2.1 Basics of ergodic theory

The probability space $(X,\mathcal {A},\mu )$ is assumed to be standard and atomless. Such a space is isomorphic to $([0,1],\mathcal {B}([0,1]),\mathrm {Leb})$ , that is, there exists a bimeasurable bijection $\Psi \colon X\to [0,1]$ (defined almost everywhere) such that $\Psi _{\star }\mu =\mathrm {Leb}$ , where $\Psi _{\star }\mu $ is defined by $\Psi _{\star }\mu (A)=\mu (\Psi ^{-1}(A))$ for every measurable set A. We consider maps $T\colon X\to X$ acting on this space and which are bijective, bimeasurable and probability measure-preserving (p.m.p.), meaning that $\mu (T^{-1}(A))=\mu (A)$ for all measurable sets $A\subseteq X$ , and the set of these transformations is denoted by $\mathrm {Aut}(X,\mathcal {A},\mu )$ , or simply $\mathrm {Aut}(X,\mu )$ , two such maps being identified if they coincide on a measurable set of full measure. In this paper, elements of $\mathrm {Aut}(X,\mu )$ are called transformations or (dynamical) systems.

A measurable set $A\subseteq X$ is T-invariant if $\mu (T^{-1}(A)\Delta A)=0$ , where $\Delta $ denotes the symmetric difference. A transformation $T\in \mathrm {Aut}(X,\mu )$ is said to be ergodic if every T-invariant set is of measure $0$ or $1$ . If T is ergodic, then T is aperiodic, that is, $T^n(x)\not =x$ for almost every $x\in X$ and for every $n\in \mathbb {Z}\setminus \{0\}$ , or equivalently the T-orbit of x, denoted by , is infinite for almost every $x\in X$ .

Here, T is weakly mixing if

$$ \begin{align*} \frac{1}{n}\sum_{k=0}^{n}{ |\mu(A\cap T^{-n}(B))-\mu(A)\mu(B) |}\underset{n\to +\infty}{\to}0 \end{align*} $$

for every measurable sets $A,B$ . Additionally, T is strongly mixing if

$$ \begin{align*} |\mu(A\cap T^{-n}(B))-\mu(A)\mu(B) |\underset{n\to +\infty}{\to}0\end{align*} $$

for every measurable set $A,B$ . It is not difficult to prove that strong mixing implies weak mixing and that the latter implies ergodicity.

The notions of weak mixing and ergodicity can be translated in terms of eigenvalues. Denoting by $\mathrm {L}^2(X,\mathcal {A},\mu )$ the space of complex-valued and square-integrable functions defined on X, a complex number $\unicode{x3bb} $ is an eigenvalue of T if there exists $f\in \mathrm {L}^2(X,\mathcal {A},\mu )\setminus \{0\}$ such that $f\circ T=\unicode{x3bb} f$ almost everywhere (f is then called an eigenfunction). An eigenvalue $\unicode{x3bb} $ is automatically an element of the unit circle . The point spectrum of T is then the set of all its eigenvalues. Notice that $\unicode{x3bb} =1$ is always an eigenvalue since the constant functions are in its eigenspace. Finally, T is ergodic if and only if the constant functions are the only eigenfunctions with eigenvalue one, in other words, the eigenspace of $\unicode{x3bb} =1$ is the line of constant functions (we say that it is a simple eigenvalue). If T is ergodic, it is weakly mixing if and only if the only eigenvalue of T is  $1$ . For a complete survey on spectral theory for dynamical systems, the reader may refer to [Reference Viana and OliveiraVO16].

All the properties that we have introduced are preserved under conjugacy. Two transformations $S\in \mathrm {Aut}(X,\mu )$ and $T\in \mathrm {Aut}(Y,\nu )$ are conjugate if there exists a bimeasurable bijection $\Psi \colon X\to Y$ such that $\Psi _{\star }\mu =\nu $ and $\Psi \circ S=T\circ \Psi $ almost everywhere. Some classes of transformations have been classified up to conjugacy, the two examples to keep in mind are the following. By Ornstein [Reference OrnsteinOrn70], entropy is a total invariant of conjugacy among Bernoulli shifts, and Ornstein and Weiss [Reference Ornstein and WeissOW87] generalized this result for Bernoulli shifts of amenable groups. For more details about entropy, see [Reference DownarowiczDow11] for non-necessarily invertible transformations $T\colon X\to X$ , and [Reference Kerr and LiKL17] more generally for actions of amenable groups. Finally, Halmos and von Neumann [Reference Halmos and Von NeumannHVN42] showed that two systems with discrete spectra are conjugate if and only if they have equal point spectra (a system has a discrete spectrum if the span of all its eigenfunctions is dense in $\mathrm {L}^2(X,\mathcal {A},\mu )$ ).

2.2 Quantitative orbit equivalence

The conjugacy problem in full generality is very complicated (see [Reference Foreman, Rudolph and WeissFRW11]). We now give the formal definition of orbit equivalence, which is a weakening of the conjugacy problem.

Definition 2.1. Two aperiodic transformations $S\in \mathrm {Aut}(X,\mu )$ and $T\in \mathrm {Aut}(Y,\nu )$ are orbit equivalent if there exist a bimeasurable bijection $\Psi \colon X\to Y$ satisfying $\Psi _{\star }\mu =\nu $ , such that $\mathrm {Orb}_S(x)=\mathrm {Orb}_{\Psi ^{-1}T\Psi }(x)$ for almost every $x\in X$ . The map $\Psi $ is called an orbit equivalence between S and T.

We can then define the cocycles associated to this orbit equivalence. These are measurable functions $c_S\colon X\to \mathbb {Z}$ and $c_T\colon Y\to \mathbb {Z}$ defined almost everywhere by

$$ \begin{align*}Sx=\Psi^{-1}T^{c_S(x)}\Psi(x)\quad\text{and}\quad Ty=\Psi S^{c_T(y)}\Psi^{-1}(y)\end{align*} $$

( $c_S(x)$ and $c_T(y)$ are uniquely defined by aperiodicity).

Given a function $\varphi \colon \mathbb {R}_+\to \mathbb {R}_+$ , a measurable function $f\colon X\to \mathbb {Z}$ is said to be $\varphi $ -integrable if

$$ \begin{align*} \int_{X}{\varphi(|f(x)|)\,\mathit{d}\mu}<+\infty. \end{align*} $$

For example, integrability is exactly $\varphi $ -integrability when $\varphi $ is non-zero and linear. Then, a weaker quantification on cocycles is the notion of $\varphi $ -integrability for a sublinear map $\varphi $ , meaning that $\lim _{t\to +\infty }{\varphi (t)/t}=0$ . Two transformations in $\mathrm {Aut}(X,\mu )$ are said to be $\varphi $ -integrably orbit equivalent if there exists an orbit equivalence between them whose associated cocycles are $\varphi $ -integrable. Another form of quantitative orbit equivalence is Shannon orbit equivalence. We say that a measurable function $f\colon X\to \mathbb {Z}$ is Shannon if the associated partition $\{f^{-1}(n)\mid n\in \mathbb {Z}\}$ of X has finite entropy. Two transformations in $\mathrm {Aut}(X,\mu )$ are Shannon orbit equivalent if there exists an orbit equivalence between them whose associated cocycles are Shannon.

3.1 The cutting-and-stacking method

Before the definition of a rank-one system (Definition 3.2), and for the definition of flexible classes (Definition 3.7), we need to define sequences of integers which will provide the combinatorial data of a rank-one system, namely the cutting and spacing parameters.

Definition 3.1. By a cutting and spacing parameter, we mean a tuple of the form

$$ \begin{align*}(q,(\sigma_{.,0},\ldots,\sigma_{.,q}))\end{align*} $$

with an integer $q\geq 2$ (the cutting parameter) and non-negative integers $\sigma _{.,0},\ldots ,\sigma _{.,q}$ (the spacing parameters), and we denote by $\mathcal {P}$ the set of all cutting and spacing parameters. We also define the set of finite sequences of cutting and spacing parameters:

Given a sequence of cutting and spacing parameters $\boldsymbol {p}=(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{k\geq 0}\in \mathcal {P}^{\mathbb {N}}$ and an integer $n\geq 0$ , the tuple $(q_n,(\sigma _{n,0},\ldots ,\sigma _{n,q_n}))$ in $\mathcal {P}$ is the nth cutting and spacing parameter of $\boldsymbol {p}$ , and the tuple $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n}$ is the projection of $\boldsymbol {p}$ on $\mathcal {P}^{n+1}$ (it gives the first $n+1$ cutting and spacing parameters). From $\boldsymbol {p}$ , we also define three sequences:

• • $(h_n)_{n\geq 0}$ the height sequence of $\boldsymbol {p}$ , inductively defined by

  $$ \begin{align*}\begin{cases} h_0=1,\\ h_{n+1}=q_nh_n+\sigma_{n}, \end{cases}\end{align*} $$
  where $h_n$ is called the height of the nth tower;

• • $(\sigma _n)_{n\geq 0}$ , with (the number of new spacers at step n);

• • $(Z_n)_{n\geq 0}$ , with ,

and it is also possible to consider the finite sequences $(h_k)_{0\leq k\leq n+1}$ , $(\sigma _k)_{0\leq k\leq n}$ and $(Z_k)_{0\leq k\leq n}$ associated to a finite sequence of cutting and spacing parameters in $\mathcal {P}^{n+1}$ .

The terminology ‘cutting’, ‘spacing’, ‘tower’, ‘height’, etc., is justified by Definition 3.2 and Figure 4. There are many definitions of rank-one systems (see [Reference FerencziFer97] for a complete survey and various facts in this section). In this paper, the goal is to use the combinatorial structure given by the cutting-and-stacking method (see Figure 4).

Figure 4

An example of cutting-and-stacking construction with $h_n=5$ , $q_n=3$ , $\sigma _{n,0}=1$ , $\sigma _{n,1}=2$ , $\sigma _{n,2}=0$ , $\sigma _{n,3}=1$ . We then have $h_{n+1}=19$ .

Since $X_n$ is increasing and $\mathcal {R}_n$ increases to the atomless $\sigma $ -algebra $\mathcal {A}$ , we have $\mu (X_n)\underset {n\to +\infty }{\to }1$ . In other words $\varepsilon _n$ tends to $0$ .

Before giving examples, the following lemmas give some easy properties on these systems to understand their combinatorial structure and the hypotheses required in the definition.

Proof of Lemma 3.3

If the series $\sum \!{({\sigma _n}/{q_0\ldots q_n})}$ converges, so does the series $\sum \!{({\sigma _n}/{h_{n+1}})}$ since $h_{n+1}$ is greater or equal to $q_0\ldots q_n$ . Now assume that the series $\sum \!{({\sigma _n}/{h_{n+1}})}$ converges. Notice that we have

$$ \begin{align*}\frac{\sigma_n}{h_{n+1}}=\frac{h_{n+1}-q_nh_{n}}{h_{n+1}}=1-q_n\frac{h_n}{h_{n+1}}\end{align*} $$

and since the series is convergent, the product $\prod {q_n({h_n}/{h_{n+1}})}$ converges to some $M_0>0$ , that is, $q_0\ldots q_n/h_{n+1}\to M_0$ . The constant $M_0$ is less than or equal to $1$ since we have $h_{n+1}\geq q_nh_n$ for every $n\geq 0$ . Finally, let us assume $q_0\ldots q_n/h_{n+1}\to M_0$ . Notice that we have

$$ \begin{align*}\frac{\sigma_n}{q_0\ldots q_n}=\frac{h_{n+1}-q_nh_n}{q_0\ldots q_n}=\frac{h_{n+1}}{q_0\ldots q_n}-\frac{h_n}{q_0\ldots q_{n-1}},\end{align*} $$

so by telescoping consecutive terms, we get $\sum _{n\geq 0}{({\sigma _n}/{q_0\ldots q_n})}=\lim _{n\to \infty }({h_{n+1}}/{q_0\ldots q_n})-h_0=({1}/{M_0})-1$ and we are done for the equivalence between the three assumptions.

