Distal higher rank lattice actions on surfaces

Distal strongly ergodic actions

Let $η$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting an invariant probability measure $μ$, such that the resulting dynamical system $(X, μ, F_3)$ is strongly ergodic and distal of rank $η$. In particular this shows that there is a $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.

Distality of Certain Actions on $p$-adic Spheres

Consider the action of $GL(n,\mathbb{Q_p})$ on the $p$-adic unit sphere $\mathcal{S}_n$ arising from the linear action on $\mathbb{Q}_p^n\setminus\{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $GL(n,\mathbb{Q}_p)$ on $\mathcal{S}_n$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on $\mathcal{S}_n$. (2) the closure of the image of $\mathfrak{S}$ in $PGL(n,\mathbb{Q}_p)$ is a compact group. On $\mathcal{S}_n$, we consider the `affine' maps $\overline{T}_a$ corresponding to $T$ in $GL(n,\mathbb{Q}_p)$ and a nonzero $a$ in $\mathbb{Q}_p^n$ satisfying $\|T^{-1…

交换作用的联合点远端性与结构理论

Joint Point-Distality and structure theory for commuting actions

本文证明交换极小作用的点远端性在联合作用下保持，构造公共高近端扩张，比较子作用与联合作用的典范结构，并给出联合Furstenberg塔的递归描述及严格性示例。

Distal Actions of Automorphisms of Lie Groups $G$ on $\rm Sub_{G}$

For a locally compact metrizable group $G$, we study the action of $\rm Aut(G)$ on $\rm Sub_G$, the set of closed subgroups of $G$ endowed with the Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of the $T$-action on $\rm Sub_G$ with that of the $T$-action on $G$ under a certain condition. If $G$ is a connected Lie group, we characterise the distality of the $T$-action on $\rm Sub_G$ in terms of compactness of the closed group generated by $T$ in $\rm Aut(G)$ under certain conditions on the center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of …

Weak containment rigidity for distal actions

We prove that if a measure distal action $α$ of a countable group $Γ$ is weakly contained in a strongly ergodic probability measure preserving action $β$ of $Γ$, then $α$ is a factor of $β$. In particular, this applies when $α$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.

Some Properties of Distal Actions on Locally Compact Groups

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We also show that a compactly generated locally compact group of polynomial growth has a compact normal subgroup $K$ such that $G/K$ is distal and the conjugacy action of $G$ on $K$ is ergodic; moreover, if $G$ itself is (pointwise) distal then $G$ is Lie projective. We prove a decomposition theorem for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient cond…

A model-theoretic approach to rigidity of strongly ergodic, distal actions

We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other di…

远端作用与移位卷积性质

Distal actions and shifted convolution property

本文定义了局部紧群的移位卷积性质（SCP），证明了远端作用的商作用在特定条件下保持远端性，并利用共轭作用的逐点远端性条件刻画了具有 SCP 的群，揭示了其与 Choquet Deny 测度及概率测度空间扩张远端性的联系。

自同构在Lie群单参数子群空间上的远端作用刻画

Characterisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups

本文刻画了Lie群自同构在单参数子群空间上的远端作用，将其与群上作用及自同构生成子群的紧性关联，并推广至子群作用，证明共轭远端作用等价于群为紧或紧与向量群直积。

Distal actions on coset spaces in totally disconnected, locally compact groups

Let $G$ be a totally disconnected, locally compact group and let $H$ be an equicontinuously (for example, compactly) generated group of automorphisms of $G$. We show that every distal action of $H$ on a coset space of $G$ is a SIN action, with the small invariant neighbourhoods arising from open $H$-invariant subgroups. We obtain a number of consequences for the structure of the collection of open subgroups. For example, it follows that for every compactly generated subgroup $K$ of $G$, there is a compactly generated open subgroup $E$ of $G$ such that $K \le E$ and such that every open subgrou…

幂零群 $G$ 的自同构在 Sub_$G$ 上的远距作用及其在李群中格的应用

Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups

本文研究了局部紧群自同构在闭子群空间上的远距作用，证明了幂零群和可解李群格上的远距性等价条件，并刻画了紧生成幂零群的结构。

环面单参数子群空间上自同构作用的动力学及其应用

Dynamics of actions of automorphisms on the space of one-parameter subgroups of a torus and applications

本文研究连通李群自同构在闭单参数子群紧空间 Sub$^p_G$ 上的 distal 与扩张动力学，证明环面上仅有限阶自同构 distal 作用，并证明中心环面维数至少为 2 的连通李群不存在扩张自同构，推广了若干已有结果。

