Borel Action Research Articles

Structural, point-free, non-Hausdorff topological realization of Borel groupoid actions

Abstract We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures. Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

Absorbing Markov decision processes

In this paper, we study discrete-time absorbing Markov Decision Processes (MDP) with measurable state space and Borel action space with a given initial distribution. For such models, solutions to the characteristic equation that are not occupation measures may exist. Several necessary and sufficient conditions are provided to guarantee that any solution to the characteristic equation is an occupation measure. Under the so-called continuity-compactness conditions, we first show that a measure is precisely an occupation measure if and only if it satisfies the characteristic equation and an additional absolute continuity condition. Secondly, it is shown that the set of occupation measures is compact in the weak-strong topology if and only if the model is uniformly absorbing. Several examples are provided to illustrate our results.

Probabilistic constructions in continuous combinatorics and a bridge to distributed algorithms

The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics.•Seward and Tucker-Drob showed that every free Borel action Γ↷X of a countable group Γ admits an equivariant Borel map π:X→Y to a free subshift Y⊂2Γ. We give a new simple proof of this result.•We show that for a countable group Γ, Free(2Γ) is weakly contained, in the sense of Elek, in every free continuous action of Γ on a zero-dimensional Polish space. This fact is analogous to the theorem of Abért and Weiss for probability measure-preserving actions and has a number of consequences in continuous combinatorics. In particular, we deduce that a coloring problem admits a continuous solution on Free(2Γ) if and only if it can be solved on finite subgraphs of the Cayley graph of Γ by an efficient deterministic distributed algorithm (this fact was also proved independently and using different methods by Seward). This establishes a formal correspondence between questions that have been studied independently in continuous combinatorics and in distributed computing.

Discrete-time zero-sum games for Markov chains with risk-sensitive average cost criterion

We study zero-sum stochastic games for controlled discrete time Markov chains with risk-sensitive average cost criterion with countable/compact state space and Borel action spaces. The payoff function is nonnegative and possibly unbounded for countable state space case and for compact state space case it is a real-valued and bounded function. For countable state space case, under a certain Lyapunov type stability assumption on the dynamics we establish the existence of the value and a saddle point equilibrium. For compact state space case we establish these results without any Lyapunov type stability assumptions. Using the stochastic representation of the principal eigenfunction of the associated optimality equation, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. Also, we present and analyze an illustrative example.

Lifts of Borel actions on quotient spaces

Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.

Abelian group actions and hypersmooth equivalence relations

We show that a Borel action of a standard Borel group which is isomorphic to a sum of a countable abelian group with a countable sum of real lines and circles induces an orbit equivalence relation which is hypersmooth, i.e., Borel reducible to eventual agreement on sequences of reals, and it follows from this result along with the structure theory for locally compact abelian groups that Borel actions of Polish LCA groups induce orbit equivalence relations which are essentially hyperfinite, extending a result of Gao and Jackson and answering a question of Ding and Gao.

Cohomology of hyperfinite Borel actions

We study cocycles of countable groups \Gamma of Borel automorphisms of a standard Borel space (X, \mathcal{B}) taking values in a locally compact second countable group G . We prove that for a hyperfinite group \Gamma the subgroup of coboundaries is dense in the group of cocycles. We describe all Borel cocycles of the 2 -odometer and show that any such cocycle is cohomologous to a cocycle with values in a countable dense subgroup H of G . We also provide a Borel version of Gottschalk–Hedlund theorem.

Maximizing the probability of visiting a set infinitely often for a countable state space Markov decision process

We consider a Markov decision process with countable state space and Borel action space. We are interested in maximizing the probability that the controlled Markov chain visits some subset of the state space infinitely often. We provide sufficient conditions, based on continuity and compactness requirements, together with a stability condition on a parametrized family of auxiliary control models, which imply the existence of an optimal policy that is deterministic and stationary. We compare our hypotheses with those existing in the literature.

A generalization of the 𝔾0 dichotomy and a strengthening of the 𝔼0ℕ dichotomy

We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.

