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Konstantin Slutsky

Publications and source records attributed to Konstantin Slutsky.

18 recordsLinked to original sources
arXiv2026

The analogue of Belinskaya's theorem for measure-preserving flows

We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose $\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by Fran\c{c}ois Le Ma\^itre 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 $\mathrm{L}^1$ 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.

math.DS↗
arXiv2026

A measurable equivariant Weierstrass theorem

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 $\Lambda$ an entire function $F_\Lambda$ such that the divisor of zeroes of $F_\Lambda$ is $\Lambda$ and such that $F_{\Lambda-w}(z) = F_\Lambda (z+w)$, $w\in\mathbb{C}$. 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.

math.CV↗
arXiv2026

Separating Orbits by Entire Functions

We show that for any free probability measure-preserving action of $\mathbb{C}^{d}$ on a standard probability space, there exists a Borel entire function $F$ such that the factor map $x \mapsto F_{x}$, where $F_{x}(z) = F(z \cdot x)$, 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.

math.DS↗
arXiv2025

Equivariant Borel liftings in complex analysis and PDE

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 $d$ an entire function $f_d$ with divisor of zeros $\mathrm{div}(f_d)=d$ and which commutes with translation, $f_{d-w}(z)=f_d(z+w)$. We also examine the existence of equivariant Borel right inverses for the distributional Laplacian, the heat operator, and the $\bar{\partial}$-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.

math.DS↗
arXiv2025

$\mathrm{L}^1$ full groups of flows

We introduce the concept of an $\mathrm{L}^{1}$ 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 $\mathrm{L}^1$ norm. Our construction generalizes $\mathrm{L}^{1}$ 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 $\mathrm{L}^{1}$ full groups are topologically simple and -- when the acting group is locally compact and amenable -- are whirly amenable and generically two-generated. $\mathrm{L}^{1}$ full groups of actions of compactly generated locally compact Polish groups are shown to remember the $\mathrm{L}^{1}$ 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 $\mathrm{L}^{1}$ full groups is shown to coincide with the kernel of the index map, which implies that $\mathrm{L}^{1}$ 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 $\mathrm{L}^{1}$ full group completely characterizes the associated ergodic flow up to flip Kakutani equivalence. Finally, we study the coarse geometry of the $\mathrm{L}^{1}$ full groups. The $\mathrm{L}^{1}$ norm on the derived subgroup of the $\mathrm{L}^{1}$ 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 $\mathrm{L}^{1}$ norm on all of the $\mathrm{L}^{1}$ full group.

math.DS↗
arXiv2023

Katok's special representation theorem for multidimensional Borel flows

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.

math.DS↗
arXiv2021

Smooth orbit equivalence of multidimensional Borel flows

Free Borel $\mathbb{R}^{d}$-flows are smoothly equivalent if there is a Borel bijection between the phase spaces that maps orbits onto orbits and is a $C^{\infty}$-smooth orientation preserving diffeomorphism between orbits. We show that all free non-tame Borel $\mathbb{R}^{d}$-flows are smoothly equivalent in every dimension $d \ge 2$. This answers a question of B. Miller and C. Rosendal.

math.DS↗
arXiv2020

Minimum-Violation Planning for Autonomous Systems: Theoretical and Practical Considerations

This paper considers the problem of computing an optimal trajectory for an autonomous system that is subject to a set of potentially conflicting rules. First, we introduce the concept of prioritized safety specifications, where each rule is expressed as a temporal logic formula with its associated weight and priority. The optimality is defined based on the violation of such prioritized safety specifications. We then introduce a class of temporal logic formulas called $\textrm{si-FLTL}_{\mathsf{G_X}}$ and develop an efficient, incremental sampling-based approach to solve this minimum-violation planning problem with guarantees on asymptotic optimality. We illustrate the application of the proposed approach in autonomous vehicles, showing that $\textrm{si-FLTL}_{\mathsf{G_X}}$ formulas are sufficiently expressive to describe many traffic rules. Finally, we discuss practical considerations and present simulation results for a vehicle overtaking scenario.

cs.RO↗
arXiv2019

Liability, Ethics, and Culture-Aware Behavior Specification using Rulebooks

The behavior of self-driving cars must be compatible with an enormous set of conflicting and ambiguous objectives, from law, from ethics, from the local culture, and so on. This paper describes a new way to conveniently define the desired behavior for autonomous agents, which we use on the self-driving cars developed at nuTonomy. We define a "rulebook" as a pre-ordered set of "rules", each akin to a violation metric on the possible outcomes ("realizations"). The rules are partially ordered by priority. The semantics of a rulebook imposes a pre-order on the set of realizations. We study the compositional properties of the rulebooks, and we derive which operations we can allow on the rulebooks to preserve previously-introduced constraints. While we demonstrate the application of these techniques in the self-driving domain, the methods are domain-independent.

cs.AI↗
arXiv2016

On time change equivalence of Borel flows

This paper addresses the notion of time change equivalence for Borel multidimensional flows. We show that all free flows are time change equivalent up to a compressible set. An appropriate version of this result for non-free flows is also given.

math.DS↗
arXiv2015

Lebesgue Orbit Equivalence of Multidimensional Borel Flows

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth Euclidean flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures.

math.DS↗
arXiv2015

Regular cross sections of Borel flows

Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

math.DS↗
arXiv2013

Graev ultrametrics and free products of Polish groups

We construct Graev ultrametrics on free products of groups with two-sided invariant ultrametrics and HNN extensions of such groups. We also introduce a notion of a free product of general Polish groups and prove, in particular, that two Polish groups G and H can be embedded into a Polish group T in such a way that the subgroup of T generated by G and H is isomorphic to the free product G*H.

math.GR↗
arXiv2013

Automatic continuity for homomorphisms into free products

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.

math.GR↗
arXiv2013

Graev metrics on free products and HNN extensions

We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces

math.LO↗
arXiv2010

Non-genericity phenomenon in some ordered Fraisse classes

We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the randomly ordered rational Urysohn space.

math.LO↗
arXiv2006

Riemann Rearrangement Theorem for some types of convergence

We reexamine the Riemann Rearrangement Theorem for different types of convergence. We consider series convergence with respect to a filter. We describe the Sum Range (SR) of a series along the 2n-filter and for statistically convergent series.

math.FA↗