Huichi Huang, Hanfeng Li, Enhui Shi, Hui Xu
We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over rational numbers to obtain a measure rigidity result for endomorphisms on $n$-torus.
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We introduce the notion of strongly independent matrices and show the existence of strongly independent matrices in $GL(n,\mathbb{Z})$ over $\mathbb{Z}^n\setminus\{0\}$ when $2n+1$ is a prime number. As an application of strong independence, we give a measure rigidity result for endomorphisms on $n$-torus $\mathbb{T}^n$.
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…
For a nonlinear Anosov diffeomorphism of the 2-torus, we present examples of measures so that the group of $μ$-preserving diffeomorphisms is, up to zero-entropy transformations, cyclic. For families of equilibrium states $μ$, we strengthen this to show that the group of $μ$-preserving diffeomorphism is virtually cyclic.
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.
Rigidity properties for some isometric extensions of partially hyperbolic actions on the torus
本文研究环面上部分双曲作用的等距扩张的局部刚性，证明了在扰动满足相交性质时的$C^\infty$局部刚性及体积保持作用类中的局部刚性，方法基于KAM迭代格式的推广。
We obtain smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. Among other results, we prove that a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ and its linearization are smoothly conjugated since they have the same periodic data. Assuming that for a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ every point is regular (in Oseledec's Theorem sense), then we obtain again smooth conjugacy with its linearization. We also obtain some results on local rigidity of linear Anosov endomorphisms of $d-$torus, where $d…
We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $λ^{wu}_f(x) = λ^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Unifo…
Rigidity of center Lyapunov exponents for Anosov diffeomorphisms on 3-torus
该研究针对3-环面上具三子束部分双曲分裂的Anosov微分同胚，证明若两微分同胚共轭且共轭保持强稳定叶状结构，则对应周期点的中心Lyapunov指数相等，给出已有结果的逆命题，并将结论推广至由Anosov导出的部分双曲微分同胚。
Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity
本文研究二维环面上Anosov自同态的光滑共轭，通过叶层正则性给出充分条件，并刻画了特殊自同态与其线性化的共轭关系。
Quantitative Rigidity of pseudo-rotations on the two-torus and Sarnak's conjecture
该研究针对二维环面的伪旋转，通过定量自由圆盘估计建立了满足$(C,\delta)$偏差条件的不同类型伪旋转的刚性结果，进而证明这些类均满足萨纳克猜想。
For any $C^\infty$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^\infty$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the volume measure is the measure with maximal entropy. We also show that the the metric entropy with respect to the volume measure and the topological entropy of suspension flow over Anosov diffeomorphisms on torus achieve all possible values. Our results fit into two programs related to entropy rigidity and flexibility of Anosov systems.
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 show that under certain boundedness condition, a $C^{r}$ conservative irrational pseudo-rotations on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid. We also obtain $C^0$-rigidity for Hölder pseudo-rotations with similar properties. These provide a partial generalisation of the main results in [B. Bramham, Invent. Math. (2015), no. 2, 561-580; A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, arXiv: 1509.06906v1]. We then use these results to study conservative irrational pseudo-rotations on $\mathbb{T}^2$ with a generic rotation vector that is semi-conjugate to a trans…
Examples beyond Bounded Mean Motion for Quantitative Rigidity on the Two-Torus
该研究为双环面上定量刚性的Sarnak猜想相关论文，构造了不满足有界平均运动的半无理及全无理微分同胚例子，证明其数量为连续统多个。
Semi-conjugacy rigidity for endomorphisms derived from Anosov on the 2-torus
研究2-环面上由非扩张Anosov自映射导出的非可逆部分双曲自映射的半共轭刚性，证明了存在部分双曲分裂当且仅当半共轭于线性化，并进一步讨论了不同子丛条件下的共轭性质与Lyapunov指数刚性。
We prove that every smooth action of Z^k, k>1, on the (k+1)-dimensional torus homotopic to an action by hyperbolic linear maps preserves an absolutely continuous measure. This is a first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
Deformational rigidity of integrable metrics on the torus
针对二维环面仅Liouville度量为可积度量的猜想，研究其形变版本，证明共形类中非平坦Liouville度量的一大类可积形变后仍为Liouville度量，且方法可推广至高维环面得到类似结论，并回顾了相关现有成果。
U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfaces
本文研究了部分超双曲映射在二维流形上的U-Gibbs测度刚性问题，证明了在特定条件下测度的唯一性或映射的特殊性，为相关系统提供了新的理论结果。
We show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This generalizes recent result of Karaliolios [4] of the action generators to higher rank abelian actions, and the result of Moser [8] to higher dimensional tori.
Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations
本文通过沿幂单方向的定量回复方法，给出了高阶对角化作用在 $SL_2$ 形式算术商上正熵测度的分类及其 Adelic 版本，并应用于整数倍丢番图逼近问题。
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 $\mathbb{Q}.$
We show a Hopf type rigidity for thermostats without conjugate points on a 2-torus
Deformation and perturbative rigidity near de la Llave examples
研究证明德拉·拉维亚例子附近的微分同胚存在变形和局部刚性，揭示了其在动力系统中的特殊性质。
Smooth rigidity for 3-dimensional dissipative Anosov flows
本文证明三维耗散Anosov流在连续共轭下，除例外情形外，共轭或光滑或交换正负SRB测度，并由此建立局部刚性、改进经典定理并揭示Teichmüller空间的分层结构。
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.
Given an integer $k \geq 5$, and a $C^k$ Anosov flow $Φ$ on some compact connected $3$-manifold preserving a smooth volume, we show that the measure of maximal entropy (MME) is the volume measure if and only if $Φ$ is $C^{k-\varepsilon}$-conjugate to an algebraic flow, for $\varepsilon>0$ arbitrarily small. Besides the rigidity, we also study the entropy flexibility, and show that the metric entropy with respect to the volume measure and the topological entropy of suspension flows over Anosov diffeomorphisms on the $2$-torus achieve all possible values subject to natural normalizations. Moreov…
We study the motion of a charge on a conformally flat Riemannian torus in the presence of magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ``twisted'' version of the recent generalisation of Hopf's rigidity result obtained by Burago and Ivanov.
Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$
本文证明了三维环面上保守 Anosov 微分同胚的 $C^1$ 邻域内，任意 $C^2$ 微分同胚要么强稳定与强不稳定丛联合可积，要么其所有全支撑 $u$-Gibbs 测度均为 SRB 测度。
We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of KAM tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily small re…
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。
Invariant sets and spectral rigidity of Hamiltonian diffeomorphisms
该研究解决了Polterovich2002年关于二维哈密顿微分同胚Hofer度量刚性的问题，在高维自治哈密顿流等情形取得进展，相关结果适用于Viterbo谱度量等，运用定量拉格朗日Floer理论完成论证。
Contact structures and Beltrami fields on the torus and the sphere
本文在3-球面和3-环面上构造了新的显式紧和过度扭转接触结构，证明了弱相容度量下接触球面定理不成立，并建立了紧接触结构的几何刚性。
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