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 nilmanifolds associated to step 2 nilpotent Lie groups.
We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $ρ_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $ρ_L$ has a $\mathbb Z$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine $\mathbb Z^2$-action with linear part $ρ_L$ is KAM-rigid under volume preserving perturbations.
In this paper, we study a local rigidity property of $\mathbb Z \ltimes_λ\mathbb R$ affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbations close enough to an affine action is smoothly conjugate to the affine action up to proportional time change.
Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them a…
In this paper, we study a natural class of groups that act as affine transformations of $\mathbb T^N$. We investigate whether these solvable, "abelian-by-cyclic," groups can act smoothly and nonaffinely on $\mathbb T^N$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine condition…
This article discusses two recent works by the author, one with Brown and Hurtado on Zimmer's conjecture and one with Bader, Miller and Stover on totally geodesic submanifolds of real and complex hyperbolic manifolds. The main purpose of juxtaposing these two very disparate sets of results in one article is to emphasize a common aspect: that the study of invariant and partially invariant measures outside the homogeneous setting is important to questions about rigidity in geometry and dynamics. I will also discuss some open questions including some that seem particularly compelling in light of …
We prove a rigidity result for foliations on surfaces of genus two, which can be seen as a generalization to higher genus of Herman's theorem on circle diffeomorphisms and, correspondingly, flows on the torus. We prove in particular that, if a smooth, orientable foliation with non-degenerate (Morse) singularities on a closed surface of genus two is minimal, then, under a full measure condition for the rotation number, it is differentiably conjugate to a linear foliation. The corresponding result at the level of Poincaré sections is that, for a full measure set of interval exchange transformati…
We classify locally finite joinings with respect to the Burger-Roblin measure for the action of a horospherical subgroup $U$ on $Γ\backslash G$, where $G = \operatorname{SO}(n,1)^\circ$ and $Γ$ is a convex cocompact and Zariski dense subgroup of $G$, or geometrically finite with restrictions on critical exponent and rank of cusps. We also prove in the more general case of $Γ$ geometrically finite and Zariski dense that certain $U$-equivariant set-valued maps are rigid.
考虑复射影平面上一类由二次向量场在固定仿射邻域内确定的叶理。这样的叶理一般在无穷远处有一条不变的无穷远直线。两个在$\mathbb C P^2$上具有奇异性的叶理是拓扑等价的，只要存在一个射影平面到自身上的同胚，该同胚保持叶上和$\mathbb C P^2$中的方向，并将第一个叶理的叶带到第二个叶理的叶上。我们证明了这个类的一般叶理可能是拓扑等价的，但有限数量的同一类的叶理，模仿射等价。此属性称为\emph{total rigidity}。Lins Neto最近的结果表明，上述有限数不超过240。这是两篇密切相关的论文中的第一篇。它涉及二次叶理的刚性性质，而第二个研究更高程度的叶理。
We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are h…
We show that for Lebesgue almost all $d$-tuples $(θ_1,\ldots,θ_d)$, with $|θ_j|>1$, any self-affine measure for a homogeneous non-degenerate iterated function system $\{Ax+a_j\}_{j=1}^m$ in ${\mathbb R}^d$, where $A^{-1}$ is a diagonal matrix with the entries $(θ_1,\ldots,θ_d)$, has power Fourier decay at infinity.
