Related papers: A survey of Measured Group Theory
We look at group actions on metric spaces, particularly at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the…
Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on…
In this article, we define amorphic complexity for actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. Amorphic complexity, originally introduced for $\mathbb Z$-actions, is a topological invariant which…
We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests…
We describe recent work that extends some of the measure and topological rigidity results in dynamical systems from situations homogeneous under a Lie group to quite general manifolds.
This is a short survey paper, partly meant as a research announcement. Its purpose is to highlight some aspects of the interplay between quantales, inverse semigroups, and groupoids. Many of the results mentioned have not yet been presented…
In this paper the metric on the set of mixing actions of a countable infinite group is introduced so that the corresponding space is complete and separable. Keywords and phrases. Monotilable group, measure preserving transformations, mixing…
This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key…
Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group…
For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications…
We prove the existence of a successful coupling for $n$ particles in the symmetric inclusion process. As a consequence we characterize the ergodic measures with finite moments, and obtain sufficient conditions for a measure to converge in…
We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover,…
We study invariant ergodic measures for quasiperiodically forced circle homeomorphisms and derive that either the system is uniquely ergodic or any such measure is associated to some invariant multigraph.
A survey of recent results about profinite groups, and results about infinite and finite groups where the theory of profinite groups plays a leading role.
In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…
Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena…
The purpose of this article is to discuss the circle method and its quantitative role in understanding pointwise almost everywhere convergence phenomena for polynomial ergodic averaging operators. Specifically, we will use the circle method…
The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms…