Related papers: Isomorphism rigidity in entropy rank two

We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various…

An algebraic $Z^{d}$-action is an action of $Z^{d}$ on a compact abelian group $X$ by automorphisms of $X$. We prove that for $d \ge 8$, there exist mixing zero entropy algebraic $Z^{d}$-actions which do not exhibit isomorphism rigidity…

We show that for certain classes of actions of Z^d, d >= 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and…

Assuming positive entropy we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for…

We prove that any smooth action of $\mathbb Z^{m-1}, m\ge 3$ on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e. isomorphic up to…

We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions $\alpha$ of $\mathbb{R}^k$ on an anonymous manifold $M$ of dimension $2k+1$ provided that there is an ergodic invariant Borel probability measure…

For a fixed topological Markov shift, we consider measure-preserving dynamical systems of Gibbs measures for 2-locally constant functions on the shift. We also consider isomorphisms between two such systems. We study the set of all…

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on…

We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions…

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic $\mathbb{Z}^k$ ($k\geqslant 2$) actions on the torus. We prove a $C^\infty$ local rigidity result for such actions, provided that the smooth…

Consider homogeneous G/H and G/F, for an S-algebraic group G. A lattice {\Gamma} acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors and joinings defined apriori only in the measurable…

Let $d > 1$, and let $(X,\alpha)$ and $(Y,\beta)$ be two zero-entropy ${\mathbb{Z}}^d$-actions on compact abelian groups by $d$ commuting automorphisms. We show that if all lower rank subactions of $\alpha$ and $\beta$ have completely…

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of…

In this article we propose a metric variation on the C^0-version of the Zimmer program for three manifolds. After a reexamination of the isometry groups of geometric three-manifolds, we consider homomorphisms defined on higher rank lattices…

We investigate the rigidity of the $\ell^p$ analog of Roe-type algebras. In particular, we show that if $p\in[1,\infty)\setminus\{2\}$, then an isometric isomorphism between the $\ell^p$ uniform Roe algebras of two metric spaces with…

We prove that every smooth action of Z^k, k>1, on the (k+1)-dimensional torus homotopic to an action by hyperbolic linear maps preserves an absolutely continuous measure. This is a first known result concerning abelian groups of…

We consider an ergodic invariant measure $\mu$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $\mu$ is…

Given a $\mathbb Z^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the…

We present a new argument in the study of positive entropy measures for higher rank diagonalisable actions. The argument relies on a quantitative form of recurrence along unipotent directions (that are not known to preserve the measure).…

We study smooth actions by lattices in higher-rank simple Lie groups. Assuming one element of the action acts with positive topological entropy, we prove a number of new rigidity results. For lattices in $\mathrm{SL}(n,\mathbb{R})$ acting…