Katok’s special representation theorem for multidimensional Borel flows

Abstract

Katok’s special representation theorem states that any free ergodic measure- preserving $\mathbb {R}^{d}$-flow can be realized as a special flow over a $\mathbb {Z}^{d}$-action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel $\mathbb {R}^{d}$-flows emerge from $\mathbb {Z}^{d}$-actions through the special flow construction using bi-Lipschitz cocycles.

1 Introduction

1.1 Overview

Theorems of Ambrose and Kakutani [1, 2] established a connection between measure-preserving $\mathbb {Z}$ -actions and $\mathbb {R}$ -flows by showing that any flow admits a cross-section and can be represented as a ‘flow under a function’. Their construction provides a foundation for the theory of Kakutani equivalence (also called monotone equivalence) [7, 11] on the one hand and the study of the possible ceiling functions in the ‘flow under a function’ representation [16, 21] on the other.

The intuitive geometric picture of a ‘flow under a function’ does not generalize to $\mathbb {R}^{d}$ -flows for $d \ge 2$ . However, Katok [12] re-interpreted it in a way that can readily be adapted to the multidimensional set-up, calling flows appearing in this construction special flows. Despite their name, they are not so special, because, as shown in the same paper, every free ergodic measure-preserving $\mathbb {R}^{d}$ -flow is metrically isomorphic to a special flow. Similarly to the works of Ambrose and Kakutani, it opened up the study of multidimensional concepts of Kakutani equivalence [5] and stimulated research on tilings of flows[15, 22].

Borel dynamics as a separate field goes back to the work of Weiss [31] and has blossomed into a versatile branch of dynamical systems. The phase space here is a standard Borel space $(X, \mathcal {B})$ , that is, a set X with a $\sigma $ -algebra $\mathcal {B}$ of Borel sets for some Polish topology on X. Some of the key ergodic theoretical results have their counterparts in Borel dynamics, although others do not generalize. For example, the Borel version of the Ambrose–Kakutani theorem on the existence of cross-sections in $\mathbb {R}$ -flows was proved by Wagh in [30] showing that, just as in ergodic theory, all free Borel $\mathbb {R}$ -flows emerge as ‘flows under a function’ over Borel $\mathbb {Z}$ -actions. Likewise, Rudolph’s two-valued theorem [21] generalizes to the Borel framework [25]. The theory of Kakutani equivalence, on the other hand, exhibits a different phenomenon. Although it is a highly non-trivial equivalence relation among measure-preserving flows [9, 20], a descriptive set theoretical version of Kakutani equivalence collapses entirely [19].

A considerable amount of work has been carried out to understand the Borel dynamics of $\mathbb {R}$ -flows, but relatively few things are known about multidimensional actions. This paper makes a contribution in this direction by showing that the analog of Katok’s special representation theorem does hold for free Borel $\mathbb {R}^{d}$ -flows.

1.2 Structure of the paper

Constructions of orbit equivalent $\mathbb {R}^{d}$ -actions often rely on (essential) hyperfiniteness and use covers of orbits of the flow by coherent and exhaustive regions. This is the case for the aforementioned paper of Katok [12], and related approaches have been used in the descriptive set theoretical set-up as well (for example, [26]). Particular assumptions on such coherent regions, however, depend on the specific application. Section 2 summarizes a general language of partial actions, in which many of the aforementioned constructions can be formulated. As an application, we show that the orbit equivalence relation generated by a free $\mathbb {R}$ -flow can also be generated by a free action of any non-discrete and non-compact Polish group (see Theorem 2.6). This is in striking contrast with the actions of discrete groups, where a probability measure-preserving free $\mathbb {Z}$ -action can be generated only by a free action of an amenable group.

Section 3 does the technical work of constructing Lipschitz maps that are needed for Theorem 3.12, which shows, roughly speaking, that, up to an arbitrarily small bi-Lipschitz perturbation, any free $\mathbb {R}^{d}$ -flow admits an integer grid—a Borel cross-section invariant under the $\mathbb {Z}^{d}$ -action.

Finally, §4 discusses the descriptive set theoretical version of Katok’s special flow construction and shows, in Theorem 4.3, that, indeed, any free $\mathbb {R}^{d}$ -flow can be represented as a special flow generated by a bi-Lipschitz cocycle with Lipschitz constants arbitrarily close to $1$ . This provides a Borel version of Katok’s special representation theorem.

2 Sequences of partial actions

We begin by discussing the framework of partial actions suitable for constructing orbit equivalent actions. Throughout this section, X denotes a standard Borel space.

2.1 Partial actions

Let G be a standard Borel group, that is, a group with a structure of a standard Borel space that makes group operations Borel. A partial G-action is a pair $(E, \phi )$ , where E is a Borel equivalence relation on X and $\phi : X \to G$ is a Borel map that is injective on each E-class: $\phi (x) \ne \phi (y)$ , whenever $x E y$ . More precisely, we should call such $(E, \phi )$ a partial free action. However, as we are mainly concerned with free actions in what follows, we choose to omit the adjective ‘free’ in the definition. The map $\phi $ itself may occasionally be referred to as a partial action when the equivalence relation is clear from the context.

The motivation for the name comes from the following observation. Consider the set

$$ \begin{align*}A_{\phi} = \{(g,x, y) \in G \times X \times X : x E y \textrm{ and } g\phi(x) = \phi(y)\}.\end{align*} $$

Injectivity of $\phi $ on E-classes ensures that, for each $x \in X$ and $g \in G$ , there is at most one $y \in X$ such that $(g,x,y) \in A_{\phi }$ . When such a y exists, we say that the action of g on x is defined and set $gx = y$ . Clearly, $(e,x, x) \in A_{\phi }$ for all $x \in X$ , and thus $ex=x$ ; also, $g_{2}(g_{1}x) = (g_{2}g_{1})x$ whenever all the terms are defined. The set $A_{\phi }$ is a graph of a total action $G \curvearrowright X$ if and only if, for each $x \in X$ and $g \in G$ , there exists some $y \in X$ such that $(g,x,y) \in A_{\phi }$ ; in this case, the orbit equivalence relation generated by the action coincides with E.

Example 2.1. An easy way of getting a partial action is by restricting a total one. Suppose we have a free Borel action $G \curvearrowright X$ with the corresponding orbit equivalence relation $E_{G}$ and suppose that a Borel equivalence sub-relation $E \subseteq E_{G}$ admits a Borel selector—a Borel E-invariant map $\pi : X \to X$ such that $x E \pi (x)$ for all $x \in X$ . If $\phi : X \to G$ is the map specified uniquely by the condition $\phi (x) \pi (x) = x$ , then $(E, \phi )$ is a partial G-action.

Sub-relations E as in Example 2.1 are often associated with cross-sections of actions of locally compact second countable (lcsc) groups.

2.2 Tessellations of lcsc group actions

Consider a free Borel action $G \curvearrowright X$ of an lcsc group. A cross-section of the action is a Borel set $\mathcal {C} \subseteq X$ that intersects every orbit in a countable non-empty set. A cross-section $\mathcal {C} \subseteq X$ is:

• • discrete if $(Kx) \cap \mathcal {C}$ is finite for every $x \in X$ and compact $K \subseteq G$ ;

• • U-lacunary, where $U \subseteq G$ is a neighborhood of the identity, if $Uc \cap \mathcal {C} = \{c\}$ for all $c \in \mathcal {C}$ ;

• • lacunary if it is U-lacunary for some neighborhood of the identity U; and

• • cocompact if $K\mathcal {C} = X$ for some compact $K \subseteq G$ .

Let $\mathcal {C}$ be a lacunary cross-section for $G \curvearrowright X$ , which exists by [13, Corollary 1.2]. Any lcsc group G admits a compatible left-invariant proper metric [28], and any left-invariant metric d can be transferred to orbits owing to freeness of the action via $\mathrm {dist}(x,y) = d(g,e)$ for the unique $g \in G$ such that $gx = y$ . One can now define the so-called Voronoi tessellation of orbits by associating with each $x \in X$ the closest point $\pi _{\mathcal {C}}(x) \in \mathcal {C}$ of the cross-section $\mathcal {C}$ , as determined by $\mathrm {dist}$ . Properness of the metric ensures that, for a ball $B_{R} \subseteq G$ of radius R, $B_{R} = \{g \in G : d(g,e) \le R\}$ , and, for any $x \in X$ , the set $\mathcal {C} \cap B_{R}x$ is finite. Indeed, there can be at most $\unicode{x3bb} (B_{R +r})/ \unicode{x3bb} (B_{r})$ points in the intersection, where $\unicode{x3bb} $ is a Haar measure on the group and $r> 0$ is so small that $B_{r}c \cap B_{r} c' = \varnothing $ whenever $c, c' \in \mathcal {C}$ are distinct.

Some care needs to be taken to address the possibility of having several closest points. For example, one may pick a Borel linear order on $\mathcal {C}$ and associated each x with the smallest closest point in the cross-section (see [23, §4] or [17, §B.2] for the specifics). In this way, we get a Borel equivalence relation $E_{\mathcal {C}} \subseteq E_{G}$ whose equivalence classes are the cells of the Voronoi tessellation: $x E_{\mathcal {C}} y$ if and only if $\pi _{\mathcal {C}}(x) = \pi _{\mathcal {C}}(y)$ .