RECURRENCE AND THE EXISTENCE OF INVARIANT MEASURES

AbstractWe show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

On Polish groups admitting non-essentially countable actions

AbstractIt is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

Dynamical simplices and Borel complexity of orbit equivalence

We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence relation induced by a Borel action of S∞.

$N$-Person Nonzero-Sum Games for Continuous-Time Jump Processes With Varying Discount Factors

This paper is concerned with the first passage discounted payoff criterion for continuous-time $N$ -person nonzero-sum stochastic games in countable states and Borel action spaces, where the discount factor is nonconstant. For the game with unbounded reward functions and transition rates, the optimality is over the set of all randomized history-dependent policies. After showing the uniqueness of the solution to the Shapley equation, we give some suitable conditions imposed on the primitive data for the existence of Nash equilibria. Then, a financial system is introduced to illustrate the applications of the main result here.

Nonzero-Sum Expected Average Discrete-Time Stochastic Games: The Case of Uncountable Spaces

In this paper we study nonzero-sum discrete-time stochastic games with an uncountable state space and Borel action spaces under the expected average payoff criterion. The reward functions can be possibly unbounded and the transition law is a convex combination of finitely many probability measures dependent on the state variable and dominated by some probability measure on the state space. We introduce several auxiliary static game models and obtain their properties. Moreover, by a technique of extending the space state, we introduce auxiliary stochastic game models and derive the uniform geometric ergodicity of Markov chains taking values in the extended state space. Furthermore, we show the existence of a stationary almost Markov Nash equilibrium via an approximation method. Finally, we use a resource extraction model to illustrate the main results.

Computable approximations for average Markov decision processes in continuous time

Abstract In this paper we study the numerical approximation of the optimal long-run average cost of a continuous-time Markov decision process, with Borel state and action spaces, and with bounded transition and reward rates. Our approach uses a suitable discretization of the state and action spaces to approximate the original control model. The approximation error for the optimal average reward is then bounded by a linear combination of coefficients related to the discretization of the state and action spaces, namely, the Wasserstein distance between an underlying probability measure μ and a measure with finite support, and the Hausdorff distance between the original and the discretized actions sets. When approximating μ with its empirical probability measure we obtain convergence in probability at an exponential rate. An application to a queueing system is presented.

On the Popov–Pommerening conjecture for linear algebraic groups

Let $G$ be a reductive group over an algebraically closed subfield $k$ of $\mathbb{C}$ of characteristic zero, $H\subseteq G$ an observable subgroup normalised by a maximal torus of $G$ and $X$ an affine $k$-variety acted on by $G$. Popov and Pommerening conjectured in the late 1970s that the invariant algebra $k[X]^{H}$ is finitely generated. We prove the conjecture for: (1) subgroups of $\operatorname{SL}_{n}(k)$ closed under left (or right) Borel action and for: (2) a class of Borel regular subgroups of classical groups. We give a partial affirmative answer to the conjecture for general regular subgroups of $\operatorname{SL}_{n}(k)$.

Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.

A Polish metric space whose group of isometries induces a universal relation for Polish group actions

We show that there exists a Polish metric space (X, d) such that the action of its isometry group on X produces an equivalence relation which is universal for relations induced by a Borel action of a Polish group on a standard Borel space.

The complexity of the classification problem of continua

We prove that the homeomorphism problem for connected compact metric spaces is Borel bireducible with a universal orbit equivalence relation induced by a Borel action of a Polish group.

Continuous-time Markov decision processes with risk-sensitive finite-horizon cost criterion

This paper studies continuous-time Markov decision processes with a denumerable state space, a Borel action space, bounded cost rates and possibly unbounded transition rates under the risk-sensitive finite-horizon cost criterion. We give the suitable optimality conditions and establish the Feynman–Kac formula, via which the existence and uniqueness of the solution to the optimality equation and the existence of an optimal deterministic Markov policy are obtained. Moreover, employing a technique of the finite approximation and the optimality equation, we present an iteration method to compute approximately the optimal value and an optimal policy, and also give the corresponding error estimations. Finally, a controlled birth and death system is used to illustrate the main results.