Siddhartha Bhattacharya
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.
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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.
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.
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。
We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and regularity of solutions of the cohomological equation for such automorphisms. Our method is based on the quantitative equidistribution results for polynomial maps combined with Diophantine estimates.
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 discuss conditions under which an affine automorphism of a compact nilmanifold is almost automorphic, and the structure of such automorphisms from a dynamical point of view.
We give a necessary and sufficient condition for $k$-step nilmanifolds associated with graphs $(k \geq 3)$ to admit Anosov automorphisms. We also prove nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds.
We study 9 and 10-dimensional Anosov Lie algebras, by using the properties of very special algebraic numbers. We classify k-step complex Anosov Lie algebras for $k>2$ and in the two step case, we give an example in each possible type, or we give a non-existence result.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
Smooth Models for Certain Fibered Partially Hyperbolic Systems
本文证明了在纤维限制下，nilmanifold上的纤维化部分双曲系统叶共轭于纤维等距且底空间为双曲自同构的光滑模型，并推广了Hiraide关于Anosov同胚拓扑共轭的结果。
Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $μ.$ We characterize the countable groups $Γ$ of automorphisms of $X$ for which the Koopman representation $κ$ on $L^2(X,μ)$ has a spectral gap. More specifically, we show that $κ$ does not have a spectral gap if and only if there exists a non-trivial $Γ$-invariant quotient solenoid (that is, a finite-dimensional, connected, compact abelian group) on which $Γ$ acts as a virtually abelian group.
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。
Exponential multiple mixing for commuting automorphisms of a nilmanifold
本文研究了紧致幂零流形上 $\mathbb{Z}^l$ 的自同构作用，在非零元素遍历性假设下，证明了该作用对任意 $n\geq 2$ 满足指数 $n$-混合，推广了 Gorodnik 和 Spatzier 的已有结果。
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.
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 these actions.
We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let $L\colon\mathbb T^3\to\mathbb T^3$ be a hyperbolic automorphism of the 3-torus with real spectrum and let $f$ be a $C^1$ small perturbation of $L$. Then $f$ is smoothly ($C^\infty$) conjugate to $L$ if and only if obstructions to $C^1$ conjugacy given by the eigenvalues at periodic points of $f$ vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya for conformal automorphisms this completes the local rigidity program for hyperbolic automorphis…
We classify complex projective surfaces with an automorphism of positive entropy for which the unique invariant measure of maximal entropy is absolutely continuous with respect to Lebesgue measure.
Polynomial rigidity and spectrum of Sidon automorphisms
本文提出谱不相交的Sidon自同构，利用Kolmogorov线性决定性算子的多项式刚性检验谱奇异性，并在混合Gaussian与Poisson悬浮类中实现新的谱重数集合。
After more than thirty years, the only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. It is also important to note that the existence of an Anosov automorphism is a really strong condition on an infranilmanifold. Any Anosov automorphism determines an automorphism of the (rational) Lie algebra of the Mal'cev completion of the corresponding lattice which is hyperbolic and unimodular. These two conditions together are strong enough to make of such rational nilpotent Lie algebras (called Anosov Lie algebras) very distinguished objects. In this paper, we…
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any vo…
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.
Centralizer classification and rigidity for some partially hyperbolic toral automorphisms
本文研究环面自同构邻近微分同胚的中心化子分类与刚性，在低维及高维情形下证明了中心化子较大时的光滑共轭刚性，并对辛情形给出有限指数分类。
Periodic data rigidity of Anosov automorphisms with Jordan blocks
本文证明带 Jordan 块的 Anosov 自同构不满足周期数据刚性，并提出精细化周期数据，用以刻画四维环面上此类系统的 $C^{1+}$ 共轭。
Let $d > 1$, and let $(X,α)$ and $(Y,β)$ 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 $α$ and $β$ have completely positive entropy, then any measurable equivariant map from $X$ to $Y$ is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.
We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$ We show that if $H$ is weakly differentiable then it is $C^{1+Hölder}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local rigidity results.
Dominated splitting from constant periodic data and global rigidity of Anosov automorphisms
本文研究双曲系统上 GL(d,R) 上循环的支配分裂判据，并据此证明一般环面 Anosov 自同构及邻近某些非线性 Anosov 微分同胚的全局周期数据刚性。
Global smooth rigidity for toral automorphisms
本文证明非常弱不可约双曲或遍历部分双曲环面自同构的$C^1$或$C^{1+\text{Hölder}}$共轭自动提升为$C^\infty$，从而改进局部与全局刚性结果。
We consider an irreducible Anosov automorphism L of a torus T^d such that no three eigenvalues have the same modulus. We show that L is locally rigid, that is, L is C^1 conjugate to any C^1-small perturbation f with the same periodic data. We also prove that toral automorphisms satisfying these assumptions are generic in SL(d,Z). Examples constructed in the Appendix by Rafael de la Llave show importance of the assumption on the eigenvalues.
We study the regularity of the conjugacy between an Anosov automorphism $L$ of a torus and its small perturbation. We assume that $L$ has no more than two eigenvalues of the same modulus and that $L^4$ is irreducible over $\mathbb Q$. We consider a volume-preserving $C^1$-small perturbation $f$ of $L$. We show that if Lyapunov exponents of $f$ with respect to the volume are the same as Lyapunov exponents of $L$, then $f$ is $C^{1+\text{Hölder}}$ conjugate to $L$. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.
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。
We consider a hyperbolic toral automorphism $L$ and its $C^1$-small perturbation $f$. It is well-known that $f$ is Anosov and topologically conjugate to $L$, but a conjugacy $H$ is only Hölder continuous in general. We discuss conditions for smoothness of $H$, such as conjugacy of the periodic data of $f$ and $L$, coincidence of their Lyapunov exponents, and weaker regularity of $H$, and we summarize questions, results, and techniques in this area. Then we introduce our new results: if $H$ is weakly differentiable then it is $C^{1+\text{Hölder}}$ and, if $L$ is also weakly irreducible, then $H…
Multiple Fractional Cohomological Equations and Quantitative Mixing on Nilmanifolds
本文提出了一种新的分析方法，用于研究nilmanifolds上自同构的定量混合性，通过引入并求解I型多重分数上同调方程（求和型），证明了这些方程在由边缘$0$处谱行为支配的cohomology-free范围内可解，并在弱稳定/不稳定子群方向上给出了部分Sobolev/Hölder范数的估计。作为结果，我们获得了在部分正则性下二阶相关函数的指数衰减，无需横截导数，并在所有阶数上实现了定量混合（定量Rokhlin定理），其速率明确体现在动力学数据中。特别是，我们证明了无理自同构在$C^\infty$可观察量下表现出所有阶数的超指数混合性。到目前为止，这些是超指数混合性超越torus的例子，以及所有阶数超指数混合性的首次例子。
In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. We give a different proof that the existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier. Similarly, we prove that given a grading of one these types and a finite group of automorphisms, there always exist a positive grading which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on th…
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。
Berend gives necessary and sufficient conditions on a $Z^r$-action $α$ on a torus $T^d$ by toral automorphisms in order for every orbit be either finite or dense. One of these conditions is that on every common eigendirection of the $Z^r$-action there is an element $\mathbf n \in Z^r$ so that $α^\mathbf n$ expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit $\{α^\mathbf n.x : \mathbf n \in Ω\}$ where $Ω$ is a set of elements which acts approximately as the identity on a given set of eigendirections. This…
We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\T^N$.
In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus $\mathbb{T}^d, d \geq 3,$ $C^1-$close to a linear automorphism diagonalizable over $\mathbb{R}$ and such that its characteristic polynomial is irreducible over $\mathbb{Q}.$
Periodic data rigidity for cocycles and hyperbolic automorphisms
研究双曲动力系统上Hölder连续线性协同映射的上同调及Anosov系统 conjugacy的正则性，证明周期数据 conjugacy 下的Hölder上同调条件，并得出弱不可约双曲自同构的全局周期数据刚性结果。
Global periodic-data rigidity for irreducible toral automorphisms
针对线性化不可约的环面Anosov微分同胚，证明若所有周期点的导数与线性化共轭，则Franks-Manning共轭是C^{1+Hölder}的，即周期数据完全刻画全局刚性。
In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute…
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…
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 property.
Some rigidity results for polynomial automorphisms of C^2
本文研究复平面上多项式自同构的刚性性质，证明了Julia集的复切片不可能是光滑曲线，并探讨了自同构与叶系的关系，以及数域上鞍点周期轨道乘数的定义域。
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 prove that topologically isomorphic linear cellular automaton shifts are algebraically isomorphic. Using this, we show that two distinct such shifts cannot be isomorphic. We conclude that the automorphism group of a linear cellular automaton shift is a finitely generated abelian group.
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.
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 algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we consruct various examples of Z^d-actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomor…
We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.
We prove that the Julia set of a Henon type automorphism on C^2 is very rigid: it supports a unique positive ddc-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kaehler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Henon type maps on C^k and of automorphisms of compact Kaehler manifolds.
For a Hénon map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a Hénon map turns out to be a Short $\mathbb{C}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short $\mathbb{C}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of …
We study nilmanifolds admitting Anosov automorphisms by applying elementary properties of algebraic units in number fields to the associated Anosov Lie algebras. We identify obstructions to the existence of Anosov Lie algebras. The case of 13-dimensional Anosov Lie algebras is worked out as an illustration of the technique. Also, we recapture the following known results: (i) Every 7-dimensional Anosov nilmanifold is toral, and (ii) every 8-dimensional Anosov Lie algebra with 3 or 5-dimensional derived algebra contains an abelian factor.
Global rigidity of smooth ${\mathbb Z}\ltimes_λ{\mathbb R}$-actions on ${\mathbb T}^2$
本文研究环面上局部自由 ${\mathbb Z}\ltimes_\lambda{\mathbb R}$ 作用的整体刚性，证明 $C^r(r\geq2)$ 作用 $C^{r-\varepsilon}$-共轭于仿射作用，并给出反例说明 $C^{1+\alpha}$ 情形仅半共轭。
Rigidity of periodic points for loxodromic automorphisms of affine surfaces
本文证明仿射曲面上动力学次数大于1的自同构若共享稠密周期点集则周期点完全相同，通过构造典范高度并利用算术等分布理论（含Moriwaki高度情形）完成证明。
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.
Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$ algebraic dynamical system is isomorphic to the dual of the phase group. We show that this duality result is part of an exact sequence if $k=1$. The exact sequence is a well known algebraic object, which has been applied by Schmidt in his work on rigidity. We show that it can be derived from dynamical considerations only. The constructions naturally lead to an almost 1-1-coding of certain Pisot automorphisms by their associated $β$-shift, generalizing similar results for Pisot automorphisms of the torus.
Kummer Rigidity for Hyperkähler Automorphisms
本研究针对射影超Kähler流形上具有正拓扑熵且以体积测度为最大熵测度的全纯自同构，通过Jensen不等式证明分布的一致伸缩性及流形平坦性，结合已有结果推出其为Kummer型例子，部分推广了复曲面对应结论。
Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$
研究多项式 $\mathbb{C}^2$ 自动同构的逃逸集刚性，证明其自同构群为有限循环群。
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