Paradoxical decomposition of Group Actions and Configuration

有限维非阿基米德赋范空间的悖论分解

Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

本文证明了在非阿基米德赋值域上，满足特定范数条件的有限维赋范空间（维数≥2）在其仿射等距群作用下可分解为四个悖论块。

Paradoxical decompositions and finitary rules

We colour every point x of a probability space X according to the colours of a finite list x_1, ...., x_k of points such that each of the x_i, as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule (1) does there exist a finitely additive extension of the probability measure for which the x_i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? if the answer to the first question is no and to…

Computable paradoxical decompositions

We prove a computable version of Hall's Harem Theorem and apply it to computable versions of Tarski's alternative theorem.

A paradoxical decomposition of the real line

In this paper we demonstrate how to partition the real number line into four subsets which may be reassembled, via "piecewise rigid functions" that preserve Lebesgue measure, into two copies of the line. We then employ a similar process to split the line into $2k$ pieces that yield $k$ copies of the line, or even into countably many subsets to obtain countably many copies of $\mathbb{R}$.

Baire measurable paradoxical decompositions via matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical…

Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces

This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski, according to which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and to defining a Følner condition. Using a Hall-Rado Theorem on matchings in graphs, we show then for pseudogroups that existence of an invariant mean is equivalent to the Følner condition; in the case of the pseudogroup of bounded perturbations of the identity …