Danijela Damjanovic, James Tanis
In this paper we prove a perturbative result for a class of $\mathbb Z^2$ actions on Heisenberg nilmanifolds, which have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.
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Stable Lyapunov Spectrum Rigidity of Nilmanifold Endomorphisms
本文研究幂零流形阿诺索夫映射，在非可逆和不可约条件下，证明不变不稳定丛、拓扑共轭与常数周期稳定李雅普诺夫指数三者等价，即稳定李雅普诺夫谱刚性。
Suppose $X_{1}, X_{2}$ are nilmanifolds and $ρ, σ$ are automorphism actions of a discrete group $Γ$ on $X_{1}$ and $X_{2}$ respectively. We show that if $(X_{1},ρ)$ and $(X_{2}, σ)$ satisfy certain additional conditions then every $Γ$-equivariant continuous map from $(X_{1},ρ)$ to $(X_{2},σ)$ is an affine map.
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Γ$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
We study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds. Assuming that every nontrivial element acts ergodically, we prove that such actions are mixing of all orders. We further show exponential 2-mixing and 3-mixing. As an application we prove smooth cocycle rigidity for higher-rank abelian groups of nilmanifold automorphisms.
Given a $\mathbb Z^r$-action $α$ on a nilmanifold $X$ by automorphisms and an ergodic $α$-invariant probability measure $μ$, we show that $μ$ is the uniform measure on $X$, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action: 1. The factor measure has zero entropy under every element of the action; 2. The factor action is virtually cyclic. We also deduce a rigidity property for invariant closed subsets.
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.
Partially hyperbolic lattice actions on 2-step nilmanifolds
本文研究高阶秩格在2步幂零流形上包含部分双曲元素的作用，在技术假设下证明其必为仿射映射，从而推广了Brown等人的刚性结果。
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…
We prove several results concerning smooth $\mathbb R^k$ actions with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and cohomology is often finite-dimensional, even trivial. Further we consider a class of examples of $\mathbb R^2$ actions on 2-step nilmanifolds, which have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.
Consider actions of $\Z ^r$ by ergodic automorphisms on a compact nilmanifolds for $r \geq 2$. We show that small $C^k$ perturbations of such higher rank partially hyperbolic actions are smoothly conjugate to the original action, using a KAM scheme. The driving force for convergence of this iteration is the exponential mixing of the original action.
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。
In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's rigidity theorem is that the curves C_{1,n} and C_{2,m} intersect transversally. We give a purely algebraic proof that the intersection points are 3-adically integral and use this to prove transversality. We also prove an analogous result when c_1 or c_2 or both are taken to be preperiodic with tail length exactly 1.
Global rigidity for some partially hyperbolic abelian actions with 1-dimensional center
本文证明了某些2步幂流形上阿贝尔部分双曲高秩作用的全局刚性，即所有此类作用均光滑共轭于仿射模型，并由此分类了小扰动下的中心化子。
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…
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.
We define globally hypoelliptic smooth $\mathbb R^k$ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $k=1$, strong global rigidity is conjectured for such actions by Greenfield-Wallach and Katok: every such action is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces \cite{FFRH}. In this paper we conjecture that among homogeneous $\mathbb R^k$ actions ($k\ge 2$) on homogeneous spaces globally hypoelliptic actions exist only on nilman…
Factor-Closure of Pro-Nilsystems via Local Rigidity of Nilsystems
本文通过幂零系统的局部刚性结果，独立于Host-Kra结构定理证明了幂零系统的因子闭包，还推导了拓扑幂零系统的因子闭包，结合Tao的推导给出了Host-Kra结构定理的新证明。
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