Proof of Lemma 3.4

For a fixed n, the levels of $\mathcal {R}_n$ have the same measure by T-invariance of the measure $\mu $ . Moreover, the first level $B_n$ is a disjoint union of $q_n$ levels $B_{n,0}, \ldots , B_{n,q_n-1}$ of $\mathcal {R}_{n+1}$ . Then, it is clear by induction that a level of $\mathcal {R}_n$ has measure ${\mu (B_0)}/{q_0\ldots q_{n-1}}$ . Since the sequence $(X_n)_{n\geq 0}$ is increasing to X, and $X_{n+1}$ is obtained from $X_n$ by adding $\sigma _n$ spacers, which are levels of $\mathcal {R}_{n+1}$ , we get

(1) $$ \begin{align} \mu(X)=\mu(X_0)+\sum_{n\geq 0}{\mu(X_{n+1}\!\setminus\! X_n)}=\mu(B_0)+\sum_{n\geq 0}{\frac{\mu(B_0)\sigma_n}{q_0\ldots q_n}}, \end{align} $$

so $\mu (B_0)$ is non-zero, and $\mu (X)$ is finite if and only if the sum $\sum _{n\geq 0}{({\sigma _n}/{q_0\ldots q_n})}$ is finite. Finally, let us assume that $\mu $ is a probability measure. This implies $\sum _{n\geq 0}{({\sigma _n}/{q_0\ldots q_n})}=({1}/{M_0})-1$ and, using equation (1), we get $\mu (B_0)=M_0$ . The measurable set $X_n$ is the disjoint union of $h_n$ levels of $\mathcal {R}_n$ , so the inequality $h_n\leq ({q_0\ldots q_{n-1}}/{M_1})$ follows from the fact that $\mu $ is a probability measure.

It is possible to build a finite measure-preserving transformation T of rank one with a given combinatorial setting $(q_n,(\sigma _{n,0},\ldots ,\sigma _{n,q_n}))_{n\geq 0}\in \mathcal {P}^{\mathbb {N}}$ satisfying the hypothesis (F). For instance, it suffices to build $(X_n)$ as an increasing sequence of intervals of $\mathbb {R}_+$ , with $B_{n,i}$ and $\Sigma _{n,i,j}$ being subintervals of equal length and disjoint (for a fixed n), each on which T is defined as an affine map, and with $B_{0}=[0,M_0]$ . The convergence of the series $\sum \!{({\sigma _n}/{h_{n+1}})}$ and Lemma 3.3 ensure that is equal to $[0,1]$ (up to a null set), so the Lebesgue measure on $[0,1]$ is a probability measure preserved by T. Notice that if the series is divergent, we can set $B_0=[0,1]$ and this defines T on the set of positive real numbers endowed with the Lebesgue measure, so this is an infinite measure-preserving transformation.

Therefore, for every $(q_n,(\sigma _{n,0},\ldots ,\sigma _{n,q_n}))_{n\geq 0}\in \mathcal {P}^{\mathbb {N}}$ satisfying condition (F), there exists a rank-one system having a cutting-and-stacking construction with these cutting and spacing parameters, this fact will be used in this paper since it enables us to only take into account the combinatorics behind the systems.

The hypothesis on the Rokhlin towers $\mathcal {R}_n$ aims not only to have $\varepsilon _n\to 0$ but also to define two isomorphic systems when they admit cutting-and-stacking constructions with the same cutting and spacing parameters. Moreover, if T admits such a construction with Rokhlin towers increasing to a sub- $\sigma $ -algebra $\mathcal {B}$ of $\mathcal {A}$ , then T, seen as an element of $\mathrm {Aut}(X,\mathcal {A},\mu )$ , is not necessarily a rank-one system but admits a rank-one system (T on the sub- $\sigma $ -algebra $\mathcal {B}$ ) as a factor.

Two different families of cutting and spacing parameters do not necessarily define non-isomorphic systems. Indeed, in a construction of a rank-one system with parameters $q_n$ and $\sigma _{n,i}$ , one can decide to only consider a subsequence $\mathcal {R}_{n_k}$ of Rokhlin towers. For example, the new cutting parameters will be $q_{n_k}q_{n_k+1}\ldots q_{n_{k+1}-1}$ for $k\geq 0$ .

The rank-one systems form a class of ergodic and zero entropy systems. The easiest examples of rank-one systems are the irrational rotations

$$ \begin{align*}R_{\theta}\colon z\in\mathbb{T}\mapsto e^{2i\pi\theta}z\in\mathbb{T}\end{align*} $$

for every irrational number $\theta $ , where $\mathbb {T}$ is the unit circle endowed with its Haar measure. These systems are not weakly mixing. Moreover, they have discrete spectrum and the point spectrum of $R_{\theta }$ is $\{e^{in\theta }\mid n\in \mathbb {Z}\}$ , so by the Halmos–von Neumann theorem [Reference Halmos and Von NeumannHVN42], $R_{\theta }$ and $R_{\theta '}$ are isomorphic if and only if $\theta =\theta '\text { mod }\mathbb {Z}$ or $\theta =-\theta '\text { mod }\mathbb {Z}$ .

The odometers are rank one. These are exactly the rank-one systems without spacers (that is, $\sigma _{n,i}=0$ ), so the Rokhlin towers are partitions of the space. Such a system is isomorphic to the adding machine S in the space $\prod _{n\geq 0}{\{0,1,\ldots ,q_n-1\}}$ , namely the addition by $(1,0,0,0,\ldots )$ with carry over to the right, defined for every ${x\in \prod _{n\geq 0}{\{0,1,\ldots ,q_n-1\}}}$ by

and it preserves the product of uniform probability measures on each finite set $\{0,1,\ldots ,q_n-1\}$ . Denote the cylinders of length k by

If S is the odometer on the space $\prod _{n\geq 0}{\{0,1,\ldots ,q_n-1\}}$ , we can also set a partially defined map

$$ \begin{align*}\zeta_n\colon X\setminus [q_0-1,\ldots,q_{n-1}-1]_{n}\to X\setminus [0,\ldots,0]_{n}\end{align*} $$

which is the addition by

$$ \begin{align*}(\underbrace{0,\ldots ,0}_{n-1\text{ times}},1,0,0,\ldots)\end{align*} $$

(so S and $\zeta _1$ coincide on $X\setminus [q_0-1]_1$ ). Then, we have

$$ \begin{align*}B_n=[\underbrace{0,\ldots ,0}_{n\text{ times}}]_{n},\end{align*} $$

$$ \begin{align*}B_{n,i}=[\underbrace{0,\ldots ,0}_{n\text{ times}},i]_{n+1}\end{align*} $$

and $B_{n,i}=\zeta _{n+1}^i(B_{n,0})$ for every $0\leq i\leq q_n-1$ , so it provides a scale in $B_n$ . Note that it is possible to recover the odometer S from these partially defined maps $\zeta _n$ (see Figure 5). In §5.1, the strategy will be to build S from partially defined maps $\zeta _n$ .

Figure 5

Example of odometer with $q_0=3$ , $q_1=2$ , $q_2=3$ .

In the class of odometers, the number of occurrences of every prime factors in the set $\{q_n\mid n\geq 0\}$ form a total invariant of conjugacy. As for irrational rotations, it is a consequence of the Halmos–von Neumann theorem since odometers have a discrete spectrum and their eigenvalues are given by these occurrences. In particular, odometers have eigenvalues non-equal to $1$ and are not weakly mixing; moreover, odometers and irrational rotations are not isomorphic. Notice that the Halmos–von Neumann theorem implies that the conjugacy classes among systems with a discrete spectrum coincide with the flip-conjugacy classes since the point spectrum of a system is a subgroup of $\mathbb {T}$ . If every prime number has infinite multiplicity in the set $\{q_n\mid n\geq 0\}$ , then the odometer is said to be universal. An odometer is dyadic if $2$ is the only prime factor.

Chacon’s map is the first example of weakly mixing system which is not strongly mixing [Reference ChaconCha69] and was the starting point for the notion of rank-one systems. It is a rank-one transformation defined with cutting and spacing parameters $q_n=3$ , $\sigma _{n,0}=\sigma _{n,1}=\sigma _{n,3}\,{=}\,0$ , $\sigma _{n,2}=1$ .

3.2 Flexible classes

Now we introduce classes of rank-one systems to which the main result of this paper applies. First, let us consider cutting-and-stacking constructions whose spacing parameters have controlled asymptotics. Recall that $\mathcal {P}^{\mathbb {N}}$ is the set of sequences of cutting and spacing parameters. As introduced in Definition 3.1, $(h_n)$ , $(\sigma _n)$ and $(Z_n)$ denote the sequences associated to a sequence in $\mathcal {P}^{\mathbb {N}}$ : $h_n$ is the height of the nth tower, $\sigma _n$ the number of new spacers at step n and $Z_n$ is the maximum number of spacers between two consecutive towers, over the first n steps.

Odometers and Chacon’s map are examples of BSP rank-one systems. Moreover, BSP implies CSP. The interest in the BSP property is its stability after skipping steps in the cutting-and-stacking process, as stated in the following lemma.

If the parameters $\sigma _{n,q_n}$ are non-zero, then skipping steps in the cutting-and-stacking process will cause an accumulation of spacers above the last columns and the new spacing parameters will not be bounded if the subsequence is properly chosen so that the jumps $n_{k+1}-n_k$ increase quickly enough. We have the same problem for $\sigma _{n,0}$ (accumulation of spacers at the bottom of the first columns), and hence the conditions $\sigma _{n,0}=\sigma _{n,q_n}=0$ in the definition of BSP.

Figure 6

Illustration of the proof of Lemma 3.6, spacing parameters from $\mathcal {R}_n$ to $\mathcal {R}_{n+2}$ with $q_n=q_{n+1}=2$ (the coloured levels are the base and the roof of the towers).

Lemma 3.6 has no reason to hold for CSP construction that are not BSP. Indeed, the spacing parameters from $\mathcal {R}_{n_k}$ to $\mathcal {R}_{n_{k+1}}$ have to be compared with $h_{n_k}$ , the height of $\mathcal {R}_{n_k}$ . The comparison is easily obtained for the spacing parameters $\sigma _{n_k,i}$ , $0\leq i\leq q_{n_k}$ , but for the other spacing parameters, we only know that they are bounded above by $Ch_{n_k+1}, Ch_{n_k+2}, \ldots , Ch_{n_{k+1}-1}$ .

In the following, we will see other important CSP examples by considering classes containing ‘nice’ cutting-and-stacking constructions, meaning that we will be able to properly choose the parameters to have the desired quantification of the cocycles for the orbit equivalence built in §5.1. By definition, every flexible class $\mathcal {C}$ will be associated to some subset $\mathcal {F}_{\mathcal {C}}$ of $\mathcal {P}^{*}$ , which can be considered as sufficient conditions that the cutting and spacing parameters have to satisfy at each step for the underlying transformation to belong to $\mathcal {C}$ . Recall that $\mathcal P^*$ denotes the set of all finite sequences of cutting and stacking parameters.

