Related papers: Strongly ergodic actions have local spectral gap

Consider a countable group Gamma acting ergodically by measure preserving transformations on a probability space (X,mu), and let R_Gamma be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there…

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an…

We consider a non-uniquely ergodic dynamical system given by a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) $\tau$ on a non-empty compact metrisable space $\Omega$, for some $l\in\N$. Let (D) denote the following property: The…

For actions with a dense orbit of a connected noncompact simple Lie group $G$, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a $G$-action to be equivalent to…

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $\Gamma<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $\Gamma$. This shows that almost all so-called "standard…

We prove a generalization of the orthogonality relations of Duflo and Moore for ergodic, trace-preserving group actions on von Neumann algebras that are integrable in a suitable sense. We also obtain convolution inequalities that generalize…

Let $\Sigma$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\Gamma$ of $\Sigma$ acts on the SU(3)-character variety of $\Sigma$. We show that the action is ergodic with respect to the…

An \textit{algebraic} action of a discrete group $\Gamma $ is a homomorphism from $\Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $\Gamma $ is determined by a module…

It is known that if each point $x$ of a dynamical system is generic for some invariant measure $\mu_x$, then there is a strong connection between certain ergodic and topological properties of that system. In particular, if the acting group…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More…

An algebraic $\Gamma$-action is an action of a countable group $\Gamma$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $\Gamma$ on a…

The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural…

We show that for any co-amenable compact quantum group A=C(G) there exists a unique compact Hausdorff topology on the set EA of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic…

Given a transformation $T$ of a standard measure space $(X,\mu)$, let $\Cal M(T)$ denote the set of spectral multiplicities of the Koopman operator $U_T$ defined in $L^2(X,\mu)\ominus\Bbb C$ by $U_Tf:=f\circ T$. It is discussed in this…

We investigate how spectral properties of a measure preserving system $(X,\mathcal{B},\mu,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we provide natural…

We prove that any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $\Gamma$ with positive first $\ell^2$-Betti number, has a unique group measure…

The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

Let $\Gamma$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $\Gamma\curvearrowright [0,1]^{\Gamma}$, where $[0,1]^{\Gamma}$ is endowed with the product Lebesgue measure. We prove that…

We prove that for a fairly large family of HNN extensions \Gamma, the group measure space II_1 factor L^\infty(X) \rtimes \Gamma given by an arbitrary free ergodic probability measure preserving action of \Gamma, has a unique group measure…