Assumed freeness of the action $G \curvearrowright X$ allows a natural identification of each Voronoi cell with a subset of the acting group via the map $\pi _{\mathcal {C}}^{-1}(c) \ni x \mapsto \phi _{\mathcal {C}}(x) \in G$ such that $\phi _{\mathcal {C}}(x)c = x$ , which is exactly what the corresponding partial action from Example 2.1 does.

Our intention is to use partial actions to define total actions, and the example above may seem like going ‘in the wrong direction’. The point, however, is that once we have a partial action $\phi : X \to G$ , we can compose it with an arbitrary Borel injection $f : G \to G$ to get a different partial action $f\circ \phi $ . This pattern is typical in the sense that new partial actions are often constructed by modifying those obtained as restrictions of total actions.

2.3 Convergent sequences of partial actions

A total action can be defined whenever we have a sequence of partial actions that cohere in the appropriate sense. Let G be a standard Borel group. A sequence $(E_{n}, \phi _{n})$ , $n \in \mathbb {N}$ , of partial G-actions on X is said to be convergent if it satisfies the following properties.

• • Monotonicity: equivalence relations $E_{n}$ form an increasing sequence, that is, $E_{n} \subseteq E_{n+1}$ for all n.

• • Coherence: for each n, the map $x \mapsto (\phi _{n}(x))^{-1}\phi _{n+1}(x)$ is $E_{n}$ -invariant.

• • Exhaustiveness: for all $x \in X$ and all $g \in G$ , there exist n and $y \in X$ such that $xE_{n}y$ and $g\phi _{n}(x) = \phi _{n}(y)$ .

With such a sequence, one can associate a free Borel (left) action $G \curvearrowright X$ , called the limit of $(E_{n}, \phi _{n})_{n}$ , whose graph is $\bigcup _{n}A_{\phi _{n}}$ . Coherence ensures that the partial action defined by $\phi _{n+1}$ is an extension of the one given by $\phi _{n}$ . Indeed, if $xE_{n}y$ are such that $g\phi _{n}(x) = \phi _{n}(y)$ , then also $x E_{n+1} y$ by monotonicity, and, using coherence,

$$\begin{align*}g \phi_{n+1}(x) = g \phi_{n}(x)(\phi_{n}(x))^{-1}\phi_{n+1}(x) = \phi_{n}(y) (\phi_{n}(y))^{-1}\phi_{n+1}(y) = \phi_{n+1}(y), \end{align*}$$

from which we get $A_{\phi _{n}} \subseteq A_{\phi _{n+1}}$ . If C is an $E_{n}$ -class and $s = (\phi _{n}(x))^{-1}\phi _{n+1}(x)$ for some $x \in C$ , then $\phi _{n+1}(C) = \phi _{n}(C)s$ , so the image $\phi _{n}(C)$ gets shifted on the right inside $\phi _{n+1}(C)$ . If we want to build a right action of the group, then $\phi _{n}(C)$ should be shifted on the left instead.

Finally, exhaustiveness guarantees that $gx$ gets defined eventually: for all $g \in G$ and $x \in X$ , there are n and $y \in X$ such that $(g,x,y) \in A_{\phi _{n}}$ . It is straightforward to check that $\bigcup _{n}A_{\phi _{n}}$ is a graph of a total Borel action $G \curvearrowright X$ . It is equally easy to check that the action is free, and that its orbits are precisely the equivalence classes of $\bigcup _{n}E_{n}$ .

This framework, general as it is, delegates most of the complexity to the construction of maps $\phi _{n}$ . We illustrate these concepts on essentially hyperfinite actions of lcsc groups.

2.4 Hyperfinite tessellations of lcsc group actions

In the context of §2.2, suppose that, furthermore, the restriction of the orbit equivalence relation $E_{G}$ onto the cross-section $\mathcal {C}$ is hyperfinite, that is, there is an increasing sequence of finite Borel equivalence relations $F_{n}$ on $\mathcal {C}$ such that $\bigcup _{n}F_{n} = E_{G}|_{\mathcal {C}}$ . We can use this sequence to define $xE_{n}y$ whenever $\pi _{\mathcal {C}}(x)F_{n}\pi _{\mathcal {C}}(y)$ , which yields an increasing sequence of Borel equivalence relations $E_{n}$ such that $E_{G} = \bigcup _{n}E_{n}$ .

The equivalence relations $F_{n}$ admit Borel transversals, that is, there are Borel sets $\mathcal {C}_{n}$ that pick exactly one point from each $F_{n}$ -class. As in §2.2, we may define $\phi _{n}(x) $ to be an element $g \in G$ such that $gc = x$ for the unique $c \in \mathcal {C}_{n}$ satisfying $x E_{n}c$ . This gives a convergent sequence of partial G-actions $(E_{n}, \phi _{n})_{n}$ whose limit is the original action $G \curvearrowright X$ .

2.5 Partial actions revisited

In practice, it is often more convenient to allow equivalence relations $E_{n}$ to be defined on proper subsets of X. Let $X_{n} \subseteq X$ , $n \in \mathbb {N}$ , be Borel subsets and suppose that, for each n, $E_{n}$ is a Borel equivalence relation on $X_{n}$ . We say that the sequence $(E_{n})_{n}$ is monotone if the following conditions are satisfied for all $m \le n$ .

• • $E_{m}|_{X_{m} \cap X_{n}} \subseteq E_{n}|_{X_{m} \cap X_{n}}$ .

• • If $x \in X_{m}\cap X_{n}$ , then the whole $E_{m}$ -class of x is in $X_{n}$ .

Partial action maps $\phi _{n} : X_{n} \to G$ , where, as earlier, G is a standard Borel group, need to satisfy the appropriate versions of coherence and exhaustiveness.

• • Coherence: $X_{m} \cap X_{n} \ni x \mapsto (\phi _{m}(x))^{-1}\phi _{n}(x)$ is $E_{m}$ -invariant for each $m < n$ .

• • Exhaustiveness: for each $x \in X$ and $g \in G$ , there exist n and $y \in X_{n}$ such that $x \in X_{n}$ , $x E_{n}y$ and $g\phi _{n}(x) = \phi _{n}(y) $ .

A sequence of partial G-actions $(X_{n},E_{n},\phi _{n})_{n}$ will be called convergent if it satisfies the above properties of monotonicity, coherence and exhaustiveness. Note that the condition $\bigcup _{n}X_{n} = X$ follows from exhaustiveness, so sets $X_{n}$ must cover all of X.

Convergent sequences $(X_{n},E_{n},\phi _{n})_{n}$ define total actions, which can be easily seen by reducing this set-up to the notationally simpler one given in §2.3. To this end, extend $E_{n}$ to the equivalence relation $\hat {E}_{n}$ on all of X by

$$ \begin{align*}x\hat{E}_{n}y \iff \text{ there exists } m \le n\ xE_{m}y \textrm { or } x = y,\end{align*} $$

and also extend $\phi _{n}$ to $\hat {\phi }_{n} : X \to G$ by setting $\hat {\phi }_{n}(x) = \phi _{m}(x)$ for the maximal $m \le n$ such that $x \in X_{m}$ or $\hat {\phi }_{n}(x) = e$ if no such m exists. It is straightforward to check that $(\hat {E}_{n}, \hat {\phi }_{n})_{n}$ is a convergent sequence of partial G-actions in the sense of §2.3. By the limit of the sequence of partial actions $(X_{n}, E_{n}, \phi _{n})_{n}$ we mean the limit of $(\hat {E}_{n}, \hat {\phi }_{n})_{n}$ as defined earlier.

Remark 2.2. A variant of this generalized formulation, which we encounter in Proposition 2.4 below, occurs when sets $X_{n}$ are nested: that is, $X_{0} \subseteq X_{1} \subseteq X_{2} \subseteq \cdots $ . Monotonicity of equivalence relations then simplifies to $E_{0} \subseteq E_{1} \subseteq E_{2} \subseteq \cdots $ and coherence becomes equivalent to the $E_{n}$ -invariant of maps $ X_{n} \ni x \mapsto (\phi _{n}(x))^{-1}\phi _{n+1}(x) \in G$ .

As was mentioned above, it is easy to create new partial actions simply by composing a partial action $\phi : X \to G$ with some Borel bijection $f : G \to G$ (or $f : G \to H$ , if we choose to have values in a different group). However, an arbitrary bijection has no reason to preserve coherence and extra care is necessary to maintain it.

Furthermore, in general, we need to apply different modifications f to different $E_{n}$ -classes, which naturally raises concern about how to ensure that construction is performed in a Borel way. In applications, the modification f applied to an $E_{n}$ -class C usually depends on the ‘shape’ of C and the $E_{m}$ -classes it contains, but it does not depend on other $E_{n}$ -classes. If there are only countably many such ‘configurations’ of $E_{n}$ -classes, the resulting partial actions $f \circ \phi $ will be Borel as long as we consistently apply the same modification whenever ‘configurations’ are the same. This idea can be formalized as follows.

2.6 Rational sequences of partial actions

Let $(E_{n}, \phi _{n})_{n}$ be a convergent sequence of partial actions on X. For an $E_{n}$ -class C, let $\mathcal {E}_{m}(C)$ denote the collection of $E_{m}$ -classes contained in C. Given two $E_{n}$ -classes C and $C'$ , we denote by $\phi _{n}(C) \equiv \phi _{n}(C')$ the existence for each $m \le n$ of a bijection $\mathcal {E}_{m}(C) \ni D \mapsto D' \in \mathcal {E}_{m}(C') $ such that $\phi _{n}(D) = \phi _{n}(D')$ for all $D \in \mathcal {E}_{m}(C)$ . The collection of images $\{\phi _{n}(D) : D \in \bigcup _{m\le n} \mathcal {E}_{m}(C) \}$ constitutes the ‘configuration’ of C referred to earlier.

We say that the sequence $(E_{n}, \phi _{n})_{n}$ of partial actions is rational if, for each n, there exists a Borel $E_{n}$ -invariant partition $X = \bigsqcup _{k} Y_{k}$ such that, for each k, one has $\phi _{n}(C) \equiv \phi _{n}(C')$ for all $E_{n}$ -classes $C,C' \subseteq Y_{k}$ .

Remark 2.3. This concept of rationality applies verbatim to convergent sequences of partial actions $(X_{n},E_{n}, \phi _{n})_{n}$ , as described in §2.5. One can check that such a sequence is rational if and only if the sequence $(\hat {E}_{n},\hat {\phi }_{n})$ is rational.

2.7 Generating the flow equivalence relation

As an application of the partial actions formalism, we show that any orbit equivalence relation given by a free Borel $\mathbb {R}$ -flow can also be generated by a free action of any non-discrete and non-compact Polish group. For this, we need the following representation of an $\mathbb {R}$ -flow as a limit of partial $\mathbb {R}$ -actions.

Proposition 2.4. Any free Borel $\mathbb {R}$ -flow on X can be represented as a limit of a convergent rational sequence of partial $\mathbb {R}$ -actions $(X_{n},E_{n}, \phi _{n})_{n}$ such that:

1. (1) both $X_{n}$ and $E_{n}$ are increasing: $X_{0} \subseteq X_{1} \subseteq \cdots $ and $E_{0} \subseteq E_{1} \subseteq \cdots $ (see Remark 2.2);

2. (2) each $E_{n+1}$ -class contains finitely many $E_{n}$ -classes;

3. (3) each $E_{0}$ -class has cardinality of continuum; and

4. (4) for each $E_{n+1}$ -class C, the set $C \setminus X_{n}$ has cardinality of continuum.

Proof. Any $\mathbb {R}$ -flow admits a rational $(-4,4)$ -lacunary cross-section (see [24, §2]), which we denote by $\mathcal {C}$ . Rationality of the cross-section here means that the distance between any two points of $\mathcal {C}$ is a rational number. More generally, rationality of a cross-section $\mathcal {C}$ for an $\mathbb {R}^{d}$ -action means that $r \in \mathbb {Q}^{d}$ whenever $c + r = c'$ for some $c, c' \in \mathcal {C}$ . Let $(E_{\mathcal {C}}, \phi _{\mathcal {C}})$ be the partial $\mathbb {R}$ -action as defined in §2.2. If D is an $E_{\mathcal {C}}$ -class, then $\phi _{\mathcal {C}}(D)$ is an interval. For $\epsilon> 0$ , let $D^{\epsilon }$ consist of those $ x \in D$ such that $\phi _{\mathcal {C}}(x)$ is at least $\epsilon $ away from the boundary points of $\phi _{\mathcal {C}}(D)$ . In other words, $D^{\epsilon }$ is obtained by shrinking the class D by $\epsilon $ from each side.

The restriction of the orbit equivalence relation onto $\mathcal {C}$ is hyperfinite. This fact is true in the much wider generality of actions of locally compact Abelian groups [4]. Specifically for $\mathbb {R}$ -flows, $E|_{\mathcal {C}}$ is generated by the first return map—a Borel automorphism of $\mathcal {C}$ that sends a point in $\mathcal {C}$ to the next one according to the order of the $\mathbb {R}$ -flow. The first return map is well defined and is invertible, except for the orbits, where $\mathcal {C}$ happens to have the maximal or the minimal point. The latter part of the space evidently admits a Borel selector and is, therefore, smooth; hence, it won’t affect hyperfiniteness of the equivalence relation. It remains to recall the standard fact that orbit equivalence relations of $\mathbb {Z}$ -actions are hyperfinite (see, for example, [6, Theorem 5.1]), and thus so is the restriction $E|_{\mathcal {C}}$ .

In particular, we can represent the $\mathbb {R}$ -flow as the limit of a convergent sequence of partial actions $(E^{\prime }_{n}, \phi ^{\prime }_{n})_{n}$ , as described in §2.4. Note that $(E^{\prime }_{n}, \phi ^{\prime }_{n})_{n}$ is necessarily rational by rationality of $\mathcal {C}$ . Such a sequence satisfies items (2) and (3), but fails (4). We fix this by shrinking equivalence classes to achieve proper containment. Let $(\epsilon _{n})_{n}$ be a strictly decreasing sequence of positive reals such that $ 1> \epsilon _{0}$ and $\lim _{n}\epsilon _{n} = 0$ . Put $X^{\prime }_{n} = \bigcup D^{\epsilon _{n}}$ , where the union is taken over all $E_{\mathcal {C}}$ -classes D. Note that sets $X^{\prime }_{n}$ fail to cover X, because the boundary points of any $E_{\mathcal {C}}$ -class do not belong to any of $X^{\prime }_{n}$ . Put $Y = X \setminus \bigcup _{n} X^{\prime }_{n}$ and let $X_{n} = X^{\prime }_{n} \cup Y$ . Clearly, $(X_{n})_{n}$ is an increasing sequence of Borel sets and $\bigcup _{n} X_{n} = X$ .

Finally, set $E_{n} = E^{\prime }_{n}|_{X_{n}}$ and $\phi _{n} : X_{n} \to \mathbb {R}$ to be $\phi ^{\prime }_{n}|_{X_{n}}$ . The sequence $(X_{n},E_{n},\phi _{n})_{n}$ of partial $\mathbb {R}$ -actions satisfies the conditions of the proposition.

All non-smooth orbit equivalence relations produced by free Borel $\mathbb {R}$ -flows are Borel isomorphic to each other [14, Theorem 3]. Theorem 2.6 shows that this orbit equivalence relation can also be generated by a free action of any non-compact and non-discrete Polish group.

Let G be a group. We say that a set $A \subseteq G$ admits infinitely many disjoint right translates if there is a sequence $(g_{n})_{n}$ of elements of G such that $Ag_{m} \cap Ag_{n} = \emptyset $ for all $m \ne n$ .

Lemma 2.5. Let G be a non-compact Polish group. There exists a neighborhood of the identity $V \subseteq G$ such that, for any finite $F \subseteq G$ , the set $VF$ admits infinitely many disjoint right translates.

Proof. We begin with the following characterization of compactness established independently by Solecki [27, Lemma 1.2] and Uspenskij [29]: a Polish group G is non-compact if and only if there exists a neighborhood of the identity $U \subseteq G$ such that $F_{1}UF_{2} \ne G$ for all finite $F_{1}, F_{2} \subseteq G$ . Let $V \subseteq G$ be a symmetric neighborhood of the identity such that $V^{2} \subseteq U$ . We claim that such a set V has the desired property. Pick a finite $F \subseteq G$ , set $g_{0} = e$ and choose $g_{n}$ inductively as follows. Let $F_{1} = F^{-1}$ and $F_{2,n} = F\cdot \{g_{k} : k < n\}$ . The defining property of U ensures existence of $g_{n} \not \in F_{1}UF_{2,n}$ . Translates $(VFg_{n})_{n}$ are then pairwise disjoint, because if $VFg_{m} \cap VFg_{n} \ne \emptyset $ for $m < n$ , then $g_{n} \in F^{-1}V^{-1}VFg_{m} \subseteq F_{1}UF_{2,n}$ , which contradicts the construction.

Theorem 2.6. Let E be an orbit equivalence relation given by a free Borel $\mathbb {R}$ -flow on X. Any non-discrete, non-compact Polish group G admits a free Borel action $G \curvearrowright X$ such that $E_{G} = E$ .

Proof. Let $(X_{n}, E_{n}, \phi _{n})_{n}$ be a convergent sequence of partial $\mathbb {R}$ -actions, as in Proposition 2.4, and let $V \subseteq G$ be given by Lemma 2.5. Choose a countable dense $(h_{n})_{n}$ in G so that $\bigcup _{n}Vh_{n} = G$ . Since the sequence of partial $\mathbb {R}$ -actions is rational, one may pick, for each n, a Borel $E_{n}$ -invariant partition $X_{n} = \bigsqcup _{k}Y_{n,k}$ such that $\phi _{n}(C) \equiv \phi _{n}(C')$ for all $E_{n}$ -classes $C,C' \subseteq Y_{n,k}$ . We construct a convergent sequence of partial G-actions $(X_{n}, E_{n}, \psi _{n})_{n}$ such that, for each n and k, there exists a finite set $F \subseteq G$ such that $\{h_{i} : i < n\} \subseteq F$ and $\psi _{n}(C) = VF$ for all $E_{n}$ -classes $C \subseteq Y_{n,k}$ .

For any $E_{0}$ -class C, both $\phi _{0}(C) \subseteq \mathbb {R}$ and $V \subseteq G$ are Borel sets of the same cardinality. Therefore, we may pick a Borel bijection $f_{k} : \phi _{0}(C) \to V$ , where $C \subseteq Y_{0,k}$ . For the base of the inductive construction, we set $\psi _{0}|_{Y_{k}} = f_{k}\circ \phi _{0}$ . Suppose that $\psi _{m} : X_{m} \to G$ , $m \le n$ , have been constructed.

We now construct $\psi _{n+1}$ . Let C be an $E_{n+1}$ -class and let $D_{1}, \ldots , D_{l}$ be a complete list of $E_{n}$ -classes contained in C. By the inductive assumption, there are finite sets $F_{1}, \ldots , F_{l} \subseteq G$ such that $\psi _{n}(D_{i}) = VF_{i}$ . Let $\tilde {F} = \bigcup _{i \le l}F_{i}$ . By the choice of V, there are elements $g_{1}, \ldots , g_{l} \in G$ such that $V\tilde {F}g_{i}$ are pairwise disjoint for $1 \le i \le l$ . Pick a finite $F \subseteq G$ large enough so that $\tilde {F}g_{i} \subseteq F$ , $\{h_{i} : i < n+1\} \subseteq F$ and $VF \setminus \bigcup _{i\le l}V\tilde {F}g_{i}$ has cardinality of continuum (the latter can be achieved, for example, by ensuring that one more disjoint translate of $V\tilde {F}$ is inside $VF$ ). Note that $\phi _{n+1}(C \setminus X_{n}) = \phi _{n+1}(C) \setminus \bigcup _{i\le l}\phi _{n+1}(D_{i})$ has cardinality of continuum by the properties guaranteed by Proposition 2.4. Pick any Borel bijection

$$ \begin{align*}f : \phi_{n+1}(C) \setminus \bigcup_{i\le l}\phi_{n+1}(D_{i}) \to VF \setminus \bigcup_{i} \psi_{n}(D_{i})g_{i}\end{align*} $$

and define $\psi _{n+1}$ by the conditions $\psi _{n+1}|_{D_{i}} = \psi _{n}|_{D_{i}} \cdot g_{i}$ and $\psi _{n+1}|_{C \setminus \bigcup _{i\le l}D_{i}} = f \circ \phi _{n+1}$ . As in the base case, the same modification f works for all $E_{n+1}$ -classes $C, C'$ such that $\phi _{n+1}(C) \equiv \phi _{n+1}(C')$ , which ensures Borelness of the construction.

It is now easy to check that $(X_{n}, E_{n},\psi _{n})_{n}$ is a convergent sequence of partial G-actions, and hence its limit is a free Borel action $G \curvearrowright X$ such that $E_{G} = E$ .

Remark 2.7. Theorem 2.6 highlights the difference with actions of discrete groups, since a free Borel $\mathbb {Z}$ -action that preserves a finite measure cannot be generated by a free Borel action of a non-amenable group (see, for example, [32, Proposition 4.3.3] or [10, Proposition 2.5(ii)]).

However, if we consider hyperfinite equivalence relations without any finite invariant measures, then we do have the analog for $\mathbb {Z}$ -actions. There exists a unique up to isomorphism, non-smooth, hyperfinite Borel equivalence relation without any finite invariant measures and it can be realized as an orbit equivalence relation of a free action of any infinite countable group [6, Proposition 11.2].

3 Lipschitz maps

Our goal in this section is to prove Theorem 3.12, which shows that any free Borel $\mathbb {R}^{d}$ -flow is bi-Lipschitz orbit equivalent to a flow with an integer grid. Sections 3.1–3.3 build the necessary tools to construct such an orbit equivalence. Verification of the Lipschitz conditions stated in the lemmas within these sections is straightforward and routine. Therefore, we omit the arguments for brevity.

Recall that a map $f : X \to Y$ between metric spaces $(X,d_{Y})$ and $(Y,d_{Y})$ is K-Lipschitz if $d_{Y}(f(x_{1}), f(x_{2})) \le K d_{X}(x_{1}, x_{2})$ for all $x_{1}, x_{2} \in X$ , and it is $(K_{1},K_{2})$ -bi- Lipschitz if f is injective, $K_{2}$ -Lipschitz and $f^{-1}$ is $K^{-1}_{1}$ -Lipschitz, which can equivalently be stated as

$$ \begin{align*}K_{1}d_{X}(x_{1},x_{2}) \le d_{Y}(f(x_{1}), f(x_{2})) \le K_{2} d_{X}(x_{1}, x_{2}) \quad \textrm{ for all } x_{1},x_{2} \in X.\end{align*} $$

The Lipschitz constant of a Lipschitz map f is the smallest K with respect to which f is K-Lipschitz.

3.1 Linked sets

Given two Lipschitz maps $ f : A \to A' $ and $ g : B \to B' $ that agree on the intersection $ A \cap B $ , the map $ f \cup g : A \cup B \to A' \cup B' $ , in general, may not be Lipschitz. The following condition is sufficient to ensure that $ f \cup g $ is Lipschitz with the Lipschitz constant bounded by the maximum of the constants of f and g.

Definition 3.1. Let $ (X, d) $ be a metric space and let $ A, B \subseteq X $ be its subsets. We say that $ A $ and $ B $ are linked if, for all $ x \in A $ and $ y \in B $ , there exists $ z \in A \cap B $ such that $ d(x,y) = d(x,z) + d(z, y) $ .

Lemma 3.2. Let $ (X, d) $ be a metric space, let $ f : A \to A' $ , $ g : B \to B' $ be $ K $ -Lipschitz maps between subsets of X and suppose that $ f |_{A \cap B} = g |_{A \cap B} $ . If $ A $ and $ B $ are linked, then $ f \cup g : A \cup B \to A' \cup B' $ is $ K $ -Lipschitz.

Recall that a metric space $(X,d)$ is geodesic if, for all points $x, y \in X$ , there exists a geodesic between them—an isometry $\tau : [0, d(x,y)] \to X$ such that $\tau (0) = x$ and $\tau (d(x,y)) = y$ . For geodesic metric spaces, closed sets $ A, B \subseteq X $ are always linked whenever the boundary of one of them is contained in the other. The boundary of a set A will be denoted by $\partial A$ and $\operatorname *{{\mathrm {int}}} A$ will stand for the interior of A.

Lemma 3.3. Suppose that $ (X,d) $ is a geodesic metric space. If $ A, B \subseteq X $ are closed and satisfy $ \partial A \subseteq B $ , then $ A $ and $ B $ are linked.

3.2 Inductive step

The following lemma encompasses the inductive step in the construction of the forthcoming Theorem 3.12.

Lemma 3.4. Let $(X,d)$ be a geodesic metric space and let $A \subseteq X$ be a closed set. Suppose that $(A_{i})_{i=1}^{n}$ are pairwise disjoint closed subsets of A and $h_{i} : A_{i} \to A_{i}$ are $(K_{1},K_{2})$ -bi-Lipschitz maps such that $h_{i}|_{\partial A_{i}}$ is the identity map for each $1 \le i \le n$ . The map $g : A \to A$ given by

$$\begin{align*}g(x) = \begin{cases} h_{i}(x) & \textrm{ if}\ x \in A_{i},\\ x & \textrm{otherwise} \end{cases} \end{align*}$$

is $(K_{1},K_{2})$ -bi-Lipschitz.

3.3 Lipschitz shifts

Let $ (X, \lVert \cdot \rVert ) $ be a normed space and let $A \subseteq X$ be a closed bounded subset. We begin with the following elementary and well-known observation regarding Lipschitz perturbations of the identity map.

Lemma 3.5. If $\xi : A \to X$ is a K-Lipschitz map, $ K < 1$ , then $ A \ni x \mapsto x + \xi (x) \in X$ is $(1-K, 1+K)$ -bi-Lipschitz.

For the rest of §3.3, we fix a vector $ v \in X $ and a real $ K> \|v\| $ . Let the function $ f_{A, K, v} : A \to X $ be given by

$$ \begin{align*} f_{A, K, v}(x) = x + \frac{d(x, \partial A)}{K} v, \end{align*} $$

where $d(x, \partial A)$ denotes the distance from x to the boundary of A. This function (as well as its variant, which will be introduced shortly) is $(1-K^{-1}\|v\|, 1+K^{-1}\|v\|)$ -bi-Lipschitz. To simplify the notation, we set $\alpha ^{+} = 1 + K^{-1}\|v\|$ and $\alpha ^{-} = 1 - K^{-1}\|v\|$ .

Lemma 3.6. The function $f_{A,K,v}$ is an $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz homeomorphism onto A.

Fix a real $L> 0$ and let $ A^{L} = \{x \in A : d(x, \partial A) \ge L \} $ be the set of those elements that are at least $ L $ units of distance away from the boundary of A.

Lemma 3.7. $ f_{A,K,v}|_{A^{L}} = f_{A^{L},K,v} + LK^{-1} v $ and $ f_{A,K,v}(A^{L}) = A^{L} + LK^{-1}v $ .

A truncated shift function $h_{A,K,v,L} : A \to X$ is defined by

$$\begin{align*}h_{A,K,v,L}(x) = \begin{cases} f_{A,K,v}(x) & \text{for}\ x \in A \setminus A^{L}, \\ x + LK^{-1} v & \text{for}\ x \in A^{L}. \end{cases} \end{align*}$$

Lemma 3.8. The function $h_{A,K,v,L}$ is an $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz homeomorphism onto A.

3.4 Lipschitz equivalence to grid flows

The maps $h_{A,K,v,L}$ can be used to show that any free Borel $\mathbb {R}^{d}$ -flow is bi-Lipschitz equivalent to a flow admitting an integer grid. This is the content of Theorem 3.12, but first we formulate the properties of partial actions needed for the construction. This is an adaption of the so-called unlayered toast construction announced in [8]. The proof given in [18, Appendix A] for $\mathbb {Z}^{d}$ -actions transfers to $\mathbb {R}^{d}$ -flows.

For the rest of the paper, we fix a norm $\| \cdot \|$ on $\mathbb {R}^{d}$ and let $d(x,y) = \|x-y\|$ be the corresponding metric on $\mathbb {R}^{d}$ . Recall that $B_{R}(r) \subseteq \mathbb {R}^{d}$ denotes a closed ball of radius R centered at $r \in \mathbb {R}^{d}$ .

Lemma 3.9. Let $K> 0$ be a positive real. Any free $\mathbb {R}^{d}$ -flow on a standard Borel space X is a limit of a rational convergent sequence of partial actions $(X_{n}, E_{n}, \phi _{n})_{n}$ (see §2.5) such that, for each $E_{n}$ -class C:

1. (1) $\phi _{n}(C)$ is a closed and bounded subset of $\mathbb {R}^{d}$ and $B_{K}(0) \subseteq \phi _{n}(C)$ ;

2. (2) the set of $E_{m}$ -classes, $m \le n$ , contained in C is finite; and

3. (3) $d(\phi _{n}(D), \partial \phi _{n}(C)) \ge K$ for any $E_{m}$ -class $D $ such that $D \subseteq C$ .

Before outlining the proof, we need to introduce some notation. Let $E_{1}, \ldots , E_{n}$ be equivalence relations on $X_{1}, \ldots , X_{n}$ , respectively. By $E_{1} \vee \cdots \vee E_{n}$ we mean the equivalence relation E on $\bigcup _{i \le n} X_{i}$ generated by $E_{i}$ , that is, $x E y$ whenever there exist $x_{1}, \ldots , x_{m}$ and, for each $ 1\le i \le m$ , there exists $ 1 \le j(i) \le n$ such that $x_{1} = x$ , $x_{m} = y$ and $x_{i} E_{j(i)} x_{i+1}$ for all $1 \le i < m$ .

If E is an equivalence relation on $Y \subseteq X$ and $K> 0$ , we define the relation $E^{+K}$ on $Y^{+K} = \{x \in X : \mathrm {dist}(x, y) \le K \textrm { for some } y \in Y\}$ by $x_{1} E^{+K} x_{2}$ if and only if there are $y_{1}, y_{2} \in Y$ such that $\mathrm {dist}(x_{1},y_{1}) \le K $ , $\mathrm {dist}(x_{2},y_{2}) \le K$ and $y_{1} E y_{2}$ . Note that, in general, $E^{+K}$ may not be an equivalence relation if two E-classes get connected after the ‘fattening’. However, $E^{+K}$ is an equivalence relation if $\mathrm {dist}(C_{1},C_{2})> 2K$ holds for all distinct E-classes $C_{1}, C_{2}$ .

Proof of Lemma 3.9

One starts with a sufficiently fast-growing sequence of radii $a_{n}$ (for example, $a_{n} = K1000^{n+1}$ is fast enough) and by choosing using [3] (see also [18, Lemma A.2]) a sequence of Borel $B_{a_{n}}(0)$ -lacunary cross-sections $\mathcal {C}_{n} \subseteq X$ such that

(1) $$ \begin{align} \text{ for all } x \in X \text{ for all } \epsilon> 0 \text{ there exists }^{\infty} n \textrm{ such that } \mathrm{dist}(x,\mathcal{C}_{n}) < \epsilon a_{n}, \end{align} $$

where $\mathrm {dist}(x, \mathcal {C}_{n}) = \inf \{ \mathrm {dist}(x,c) : c \in \mathcal {C}_{n}\}$ and $\text { there exists }^{\infty }$ stands for ‘there exist infinitely many’. We may assume, without loss of generality, that cross-sections $\mathcal {C}_{n}$ are rational in the sense that if $c_{1} + r = c_{2}$ for some $c_{1}, c_{2} \in \bigcup _{n}\mathcal {C}_{n}$ , then $r \in \mathbb {Q}^{d}$ . This can be achieved by moving elements of $\mathcal {C}_{n}$ by an arbitrarily small amount (see [24, Lemma 2.4]), which maintains the property given in equation (1). Rationality of cross-sections guarantees that the sequence of partial actions constructed below is rational.

One now defines $X_{n}$ and $E_{n}$ inductively with the base $X_{0} = \mathcal {C}_{0} + B_{a_{0}/10}(0)$ and with $x E_{0} y$ if and only if there is $c \in \mathcal {C}_{0}$ such that $x, y \in c + B_{a_{0}/10}(0)$ . For the inductive step, begin with $\tilde {X}_{n} = \mathcal {C}_{n} + B_{a_{n}/10}(0) $ and $\tilde {E}_{n}$ being given analogously to the base case: $x \tilde {E}_{n} y$ if and only if there is some $c \in \mathcal {C}_{n}$ such that $\mathrm {dist}(x,c) \le a_{n}/10$ and $\mathrm {dist}(y,c) \le a_{n}/10$ . Set $E^{\prime }_{n} = \tilde {E}_{n} \vee E^{+K}_{n-1} \vee \cdots \vee E_{0}^{+K}$ and let $X^{\prime }_{n} = \tilde {X}_{n} \cup \bigcup _{i=0}^{n-1}X_{i}^{+K}$ be the domain of $E^{\prime }_{n}$ . Finally, let $X_{n}$ be the $E^{\prime }_{n}$ -saturation of $\tilde {X}_{n}$ , that is, $x \in X_{n}$ if and only if there exists $y \in \tilde {X}_{n}$ such that $x E^{\prime }_{n} y$ . Put $E_{n} = E^{\prime }_{n}|_{X_{n}}$ .

An alternative description of an $E_{n}$ -class is as follows. We start with an $\tilde {E}_{n}$ -class $C_{n}$ and join it first with all $E^{+K}_{n-1}$ -classes D that intersect $C_{n}$ . Let the resulting $\tilde {E}_{n}\vee E^{+K}_{n-1}$ -class be denoted by $C_{n-1}$ . Next, we add all $E^{+K}_{n-2}$ -classes that intersect $C_{n-2}$ to produce an $\tilde {E}_{n}\vee E^{+K}_{n-1} \vee E^{+K}_{n-2}$ -class $C_{n-2}$ . The process terminates with an $E_{n}$ -class $C_{0}$ .

It is easy to check inductively that the diameter of any $E_{n}$ -class C satisfies $\mathrm {diam}(C) \le a_{n}/3$ and, therefore, $\mathrm {dist}(C_{1}, C_{2}) \ge a_{n}/3 \gg 2K$ for all distinct $E_{n}$ -classes $C_{1},C_{2}$ by the lacunarity of $\mathcal {C}_{n}$ . The latter shows that $E^{+K}_{n}$ is an equivalence relation on $X_{n}^{+K}$ .

Monotonicity of the sequence $(X_{n},E_{n})_{n}$ is evident from the construction. Equation (1) is crucial for establishing the fact that $\bigcup _{n} X_{n} = X$ . Indeed, for each $x \in X$ , there exists some n such that $\mathrm {dist}(x, \mathcal {C}_{n}) < a_{n}/10$ and thus also $x \in \tilde {X}_{n} \subseteq X_{n}$ .

The maps $\phi _{n} : X_{n} \to \mathbb {R}^{d}$ , needed to specify partial $\mathbb {R}^{d}$ -actions, are defined by the condition $\phi _{n}(x)c = x$ for the unique $c \in \mathcal {C}_{n}$ such that $c E_{n} x$ . Note that $d(\phi _{n}(D), \partial \phi _{n}(C)) \ge K$ for any $E_{m}$ -class D, $m < n$ , that is contained in an $E_{n}$ -class C is a consequence of the fact that $D^{+K} \subseteq C $ by the construction. Therefore, the convergent sequence of partial actions $(X_{n},E_{n}, \phi _{n})_{n}$ satisfies the desired properties.

Let $\mathfrak {F}_{1}$ and $\mathfrak {F}_{2}$ be free $\mathbb {R}^{d}$ -flows on X that generate the same orbit equivalence relation, $E_{\mathfrak {F}_{1}} = E_{\mathfrak {F}_{2}}$ , and let $\rho = \rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}} : \mathbb {R}^{d} \times X \to \mathbb {R}^{d}$ be the associated cocycle map, defined for $x \in X$ and $r \in \mathbb {R}^{d}$ by the condition $x +_{2} r = x +_{1} \rho (r,x)$ . We say that the cocycle $\rho $ is $(K_{1},K_{2})$ -bi-Lipschitz if such is the map $\rho (\,\cdot \, , x) : \mathbb {R}^{d} \to \mathbb {R}^{d}$ for all $x \in X$ : that is,

(2) $$ \begin{align} K_{1}\|r_{2} -r_{1}\| \le \|\rho(r_{2},x) - \rho(r_{1},x)\| \le K_{2} \|r_{2}-r_{1}\|. \end{align} $$

Since $\rho (r_{2},x) - \rho (r_{1},x) = \rho (r_{2} - r_{1},x +_{1} r_{1})$ , Lipschitz condition (2) for a cocycle can be equivalently and more concisely stated as

(3) $$ \begin{align} K_{1} \le \dfrac{\|\rho(r,x)\|}{\|r\|} \le K_{2} \quad \textrm{for all }x \in X \textrm{ and } r \in \mathbb{R}^{d} \setminus \{0\}. \end{align} $$

Remark 3.10. Note that cocycles $\rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}}$ and $\rho _{\mathfrak {F}_{2}, \mathfrak {F}_{1}}$ are connected via the identities

$$ \begin{align*}\rho_{\mathfrak{F}_{1}, \mathfrak{F}_{2}} (\rho_{\mathfrak{F}_{2}, \mathfrak{F}_{1}}(r,x),x) = r \quad \textrm{and} \quad \rho_{\mathfrak{F}_{2}, \mathfrak{F}_{1}} (\rho_{\mathfrak{F}_{1}, \mathfrak{F}_{2}}(r,x),x) = r.\end{align*} $$

In particular, if $\rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}}$ is $(K_{1},K_{2})$ -bi-Lipschitz, then $\rho _{\mathfrak {F}_{2}, \mathfrak {F}_{1}}$ is $(K_{2}^{-1}, K_{1}^{-1})$ -bi-Lipschitz.

Definition 3.11. Let $\mathfrak {F}$ be a free $\mathbb {R}^{d}$ -flow on X. An integer grid for the flow $\mathfrak {F}$ is a $\mathbb {Z}^{d}$ -invariant Borel subset $Z \subseteq X$ whose intersection with each orbit of the flow is a $\mathbb {Z}^{d}$ -orbit. In other words, $Z + \mathbb {R}^{d} = X$ , $Z + \mathbb {Z}^{d} = Z$ and $z_{1}+ \mathbb {Z}^{d} = z_{2} + \mathbb {Z}^{d}$ for all $z_{1}, z_{2} \in Z$ such that $z_{1}E_{\mathfrak {F}}z_{2}$ .

Not every flow admits an integer grid, but, as the following theorem shows, each flow is bi-Lipschitz equivalent to the one that does.

Theorem 3.12. Let $\mathfrak {F}_{1}$ be a free Borel $\mathbb {R}^{d}$ -flow on X. For any $\alpha> 1$ , there exists a free Borel $\mathbb {R}^{d}$ -flow $\mathfrak {F}_{2}$ on X that admits an integer grid, induces the same orbit equivalence as does $\mathfrak {F}_{1}$ , that is, $E_{\mathfrak {F}_{1}} = E_{\mathfrak {F}_{2}}$ , and whose associated cocycle $\rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}}$ is $(\alpha ^{-1}, \alpha )$ -bi-Lipschitz.

Proof. Let R be so big that the ball $B_{R}(0) \subseteq \mathbb {R}^{d}$ satisfies $\mathbb {Z}^{d} + B_{R}(0) = \mathbb {R}^{d}$ . Choose $K> 0$ large enough to ensure that $\alpha ^{-} = 1 - K^{-1}R> \alpha ^{-1}$ and, therefore, also that $\alpha ^{+} = 1 + K^{-1}R < \alpha $ . Let $(X_{n}, E_{n}, \phi _{n})_{n}$ be a rational convergent sequence of partial actions produced by Lemma 3.9 for the chosen value of K. For an $E_{n}$ -class C, let $C'$ denote the collection of all $x \in C$ that are at least K-distance away from the boundary of C: that is,

$$ \begin{align*}C' = \{x \in C : d(\phi_{n}(x), \partial \phi_{n}(C)) \ge K\}.\end{align*} $$

If D is an $E_{m}$ -class such that $D \subseteq C$ , then item (3) of Lemma 3.9 guarantees the inclusion $D \subseteq C'$ . Let $X^{\prime }_{n} = \bigcup C'$ , where the union is taken over all $E_{n}$ -classes C, and set $E_{n}' = E_{n}|_{X^{\prime }_{n}}$ , $\phi ^{\prime }_{n}= \phi _{n}|_{X^{\prime }_{n}}$ . Note that $(X^{\prime }_{n}, E^{\prime }_{n}, \phi ^{\prime }_{n})_{n}$ is a rational convergent sequence of partial actions whose limit is the flow $\mathfrak {F}_{1}$ . The flow $\mathfrak {F}_{2}$ will be constructed as the limit of partial actions $(X^{\prime }_{n},E^{\prime }_{n},\psi _{n})$ , where maps $\psi _{n}$ will be defined inductively and will satisfy $\psi _{n}(C') = \phi _{n}(C')$ for all $E_{n}$ -classes C. The sets $Z_{n} = \psi _{n}^{-1}(\mathbb {Z}^{d})$ will satisfy $Z_{m}\cap X^{\prime }_{n} \subseteq Z_{n}$ for $m \le n$ , and $Z = \bigcup _{n}Z_{n}$ will be an integer grid for $\mathfrak {F}_{2}$ .

For the base of the construction, set $\psi _{0} = \phi ^{\prime }_{0}$ and $Z_{0} = \psi _{0}^{-1}(\mathbb {Z}^{d})$ . Next, consider a typical $E_{1}$ -class C with $D_{1}, \ldots , D_{l}$ being a complete list of $E_{0}$ -classes contained in it (see Figure 1). Consider the set $\tilde {Z}_{C'} = \phi _{1}^{-1}(\mathbb {Z}^{d}) \cap C'$ , which is the integer grid inside $C'$ (marked by dots in Figure 1). Each of the $D_{i}$ -classes comes with the grid $\tilde {Z}_{D^{\prime }_{i}} = \psi _{0}^{-1}(\mathbb {Z}^{d})\cap D^{\prime }_{i}$ constructed at the previous stage (depicted by crosses in Figure 1). The coherence condition for partial actions guarantees existence of some $s_{i} \in \mathbb {R}^{d}$ , $i \le l$ , such that

$$ \begin{align*}\phi_{1}(D^{\prime}_{i})= \phi_{0}(D^{\prime}_{i}) + s_{i} = \psi_{0}(D^{\prime}_{i}) + s_{i}.\end{align*} $$

Figure 1 Construction of the integer grid.

In general, the grid $\tilde {Z}_{C'}$ does not contain $\tilde {Z}_{D^{\prime }_{i}} $ , but, for each $i \le l$ , we can find a vector $v_{i} \in \mathbb {R}^{d}$ of norm $\|v_{i}\|\le R$ such that $\tilde {Z}_{D^{\prime }_{i}} +_{1} v_{i} \subseteq \tilde {Z}_{C'}$ . More precisely, we take for $v_{i}$ any vector in $B_{R}(0)$ such that $s_{i} + v_{i} \in \mathbb {Z}^{d}$ , which exists by the choice of R. Let $h_{i} : \phi _{1}(D_{i}) \to \phi _{1}(D_{i})$ be the function $h_{\phi _{1}(D_{i}), K, v_{i}, K}$ , which is $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz by Lemma 3.8. Finally, define $g_{1} : \phi _{1}(C') \to \phi _{1}(C')$ to be

$$\begin{align*}g_{1}(r) = \begin{cases} h_{i}(r) & \textrm{if}\ r \in \phi_{1}(D_{i}),\\ r & \textrm{otherwise}. \end{cases} \end{align*}$$

Lemma 3.4 has been tailored specifically to show that $g_{1}$ is $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz. We set $\psi _{1}|_{C'} = g_{1} \circ \phi _{1}|_{C'}$ . Note that

(4) $$ \begin{align}\begin{aligned} \psi_{1}(D^{\prime}_{i}) &= g_{1} \circ \phi_{1}(D^{\prime}_{i}) = h_{i} \circ \phi_{1}(D^{\prime}_{i}) = \phi_{1}(D^{\prime}_{i}) + KK^{-1}v_{i} \\ &= \phi_{0}(D^{\prime}_{i}) + s_{i} + v_{i} = \psi_{0}(D^{\prime}_{i}) + s_{i} + v_{i},\end{aligned} \end{align} $$

which validates coherence and, in view of $s_{i} + v_{i} \in \mathbb {Z}^{d}$ , gives $\psi ^{-1}_{1}(\mathbb {Z}^{d}) \cap D^{\prime }_{i} = \psi _{0}^{-1}(\mathbb {Z}^{d}) \cap D^{\prime }_{i} $ for all $i \le l$ .

Although we have provided the definition of $\psi _{1}$ on a single $E_{1}$ -class C, the same construction can be done in a Borel way across all $E_{1}$ -classes C using rationality of the sequence of partial actions in the same way as we did in Theorem 2.6. If we let $Z_{1} = \psi _{1}^{-1}(\mathbb {Z}^{d})$ , then $Z_{0} \cap X_{1} \subseteq Z_{1}$ by equation (4).

The general inductive step is analogous. Suppose that we have constructed maps $\psi _{k}$ for $k \le n$ . An $E_{n+1}$ -class C contains finitely many subclasses $D_{1}, \ldots , D_{l}$ , where $D_{i}$ is an $E_{m_{i}}$ -class, $m_{i} < n$ and no $D_{i}$ is contained in a bigger $E_{m}$ -class for some $m_{i} < m < n$ . By coherence and the inductive assumption, there exist $s_{i} \in \mathbb {R}^{d}$ , $i \le l$ , such that

$$ \begin{align*}\phi_{n+1}(D^{\prime}_{i}) = \phi_{m_{i}}(D^{\prime}_{i}) + s_{i} = \psi_{m_{i}}(D^{\prime}_{i}) + s_{i}. \end{align*} $$

Choose vectors $v_{i} \in B_{R}(0)$ to satisfy $s_{i} + v_{i} \in \mathbb {Z}^{d}$ , set $h_{i} : \phi _{n+1}(D_{i}) \to \phi _{n+1}(D_{i})$ to be $h_{\phi _{n+1}(D_{i}), K, v_{i}, K}$ and define an $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz function $g_{n+1}$ by

$$\begin{align*}g_{n+1}(r) = \begin{cases} h_{i}(r) & \textrm{if}\ r \in \phi_{n+1}(D_{i}),\\ r & \textrm{otherwise}. \end{cases} \end{align*}$$

Finally, set $\psi _{n+1}|_{C'} = g_{n+1} \circ \phi _{n+1}|_{C'} $ and extend this definition to a Borel map $\psi _{n+1} : X^{\prime }_{n+1} \to \mathbb {R}^{d}$ using the rationality of the sequence of partial actions. Coherence of the maps $(\psi _{k})_{k \le n+1}$ and the inclusion $Z_{m}\cap X^{\prime }_{n+1} \subseteq Z_{n+1}$ for $m \le n+1$ follow from the analog of equation (4).

It remains to check the bi-Lipschitz condition for the resulting cocycle $\rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}}$ . It is easier to work with the cocycle $\rho _{\mathfrak {F}_{2}, \mathfrak {F}_{1}}$ , which, for $x, x+r \in X^{\prime }_{n}$ , satisfies

$$ \begin{align*}\rho_{\mathfrak{F}_{2}, \mathfrak{F}_{1}}(r,x) = g_{n}(\phi_{n}(x) + r) - g_{n}(\phi_{n}(x)), \end{align*} $$

and is therefore $(\alpha ^{-}, \alpha ^{+})$ -bi-Lipschitz, because so is $g_{n}$ . Hence, $\rho _{\mathfrak {F}_{2}, \mathfrak {F}_{1}}$ is also $(\alpha ^{-1}, \alpha )$ -bi-Lipschitz, because $\alpha ^{-1} < \alpha ^{-} < \alpha ^{+} < \alpha $ by the choice of K. Finally, we apply Remark 3.10 to conclude that $\rho _{\mathfrak {F}_{1}, \mathfrak {F}_{2}}$ is also $(\alpha ^{-1}, \alpha )$ -bi-Lipschitz.

By restricting the action of $\mathfrak {F}_{2}$ onto the integer grid Z, we get the following corollary.

Corollary 3.13. Let $\mathfrak {F}$ be a free Borel $\mathbb {R}^{d}$ -flow on X. For any $\alpha> 1$ , there exist a cross-section $Z \subseteq X$ and a free $\mathbb {Z}^{d}$ -action T on Z such that the cocycle $\rho = \rho _{\mathfrak {F}, T}: \mathbb {Z}^{d} \times X \to \mathbb {R}^{d}$ given by $T_{n}x = x + \rho (n,x)$ is $(\alpha ^{-1}, \alpha )$ -bi-Lipschitz.

4 Special representation theorem

The main goal of this section is to formulate and prove a Borel version of Katok’s special representation theorem [12] that connects free $\mathbb {R}^{d}$ -flows with free $\mathbb {Z}^{d}$ -actions. We have already done most of the work in proving Theorem 3.12, and it is now a matter of defining special representations in the Borel context and connecting them to our earlier set-up.

4.1 Cocycles

Given a Borel action $G \curvearrowright X$ , a (Borel) cocycle with values in a group H is a (Borel) map $\rho : G \times X \to H$ that satisfies the cocycle identity: that is,

$$ \begin{align*}\rho(g_{2}g_{1},x) = \rho(g_{2},g_{1}x) \rho(g_{1},x) \quad \textrm{for all }g_{1},g_{2} \in G \textrm{ and } x \in X.\end{align*} $$

We are primarily concerned with the Abelian groups $\mathbb {Z}^{d}$ and $\mathbb {R}^{d}$ in this section, so the cocycle identity will be written additively. A cocycle $\rho : G \times X \to H$ is said to be injective if $\rho (g,x) \ne e_{H}$ for all $g \ne e_{G}$ and all $x \in X$ , where $e_{G}$ and $e_{H}$ are the identity elements of the corresponding groups. Furthermore, suppose that the groups G and H are locally compact. We say that $\rho $ escapes to infinity if, for all $x \in X$ , $\lim _{g \to \infty }\rho (g,x) = +\infty $ in the sense that, for any compact $K_{H} \subseteq H$ , there exists a compact $K_{G} \subseteq G$ such that $\rho (g,x) \not \in K_{H}$ whenever $g \not \in K_{G}$ .

Example 4.1. Suppose that $a_{H} : H \curvearrowright X$ and $a_{G} : G \curvearrowright Y$ , $Y \subseteq X$ , are free actions of groups G and H on standard Borel spaces, and suppose that we have containment of orbit equivalence relations $E_{G} \subseteq E_{H}$ . For each $y \in Y$ and $g \in G$ , there exists a unique $\rho _{a_{H},a_{G}}(g,y) \in H$ such that $a_{H}(\rho _{a_{H},a_{G}}(g,y),y) = a_{G}(g,y) $ . The map $(g,y) \mapsto \rho _{a_{H},a_{G}}(g,y)$ is an injective Borel cocycle. We have already encountered two instances of this idea in §3.4.

4.2 Flow under a function

Borel $ \mathbb {R} $ -flows and $ \mathbb {Z} $ -actions are tightly connected through the ‘flow under a function’ construction. Let $ T : Z \to Z $ be a free Borel automorphism of a standard Borel space and let $ f : Z \to \mathbb {R}^{>0} $ be a positive Borel function. There is a natural definition of a flow $ \mathfrak {F} : \mathbb {R} \curvearrowright X $ on the space $ X = \{ (z, t) : z \in Z, 0 \le t < f(z)\} $ under the graph of $ f $ . The action $ (z, t) + r $ for a positive $ r $ is defined by shifting the point $ (z,t) $ by $ r $ units upwards until the graph of $ f $ is reached, jumping to the point $(Tz, 0)$ and then continuing to flow upwards until the graph of f at $Tz$ is reached, etc. More formally,

$$ \begin{align*} (z, t) + r = \bigg(T^{k}z, t + r - \sum\limits_{i=0}^{k-1}f(T^{i}z)\bigg) \end{align*} $$

for the unique $ k \ge 0 $ such that $ \sum _{i=0}^{k-1}f(T^{i}z) \le t + r < \sum _{i=0}^{k}f(T^{i}z) $ . For $ r \le 0 $ , the action is defined by ‘flowing backward’, that is,

$$ \begin{align*} (z, t) + r = \bigg(T^{-k}z, t + r + \sum\limits_{i=1}^{k}f(T^{-i}z)\bigg) \end{align*} $$

for $ k \ge 0 $ such that $ 0 \le t + r + \sum \nolimits _{i=1}^{k}f(T^{-i}z) < f(T^{-k}z) $ . The action is well defined provided that the fibers within the orbits of $ T $ have infinite cumulative lengths: that is,

(5) $$ \begin{align} \sum_{i=0}^{\infty}f(T^{i}z) = +\infty\quad \textrm{and} \quad \sum_{i=0}^{\infty}f(T^{-i}z) = +\infty\quad \textrm{for all } z \in Z. \end{align} $$

The appealing geometric picture of the ‘flow under a function’ does not generalize to higher dimensions, but it admits an interpretation as the so-called special flow construction suggested in [12].

4.3 Special flows

Let T be a free $\mathbb {Z}^{d}$ -action on a standard Borel space Z and let $\rho : \mathbb {Z}^{d} \times Z \to \mathbb {R}^{d}$ be a Borel cocycle. One can construct a $\mathbb {Z}^{d}$ -action $\hat {T}$ , the so-called principal $\mathbb {R}^{d}$ -extension, defined on $Z \times \mathbb {R}^{d}$ via $\hat {T}_{n}(z, r) = (T_{n}z, r + \rho (n,z))$ . An easy application of the cocycle identity verifies axioms of the action. Although the action T will, typically, have complicated dynamics, the action $\hat {T}$ admits a Borel transversal as long as the cocycle $\rho $ escapes to infinity.

Lemma 4.2. If the cocycle $\rho $ satisfies $\lim _{n \to \infty }\|\rho (n,z)\| = +\infty $ for all $z \in Z$ , then the action $\hat {T}$ has a Borel transversal.

Proof. Let $Y_{k} = \{ (z, r) \in Z \times \mathbb {R}^{d} : \|r\| \le k\}$ . We claim that each orbit of $\hat {T}$ intersects $Y_{k}$ in a finite (possibly empty) set. Indeed, cocycle values escaping to infinity yield, for any $(z,r) \in Z \times \mathbb {R}^{d}$ , a number N so large that $\|r + \rho (n,z)\|> k$ whenever $\|n\| \ge N$ . In particular, $\|n\| \ge N$ implies that $\hat {T}_{n}(z,r) = (T_{n}z, r + \rho (n,z)) \not \in Y_{k}$ . Hence, the intersection of the orbit of $(z,r)$ with $Y_{k}$ is finite.

Set $Y = \bigsqcup _{k \in \mathbb {N}} (Y_{k} \setminus \bigcup _{n \in \mathbb {Z}^{d}} \hat {T}_{n}Y_{k-1})$ . Each orbit of $\hat {T}$ intersects Y in a finite and necessarily non-empty set, so $E_{\hat {T}}|_{Y}$ is a finite Borel equivalence relation. A Borel transversal for $E_{\hat {T}}|_{Y}$ is also a transversal for the action of $\hat {T}$ .

Now, we assume that the cocycle $\rho $ satisfies the assumptions of Lemma 4.2 and $X = (Z \times \mathbb {R}^{d})/E_{\hat {T}}$ , therefore, carries the structure of a standard Borel space as a push-forward of the factor map $\pi : Z \times \mathbb {R}^{d} \to X$ , which sends a point to its $E_{\hat {T}}$ -equivalence class.

There is a natural $\mathbb {R}^{d}$ -flow $\hat {\mathfrak {F}}$ on $Z \times \mathbb {R}^{d}$ , which acts by shifting the second coordinate: $(z,r) +_{\hat {\mathfrak {F}}} s = (z, r + s)$ . This flow commutes with the $\mathbb {Z}^{d}$ -action $\hat {T}$ and, therefore, projects onto the flow $\mathfrak {F}$ on X given by the condition $\pi ((z,r) +_{\hat {\mathfrak {F}}} s) = \pi (z,r) +_{\mathfrak {F}} s$ . We say that $\mathfrak {F}$ is the special flow over T generated by the cocycle $\rho $ . Freeness of T implies freeness of $\mathfrak {F}$ .

The construction outlined above works just as well in the context of ergodic theory, where the space Z would be endowed with a finite measure $\nu $ preserved by the action T. The product of $\nu $ with the Lebesgue measure on $\mathbb {R}^{d}$ induces a measure $\mu $ on X, which is preserved by the flow $\mathfrak {F}$ . Furthermore, $\mu $ is finite provided the cocycle $\rho $ is integrable in the sense of [12, Condition (J), pp. 122]. Katok’s special representation theorem asserts that, up to a null set, any free ergodic measure-preserving flow can be obtained via this process. Furthermore, the cocycle can be picked to be bi-Lipschitz with Lipschitz constants arbitrarily close to $1$ .

As will be shown shortly, such a representation result continues to hold in the framework of descriptive set theory, and every free Borel $\mathbb {R}^{d}$ -flow is Borel isomorphic to a special flow over some free Borel $\mathbb {Z}^{d}$ -action. Moreover, just as in Katok’s original work, Theorem 4.3 provides some significant control on the cocycle that generates the flow, tightly coupling the dynamics of the $\mathbb {Z}^{d}$ -action with the dynamics of the flow it produces. But, first, we re-interpret the construction in different terms.

4.4 Flows generated by admissible cocycles

Let the map $ Z \ni z \mapsto (z,0) \in Z \times \{0\}$ be denoted by $\iota $ . If the cocycle $\rho $ is injective, then $\pi \circ \iota : Z \to \pi (Z\times \{0\}) = Y$ is a bijection and Y intersects every orbit of $\mathfrak {F}$ in a non-empty countable set. The $\mathbb {Z}^{d}$ -action T on Z can be transferred via $\pi \circ \iota $ to give a free $\mathbb {Z}^{d}$ -action $T' = \pi \circ \iota \circ T \circ \iota ^{-1} \circ \pi ^{-1}$ on Y. Let $\rho ' = \rho _{T', \mathfrak {F}}: \mathbb {Z}^{d} \times Y \to \mathbb {R}^{d}$ be the cocycle of the action $\pi \circ \iota \circ T \circ \iota ^{-1} \circ \pi ^{-1}$ : that is,

(6) $$ \begin{align} T^{\prime}_{n}(y) = (\pi \circ \iota \circ T_{n} \circ \iota^{-1} \circ \pi^{-1})(y) = y +_{\mathfrak{F}} \rho'(n,y) \quad \textrm{for all }n \in \mathbb{Z}^{d}\textrm{ and } y \in Y. \end{align} $$

If $y = (\pi \circ \iota )(z)$ for $z \in Z$ , then equation (6) translates into

$$ \begin{align*}\pi(T_{n}z, 0) = \pi(z, \rho'(n,y)).\end{align*} $$

Since $\pi (T_{n}z, 0) = \pi (z, \rho (-n, T_{n}z)) = \pi (z, -\rho (n,z))$ , we conclude that $\rho '(n,y) = - \rho (n,z)$ , where $y = (\pi \circ \iota ) (z)$ . In particular, Y is a discrete cross-section for the flow $\mathfrak {F}$ precisely because $\rho $ escapes to infinity.

Conversely, if $\mathfrak {F}$ is any free $\mathbb {R}^{d}$ -flow on a standard Borel space X and if $Z \subseteq X$ is a discrete cross-section with a $\mathbb {Z}^{d}$ -action T on it, then $\mathfrak {F}$ is isomorphic to the special flow over T generated by the (necessarily injective) cocycle $-\rho _{T,\mathfrak {F}}$ .

We say that a cocycle $\rho $ is admissible if it is both injective and escapes to infinity. The discussion of the above two paragraphs can be summarized by saying that, up to a change of sign in the cocycles, representing a flow as a special flow generated by an admissible cocycle is the same thing as finding a free $\mathbb {Z}^{d}$ -action on a discrete cross-section of the flow.

For instance, given any free $\mathbb {Z}^{d}$ -action T on Z, we may consider the admissible cocycle $\rho (n, z) = -n$ for all $z \in Z$ and $n \in \mathbb {Z}^{d}$ . The set $Y = \pi (Z \times \{0\})$ is then an integer grid for the flow $\mathfrak {F}$ (in the sense of Definition 3.11). Conversely, any flow that admits an integer grid is isomorphic to a special flow generated by such a cocycle.

4.5 Special representation theorem

Restriction of the orbit equivalence relation of any $\mathbb {R}^{d}$ -flow onto a cross-section gives a hyperfinite equivalence relation [10, Theorem 1.16], and therefore can be realized as an orbit equivalence relation by a free Borel $\mathbb {Z}^{d}$ -action (as long as the restricted equivalence relation is aperiodic). Since any free flow admits a discrete (in fact, lacunary) aperiodic cross-section, it is isomorphic to a special flow over some action generated by some cocycle. In general, however, the structure of the $\mathbb {Z}^{d}$ -orbit and the corresponding orbit of the flow have little to do with each other. Theorem 3.12 and Corollary 3.13 allow us to improve on this and find a special representation generated by a bi-Lipschitz cocycle.

For comparison, Katok’s theorem [12] can be formulated in the parlance of this section as follows.

Theorem. (Katok)

Pick some $\alpha> 1$ . Any free ergodic measure-preserving $\mathbb {R}^{d}$ -flow on a standard Lebesgue space is isomorphic to a special flow over a free ergodic measure-preserving $\mathbb {Z}^{d}$ -action generated by an $(\alpha ^{-1},\alpha )$ -bi-Lipschitz cocycle.

As is the case with all ergodic theoretical results, isomorphism is understood to hold up to a set of measure zero. We conclude with a Borel version of Katok’s special representation theorem, which holds for all free Borel $\mathbb {R}^{d}$ -flows and establishes isomorphism on all orbits.

Theorem 4.3. Pick some $\alpha> 1$ . Any free Borel $\mathbb {R}^{d}$ -flow is isomorphic to a special flow over a free Borel $\mathbb {Z}^{d}$ -action generated by an $(\alpha ^{-1},\alpha )$ -bi-Lipschitz cocycle.

Proof. Let $\mathfrak {F}$ be a free Borel $\mathbb {R}^{d}$ -flow on X. Corollary 3.13 gives a cross-section $Z \subseteq X$ and a $\mathbb {Z}^{d}$ -action T on it such that the cocycle $\rho _{\mathfrak {F},T} : \mathbb {Z}^{d} \times X \to \mathbb {R}^{d}$ is $(\alpha ^{-1}, \alpha )$ -bi-Lipschitz. By the discussion in §4.4, this gives a representation of the flow as a special flow over T generated by the cocycle $-\rho _{\mathfrak {F},T}$ , which is also $(\alpha ^{-1},\alpha )$ -bi-Lipschitz.