The third point of the definition aims to recursively choose the cutting parameters (and we want them to increase quickly enough) with an asymptotic control on $(\sigma _n)_n$ , while the first point guarantees that it is possible to do so for the underlying system to be in the class $\mathcal {C}$ . The second point is minor, but it is required for the initialization of the recursive construction of an odometer orbit equivalent to an element of our flexible class (see Lemma 5.10 and Remark 5.11). It also ensures that $\mathcal {F}_{\mathcal {C}}$ is not an empty set.

Notice that if a construction satisfies $Z_n\leq Ch_{n-1}$ for all n, then it is in particular CSP and we get $\sigma _n\leq C(q_n+1)h_{n-1}\leq 2Cq_nh_{n-1}$ as in the third point of Definition 3.7.

We now give examples of flexible classes. The proof is given in §4.

Theorems A, C, E and F follow from Proposition 3.8 and the following theorem which is the main result.

4.1 BSP systems

Let T be a BSP rank-one system, and $q_n$ , $\sigma _{n,i}$ the parameters of a BSP construction of T, with a constant $C>0$ . For every $n\geq 0$ and $j\geq 1$ , let $\sigma ^{(n,n+j)}_{0},\ldots ,\sigma ^{(n,n+j)}_{q_{n}\ldots q_{n+j-1}}$ be the spacing parameters from $\mathcal {R}_{n}$ to $\mathcal {R}_{n+j}$ , assuming that the steps for $\mathcal {R}_{n+1},\ldots ,\mathcal {R}_{n+j-1}$ are skipped during the construction (we then have $\sigma ^{(n,n+1)}_{i}=\sigma _{n,i}$ and also $\sigma ^{(n,n+j)}_{i}=0$ for i equal to $0$ and $q_n\ldots q_{n+j-1}$ by Lemma 3.6). The new cutting parameters are and are large enough with huge jumps j. Now, define

Using Lemma 3.6, the new spacing parameters $\sigma ^{(n_k,n_{k+1})}_{j}$ are bounded by C and we get

$$ \begin{align*}\sum_{1\leq j\leq q^{(n_k,n_{k+1})}}{\sigma^{(n_k,n_{k+1})}_{j}}\leq Cq^{(n_k,n_{k+1})}.\end{align*} $$

The set of parameters $\mathcal {F}_{\mathcal {C}}$ thus witnesses that $\{T\}$ is flexible.

4.2 Irrational rotations

We now consider a construction from [Reference Drillick, Espinosa-Dominguez, Jones-Baro, Leng, Mandelshtam and SilvaDEJ⁺23]. For every irrational number $\theta $ , Drillick et al give an explicit cutting-and-stacking construction of a transformation T which is the irrational rotation of angle $\theta $ when the construction yields a finite measure-preserving system.

4.2.2 Proof of Proposition 3.8 for these systems

Let $\mathcal {V}$ be a non-empty open subset of $\mathbb {R}$ and

We now prove that $\mathcal {C}$ is a flexible class.

We first use the following basic fact from the theory of continued fractions: if A denotes the set of sequences $(q_i)_{i\geq -1}$ of integers such that $q_0,q_1,\ldots $ are positive, and if A is equipped with the induced product topology, then the map

$$ \begin{align*}(q_i)_{i\geq -1}\in A\mapsto [q_{-1};q_0,q_1,\ldots]\in\mathbb{R}\setminus\mathbb{Q}\end{align*} $$

is a homeomorphism (see [Reference Einsiedler and WardEW11, Lemma 3.4] for instance). We can then fix integers $n_0\geq 0$ and $Q_{-1},Q_{0},\ldots ,Q_{n_0}$ (where $Q_{0},\ldots ,Q_{n_0}$ are positive) such that $Q_{0}\ldots Q_{n_0}$ is greater than or equal to $3$ and the following holds: for every irrational number $\theta $ , if the first coefficients of its continued fraction expansion are $Q_{-1},Q_{0},\ldots ,Q_{n_0}$ , then $\theta $ is in  $\mathcal {V}$ .

We write and we consider the set $\tilde {\mathcal {F}}(\boldsymbol {Q})$ of finite sequences $(\tilde {q}_k,(\tilde {\sigma }_{k,0},\ldots ,\tilde {\sigma }_{k,\tilde {q}_k}))_{0\leq k\leq n}$ such that $n\geq n_0$ and for all $k\in \{0,\ldots ,n\}$ ,

$$ \begin{align*}\begin{array}{ll} \tilde{q}_k=Q_k&\text{if }k\leq n_0,\\ \tilde{q}_k\geq 2&\text{if }k>n_0, \end{array}\end{align*} $$

and

$$ \begin{align*}\begin{array}{ll} \tilde{\sigma}_{k,i}=0&\text{for }i\in\{0,\ldots,\tilde{q}_k-1\},\\ \tilde{\sigma}_{k,\tilde{q}_k}=\tilde{h}_{k-1}& \end{array}\end{align*} $$

(where $(\tilde {h}_k)_{0\leq k\leq n}$ is the associated height sequence and ). The finite sequences of $\tilde {\mathcal {F}}(\boldsymbol {Q})$ may not be finite sequences of cutting and stacking parameters in the sense of Definition 3.1, since the integers $Q_0,\ldots ,Q_{n_0}$ may be equal to $1$ . Moreover, even if the integers $Q_0,\ldots ,Q_{n_0}$ were greater than or equal to $2$ , we could not prove that $\mathcal {C}$ is a flexible class with $\mathcal {F}_{\mathcal {C}}=\tilde {\mathcal {F}}(\boldsymbol {Q})$ , since the first cutting parameters $\tilde {q}_1,\ldots ,\tilde {q}_{n_0}$ cannot be chosen large enough. Notice that, although we may have $\tilde {q}_k=1$ for some $k\in \{0,1,\ldots ,n_0\}$ , a finite sequence $(\tilde {q}_k,(\tilde {\sigma }_{k,0},\ldots ,\tilde {\sigma }_{k,q_k}))_{0\leq k\leq n}$ can still define the first $(n+1)$ steps of a cutting-and-stacking construction, and we associate to it another finite sequence $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n-n_0}$ corresponding to the cutting-and-stacking construction obtained from the previous one by skipping the steps $1, 2,\ldots ,n_0$ . We get $h_0=\tilde {h}_0=1$ , $q_0=Q_0\ldots Q_{n_0}\geq 3$ and $\text { for all } k\in \{1,\ldots ,n-n_0\}$ ,

$$ \begin{align*}\begin{array}{ll} q_k=\tilde{q}_{n_0+k}\geq 2,&\\ h_k=\tilde{h}_{n_0+k},&\\ \sigma_{k,i}=\tilde{\sigma}_{n_0+k,i}=0&\text{for }i\in\{0,\ldots,q_k-1\},\\ \sigma_{k,q_k}=\tilde{\sigma}_{n_0+k,\tilde{q}_{n_0+k}}=\tilde{h}_{n_0+k-1}=h_{k-1}&\text{if }k\geq 2. \end{array}\end{align*} $$

For $k=1$ , we have $\sigma _{1,q_1}=\tilde {h}_{n_0}$ , where $\tilde {h}_{n_0}$ is not equal to $h_0$ . Setting (this constant only depends on $Q_0,\ldots ,Q_{n_0}$ ), we have $Z_1\leq Ch_1$ and $\sigma _{1}\leq C'h_0$ . We immediately get the inequalities $Z_k\leq Ch_k$ and $\sigma _{k}\leq C'h_{k-1}$ for $k\in \{2,\ldots ,n-n_0\}$ .

Let $\mathcal {F}(\boldsymbol {Q})$ be the set of finite sequences $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n-n_0}$ obtained from finite sequences $(\tilde {q}_k,(\tilde {\sigma }_{k,0},\ldots ,\tilde {\sigma }_{k,q_k}))_{0\leq k\leq n}\in \tilde {\mathcal {F}}(\boldsymbol {Q})$ . It is now easy to check that $\mathcal {C}$ is a flexible class, with the set of parameters and the constants C and $C'$ .

4.3 Systems with a given eigenvalue

Let $\theta $ be an irrational number and $\mathcal {C}$ the class of rank-one systems which has as an eigenvalue. In [Reference Danilenko and VieprikDV23], Danilenko and Vieprik present an explicit cutting-and-stacking construction of a system in $\mathcal {C}$ . The parameters are chosen in the following way (see [DV23, proof of Theorem 4.1]).

4.3.1 The construction of Danilenko and Vieprik

For every $n\geq 1$ , we fix a number $j_n\in \{1,\ldots ,n\}$ such that is less than $2\pi /n$ . Fix $n\geq 1$ , assume that $ (q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}) )_{0\leq k\leq n-1}$ has already been constructed with an auxiliary condition

(2) $$ \begin{align} h_{n}>\frac{n^4}{\delta_{n^2}}. \end{align} $$

Danilenko and Vieprik show the existence of a sequence $(\ell ^{(n)}_m)_{m\geq 1}$ of positive integers less than or equal to $2\pi /\delta _{n^2}$ , such that for every $m\geq 1$ ,

(3) $$ \begin{align} |1-\unicode{x3bb}^{mh_{n}+(\ell^{(n)}_1+\cdots +\ell^{(n)}_{m})j_{n^2}}|<\frac{2\pi}{n^2}. \end{align} $$

Next, let $q_n$ be an integer large enough so that the auxiliary condition (2) holds at the next step, namely

(in [Reference Danilenko and VieprikDV23], $q_n$ is chosen as the smallest integer satisfying the property, but it is not needed, so there are infinitely many choices). Finally, the spacing parameters at this step are defined by $\sigma _{n,0}=\sigma _{n,q_n}=0$ and $\sigma _{n,m}=\ell ^{(n)}_mj_{n^2}$ for $1\leq m\leq q_n-1$ .

With these parameters satisfying conditions (2) and (3), $\unicode{x3bb} $ is an eigenvalue of the underlying rank-one system (see [DV23, proof of Theorem 4.1] for details).

4.3.2 Proof of Proposition 3.8 for these systems

Let us consider the same construction as above, but with the following auxiliary condition:

(4) $$ \begin{align} h_{n}>\max{\bigg (\frac{n^4}{\delta_{n^2}},\frac{(n+1)^4}{\delta_{(n+1)^2}}\bigg )}, \end{align} $$

which is stronger than the previous auxiliary condition (2). Note that the real numbers $\delta _i$ have been fixed before setting the parameters.

The subset $\mathcal {F}_{\mathcal {C}}$ of $\mathcal {P}^*$ is defined to be the set of finite sequences $ (q_k,(\sigma _{k,0},\ldots , \sigma _{k,q_k}) )_{0\leq k\leq n}$ constructed in a recursive way. Any cutting and spacing parameter $(q_0,(\sigma _{0,0},\ldots ,\sigma _{q_0,0}))$ is in $\mathcal {F}_{\mathcal {C}}$ , and if $ (q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}) )_{0\leq k\leq n-1}$ is in $\mathcal {F}_{\mathcal {C}}$ , then so is $ (q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}) )_{0\leq k\leq n}$ for every $(q_n,\sigma _{n,0},\ldots ,\sigma _{n,q_n})$ that we can obtain at the next step, as described above, but with the stronger auxiliary condition (4).