离散群 $G$ 的自同构在 Sub$_G$ 上作用的动力学及其在 Lie 群格中的应用

Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups

本文研究了离散群自同构在子群空间上的动力学行为及连通 Lie 群中格的结构性质，证明了多循环群或 Lie 群格不存在膨胀自同构，并给出了自同构在 Lie 群及其格的子群空间上远距作用的等价条件。

群作用下与所有极小系统的不交性

Disjointness with all minimal systems under group actions

本文针对可数离散群作用，给出了传递系统与所有极小系统不交的充要条件，并证明该条件蕴含无穷传递性与稠密极小点，同时建立了具有稠密 distal 点的无穷传递系统与所有极小系统不交的结论。

Global rigidity of higher rank Anosov algebraic actions

We show that all $C^\infty$ Anosov $Z^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.

KAM rigidity for partially hyperbolic affine Z^k actions on the torus with a rank one factor

We show that ergodic affine abelian discrete actions on the torus, that have a rank-one factor in their linear part, are locally rigid in a KAM sense if and only if the rank one factor is trivial and the action is higher-rank transversally to this factor. Since it has been proved by Damjanovic and Katok that affine actions with higher-rank linear part are locally rigid, our result completes the local rigidity picture for affine actions on the torus.

Local rigidity of higher rank non-abelian action on tours

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms and prove the existence of such action on any torus TN for any even N ? 6. We also give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov-Arnold-Moser) iterative scheme.

Rigidity of the measurable structure for algebraic actions of higher rank abelian groups

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 rigidity theorems for isomorphisms and joinings as corollaries.

Smooth Factors of Projective Actions of Higher Rank Lattices and Rigidity

We study smooth factors of the standard actions of lattices in higher rank semisimple Lie groups on flag manifolds. Under a mild condition on existence of a single differentiable sink, we show that these factors are $C^{\infty}$-conjugate to the standard actions on flag manifolds.

On Local Rank, Joinings and Asymptotic Properties of Measure-Preserving Actions

The note contains a collection of facts and observations around locally rank one actions as well as constructions connected with some results by T.Downarowicz, A.Katok, J.King, F.Parreau, A.A.Prikhodko, E.Roy, J.Serafin, J-P.Thouvenot et al.

Local rigidity of homogeoenous actions of parabolic subgroups of rank-one Lie groups

We show the local rigidity of the natural action of the Borel subgroup of SO_+(n,1) on a cocompact quotient of SO_+(n,1) for n>2.

Rank one Z^d actions and directional entropy

We study the directional entropy of rank one Z^d actions. We show that if the sequence of towers generating the action are rectangular in shape, then there is always a direction along which the directional entropy is zero. If the rectangles satisfy a "sub-exponential growth" condition, then we show that the directional entropy is zero in every direction.

Cocycle rigidity of partially hyperbolic abelian actions with almost rank one factors

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank 1 factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

乘积作用的秩一因子的不稳定性

Instability for rank one factors of product actions

本文通过反例反驳了Katok-Spatzier猜想的标准解释，并提出了替代性问题。

Actions of finite rank: weak rational ergodicity and partial rigidity

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergod…

Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups

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 an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.

Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces

Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1 \in S = S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x \in X$, where $G = \langle S \rangle$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset $U$ of $X$, there is $F \subseteq G$ finite such that $\bigcap_{g \in F}g(U)$ is $G$-invariant. All of these are equivalent to a notion of recurrence, which is a variation on…

Limit theorems for higher rank actions on Heisenberg nilmanifolds

The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct \emph{Bufetov functionals} on rectangles on $(2g+1)$-dimensional Heisenberg manifolds. We prove that deviation of the ergodic integral of higher rank actions is described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals which have variance 1, converges along certain subs…

Slow entropy of higher rank abelian unipotent actions

We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/Γ$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $\operatorname{Lie}(G)$ induced by $\operatorname{Lie}(U)$. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $G$. This generalizes the rank one results from [A. Kani…

Rigidity for higher rank lattice actions on dendrites

We study the rigidity in the sense of Zimmer for higher rank lattice actions on dendrites and show that: (1) if $Γ$ is a higher rank lattice and $X$ is a nondegenerate dendrite with no infinite order points, then any action of $Γ$ on $X$ cannot be almost free; (2) if $Γ$ is further a finite index subgroup of $SL_n(\mathbb Z)$ with $n\geq 3$, then every action of $Γ$ on $X$ has a nontrivial almost finite subsystem. During the proof, we get a new characterization of the left-orderability of a finitely generated group through its actions on dendrites.

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.

高秩格群的边界作用：低维下的分类与嵌入、局部刚性、光滑因子

Boundary actions by higher-rank lattices: Classification and embedding in low dimensions, local rigidity, smooth factors

本文研究了高秩格群在紧流形上的作用，通过分类相关不变测度推导出新的刚性结果，证明了低维作用共轭于边界作用，并验证了标准边界作用的局部刚性及光滑因子的分类。

Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds

We consider smooth actions of lattices in higher-rank semisimple Lie groups on manifolds. We define two numbers $r(G)$ and $m(G)$ associated with the roots system of the Lie algebra of a Lie group $G$. If the dimension of the manifold is smaller than $r(G)$, then we show the action preserves a Borel probability measure. If the dimension of the manifold is at most $m(G)$, we show there is a quasi-invariant measure on the manifold such that the action is measurable isomorphic to a relatively measure preserving action over a standard boundary action.

秩一$\u2124^d$作用的有限里程表因子

Finite odometer factors of rank one $\mathbb{Z}^d$-actions

本研究给出Følner秩一$\u2124^d$作用对应不同有限/任意$\u2124^d$-里程表因子及共轭情形的显式刻画条件，基于切割堆叠过程，推广了秩一$\u2124$-作用的相关结果。

可数群的秩一非奇异作用及其里程计因子

Rank-one nonsingular actions of countable groups and their odometer factors

该研究引入可数群的非奇异秩一作用，证明其与非奇异$(C,F)$作用等价，构造非奇异里程计，给出秩一作用含有限因子、里程计因子及同构于里程计的充要条件，还构造相关极小模型并揭示非交换情形与经典情形的差异。

Local Rigidity of Higher Rank Homogeneous Abelian Actions: a Complete Solution via the Geometric Method

We show local and cocycle rigidity for $\R^k \times \Z^l$ partially hyperbolic translation actions on homogeneous spaces $\mc G/ Λ$. We consider a large class of actions whose geometric properties are more complicated than previously treated cases. It is also the first time that partially hyperbolic twisted symmetric space examples have been treated in the literature. The main new ingredient in the proof is a combination of geometric method and the theory of central extensions.

高阶部分双曲代数作用的局部刚性

Local rigidity of higher rank partially hyperbolic algebraic actions

本研究结合表示论、调和分析与KAM迭代，完整解决高阶部分双曲代数作用局部分类问题，证明多类阿贝尔遍历代数作用的$C^\infty$局部刚性，首次得到非可达部分双曲作用及带非阿贝尔幂零根基扭空间的相关新结果。

Spectral properties and approximations of joinings for infinite rank-one actions

An ergodic self-joining of an infinite rank-one transformation is a part of the weak limit of off-diagonal measures. A class of uncountaible cardinality of nonisomorphic transformations with polynomial weak closure is presented. Such actions have minimal self-joinings and some unusual spectral properties. For any set $M$ of positive integers there exists an infinite transformation $T$ such that the products $T\otimes T^m$ have simple singular spectrum as $1<m\in M$, and Lebesgue spectrum as $1<m\notin M$. There are similar effects for the corresponding Gaussian and Poisson suspensions.

Diffeomorphism group valued cocycles over higher rank abelian Anosov actions

We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialisation apply, via the existing global rigidity results, to maximal Cartan actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth…

Finite ergodic index and asymmetry for infinite measure preserving actions

Given $k>0$ and an Abelian countable discrete group $G$ with elements of infinite order, we construct $(i)$ rigid funny rank-one infinite measure preserving (i.m.p.) $G$-actions of ergodic index $k$, $(ii)$ 0-type funny rank-one i.m.p. $G$-actions of ergodic index $k$, $(iii)$ funny rank-one i.m.p. $G$-actions $T$ of ergodic index 2 such that the product $T\times T^{-1}$ is not ergodic. It is shown that $T\times T^{-1}$ is conservative for each funny rank-one $G$-action $T$.

Isomorphism rigidity in entropy rank two

We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.

