Let $Γ$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $Γ$ and a "direction" in $G$ (i.e. an element $θ$ of the projective space $P(\goth g)$ of the Lie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$ along $θ$ are introduced. It is shown that the set of recurrent directions $\Cal R(T)$ and the set of rigid directions for $T$ are both $G_δ$. In the case where $G=\Bbb R^d$ and $Γ=\Bbb Z^d$, we prove that (a) for each $G_δ$-subset $Δ$ of $P(\goth g)$ and a countable subset $D\subsetΔ$, there is a rank-one action $T$ such that $D\subset\Cal R(T)\subsetΔ$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinite measure preserving action $T$ of $Γ$. This answers partly a question from a recent paper by A.~Johnson and A.~{\c S}ahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_3(\Bbb R)$ and $Γ=H_3(\Bbb Z)$, a rank-one $Γ$-action $T$ is constructed for which $\Cal R(T)$ is not invariant under the natural "adjoint" $G$-action.
Let G = SU(n,1), n >1 be the orientation-preserving isometry group of the complex hyperbolic space with an Iwasawa decomposition G = KAN. We prove local rigidity of a family of certain actions of a subgroup of AN on the imaginary boundary of the complex hyperbolic space.
We construct a set $S$ such that every translate of $S$ is a set of recurrence and a set of rigidity for a weak mixing measure preserving system. This construction generalizes or strengthens results of Katznelson, Saeki, Forrest, and Fayad and Kanigowski. The construction provides a density analogue of Julia Wolf's results on popular differences in finite abelian groups.
We estimate size of recurrence of an action of a nilpotent group by homeomorphisms of a compact space for polynomial mappings into a nilpotent group form the partial semigroup $(\mathcal{P}_{f}(\mathbb{N}),\uplus)$. To do this we have used algebraic structure of the Stone-Čech copactification partial semigroup and that of the given nilpotent group.
The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about $\mathbb{Z}$-actions extend to this setting: 1. If $(a_n)$ is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system. 2. There exists a sequence $(r_n)$ such that every translate is both a rigidity sequence and a set of recurrence. The first of these results was shown for $\mathbb{Z}$-actions by Adams, Fayad and Thouvenot, and…
Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences …
The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $Λ$ contained in an optical hypersurface of a cotangent bundle, under …
In this paper we examine the interplay between recurrence properties and the shadowing property in dynamical systems on compact metric spaces. In particular, we demonstrate that if the dynamical system $(X,f)$ has shadowing, then it is recurrent if and only if it is minimal. Furthermore, we show that a uniformly rigid system $(X,f)$ has shadowing if and only if $X$ is totally disconnected and use this to demonstrate the existence of a space $X$ for which no surjective system $(X,f)$ has shadowing. We further refine these results to discuss the dynamics that can occur in spaces with compact spa…
We generalize the IP-polynomial Szemerédi theorem due to Bergelson and McCutcheon and the nilpotent Szemerédi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.
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.
We consider expansive group actions on a compact metric space containing a special fixed point denoted by $0$, and endomorphisms of such systems whose forward trajectories are attracted toward $0$. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to $0$ in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in p…
We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.
We prove an analog for integrable measurable cocycles of Pansu's differentiation theorem for Lipschitz maps between Carnot-Carathéodory spaces. This yields an alternative, ergodic theoretic proof of Pansu's quasi-isometric rigidity theorem for nilpotent groups, answers a question of Tim Austin regarding integrable measure equivalence between nilpotent groups, and gives an independent proof and strengthening of Austin's result that integrable measure equivalent nilpotent groups have bi-Lipschitz asymptotic cones. Our main tools are a nilpotent-valued cocycle ergodic theorem and a Poincaré recur…
We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $Λ$ a lattice in $G$, and $Γ$ a subgroup of the affine group Aff$(G)$ stabilizing $Λ$. Then the action of $Γ$ on $G/Λ$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $Γ$ on $\mathbb{P}(\frak{g})$ admits no $Γ$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nil…
Let $G$ be a connected nilpotent Lie group. Given probability-preserving $G$-actions $(X_i,Σ_i,μ_i,u_i)$, $i=0,1,...,k$, and also polynomial maps $ϕ_i:\mathbb{R}\to G$, $i=1,...,k$, we consider the trajectory of a joining $λ$ of the systems $(X_i,Σ_i,μ_i,u_i)$ under the `off-diagonal' flow \[(t,(x_0,x_1,x_2,...,x_k))\mapsto (x_0,u_1^{ϕ_1(t)}x_1,u_2^{ϕ_2(t)}x_2,...,u_k^{ϕ_k(t)}x_k).\] It is proved that any joining $λ$ is equidistributed under this flow with respect to some limit joining $λ'$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, r…
We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterise continuous orbit equivalence in terms of isomorphisms of C*-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Secondly, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this …
In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between a t.d.s. $(X,T)$ and the t.d.s. $(K(X),T_K)$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a t.d.s. which has dense small periodic sets but does not have dense distal points, solving that way open questions existing in the literature.
Let $G$ be a countable abelian group. We study ergodic averages associated with configurations of the form $\{ag,bg,(a+b)g\}$ for some $a,b\in\mathbb{Z}$. Under some assumptions on $G$, we prove that the universal characteristic factor for these averages is a factor of a $2$-step nilpotent homogeneous space. As an application we derive a Khintchine type recurrence result. In particular, we prove that for every countable abelian group $G$, if $a,b\in\mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finite index in $G$, then for every $E\subset G$ and $\varepsilon>0$ the set $$\{g\in …
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers. Secondly, we characterise geometric properties of a group (coarse e…