Let be a sequence of cutting and spacing parameters. If all its projections are in $\mathcal {F}_{\mathcal {C}}$ , then $\boldsymbol {p}$ provides a CSP construction with $C=2\pi $ . Indeed, we have $\sigma _{n,m}=\ell _{m}^{(n)}j_{n^2}\leq 2\pi n^2/\delta _{n^2}<2\pi h_n$ . As mentioned above, conditions (2) and (3) imply that the sequence $\boldsymbol {p}$ provides rank-one systems which have $\unicode{x3bb} $ as an eigenvalue.

Finally, if $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n-1}$ is a finite sequence in $\mathcal {F}_{\mathcal {C}}$ , we can choose a large enough integer $q_n$ so that the following holds at the next step:

$$ \begin{align*}h_{n+1}>\max{\bigg (\frac{(n+1)^4}{\delta_{(n+1)^2}},\frac{(n+2)^4}{\delta_{(n+2)^2}}\bigg )}\end{align*} $$

(in particular, the new auxiliary condition (4) is satisfied). We use the same spacing parameters as before, namely $\sigma _{n,m}=\ell _m^{(n)}j_{n^2}$ . Using $j_{n^2}\leq n^2$ and $\ell _m^{(n)}\leq ({2\pi }/{\delta _{n^2}})$ , this gives

$$ \begin{align*}\sigma_n=(\ell^{(n)}_1+\cdots +\ell^{(n)}_{q_n-1})j_{n^2}\leq q_n\frac{2\pi n^2}{\delta_{n^2}}\leq q_n h_{n-1},\end{align*} $$

so the third point of Definition 3.7 is satisfied for $C'=1$ .

4.4 Strongly mixing systems

Let $\mathcal {C}$ be the class of strongly mixing rank-one systems. We consider the construction of Ornstein in [Reference OrnsteinOrn72]. The property the parameters have to satisfy at each step is given by the following lemma (in [Orn72, Lemma 3.2]), proven with a probabilistic method.

The set of parameters $\mathcal {F}_{\mathcal {C}}$ is defined in a recursive way, as in §4.3: any cutting and spacing parameter $(q_0,(\sigma _{0,0},\ldots ,\sigma _{q_0,0}))$ is in $\mathcal {F}_{\mathcal {C}}$ , and from a finite sequence of parameters $ (q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}) )_{0\leq k\leq n-1}$ in $\mathcal {F}_{\mathcal {C}}$ , $ (q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}) )_{0\leq k\leq n}$ is also in $\mathcal {F}_{\mathcal {C}}$ if the new parameters can be written as $q_n=m$ and $\sigma _{n,i}=a_i+h_{n-1}$ , where $m,a_1,\ldots , a_m$ are integers whose existence is granted by Lemma 4.3 with $N> 10^n$ , $K=h_{n-1}$ , $\varepsilon =10^{n-3}$ and $\alpha =5/4$ . There are infinitely many possibilities for $q_n$ as N can be arbitrarily large. It is shown in [Reference OrnsteinOrn72] that cutting-and-stacking constructions with these parameters give strongly mixing systems, and it is clear that they are CSP with $C=2$ and the third point of Definition 3.7 is satisfied for $C'=2$ .

5.1 The construction

5.1.1 Overview of the construction

We first present a natural adaptation to the case of the rank-one system of Kerr and Li’s construction of an explicit orbit equivalence between the universal odometer and any other odometers. We will then see that the quantification of the cocycles becomes more complicated due to the presence of non-zero spacing parameters.

Let $T\in \mathrm {Aut}(X,\mu )$ be a rank-one system and consider a cutting-and-stacking construction of this transformation with the same notation $q_n,\sigma _{n,i},\sigma _n,h_n,\mathcal {R}_n,\varepsilon _n,X_n$ as in Definition 3.2. From the sequence of Rokhlin towers $\mathcal {R}_n$ , new towers $\mathcal {R}^{\prime }_n$ will be built as Rokhlin towers for a new system S. These towers $\mathcal {R}^{\prime }_n$ will have no spacers, that is, $\sigma ^{\prime }_{n,i}=0$ , so they will be partitions of X. The construction will ensure that $\mathcal {R}^{\prime }_n$ increases to the $\sigma $ -algebra $\mathcal {A}$ using the fact that it is the case for $\mathcal {R}_n$ , so S will be an odometer. For the odometer S to be universal, we fix a sequence of prime numbers $(p_n)_{n\geq 0}$ such that every prime number appears infinitely many times, and every cutting parameter $q^{\prime }_n$ will be a multiple of $p_n$ .

We will recursively define S on subsets increasing to X up to a null set. More precisely, if the nth tower $\mathcal {R}^{\prime }_n$ has been built and its base and its height are denoted by $B^{\prime }_n$ and $h^{\prime }_n$ , then S is provisionally defined on all the levels of the tower except the highest one and maps the ith level to the $(i+1)$ th one. So, $\mathcal {R}^{\prime }_n$ is exactly $(B^{\prime }_n,S(B^{\prime }_n),\ldots ,S^{h^{\prime }_n-1}(B^{\prime }_n))$ and S is defined on $X\setminus S^{h^{\prime }_n-1}(B^{\prime }_{n})$ . To refine S, that is, to define it on a greater set, we have to build the next tower $\mathcal {R}^{\prime }_{n+1}$ and define S as for $\mathcal {R}^{\prime }_n$ . To do so and according to Definition 3.2, we have to determine a subdivision of the base $B^{\prime }_n$ into $q^{\prime }_n$ subsets $B^{\prime }_{n,0},\ldots ,B^{\prime }_{n,q^{\prime }_n-1}$ . We find a function $\zeta _{n+1}$ mapping bimeasurably each $B^{\prime }_{n,i}$ to $B^{\prime }_{n,i+1}$ for $0\leq i\leq q^{\prime }_n-2$ . On the subset of the roof $S^{h^{\prime }_n-1}(B_n)$ of $\mathcal {R}^{\prime }_n$ , S will coincide with $\zeta _{n+1}S^{-h^{\prime }_n}$ and will be defined on $X\setminus S^{h^{\prime }_{n+1}}(B^{\prime }_{n+1})=D_1\sqcup \cdots \sqcup D_{n+1}$ , where ${B^{\prime }_{n+1}=B^{\prime }_{n,0}}$ is the base of the new Rokhlin tower $\mathcal {R}_{n+1}$ for S. To sum up, S is successively defined by the finite approximations obtained from the maps $\zeta _n$ . Up to conjugacy, $\zeta _n$ is exactly the addition by $(\underbrace {0,\ldots ,0}_{n-1\text { times}},1,0,0,\ldots )$ with carry over to the right (as defined in §3.1), restricted to $[0,\ldots ,0]_{n-1}\setminus [0,\ldots ,0,q^{\prime }_{n-1}-1]_n$ .

The construction of the maps $\zeta _n$ is by induction on $n\geq 0$ . At step n, we will actually define

$$ \begin{align*}\zeta_{n+1}\colon B^{\prime}_{n,0}\sqcup\cdots\sqcup B^{\prime}_{n,q^{\prime}_n-2}\to B^{\prime}_{n,1}\sqcup\cdots\sqcup B^{\prime}_{n,q^{\prime}_n-1}.\end{align*} $$

To build $\zeta _{n+1}$ , a second induction on a parameter $m\geq n$ is required. Actually, $B^{\prime }_{n,i}$ will be the disjoint union of the $B^{\prime }_{n,i}(m)$ for $m\geq n$ , and this inner recursion consists in choosing m bricks to define $B^{\prime }_{n,i}(m)$ . By definition, the m bricks will be the m levels (that is, the levels of $\mathcal {R}_m$ ) explicitly chosen to constitute $B^{\prime }_{n,i}(m)$ . Using powers of T, m bricks of $B^{\prime }_{n,i}$ are mapped to those of $B^{\prime }_{n,i+1}$ (there will be as many in $B_{n,i}$ as in $B^{\prime }_{n,i+1}$ ) and this gives $\zeta _{n+1}$ whose orbits are included in those of T, implying immediately that the orbits of S satisfy the same property. The reverse inclusion between the orbits will be more difficult to prove and will be due to the choice of the bricks (see Remark 5.1 after the construction).

5.1.2 The construction

Here, T is a rank-one system in $\mathrm {Aut}(X,\mu )$ . We fix one of its cutting-and-stacking construction whose parameters are denoted as in Definition 3.2. Let $(p_n)_{n\geq 0}$ be a sequence of prime numbers such that every prime number appears infinitely many times.

In the following, we will assume that, given the cutting parameters of T, some positive integers $q^{\prime }_n$ and $t_{n,m}$ that we will introduce are well defined. In §5.3 (see Lemma 5.10), we will give conditions on the parameters of T for these quantities (and so the construction) to be well defined.

Step $\textit {n}=1$ .

We first build $\mathcal {R}^{\prime }_1$ and $\zeta _1$ by an induction over $m\geq 1$ . We could denote by $\mathcal {R}^{\prime }_0$ the trivial tower $(X)$ with its base . At the end of step $n=1$ , S is not yet defined on the roof of the tower $\mathcal {R}^{\prime }_1$ , that is, on its highest level, which is a Rokhlin tower of S.

• • $m=1$ : let $q^{\prime }_0>0$ be the largest multiple of $p_0$ such that $q^{\prime }_0\leq q_0-1$ . For every $0\leq i\leq q^{\prime }_0-1$ , we define

  
  and
  $$ \begin{align*}\zeta_1(1)\colon\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i}(1)}\to\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i+1}(1)}\end{align*} $$
  coinciding with T on its domain (hence, every subset $B^{\prime }_{0,i}(1)$ is composed of a unique $1$ -brick $T^i(B_{1})$ ).

  Remark 7.1. At this step, we simply have to assume $q_0>p_0$ for the integer $q^{\prime }_0$ to be non-zero. However, for the well definition of other quantities at other steps, the conditions on the cutting parameters of T get increasingly more technical, which is the reason why we first assume that the parameters of T are chosen so that the positive quantities are well defined and we will then state the conditions in Lemma 5.10 (as mentioned before the beginning of the construction).

• • $m>1$ : assume that the subsets $B^{\prime }_{0,i}(M)$ have been built for every $1\leq M\leq m-1$ and $0\leq i\leq q^{\prime }_0-1$ . Let

  
  be the remaining piece of X at the end of the $(m-1)$ th step (we could also define ).

  Remark 7.2. Notice that m-levels are either contained in $W_{1,m}$ or disjoint from it since $X\setminus W_{1,m}$ is composed of M-levels for $1\leq M\leq m-1$ and the Rokhlin towers are nested. This will more generally hold true for $W_{n,m}$ with $n\geq 2$ .

  Let $r_{1,m}$ be the number of integers such that $T^j(B_m)\subseteq W_{1,m}$ , denoted by

  $$ \begin{align*}0\leq j_{1}^{(1,m)} <j_{2}^{(1,m)} <\cdots <j_{r_{1,m}}^{(1,m)} <h_m.\end{align*} $$
  Let $t_{1,m}$ be the positive integer (we assume that we can choose the cutting parameters of T for this integer to be positive, see Lemma 5.10) such that $q^{\prime }_0t_{1,m}$ is the largest multiple of $q^{\prime }_0$ such that $q^{\prime }_0t_{1,m}<r_{1,m}$ . The first $q^{\prime }_0t_{1,m}$ m-levels contained in $W_{1,m}$ are now used as m-bricks; they are split in $q^{\prime }_0$ groups of $t_{1,m}$ m-bricks of the subsets $B^{\prime }_{0,i}$ in the following way and the same will be done at steps $n>1$ (the fact that the inequality $q^{\prime }_0t_{1,m}<r_{1,m}$ is strict and the way we make the $q^{\prime }_0$ groups will guarantee an easy control of the cocycles, see Lemma 5.4 used for Lemmas 5.16 and 5.19). For every $0\leq i\leq q^{\prime }_0-1$ , we define
  
  and $\zeta _1(m)$ coinciding with $T^{ (j_{i+2+tq^{\prime }_0}^{(1,m)} )-(j_{i+1+tq^{\prime }_0}^{(1,m)} )}$ on $T^{ (j_{i+1+tq^{\prime }_0}^{(1,m)} )}(B_m)$ for every $0\leq i\leq q^{\prime }_0-2$ and $0\leq t\leq t_{1,m}-1$ , so that each brick $T^{ (j_{i+1+tq^{\prime }_0}^{(1,m)} )}(B_m)$ is mapped on another $T^{ (j_{i+2+tq^{\prime }_0}^{(1,m)} )}(B_m)$ . Thus, $\zeta _1(m)$ maps each $B^{\prime }_{0,i}(m)$ on $B^{\prime }_{0,i+1}(m)$ and this gives
  $$ \begin{align*}\zeta_1(m)\colon\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i}(m)}\to\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i+1}(m)}.\end{align*} $$

End of step $n=1$ .

For every $0\leq i\leq q^{\prime }_0-1$ , we define

(the set of its m-bricks for all $m\geq 1$ ), and

$$ \begin{align*}\zeta_1\colon\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i}}\to\bigsqcup_{0\leq i\leq q^{\prime}_0-2}{B^{\prime}_{0,i+1}}\end{align*} $$

coinciding with the maps $\zeta _1(m)$ on their respective domain (see Figure 7).

Figure 7

First step of the construction (that is, $n=1$ ). In §5.4, we will define sets $E_{n,m}$ for every pair of integers $(n,m)$ satisfying $m\geq n\geq 1$ . The set $E_{1,1}$ (respectively $E_{1,2}$ ; $E_{1,3}$ ) is the union of the red areas (respectively red and blue areas; red, blue and green areas).

The universal odometer S we want to build is partially defined on X. More precisely, we define it on the domain of $\zeta _1$ so that it coincides with $\zeta _1$ . This gives the first Rokhlin tower . The next step will provide us with a refinement $\mathcal {R}^{\prime }_2$ of the tower $\mathcal {R}^{\prime }_1$ , allowing us to extend partially S on the highest level of the $\mathcal {R}^{\prime }_1$ .

Step $n>1$ .

Assume that steps $1,\ldots ,n-1$ have been achieved. There are nested towers $\mathcal {R}^{\prime }_1,\ldots ,\mathcal {R}^{\prime }_{n-1}$ . The kth tower $\mathcal {R}^{\prime }_k$ has levels and its base $B^{\prime }_k$ is partitioned in $q^{\prime }_k$ levels $B^{\prime }_{k,0},\ldots ,B^{\prime }_{k,q^{\prime }_k-1}$ . These levels belong to $\mathcal {R}^{\prime }_{k+1}$ , whose base is , with $\zeta _{k+1}$ mapping $B^{\prime }_{k,i}$ to $B^{\prime }_{k,i+1}$ . The map S is defined on $D_1\sqcup \cdots \sqcup D_{n-1}$ using the maps $\zeta _1,\ldots ,\zeta _{n-1}$ , where $D_1\sqcup \cdots \sqcup D_{n-1}$ corresponds to the union of all the levels of $\mathcal {R}^{\prime }_{n-1}$ except its roof.

The map S is not yet defined on the roof of $\mathcal {R}^{\prime }_{n-1}$ . By partitioning $B^{\prime }_{n-1}$ in subsets $B^{\prime }_{n-1,0},\ldots ,B^{\prime }_{n-1,q^{\prime }_{n-1}-1}$ , we will define $\mathcal {R}^{\prime }_n$ which refines $\mathcal {R}^{\prime }_{n-1}$ and a function $\zeta _n$ mapping $B^{\prime }_{n-1,i}$ to $B^{\prime }_{n-1,i+1}$ . The extension of S will be defined on all the levels of $\mathcal {R}^{\prime }_n$ , except its roof (which is contained in the one of $\mathcal {R}^{\prime }_{n-1}$ ). We will construct the subsets $B^{\prime }_{n-1,i}$ as was done for the subsets $B^{\prime }_{0,i}$ , except that we only use the ‘material’ in $B^{\prime }_{n-1}$ to form the m-bricks of each $B^{\prime }_{n-1,i}$ . To do so, notice that the base $B^{\prime }_{n-1}$ is exactly $B^{\prime }_{n-2,0}$ (the first subset in the subdivision of $B^{\prime }_{n-2}$ ) which is the disjoint union of subsets of the form $B^{\prime }_{n-2,0}(m)$ for $m\geq n-1$ . Moreover, for all $n-1\leq M\leq m$ , every m-level is contained in an M-level, then we will pick the new m-bricks in $B^{\prime }_{n-2,0}(n), \ldots , B^{\prime }_{n-2,0}(m)$ . This motivates the definition of each set $W_{n,m}$ (the set of the remaining material to form m-bricks with an incremented integer m). We now discuss separately the following cases.

• • $m=n$ : set

  (5)

  
  and let $r_{n,n}$ be the number of integers such that $T^j(B_n)\subseteq W_{n,n}$ (note that we could have defined $r_{1,1}=q_0$ ), denoted by
  $$ \begin{align*}0\leq j_{1}^{(n,n)} <j_{2}^{(n,n)} <\cdots <j_{r_{n,n}}^{(n,n)} <h_n. \end{align*} $$
  Let $q^{\prime }_{n-1}$ be the largest multiple (we assume that we can choose the cutting parameters of T for this integer to be positive, see Lemma 5.10) of $p_{n-1}$ such that $q^{\prime }_{n-1}< r_{n,n}$ . We then define for every $0\leq i\leq q^{\prime }_{n-1}-1$ ,
  
  meaning that among the n-levels in $W_{n,n}$ , the n-bricks at step $(n,n)$ are exactly the first $q^{\prime }_{n-1}$ ones (and set $t_{n,n}=1$ for consistency later on). Let
  $$ \begin{align*}\zeta_n(n)\colon\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i}(n)}\to\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i+1}(n)}\end{align*} $$
  be the map coinciding with $T^{ (j_{i+2}^{(n,n)} )- (j_{i+1}^{(n,n)} )}$ on each $B^{\prime }_{n-1,i}(n)$ , so that $B^{\prime }_{n-1,i}(n)$ is mapped to $B^{\prime }_{n-1,i+1}(n)$ .

• • $m>n$ : set

  (6)

  
  and let $r_{n,m}$ be the number of integers such that $T^j(B_m)\subseteq W_{n,m}$ , denoted by
  $$ \begin{align*}0\leq j_{1}^{(n,m)} <j_{2}^{(n,m)} <\cdots <j_{r_{n,m}}^{(n,m)} <h_m.\end{align*} $$
  Let $t_{n,m}$ be the positive integer (we assume that we can choose the cutting parameters of T for this integer to be positive, see Lemma 5.10) such that $q^{\prime }_{n-1}t_{n,m}$ is the largest multiple of $q^{\prime }_{n-1}$ such that $q^{\prime }_{n-1}t_{n,m}<r_{n,m}$ . The first $q^{\prime }_{n-1}t_{n,m}$ m-levels contained in $W_{n,m}$ are now used as m-bricks at step $(n,m)$ ; they are split in $q^{\prime }_{n-1}$ groups of $t_{n,m}$ m-bricks of the subsets $B^{\prime }_{n-1,i}$ in the following way. For every $0\leq i\leq q^{\prime }_{n-1}-1$ , we define
  
  and $\zeta _n(m)$ coinciding with $T^{ (j_{i+2+tq^{\prime }_{n-1}}^{(n,m)} )- (j_{i+1+tq^{\prime }_{n-1}}^{(n,m)} )}$ on $T^{ (j_{i+1+tq^{\prime }_{n-1}}^{(n,m)} )}(B_m)$ for every ${0\leq i\leq q^{\prime }_{n-1}-2}$ , $0\leq t\leq t_{n,m}-1$ , so that each m-brick $T^{ (j_{i+1+tq^{\prime }_{n-1}}^{(n,m)} )}(B_m)$ is mapped on another $T^{ (j_{i+2+tq^{\prime }_{n-1}}^{(n,m)} )}(B_m)$ . Thus, $\zeta _n(m)$ maps each $B^{\prime }_{n-1,i}(m)$ on $B^{\prime }_{n-1,i+1}(m)$ and this gives
  $$ \begin{align*}\zeta_n(m)\colon\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i}(m)}\to\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i+1}(m)}.\end{align*} $$

End of step n.

We define for every $0\leq i\leq q^{\prime }_{n-1}-1$ ,

(the set of its m-bricks for $m\geq n$ ), $B^{\prime }_n=B^{\prime }_{n-1,0}$ and

$$ \begin{align*}\zeta_n\colon\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i}}\to\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i+1}}\end{align*} $$

coinciding with the maps $\zeta _n(m)$ on their respective domain (see Figure 8 for step $n=2$ , after the first step illustrated in Figure 7).

Figure 8

From the first step (illustrated in Figure 7) to the second one. In $B^{\prime }_1$ , we inductively build $B^{\prime }_{1,i}(2), B^{\prime }_{1,i}(2), B^{\prime }_{1,i}(3), \ldots $ for every $0\leq i\leq q^{\prime }_1-1$ (in this example, we have $q^{\prime }_1=2$ ). Each set $B^{\prime }_{1,i}(2)$ is composed of a unique $2$ -level in $B^{\prime }_{0,0}(1)\sqcup B^{\prime }_{0,0}(2)$ (that is, in the pale red and pale blue areas). Then each set $B^{\prime }_{1,i}(3)$ is composed of $3$ -levels in $B^{\prime }_{0,0}(1)\sqcup B^{\prime }_{0,0}(2)\sqcup B^{\prime }_{0,0}(3)$ (that is, in the pale red, pale blue and pale green areas) and so on. The structure that we build in $B^{\prime }_1=B^{\prime }_{0,0}$ can be translated in $B^{\prime }_{0,1}$ and $B^{\prime }_{0,2}$ using the map $\zeta _1$ . In §5.4, we will define sets $E_{n,m}$ for every pair of integers $(n,m)$ satisfying $m\geq n\geq 1$ . The set $E_{2,1}$ (respectively $E_{2,2}$ ) is the union of the areas hatched in blue (respectively in blue or green).

As the base $B^{\prime }_{n-1}$ of $\mathcal {R}^{\prime }_{n-1}$ is partitioned in $B^{\prime }_{n-1,0}\sqcup \cdots \sqcup B^{\prime }_{n-1,q^{\prime }_{n-1}-1}$ , its highest level $S^{h^{\prime }_{n-1}-1}(B^{\prime }_{n-1})$ is partitioned in $S^{h^{\prime }_{n-1}-1}(B^{\prime }_{n-1,0})\sqcup \cdots \sqcup S^{h^{\prime }_{n-1}-1}(B^{\prime }_{n-1,q^{\prime }_{n-1}-1})$ . The map S is extended in the following way. On

it coincides with $\zeta _nS^{-(h^{\prime }_{n-1}-1)}$ . So S maps $S^{h^{\prime }_{n-1}-1}(B^{\prime }_{n-1,i})$ on $B^{\prime }_{n-1,i+1}$ . This gives a Rokhlin tower $\mathcal {R}^{\prime }_n$ for S, nested in the previous one, of base and height . Now S is defined on $(D_1\sqcup \cdots \sqcup D_{n-1})\sqcup D_n$ . The set $D_n$ consists in the levels of $\mathcal {R}_n$ , except the highest one, which are contained in the highest level of $\mathcal {R}^{\prime }_{n-1}$ .

Remark 5.1. Notice that the inclusion of the S-orbits in the T-orbits is easy since S is defined from maps $\zeta _n(m)$ which are ‘piecewise powers of T’.

The reverse inclusion will follow from the fact that we have $t_{n,n}=1$ for every $n\geq 1$ (at step $(n,n)$ , we form groups of only one n-level). Indeed, uniqueness implies that these chosen blocks are linked by $\zeta _n(n)$ and hence clearly by S (in contrast, an m-level, for $m>n$ , of $B^{\prime }_{n-1,i}$ is mapped by $\zeta _n(m)$ to only one of the $t_{n,m}$ m-levels of $B^{\prime }_{n-1,i+1}$ , and not to the other). Thus, ensuring that the unique n-brick of each $B^{\prime }_{n-1,i}$ is a large part of it that enables the system S to capture most of the T-orbits.

5.2 First properties of this construction

Recall that we consider a cutting-and-stacking construction of T with the same notation as in Definition 3.2 ( $q_n$ , $\sigma _{n,i}$ , $\mathcal {R}_n$ , $X_n$ , $\varepsilon _n$ , $\ldots $ ), and the sequences $(h_n)$ , $(\sigma _{n})$ and $(Z_n)$ associated to the cutting and spacing parameters, and the notation $q^{\prime }_n$ , $\mathcal {R}^{\prime }_n$ , $\ldots $ refer to the construction of S.

We state some important properties preparing for further results in §5.4. Many of them enable us to only take into account the combinatorics behind a cutting-and-stacking construction. We assume that all the ‘largest multiples’ (for every $n<m$ , the largest multiple $q^{\prime }_{n-1}$ of $p_{n-1}$ such that $q^{\prime }_{n-1}<r_{n,n}$ , and the largest multiple $q^{\prime }_{n-1}t_{n,m}$ of $q^{\prime }_{n-1}$ such that $q^{\prime }_{n-1}t_{n,m}<r_{n,m}$ ) are non-zero. In §5.3 (see Lemma 5.10), we will see how to choose the parameters for the construction to be well defined.

Proof of Lemma 5.2

Let $n\geq 1$ . The levels of $\mathcal {R}^{\prime }_n$ are pairwise disjoint by the definition of $(W_{n,m})_{m\geq n}$ . It remains to show that $\mathcal {R}^{\prime }_n$ covers the whole space. Recall that $X_n$ denotes the subset covered by the tower $\mathcal {R}_n$ , and $\varepsilon _n$ the measure of its complement.

The result holds for $n=1$ since $\mu (W_{1,m})\underset {m\to +\infty }{\to }0$ . Indeed, $W_{1,m+1}\cap X_{m}$ is the union of the m-levels which are not chosen at step $(1,m)$ . By the definition of $t_{1,m}$ , there are at most $q^{\prime }_0$ . So we have $W_{1,m+1}\leq \varepsilon _m+q^{\prime }_0\mu (B_m)\to 0$ .

For $n>1$ , it suffices to show that the levels $B^{\prime }_{n-1,0},\ldots ,B^{\prime }_{n-1,q^{\prime }_{n-1}-1}$ of $\mathcal {R}^{\prime }_n$ form a partition of the base $B^{\prime }_{n-1}$ of $\mathcal {R}^{\prime }_{n-1}$ . We have to show that the measure of

tends to $0$ as $m\to +\infty $ . However, this set $\tilde {W}_{n,m}$ is the disjoint union of $\bigsqcup _{M\geq m+1} {B^{\prime }_{n-2,0}(M)}$ and $W_{n,m}$ . It is clear that

$$ \begin{align*}\mu\bigg (\bigsqcup_{M\geq m+1}{B^{\prime}_{n-2,0}(M)}\bigg )\underset{m\to +\infty}{\to}0\end{align*} $$

since $\mu $ is a finite measure. The set $W_{n,m}$ is obtained from $W_{n,m-1}$ by adding $B^{\prime }_{n-2,0}(m)$ and removing $q^{\prime }_{n-1}t_{n,m-1}$ $(m-1)$ -levels. Thus, we have $\mu (W_{n,m} )\underset {m\to +\infty }{\to }0$ by the definition of $(t_{n,m})_{m\geq n}$ . Hence, we have $\mu (\tilde {W}_{n,m} )\underset {m\to +\infty }{\to }0$ and we are done.

As a consequence, if $(\mathcal {R}^{\prime }_n)_n$ increases to the $\sigma $ -algebra $\mathcal {A}$ (this will be proved in Corollary 5.14), then S is a rank-one system without spacer, so this is an odometer.

Lemma 5.3. Let $n\geq 1$ . On the base $B^{\prime }_n$ of the nth S-Rokhlin tower $\mathcal {R}^{\prime }_n$ , S is defined as follows. For every $0\leq i\leq h^{\prime }_n-1$ , we have

$$ \begin{align*}S^i=\zeta_1^{i_0}\ldots \zeta_n^{i_{n-1}}\quad\text{on }B^{\prime}_n\end{align*} $$

with such that

$$ \begin{align*} i=\sum\limits_{\ell=0}^{n-1}{q^{\prime}_0\cdots q^{\prime}_{\ell -1}i_{\ell}}=\sum\limits_{\ell=0}^{n-1}{h^{\prime}_{\ell}i_{\ell}}. \end{align*} $$

Therefore, the subset $D_n$ defined in the construction can be written as follows:

(7) $$ \begin{align} \begin{aligned} D_n&= \zeta_1^{q^{\prime}_0-1}\ldots \zeta_{n-1}^{q^{\prime}_{n-2}-1}\bigg (\bigsqcup_{0\leq i_n\leq q^{\prime}_{n-1}-2}{B^{\prime}_{n-1,i_n}}\bigg )\\ &= \zeta_1^{q^{\prime}_0-1}\ldots \zeta_{n-1}^{q^{\prime}_{n-2}-1}\bigg (\bigsqcup_{0\leq i_n\leq q^{\prime}_{n-1}-2}{\zeta_n^{i_n}(B^{\prime}_{n-1,0})}\bigg ) \end{aligned} \end{align} $$

and S coincides with $\zeta _n\zeta _{n-1}^{-(q^{\prime }_{n-2}-1)}\ldots \zeta _1^{-(q^{\prime }_0-1)}$ on $D_n$ .

By the cocycle of $\zeta _n(m)$ , we mean the integer-valued map defined on the domain of $\zeta _n(m)$ and which maps x to the unique integer k satisfying $\zeta _n(m)x=T^kx$ .

Proof of Lemma 5.4

By definition, for fixed integers $0\leq i\leq q^{\prime }_{n-1}-2$ and $0\leq t\leq t_{n,m}-1$ , the cocycle on takes the value . Let us recall that the integers

$$ \begin{align*}0\leq j_{1}^{(n,m)} <j_{2}^{(n,m)} <\cdots <j_{r_{n,m}}^{(n,m)} <h_m\end{align*} $$

are the sets of indices such that $T^j(B_m)\subseteq W_{n,m}$ . Thus, $\Delta j$ is obviously positive. Let us fix an $(m-1)$ -level $\mathbb {B}^{\star }$ which is not chosen at step $(n,m-1)$ , so it is contained in $W_{n,m}$ . If m is equal to n, we can choose $\mathbb {B}^{\star }=B^{\prime }_{n-2,0}(n-1)$ . For $m>n$ , the existence of $\mathbb {B}^{\star }$ is granted by the fact that we have $q^{\prime }_{n-1}t_{n,m-1}<r_{n,m-1}$ . We write it as $\mathbb {B}^{\star }=T^{k_0}(B_{m-1})$ , where $k_0$ is an integer in .

By definition, $\Delta j$ is the least positive integer j such that $T^j(\mathbb {B})$ is in $W_{n,m}$ . Moreover, the m-levels of $\mathbb {B}^{\star }$ are in $W_{n,m}$ . Therefore, the consecutive m-levels $T(\mathbb {B}),\ldots ,T^{\Delta j-1}(\mathbb {B})$ are not in $\mathbb {B}^{\star }$ .

First case. In the tower $\mathcal {R}_m$ , assume that the m-levels $T(\mathbb {B}),\ldots ,T^{\Delta j-1}(\mathbb {B})$ are before $T^{k_0}(B_{m-1,0})$ , that is, before the first m-level of $\mathbb {B}^{\star }$ . Therefore, the enumeration $\mathbb {B},\ldots ,T^{\Delta j}(\mathbb {B})$ is included in the enumeration

$$ \begin{align*}\Sigma_{m-1,0,1},\ldots,\Sigma_{m-1,0,\sigma_{m-1,0}},B_{m-1,0},\ldots,T^{k_0}(B_{m-1,0}),\end{align*} $$

implying that $\Delta j\leq \sigma _{m-1,0}+k_0\leq Z_{m-1}+h_{m-1}$ .

Second case. Now assume that $T(\mathbb {B}),\ldots ,T^{\Delta j-1}(\mathbb {B})$ are after $T^{k_0}(B_{m-1,q_{m-1}-1})$ , that is, after the last m-level of $\mathbb {B}^{\star }$ . Therefore, the enumeration $\mathbb {B},\ldots ,T^{\Delta j}(\mathbb {B})$ is included in the enumeration

$$ \begin{align*} T^{k_0}(B_{m-1,q_{m-1}-1}),\ldots,T^{h_{m-1}-1}(B_{m-1,q_{m-1}-1}),\Sigma_{m-1,q_{m-1},1},\ldots,\Sigma_{m-1,q_{m-1},\sigma_{m-1,q_{m-1}}},\end{align*} $$

and we get $\Delta j\leq (h_{m-1}-k_0-1)+\sigma _{m-1,q_{m-1}}\leq h_{m-1}+Z_{m-1}$ .

Third case. Finally, if $T(\mathbb {B}),\ldots ,T^{\Delta j-1}(\mathbb {B})$ are between $T^{k_0}(B_{m-1,i})$ and $T^{k_0}(B_{m-1,i+1})$ for some $0\leq i\leq q_{m-1}-2$ , that is, between two consecutive m-levels of $\mathbb {B}^{\star }$ , then the enumeration $\mathbb {B},\ldots ,T^{\Delta j}(\mathbb {B})$ is included in the enumeration

$$ \begin{align*}&T^{k_0}(B_{m-1,i}),\ldots,T^{h_{m-1}-1}(B_{m-1,i}),\Sigma_{m-1,i,1},\ldots,\Sigma_{m-1,i,\sigma_{m-1,i}},\\ &\quad B_{m-1,i+1},\ldots,T^{k_0}(B_{m-1,i+1}),\end{align*} $$

which gives $\Delta j\leq (h_{m-1}-k_0-1)+\sigma _{m-1,i}+(k_0+1)\leq h_{m-1}+Z_{m-1}$ .

We now present a combinatorial description of the construction.

Lemma 5.6. The quantities $r_{n,m}, q_n, q^{\prime }_n, t_{n,m}, \sigma _{n}$ satisfy the following recurrence relation:

During the construction, some integers have been defined for consistency (, ). Note that in this lemma, we also define the integers $t_{n,m}$ for $n=0$ . This enables us to extend the relations

$$ \begin{align*}r_{n,n}=q_{n-1}+t_{n-1,n}\quad\text{and}\quad r_{n,m}=q_{m-1}(r_{n,m-1}-q^{\prime}_{n-1}t_{n,m-1})+t_{n-1,m}\end{align*} $$

for $n=1$ .

Proof of Lemma 5.6

Case $n=1$ . For $m=1$ , the $r_{1,1}$ $1$ -levels potentially chosen to be $1$ -bricks are exactly the levels of $\mathcal {R}_1$ , so we have $r_{1,1}=q_0+t_{0,1}$ since . We choose $q^{\prime }_0$ of them, where $q^{\prime }_0$ is the largest multiple of $p_0$ such that $q^{\prime }_0< r_{1,1}$ , so $q^{\prime }_0$ is equal to $\lfloor (r_{1,1}-1)/p_0\rfloor p_0$ . Finally, $q^{\prime }_0$ is obviously equal to $q^{\prime }_0t_{1,1}$ since . For $m>1$ , there are $r_{n,m}$ m-levels in $W_{1,m}$ : some of them are in the $r_{1,m-1}-q^{\prime }_{0}t_{1,m-1}$ $(m-1)$ -levels which are not chosen at step $(1,m-1)$ and the other are the spacers from $\mathcal {R}_{m-1}$ to $\mathcal {R}_{m}$ . So, we have

$$ \begin{align*}r_{1,m}=q_{m-1}(r_{1,m-1}-q^{\prime}_{0}t_{1,m-1})+\sigma_{m-1}\end{align*} $$

and we set . We choose $q^{\prime }_0t_{1,m}$ of them as m-bricks, where $q^{\prime }_0t_{1,m}$ is the largest multiple of $q^{\prime }_0$ such that $q^{\prime }_0t_{1,m}<r_{1,m}$ , that is, .

Case $n>1$ . For $m=n$ , there are $r_{n,n}$ n-levels in $W_{n,n}=B^{\prime }_{n-2,0}(n-1)\sqcup B^{\prime }_{n-2,0}(n)$ . First, since we have $t_{n-1,n-1}=1$ , the set $B^{\prime }_{n-2,0}(n-1)$ is an $(n-1)$ -brick at step $n-1$ and it contains $q_{n-1}$ n-levels. Second, $B^{\prime }_{n-2,0}(n)$ is the union of $t_{n-1,n}$ n-bricks. Hence, we have $r_{n,n}=q_{n-1}+t_{n-1,n}$ . By definition, $q^{\prime }_{n-1}$ is equal to $\lfloor (r_{n,n}-1)/p_{n-1}\rfloor p_{n-1}$ and obviously to $q^{\prime }_{n-1}t_{n,n}$ with . For $m>n$ , there are $r_{n,m}$ m-levels in $W_{n,m}$ . This set is composed of

$$ \begin{align*}\bigg (\bigsqcup_{n-1\leq M\leq m-1}{B^{\prime}_{n-2,0}(M)}\bigg )\setminus\bigg (\bigsqcup_{n\leq M\leq m-1}{\bigsqcup_{0\leq i\leq q^{\prime}_{n-1}-1}{B^{\prime}_{n-1,i}(M)}}\bigg )\end{align*} $$

and

$$ \begin{align*}B^{\prime}_{n-2,0}(m).\end{align*} $$

The first one is the union of the $r_{n,m-1}-q^{\prime }_{n-1}t_{n,m-1}$ $(m-1)$ -levels which are not chosen at step $(n,m-1)$ , and the second one is built at step $(n-1,m)$ from its $t_{n-1,m}$ m-bricks. So, we have

$$ \begin{align*}r_{n,m}=q_{m-1}(r_{n,m-1}-q^{\prime}_{n-1}t_{n,m-1})+t_{n-1,m}.\end{align*} $$

We choose $q^{\prime }_{n-1}t_{n,m}$ of these m-levels as m-bricks at this step, where $q^{\prime }_{n-1}t_{n,m}$ is the largest multiple of $q^{\prime }_{n-1}$ such that $q^{\prime }_{n-1}t_{n,m}< r_{n,m}$ , that is, .

It will be more convenient to use the following slight modification of Lemma 5.6:

(8) $$ \begin{align} &\ \ \ t_{0,1}=0;\nonumber\\ \text{for }m\geq 2,\quad &\ \ \ t_{0,m}=\sigma_{m-1};\nonumber\\ \text{for }m=n\geq 1,\quad & \begin{cases} r_{n,n}=q_{n-1}+t_{n-1,n},\\ q^{\prime}_{n-1}\leq r_{n,n}-1,\\ t_{n,n}=1; \end{cases} \\ \text{for }m>n\geq 1,\quad & \begin{cases} r_{n,m}\leq q_{m-1}q^{\prime}_{n-1}+t_{n-1,m},\\ t_{n,m}\leq\dfrac{r_{n,m}-1}{q^{\prime}_{n-1}}. \end{cases}\nonumber \end{align} $$

This is a consequence of the inequalities $\lfloor x\rfloor \leq x$ and $r_{n,m-1}-q^{\prime }_{n-1}t_{n,m-1}\leq q^{\prime }_{n-1}$ (by the definition of $t_{n,m-1}$ ).

As the strategy will be to recursively choose large enough cutting parameters $q_n$ for T, we would like to understand the asymptotic behaviour of $q^{\prime }_n$ as $q_n$ increases. Then, the goal is to find bounds for $q^{\prime }_n/q_n$ .

Lemma 5.7. For every $n\geq 0$ , we have

$$ \begin{align*}q^{\prime}_{n}\geq q_{n}-(1+p_{n}).\end{align*} $$

Proof of Lemma 5.7

Using the equalities $q^{\prime }_{n}= \lfloor {r_{n+1,n+1}-1}/{p_{n}} \rfloor p_{n}$ and $r_{n+1,n+1}=q_{n}+t_{n,n+1}$ in Lemma 5.6, where the integer $t_{n,n+1}$ is non-negative, we get

$$ \begin{align*}q^{\prime}_{n}\geq\bigg (\frac{r_{n+1,n+1}-1}{p_{n}}-1\bigg )p_{n}\geq q_{n}-1-p_{n}\end{align*} $$

and we are done.

We have found a lower bound for $q^{\prime }_n/q_n$ (up to some additional term $-(1+p_n)$ ). Let us find an upper bound.

Lemma 5.8. For every $n\geq 1$ , we have

$$ \begin{align*}q^{\prime}_{n}\leq 3q_{n}+\frac{\sigma_{n}}{q^{\prime}_{0}\ldots q^{\prime}_{n-1}}.\end{align*} $$

With an asymptotic control on $\sigma _n$ , using flexible classes, we will be able to get $q^{\prime }_n\leq 4q_n$ (see Lemma 5.12).

Proof of Lemma 5.8

By induction over (with $n\geq 1$ ) and using equation (8), we show that

$$ \begin{align*}q^{\prime}_{n}\leq q_{n}\bigg (2+\sum_{j=1}^{i}{\prod_{k=1}^{j}{\frac{1}{q^{\prime}_{n-k}}}}\bigg )+t_{n-1-i,n+1}\prod_{k=1}^{i+1}{\frac{1}{q^{\prime}_{n-k}}}.\end{align*} $$

For $i=0$ , we have $q^{\prime }_{n}< r_{n+1,n+1}= q_{n}+t_{n,n+1}$ and

$$ \begin{align*}t_{n,n+1}\leq \frac{r_{n,n+1}-1}{q^{\prime}_{n-1}}\leq\frac{1}{q^{\prime}_{n-1}} (q_{n}q^{\prime}_{n-1}+t_{n-1,n+1} )=q_{n}+\frac{t_{n-1,n+1}}{q^{\prime}_{n-1}},\end{align*} $$

so we get $q^{\prime }_{n}\leq 2q_{n}+({t_{n-1,n+1}}/{q^{\prime }_{n-1}})$ . For $0\leq i\leq n-2$ , we have

$$ \begin{align*}t_{n-1-i,n+1}\leq\frac{r_{n-1-i,n+1}-1}{q^{\prime}_{n-2-i}}\leq\frac{1}{q^{\prime}_{n-2-i}} (q_{n}q^{\prime}_{n-2-i}+t_{n-2-i,n+1} )=q_{n}+\frac{t_{n-2-i,n+1}}{q^{\prime}_{n-2-i}}.\end{align*} $$

If the result holds true for i, we get

$$ \begin{align*} q^{\prime}_{n}&\leq q_{n}\bigg (2+\sum_{j=1}^{i}{\prod_{k=1}^{j}{\dfrac{1}{q^{\prime}_{n-k}}}}\bigg )+\bigg (q_{n}+\dfrac{t_{n-2-i,n+1}}{q^{\prime}_{n-2-i}}\bigg )\prod_{k=1}^{i+1}{\dfrac{1}{q^{\prime}_{n-k}}}\\ &= q_{n}\bigg (2+\sum_{j=1}^{i+1}{\prod_{k=1}^{j}{\dfrac{1}{q^{\prime}_{n-k}}}}\bigg )+t_{n-1-(i+1),n}\prod_{k=1}^{i+2}{\dfrac{1}{q^{\prime}_{n-k}}}, \end{align*} $$

so the result is also true for $i+1$ .

Taking $i=n-1$ , this gives the lemma since $q^{\prime }_{\ell }\geq 2$ for every integer $\ell \geq 1$ .

5.3 Towards flexible classes

We now explain why flexible classes fit in this construction.

First, a condition for the construction to be well defined needs an inductive choice of the cutting parameters $(q_n)_{n\geq 0}$ of T (see Lemma 5.9). Second, a control on the spacing parameters will imply useful asymptotic controls for the quantification of the cocycles (see Lemma 5.12). Note that in the proof of Theorem 3.9 (see §5.5), we will need other estimates to quantify the cocycles. It will be possible, again using the definition of a flexible class, to inductively build large enough cutting parameters to have these estimates.

If the parameters are chosen according to a set $\mathcal {F}_{\mathcal {C}}\subseteq \mathcal {P}^*$ associated to a flexible class $\mathcal {C}$ , the underlying rank-one system has the desired property, that is, it is in $\mathcal {C}$ , and is orbit equivalent to the universal odometer, with some quantification guaranteed by the control of the spacing parameters and by the fact that the cutting parameters $q_n$ have been recursively chosen and large enough.

Lemma 5.9. Let T be a rank-one system with cutting and spacing parameters

$$ \begin{align*}(q_n,(\sigma_{n,0},\ldots,\sigma_{n,q_n}))_{n\geq 0}\end{align*} $$

such that the construction in §5.1 is well defined. Then, for every $n\in \mathbb {N}$ , $q^{\prime }_n$ only depends on $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n}$ .

Then, the main novelty in this paper is to build the rank-one system T while we are building the universal odometer S. Once $(q^{\prime }_0,\ldots ,q^{\prime }_n)$ has been built from $(q_k,(\sigma _{k,0},\ldots ,\sigma _{k,q_k}))_{0\leq k\leq n}$ , we are free to choose $(q_{n+1},(\sigma _{n+1,0},\ldots ,\sigma _{n+1,q_{n+1}}))$ for the definition of $q^{\prime }_{n+1}$ . The recursive definition of the cutting parameters is one of the main ideas behind the definition of a flexible class, and it allows to find cutting parameters satisfying some assumptions, for example, the assumptions of the following lemma.