秩一作用的轨道等价的显式构造

Explicit construction of orbit equivalence for rank-one actions

针对离散可数无限群G的两个秩一保测G作用，从切割与堆叠参数出发构造Borel轨道等价，还证明了其在连续(C,F)作用下的拓扑对应，属于动力系统领域的轨道等价研究。

Exponential Mixing and Smooth Classification of Commuting Expanding Maps

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…

Symmetry of entropy in higher rank diagonalizable actions and measure classification

An important consequence of the theory of entropy of Z-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some …

Quantitative Density under Higher Rank Abelian Algebraic Toral Actions

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent one-dimensional quantitative density result to abelian algebraic actions on higher dimensional tori. Up to finite index, the group actions that we study are conjugate to the action of $U_K$, the group of units of some non-CM number field $K$, on a compact quotient of $K\otimes_{\mathbb Q}\mathbb R$. In such a setting, we investigate how fast the orbit of a generic point can become dense in the torus. This effectivizes a special case of a theorem of Berend; and is deduced from a parallel measure-theoretical statement which effectivi…

高秩阿贝尔群的 Cartan 作用及其分类

Cartan Actions of Higher Rank Abelian Groups and their Classification

本文研究紧流形上具有 Anosov 元素稠密集和 1 维粗 Lyapunov 叶状结构的全 Cartan 作用，通过引入动态定义拓扑群的新工具，将其完全分类为低维 Anosov 流、微分同胚和仿射作用的组合，验证了 Katok-Spatzier 猜想并应用于 Zimmer 纲领。

Global smooth and topological rigidity of hyperbolic lattice actions

In this article we prove global rigidity results for hyperbolic actions of higher-rank lattices. Suppose $Γ$ is a lattice in semisimple Lie group, all of whose factors have rank $2$ or higher. Let $α$ be a smooth $Γ$-action on a compact nilmanifold $M$ that lifts to an action on the universal cover. If the linear data $ρ$ of $α$ contains a hyperbolic element, then there is a continuous semiconjugacy intertwining the actions of $α$ and $ρ$, on a finite-index subgroup of $Γ$. If $α$ is a $C^\infty$ action and contains an Anosov element, then the semiconjugacy is a $C^\infty$ conjugacy. As a coro…

On classification of higher rank Anosov actions on compact manifold

We prove global smooth classification results for TNS totally Anosov Z^k actions on general compact manifolds, under each one of the following conditions: joint integrability, resonance-free or Lyapunov pinching condition. Unlike the previous results, we do not require any uniform quasiconformality or pinching condition of action elements on coarse Lyapunov distributions, nor do we have any restriction on the dimension of coarse Lyapunov distributions. The main novelty is in proving a new standard form of the derivative cocycle for any TNS totally Anosov Z^k action on general manifold. A main …

Rank-one actions, their $(C,F)$-models and constructions with bounded parameters

Let $G$ be a discrete countable infinite group. We show that each topological $(C,F)$-action $T$ of $G$ on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and Rørdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying $(C,F)$-parameters. If $G$ is linearly ordered Abelian then the topological centralizer of $T$ is trivial. If $G$ is monotileable and amenable, denote by ${\cal A}_G$ t…

Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

We prove quantitative equidistribution results for actions of Abelian subgroups of the $2g+1$ dimensional Heisenberg group acting on compact $2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study of the $C^\infty$-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalisation method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in $g$ variables, generalizing the…

On Disjointness of Mixing Rank One Actions

For flows the rank is an invariant by linear change of time. But what we can say about isomorphisms? It seems that in case of mixing flows this problem is the most difficult. However the known technique of joinings provides non-isomorphism for mixing rank-one flows under linear change of time. For automorphisms we consider another problems (with similar solutions). For example, the staircase cutting-and-stacking construction is set by a height $h_1$ of the first tower and a sequence $\{r_j\}$ of cut numbers. Let us consider two similar constructions: one is set by $(h_1, \{r_j\})$, another is …

Minimal Self-Joinings, Bounded Constructions, and Weak Closure of Ergodic Actions

For a weakly mixing bounded rank-one construction the disjointness of its powers is proved. For non-rigid constructions we get minimal self-joinings. Examples of non-mixing rank one actions with explicit weak closure are proposed.

The structures of higher rank lattice actions on dendrites

Let $Γ$ be a higher rank lattice acting on a nondegenerate dendrite $X$ with no infinite order points. We show that there exists a nondegenerate subdendrite $Y$ which is $Γ$-invariant and satisfies the following items: (1) There is an inverse system of finite actions $\{(Y_i, Γ):i=1,2,3,\cdots\}$ with monotone bonding maps $ϕ_i: Y_{i+1}\rightarrow Y_i$ and with each $Y_i$ being a dendrite, such that $(Y, Γ|Y)$ is topologically conjugate to the inverse limit $(\underset{\longleftarrow}{\lim}(Y_i, Γ), Γ)$. (2) The first point map $r:X\rightarrow Y$ is a factor map from $(X, Γ)$ to $(Y, Γ|Y)$; if…

Around King's Rank-One theorems: Flows and Z^n-actions

We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z^n-actions. We prove that these results remain valid in the case of rank-one flows. In the case of rank-one Z^n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors.