Scope of Lindenstrauss's measurable-to-continuous cocycle regularization and the characterization of strictly ergodic distal models

Lindenstrauss (Ergodic Theory and Dynamical Systems, 1999) proved that for a measure-preserving transformation of a compact metric probability space whose periodic points form a null set, every measurable cocycle into a compact connected metric group is measurably cohomologous to a continuous cocycle (his Theorem 3.1), and used this to show that every ergodic measurably distal Z-system is isomorphic to a minimal topologically distal system carrying a full-support invariant Borel measure — uniquely ergodic when the distal tower is built from connected compact groups, while certain measurably distal systems admit no uniquely ergodic distal model at all. The source paper left the extent of this regularization program open, and this problem asks for its completion along three linked directions: (P1) for which locally compact target groups beyond compact connected ones — disconnected groups, and non-compact groups beyond Derrien's connected Lie groups with bi-invariant metric — is every measurable (integrable) cocycle cohomologous to a continuous one; (P2) whether analogues hold for acting groups beyond Z, for instance Z^2; and (P3) a necessary-and-sufficient characterization of the ergodic measurably distal systems admitting a strictly ergodic topologically distal model. The follow-up literature (Derrien 2000; Glasner-Weiss 2006; Huang-Shao-Ye 2019; Gutman-Lian 2023; Edeko-Kreidler 2023; Moreira-Richter-Robertson 2023; Kra-Shalom 2025) provides new sufficient conditions and new obstructions but resolves none of the three directions; all remain open.

Background

A topological dynamical system is a homeomorphism $T$ of a compact metric space $X$. It is distal if any two distinct points stay a definite distance apart along their orbits, $\inf_{n\in\mathbb{Z}} d(T^n x, T^n y)>0$ for $x\neq y$, and minimal if every orbit is dense. A classical structure theorem of Furstenberg describes every minimal distal system as a transfinite tower of isometric extensions: systems obtained by successive skew products $(x,m)\mapsto (Tx,\alpha(x)m)$ over homogeneous spaces $M=K/L$ of compact groups $K$ via continuous cocycles $\alpha:X\to K$, closed under inverse limits. On the measure side, a measure-preserving system $(X,\mu,T)$ is measurably distal — a notion introduced by Parry and developed by Zimmer for actions of locally compact groups — if it can be assembled from a transfinite tower of measurable isometric extensions; Zimmer proved this equivalent to the existence of a separating sieve, a sequence of positive-measure sets that eventually separates almost every pair of points along orbits. By a representation theorem of Zimmer, a measurable isometric extension is again a skew product over a homogeneous space of a compact group, but now built from a measurable cocycle $m:X\to K$ together with an invariant measure on the product.

The two structure theories do not match a priori, because a topological isometric extension requires a continuous cocycle while the measurable theory only supplies a measurable one. Lindenstrauss bridged this gap for $\mathbb{Z}$-actions. The engine is Theorem 3.1 of the 1999 paper: if $X$ is a compact metric space with a Borel probability measure $\mu$, if $T$ is a $\mu$-preserving measurable transformation whose periodic points have $\mu$-measure zero, and if $K$ is a compact connected metric group, then every measurable map $m:X\to K$ is cohomologous to a continuous one: there exist a measurable $h:X\to K$ and a continuous $c:X\to K$ with $m(x)=h(Tx)\,c(x)\,h(x)^{-1}$ for $\mu$-almost every $x$ (two maps are called cohomologous precisely when they differ by such a measurable conjugacy). The proof approximates the compact connected group through its compact Lie quotients, uses one-parameter subgroups to build continuous approximants, and glues them through tower decompositions of aperiodic systems and a Borel-Cantelli argument. Applying this stepwise along the tower yields the main theorem: every ergodic measurably distal $\mathbb{Z}$-system is measurably isomorphic to a minimal topologically distal system carrying an invariant Borel probability measure of full support. When every isometric extension in the tower is by a connected compact group, the model is uniquely ergodic (admits exactly one invariant Borel probability measure); the paper also exhibits measurably distal systems admitting no uniquely ergodic topologically distal model at all, namely certain finite-to-one isometric extensions of an irrational circle rotation that are not Kronecker systems (a Kronecker system is a rotation on a compact monothetic group, that is, a compact group containing a dense cyclic subgroup).

Theorem 3.1 has a lineage: Kochergin and Rudolph had earlier proved regularization into special target groups, and Derrien subsequently extended the conclusion to integrable measurable cocycles into a connected Lie group carrying a bi-invariant Riemannian metric — a class amounting, up to local structure, to products of compact and abelian groups, so it includes non-compact abelian targets such as $\mathbb{R}^n$ but excludes disconnected target groups (finite groups, profinite groups, the Cantor group) as well as non-compact semisimple groups such as $SL_2(\mathbb{R})$, which admits no bi-invariant metric. On the model side, the Jewett-Krieger theorem guarantees every ergodic measure-preserving system some strictly ergodic model (minimal and uniquely ergodic), but not one that is simultaneously distal: the Morse system — the strictly ergodic binary subshift generated by the Morse-Thue sequence, a two-to-one group extension of the odometer (rotation on the 2-adic integers) — is measurably distal yet admits no strictly ergodic distal model (Glasner-Weiss, following the Lindenstrauss-type obstructions). Gutman and Lian later found a new sufficient condition for strictly ergodic distal models, formulated through nilcycles (measurable cocycle-like objects on cubes satisfying a glueing identity), and used it to give a new proof that the Host-Kra factors of ergodic systems — the characteristic factors governing multiple ergodic averages — are inverse limits of nilsystems (systems built from nilpotent Lie groups). Edeko and Kreidler constructed canonical minimal distal models for transformations of arbitrary probability spaces by an operator-theoretic route, explicitly leaving the strict ergodicity of the canonical model open. Recent work of Kra and Shalom still invokes Theorem 3.1 in its original form (compact connected target, $\mathbb{Z}$-action) as a working tool. The unresolved questions below are posed against this landscape.

Problem Statement

Throughout, a base system means a compact metric space $X$ with a Borel probability measure $\mu$ and a jointly measurable, $\mu$-preserving action of a countable group $G$, whose periodic points (the points fixed by some non-identity element of $G$) have $\mu$-measure zero; for $G=\mathbb{Z}$ with generator $T$ this is exactly the hypothesis of Theorem 3.1 that the $T$-periodic points be $\mu$-null. A measurable cocycle of such an action into a topological group $K$ is a measurable map $m:G\times X\to K$ satisfying $m(gh,x)=m(g,hx)\,m(h,x)$ for all $g,h\in G$; for $G=\mathbb{Z}$ it is determined by the single map $x\mapsto m(T,x)$ from $X$ to $K$, written simply $m:X\to K$. Two cocycles $m_0,m_1$ are cohomologous if for some measurable transfer function $h:X\to K$ one has $m_1(g,x)=h(gx)\,m_0(g,x)\,h(x)^{-1}$ for $\mu$-almost every $x$ and every $g\in G$; for $G=\mathbb{Z}$ with generator $T$ this is the relation $m_1(x)=h(Tx)\,m_0(x)\,h(x)^{-1}$ almost everywhere used in Theorem 3.1, which asserts that every measurable $m:X\to K$ into a compact connected metric group lies in the cohomology class of a continuous map. The problem, open since Lindenstrauss's 1999 paper and left unresolved by subsequent literature, is to determine the scope of the measurable-to-continuous cocycle regularization and to complete the resulting model theory for measurably distal systems:

(P1) Target-group scope. Fix a base system with acting group $\mathbb{Z}$ (a single aperiodic $\mu$-preserving transformation $T$ of $X$). For which locally compact second countable groups $K$ beyond compact connected ones is every measurable cocycle $m:X\to K$ — required to be integrable, when $K$ is non-compact, in the sense that the distance from the identity, for a left-invariant metric inducing the topology of $K$, is $\mu$-integrable as in Derrien's theorem — cohomologous to a continuous cocycle? Decide in particular the cases (a) of non-connected compact groups, such as finite groups, profinite groups or the Cantor group, and (b) of non-compact connected Lie groups outside Derrien's class, such as $SL_2(\mathbb{R})$ and its subgroups. The determination is sought over this entire class of target groups, the two listed families being instances of it. The regularization may hold or fail; a failure is witnessed by a measurable cocycle whose measurable cohomology class contains no continuous representative.

(P2) Acting-group scope. For a jointly measurable, measure-preserving action of a countable group $G$ beyond $\mathbb{Z}$ — for example $G=\mathbb{Z}^2$ with commuting generators — on a compact metric probability space, with the periodic points of the action (the points fixed by some non-identity element of $G$) assumed $\mu$-null, decide whether every measurable cocycle of the action into a compact connected metric group is cohomologous to a continuous cocycle $m'$ (one whose slices $x\mapsto m'(g,x)$ are continuous for every $g\in G$). Determine for which acting groups the analogue of Theorem 3.1 holds and where it fails.

(P3) Strictly ergodic distal models. Characterize, by necessary and sufficient conditions on the Furstenberg-Zimmer tower (the transfinite sequence of measurable isometric extensions and their structure groups), which ergodic measurably distal $\mathbb{Z}$-systems admit a strictly ergodic topologically distal model. The known sufficient conditions (Lindenstrauss's all-connected tower; the Gutman-Lian nilcycle conditions) and the known obstructions (the Morse system; finite-to-one non-Kronecker isometric extensions of irrational rotations) do not settle the characterization.

The three parts share one objective: to delimit the power of the regularization mechanism that converts measurable isometric extensions into topological ones, and thereby to complete the correspondence between measurable and topological distality. (P1) and (P2) are the two axes of generality of that mechanism — the target group and the acting group — and progress on either directly enlarges the class of towers to which the model constructions, and hence the analysis of (P3), apply. The three directions are linked: settling one of (P1), (P2), (P3) does not by itself settle the other two.

References

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