length substitutions.