Konstantin Slutsky

We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose $\mathrm{L}^1$L1 full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by François Le Maître and the author. We show that whenever two free ergodic flows generate the same orbit equivalence relation and one is contained in the other's $\mathrm{L}^1$L1 full group, their positive half-orbits are commensurate after possibly reversing time. Katznelson's criterion then yields conjugacy after a scalar time change. The key new ingredient is a commensuration criterion asserting that a measurable subset of the real line whose symmetric differences with its translates have finite average measure over the unit interval is commensurate with exactly one of the empty set, the whole line, and the two half-lines. This criterion and its application were discovered autonomously by a two-agent AI system. The author independently verified the proofs and prepared the final text.

This paper is a prequel to our recent work, "Equivariant Borel liftings in complex analysis and PDE" (arXiv:2507.12058). While the results presented here were established in that work in a more general and abstract setting, the purpose of this paper is to provide a direct proof of the equivariant Weierstrass theorem. It states that there exists a Borel map assigning to each non-periodic positive divisor $Λ$Λ an entire function $F_Λ$FΛ​ such that the divisor of zeroes of $F_Λ$FΛ​ is $Λ$Λ and such that $F_{Λ-w}(z) = F_Λ(z+w)$FΛ−w​(z)=FΛ​(z+w), $w\in\mathbb{C}$w∈C. In general, non-periodicity cannot be omitted, and Borel measurability cannot be strengthened to continuity. The two key ingredients are the Runge approximation theorem and the existence of "Borel toasts", which are Borel counterparts of Rokhlin towers from ergodic theory. We do not assume prior knowledge of descriptive set theory and have aimed to make the exposition self-contained, aside from several results taken from graduate textbooks.

We show that for any free probability measure-preserving action of $\mathbb{C}^{d}$Cd on a standard probability space, there exists a Borel entire function $F$F such that the factor map $x \mapsto F_{x}$x↦Fx​, where $F_{x}(z) = F(z \cdot x)$Fx​(z)=F(z⋅x), is injective. This work builds on a result of Glücksam and Weiss, who constructed non-constant measurable entire functions for such actions. The proof combines a separating cross-section whose cocycle values lie in a countable subgroup with Forstnerič's holomorphic approximation theorem with prescribed critical points.

We establish Borel equivariant analogues of several classical theorems from complex analysis and PDE. The starting point is an equivariant Weierstrass theorem for entire functions: there exists a Borel mapping which assigns to each non-periodic positive divisor $d$d an entire function $f_d$fd​ with divisor of zeros $\mathrm{div}(f_d)=d$div(fd​)=d and which commutes with translation, $f_{d-w}(z)=f_d(z+w)$fd−w​(z)=fd​(z+w). We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the $\bar{\partial}$∂ˉ-operator on the space of smooth functions. We demonstrate that Borel equivariant inverses for these maps exist on the free part of the range. In general, the freeness assumptions cannot be omitted and Borelness cannot be strengthened to continuity. Our positive results follow from a theorem establishing sufficient conditions for the existence of equivariant Borel liftings. Two key ingredients are Runge-type approximation theorems and the existence of Borel toasts, which are Borel analogues of Rokhlin towers from ergodic theory.

We introduce the concept of an $\mathrm{L}^{1}$L1 full group associated with a measure-preserving action of a Polish normed group on a standard probability space. These groups carry a natural Polish group topology induced by an $\mathrm{L}^1$L1 norm. Our construction generalizes $\mathrm{L}^{1}$L1 full groups of actions of discrete groups, which have been studied recently by the first author. We show that under minor assumptions on the actions, topological derived subgroups of $\mathrm{L}^{1}$L1 full groups are topologically simple and -- when the acting group is locally compact and amenable -- are whirly amenable and generically two-generated. $\mathrm{L}^{1}$L1 full groups of actions of compactly generated locally compact Polish groups are shown to remember the $\mathrm{L}^{1}$L1 orbit equivalence class of the action. For measure-preserving actions of the real line (also often called measure-preserving flows), the topological derived subgroup of an $\mathrm{L}^{1}$L1 full groups is shown to coincide with the kernel of the index map, which implies that $\mathrm{L}^{1}$L1 full groups of free measure-preserving flows are topologically finitely generated if and only if the flow admits finitely many ergodic components. We also prove a reconstruction-type result: the $\mathrm{L}^{1}$L1 full group completely characterizes the associated ergodic flow up to flip Kakutani equivalence. Finally, we study the coarse geometry of the $\mathrm{L}^{1}$L1 full groups. The $\mathrm{L}^{1}$L1 norm on the derived subgroup of the $\mathrm{L}^{1}$L1 full group of an aperiodic action of a locally compact amenable group is proved to be maximal in the sense of C. Rosendal. For measure-preserving flows, this holds for the $\mathrm{L}^{1}$L1 norm on all of the $\mathrm{L}^{1}$L1 full group.

Given a set of positive reals we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.

These are the notes from a semester long course taught at UIC in the Fall of 2015. They cover compressible and hyperfinite relations, including Nadkarni's characterization of compressibility, classfification of hyperfinite relations by Dougherty, Jackson, and Kechris, and some examples of groups that give rise to hyperfinite relations only, after Jackson, Kechris, and Louveau.

This lecture notes form an eight week introductory course to topological groups of Cantor systems. Some of the topics covered by the notes include: Verhsik maps and Bratteli diagrams, flip conjugacy, commutator subgroup, amenability of full groups.