Lemma 5.10. Assume that for every $n\in \mathbb {N}$ ,

(9) $$ \begin{align} q_n>\max{(p_n,q^{\prime}_0,\ldots,q^{\prime}_{n-1})}. \end{align} $$

Then, the construction is well defined, that is, all the ‘largest multiples’ are non-zero (that is, the largest multiple $q^{\prime }_{n-1}$ of $p_{n-1}$ such that $q^{\prime }_{n-1}<r_{n,n}$ , and the largest multiple $q^{\prime }_{n-1}t_{n,m}$ of $q^{\prime }_{n-1}$ such that $q^{\prime }_{n-1}t_{n,m}<r_{n,m}$ ).

The next lemma refines the estimate given by Lemma 5.8, with assumptions which will be satisfied in the context of flexible classes.

Lemma 5.12. Let $(q_n,(\sigma _{n,0},\ldots ,\sigma _{n,q_n}))_{n\geq 0}$ be the parameters of a CSP construction of T with associated constant $C>0$ . Assume that there exists a constant $C'>0$ such that

$$ \begin{align*}\text{ for all } n\geq 1,\quad \sigma_n\leq C'q_nh_{n-1}\quad\text{and}\quad q_{n}-(1+p_{n})\geq C'h_{n} \end{align*} $$

(for instance, if the third point of Definition 3.7 holds and if, given $(q_k,(\sigma _{0,k},\ldots , \sigma _{q_k,k}))_{0\leq k\leq n-1}$ , $q_n$ is chosen large enough). Then, we get the following bound:

$$ \begin{align*}\text{ for all } n\in\mathbb{N},\quad \frac{q^{\prime}_n}{q_n}\leq 4.\end{align*} $$

Proof of Lemma 5.12

For $n=0$ , this is a consequence of the inequality $q^{\prime }_0\leq q_0-1$ . Now let us prove the result for $n\geq 1$ . Using Lemma 5.8, it suffices to get

$$ \begin{align*}\text{ for all } n\geq 1,\quad \frac{\sigma_n}{q^{\prime}_0\ldots q^{\prime}_{n-1}}\leq q_n.\end{align*} $$

However, we have

$$ \begin{align*}\sigma_n\leq C'q_n h_{n-1}\leq q_n (q_{n-1}-(1+p_{n-1}) ),\end{align*} $$

and the right-hand side is bounded above by $q_nq^{\prime }_{n-1}$ (by Lemma 5.7), so the result follows from the inequality $q^{\prime }_{n-1}\leq q^{\prime }_0\ldots q^{\prime }_{n-1}$ .

5.4 Equality of the orbits, universal odometer and quantitative control of the cocycles

Recall the notation for the construction of T by cutting and stacking, where $(q_n)_n$ and $(\sigma _{n,i})_{n,i}$ are respectively the cutting and spacing parameters. The tower $\mathcal {R}_n$ is the nth T-Rokhlin tower, its height is $h_n$ , it covers the subset $X_n$ of X, $\varepsilon _n$ is the measure of its complement, $Z_n$ is the maximum of the spacing parameters over the first n steps and $M_0$ is the measure of the unique $0$ -level $B_0$ .

We use similar notation $q^{\prime }_n$ , $h^{\prime }_n$ and $\mathcal {R}^{\prime }_n$ for S. We also set

for all $n\geq 1$ , and .

The construction is assumed to be well defined, considering a cutting-and-stacking definition of T with parameters satisfying the criterion (9) (see Lemma 5.10). Since S is piecewise given by powers of T, the S-orbits are included in the T-orbits. It remains to show the reverse inclusion, to prove that $(\mathcal {R}^{\prime }_n)_{n\geq 0}$ is increasing to the $\sigma $ -algebra $\mathcal {A}$ and to quantify the cocycles.

As in [Reference Kerr and LiKL24], we set

and

Since $B^{\prime }_{n-1,0}$ is exactly the base $B^{\prime }_n$ of $\mathcal {R}^{\prime }_n$ , the subsets $S^i(B^{\prime }_{n-1,0})$ , for $0\leq i\leq h^{\prime }_{n}-1$ , are exactly the levels of $\mathcal {R}^{\prime }_n$ which is a partition of X. So the motivation behind the definition of $E_{n,m}$ is first to approximate $B^{\prime }_{n-1,0}$ by its M-bricks for $n\leq M\leq m$ , and then the set $E_{n,m}$ is actually the union of the M-bricks, for $n\leq M\leq m$ , of step n of the outer recursion, and their translates by S in the other levels in $\mathcal {R}^{\prime }_n$ (the sets $E_{1,1}$ , $E_{1,2}$ , $E_{1,3}$ , $E_{2,1}$ and $E_{2,2}$ are illustrated in Figures 7 and 8). We get a better approximation of X as m increases and notice that $E_{n,m}$ is a subset of $X_m$ since every M-brick, for $n\leq M\leq m$ , is a union of m-levels. Finally, the sets $K_n$ , for $n\geq 1$ , are introduced to show that the system S captures the T-orbits (recall Remark 5.1).

Lemma 5.13. The following holds:

$$ \begin{align*}\mu (X_m\setminus E_{n,m} )\leq\begin{cases} \dfrac{H^{\prime}_n}{h_m}&\text{for }n<m,\\[8pt] \dfrac{H^{\prime}_{n-1}+p_{n-1}h^{\prime}_{n-1}}{h_n}&\text{for }n=m. \end{cases}\end{align*} $$

Proof of Lemma 5.13

We prove the inclusions

$$ \begin{align*}E_{n,m}\subset E_{n-1,m}\subset\cdots\subset E_{2,m}\subset E_{1,m}\subset X_m,\end{align*} $$

and we bound the measures of $X_m\setminus E_{1,m}$ and each set $E_{k,m}\setminus E_{k-1,m}$ . The result follows from the decomposition

(10) $$ \begin{align} X_m\setminus E_{n,m}= (X_m\setminus E_{1,m} )\sqcup\bigsqcup_{2\leq k\leq n}{ (E_{k-1,m}\setminus E_{k,m} )} \end{align} $$

and $\sigma $ -additivity of $\mu $ .

The set $E_{1,m}$ is composed of m-levels, so it is contained in $X_m$ . If $m=1$ , then ${X_m\setminus E_{1,m}}$ is the disjoint union of $r_{1,1}-q^{\prime }_0$ $1$ -levels (see step $(1,1)$ of the construction). If $m>1$ , then $X_m\setminus E_{1,m}$ is the disjoint union of $r_{1,m}-q^{\prime }_0t_{1,m}$ m-levels (see step $(1,m)$ of the construction). By definition of $q^{\prime }_0$ (if $m=1$ ) or $t_{1,m}$ (if $m>1$ ), we thus have

$$ \begin{align*}\mu (X_m\setminus E_{1,m} )\leq\begin{cases} \dfrac{p_0}{h_m}&\text{if }m=1,\\[8pt] \dfrac{h^{\prime}_1}{h_m}&\text{if }m>1 \end{cases}\end{align*} $$

(recall that $h^{\prime }_1=q^{\prime }_0$ ).

Let . The function $\zeta _k$ has been built to map each M-brick ( $M\geq k$ ) at step k to another. However, such a brick is contained in an $M'$ -brick ( $k-1\leq M'\leq M$ ) from the previous step $k-1$ (see Lemma 5.5). We then have

$$ \begin{align*}\bigsqcup_{0\leq i_{k-1}\leq q^{\prime}_{k-1}-1}{\zeta_{k}^{i_{k-1}}\bigg (\bigsqcup_{k\leq M\leq m}{B^{\prime}_{k-1,0}(M)}\bigg )}\subseteq \bigsqcup_{k-1\leq M\leq m}{B^{\prime}_{k-2,0}(M)}.\end{align*} $$

Applying $\zeta _{1}^{i_0}\ldots \zeta _{k-1}^{i_{k-2}}$ and considering the union over $i_0,\ldots ,i_{k-2}$ , we get the inclusion $E_{k,m}\subseteq E_{k-1,m}$ and the equality

$$ \begin{align*}\begin{array}{@{}l} E_{k-1,m}\setminus E_{k,m}\\ \quad=\displaystyle\bigsqcup_{\substack{0\leq i_0\leq q^{\prime}_0-1\\ \vdots\\0\leq i_{k-2}\leq q^{\prime}_{k-2}-1}}\!{\zeta_{1}^{i_0}\ldots \zeta_{k-1}^{i_{k-2}}\!\left(\underbrace{\bigg (\bigsqcup_{k-1\leq M\leq m}{B^{\prime}_{k-2,0}(M)}\bigg)\setminus\bigg (\bigsqcup_{k\leq M\leq m}{\bigsqcup_{0\leq i_{k-1}\leq q^{\prime}_{k-1}-1}{B^{\prime}_{k-1,i_k}(M)}}\bigg )}_{=:\ [\ast]}\right )}. \end{array} \end{align*} $$

So the measure of $E_{k-1,m}\setminus E_{k,m}$ is $q^{\prime }_0\ldots q^{\prime }_{k-2}\mu ([\ast ] )=h^{\prime }_{k-1}\mu ([\ast ] )$ by T-invariance. The set $[\ast ]$ is obtained from $W_{k,m}$ (see equations (5) and (6)) by removing the m-bricks that have been chosen at step $(k,m)$ . If $m=k$ , then $[\ast ]$ is the disjoint union of $r_{k,k}-q^{\prime }_{k-1}$ m-levels (see step $(k,k)$ of the construction). If $m>k$ , then $[\ast ]$ is the disjoint union of $r_{k,m}-q^{\prime }_{k-1}t_{k,m}$ m-levels (see step $(k,m)$ of the construction). By definition of $q^{\prime }_{k-1}$ (if $m=k$ ) or $t_{k,m}$ (if $m>k$ ), we thus have

$$ \begin{align*}\mu ([\ast] )\leq\begin{cases} \dfrac{p_{k-1}}{h_m}&\text{if }m=k,\\[8pt] \dfrac{q^{\prime}_{k-1}}{h_m}&\text{if }m>k, \end{cases}\end{align*} $$

and

$$ \begin{align*}\mu (E_{k-1,m}\setminus E_{k,m} )\leq\begin{cases} \dfrac{h^{\prime}_{k-1}p_{k-1}}{h_m}&\text{if }m=k,\\[8pt] \dfrac{h^{\prime}_{k}}{h_m}&\text{if }m>k. \end{cases}\end{align*} $$

Using equation (10) and $\sigma $ -additivity of $\mu $ , we get the following inequalities. If $m>n$ , we get

$$ \begin{align*}\mu (X_m\setminus E_{n,m} )=\mu (X_m\setminus E_{1,m} )+\sum_{2\leq k\leq n}{\mu (E_{k-1,m}\setminus E_{k,m} )}\leq\sum_{1\leq k\leq n}{\frac{h^{\prime}_k}{h_m}}=\frac{H^{\prime}_n}{h_m}.\end{align*} $$

If $m=n$ , we get

$$ \begin{align*}\begin{aligned} \mu (X_m\setminus E_{n,m} )&= (\mu (X_m\setminus E_{1,m} )+\sum_{2\leq k\leq m-1}{\mu (E_{k-1,m}\setminus E_{k,m} )} )+\mu (E_{n-1,n}\setminus E_{n,n} )\\ &\leq\sum_{1\leq k\leq m-1}{\frac{h^{\prime}_k}{h_m}}+\frac{p_{n-1}h^{\prime}_{n-1}}{h_n}\\ &=\frac{H^{\prime}_{n-1}}{h_n}+\frac{p_{n-1}h^{\prime}_{n-1}}{h_m} \end{aligned}\end{align*} $$

and we are done.