Katok’s special representation theorem
for multidimensional Borel flows

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 study of the possible ceiling functions in the “flow under a function” representation [21, 16] on the other.

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

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 $(X,\mathcal{B})$, i.e., 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, while others do not generalize. For example, 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 like 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. While being a highly non-trivial equivalence relation among measure-preserving flows [20, 9], descriptive set theoretical version of Kakutani equivalence collapses entirely [19].

Considerable work has been done 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.

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 setup as well (e.g., [26]). Particular assumptions on such coherent regions, however, depend on the specific application. Section 2 distills 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 a 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, Section 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.

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 $G$-action¹¹1More precisely, we should call such $(E,\phi)$ a partial free action. Since we are mainly concerned with free actions in what follows, we choose to omit the adjective “free” in the definition. 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: $\phi(x)\neq\phi(y)$ whenever $xEy$. The map $\phi$ itself may occasionally be refer 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

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 $x\in X$, 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 does exist 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 $E$.

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

Consider a free Borel action $G\curvearrowright X$ of a locally compact second countable 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

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

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

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

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  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 due 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 $R$, $B_{R}=\{g\in G:d(g,e)\leq R\}$, and any $x\in X$, the set $\mathcal{C}\cap B_{R}x$ is finite. Indeed, there can be at most $\lambda(B_{R+r})/\lambda(B_{r})$ points in the intersection, where $\lambda$ is a Haar measure on the group and $r>0$ is so small that $B_{r}c\cap B_{r}c^{\prime}=\varnothing$ whenever $c,c^{\prime}\in\mathcal{C}$ are distinct.

Small 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, Section 4] or [17, Section B.2] for the specifics). This way we get a Borel equivalence relation $E_{\mathcal{C}}\subseteq E_{G}$ whose equivalence classes are the cells of the Voronoi tessellation: $xE_{\mathcal{C}}y$ if and only if $\pi_{\mathcal{C}}(x)=\pi_{\mathcal{C}}(y)$.

Assumed freeness of the action $G\curvearrowright X$ allows for 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.

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:

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

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  coherence: for each $n$ the map $x\mapsto(\phi_{n}(x))^{-1}\phi_{n+1}(x)$ is $E_{n}$-invariant;

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  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 $xE_{n+1}y$ by monotonicity and, using coherence,

whence $A_{\phi_{n}}\subseteq A_{\phi_{n+1}}$. 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 $G\curvearrowright X$. Equally easy is to check that the action is free, and 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 $\phi_{n}$. Let us illustrate these concepts on essentially hyperfinite actions of lcsc groups.

In the context of Section 2.2, suppose that, furthermore, the restriction of the orbit equivalence relation $E_{G}$ onto the cross-section $\mathcal{C}$ is hyperfinite, i.e., 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, i.e., there are Borel sets $\mathcal{C}_{n}$ that pick exactly one point from each $F_{n}$-class. Just as in Section 2.2, we may define $\phi_{n}(x)$ to be such an element $g\in G$ that $gc=x$ for the unique $c\in\mathcal{C}_{n}$ satisfying $xE_{n}c$. This gives a convergent sequence of partial $G$-actions $(E_{n},\phi_{n})_{n}$ whose limit is the original action $G\curvearrowright X$.

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 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 $m\leq n$:

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

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  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:

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  coherence: $X_{m}\cap X_{n}\ni x\mapsto(\phi_{m}(x))^{-1}\phi_{n}(x)$ is $E_{m}$-invariant for each $m<n$;

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  exhaustiveness: for each $x\in X$ and $g\in G$ there exist $n$ and $y\in X_{n}$ such that $x\in X_{n}$, $xE_{n}y$, 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 most easily seen by reducing this setup to the notationally simpler one given in Section 2.3. To this end, extend $E_{n}$ to the equivalence relation $\hat{E}_{n}$ on all of $X$ by

and also extend $\phi_{n}$ to $\hat{\phi}_{n}:X\to G$ by setting $\hat{\phi}_{n}(x)=\phi_{m}(x)$ for the maximal $m\leq n$ 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 Section 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.

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 reasons 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 of 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 does not depend on other $E_{n}$-classes. If there are only countably many such “configurations” of $E_{n}$-classes, 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.

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 $C^{\prime}$, we denote by $\phi_{n}(C)\equiv\phi_{n}(C^{\prime})$ the existence for each $m\leq n$ of a bijection $\mathcal{E}_{m}(C)\ni D\mapsto D^{\prime}\in\mathcal{E}_{m}(C^{\prime})$ such that $\phi_{n}(D)=\phi_{n}(D^{\prime})$ for all $D\in\mathcal{E}_{m}(C)$. Collection of images $\{\phi_{n}(D):D\in\bigcup_{m\leq n}\mathcal{E}_{m}(C)\}$ 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 $\phi_{n}(C)\equiv\phi_{n}(C^{\prime})$ for all $E_{n}$-classes $C,C^{\prime}\subseteq Y_{k}$.

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.

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 will show 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 $m\neq n$.

Given two Lipschitz maps $f:A\to A^{\prime}$ and $g:B\to B^{\prime}$ that agree on the intersection $A\cap B$, the map $f\cup g:A\cup B\to A^{\prime}\cup B^{\prime}$, 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 $g$.

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 $\tau:\bigl{[}0,d(x,y)\bigr{]}\to X$ such that $\tau(0)=x$ and $\tau\bigl{(}d(x,y)\bigr{)}=y$. 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 $A$.

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

Let $(X,\left\lVert\cdot\right\rVert)$ 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.

For the rest of Section 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

where $d(x,\partial A)$ denotes the distance from $x$ to the boundary of $A$. This function (as well as its variant to 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 $\alpha^{-}=1-K^{-1}||v||$.

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

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

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 $r\in\mathbb{R}^{d}$.

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 $\bigcup_{i\leq n}X_{i}$ generated by $E_{i}$, i.e., $xEy$ whenever there exist $x_{1},\ldots,x_{m}$ and for each $1\leq i\leq m$ there exists $1\leq j(i)\leq n$ such that $x_{1}=x$, $x_{m}=y$ and $x_{i}E_{j(i)}x_{i+1}$ for all $1\leq i<m$.

If $E$ is an equivalence relation on $Y\subseteq X$ and $K>0$, we define the relation $E^{+K}$ on $Y^{+K}=\{x\in X:\mathrm{dist}(x,y)\leq K\textrm{ for some }y\in Y\}$ by $x_{1}E^{+K}x_{2}$ if and only if there are $y_{1},y_{2}\in Y$ such that $\mathrm{dist}(x_{1},y_{1})\leq K$, $\mathrm{dist}(x_{2},y_{2})\leq K$ and $y_{1}Ey_{2}$. 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 $C_{1},C_{2}$.