Tracy L. Payne
Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms.
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A characterization of Anosov rational forms in nilpotent Lie algebras associated to graphs
本文刻画了与图相关的幂零李代数中哪些有理形式对应的幂流形允许Anosov微分同胚，首次完整描述了一类具有多个非同构有理形式的所有Anosov情形，并纠正了文献中的错误结果。
In this paper we classify algebraic Anosov actions of nilpotent Lie groups on closed manifolds, extending the previous results by P. Tomter. We show that they are all nil-suspensions over either suspensions of Anosov actions of Z^k on nilmanifolds, or (modified) Weyl chamber actions. We check the validity of the generalized Verjovsky conjecture in this algebraic context. We also point out an intimate relation between algebraic Anosov actions and Cartan subalgebras in general real Lie groups.
An infra-nilmanifold is a manifold which is constructed as a quotient space $Γ\backslash G$ of a simply connected nilpotent Lie group $G$, where $Γ$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so called affine maps. The manifold $Γ\backslash G$ is said to be modeled on the Lie group $G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism. However, it is far from obvious which of these infra--nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question fo…
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。
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.
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…
We consider Anosov actions of a Lie group $G$ of dimension $k$ on a closed manifold of dimension $k+n.$We introduce the notion of Nil-Anosov action of $G$ (which includes the case where $G$ is nilpotent) and establishes the invariance by the entire group $G$of the associated stable and unstable foliations. We then prove a spectral decomposition Theoremfor such an action when the group $G$ is nilpotent. Finally, we focus on the case where $G$ is nilpotent andthe unstable bundle has codimension one. We prove that in this case the action is a Nil-extensionover an Anosov action of an abelian Lie g…
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。
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 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.
Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups
本文为Anosov子群构造扭曲共形密度并证明其唯一性、无原子性和遍历性，进而利用商群特征分类幂零覆盖上正规子群的遍历共形密度。
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…
Anosov diffeomorphisms on closed Riemannian manifolds are a type of dynamical systems exhibiting uniform hyperbolic behavior. Therefore their properties are intensively studied, including which spaces allow such a diffeomorphism. It is conjectured that any closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold, i.e. a compact quotient of a 1-connected nilpotent Lie group by a discrete group of isometries. This conjecture motivates the problem of describing which infra-nilmanifolds admit an Anosov diffeomorphism. So far, most research was focused on the restr…
For every irreducible automorphism $ϕ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $ϕ$, whose stretch factor is the product of the expanding eigenvalues of $ϕ$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $ϕ$ has exactly two eigenvalues outside the unit cir…
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…
A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.
We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are `small' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we get also new examples of compact manifolds covered by two-step simply connected Lie groups, which do not admit Anosov automorphisms.
We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves…
Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+ν}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same co…
For algebraic Anosov diffeomorphisms we first express the reduced leafwise cohomology with respect to the unstable foliation in terms of finite dimensional Lie algebra cohomology. We then prove a dynamical Lefschetz trace formula for the induced action on this cohomology using a result of Nomizu. We also consider generalized Anosov maps for which the cohomology in question is no longer finite dimensional in general. Using the representation theory of nilpotent Lie groups we still arrive at a meaningful definition of traces on cohomology. Again we establish a dynamical Lefschetz trace formula.
We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.
We study real analytic perturbations of hyperbolic linear automorphisms on the 2-torus. The Koopman and the transfer operator are nuclear of order 0 when acting on a suitable Hilbert space. We show the generic existence of non-trivial Ruelle resonances for both operators. We prove that some of the perturbations preserve the volume and some of them do not.
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…
Box dimension of stable sub-slices of fractal graphs over Anosov automorphisms
研究特定斜积形式的分形图，利用Kaplan、Mallet-Paret和York的结果，给出斜积条件确保图的盒维数小于其稳定/不稳定子切片盒维数之和，并证明对一般函数\(p\in C^{k + 1}\)条件成立。
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.
We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a finite cover of such an action is smoothly conjugate to an action by toral automorphisms.
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 $\mathbb{Q}.$
In the paper we give an introduction to Anosov diffeomorphisms, ways to represent their chaotic properties and some historical remarks on this subject. A complete classification of hyperbolic linear automorphisms of 2-torus is presented. We introduce a notion of pre-Markov partition for such automorphisms and give their classification and an algorithm for their construction.
An Anosov diffeomorphism is topologically conjugate to an infranilmanifold automorphism if and only if it has polynomial Global Product Structure.
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.
Nonstationary Markov Partitions and Multidimensional Continued Fraction Algorithms
本文研究了非平稳马尔可夫划分与多维连续分数算法的关联，通过皮索型条件证明了非平稳动力系统的双曲性，并构建了基于符号动力系统的非平稳边移位模型。
An algebraic $Γ$-action is an action of a countable group $Γ$ on a compact abelian group $X$ by continuous automorphisms of $X$. We prove that any expansive algebraic action of a finitely generated nilpotent group $Γ$ on a connected group $X$ is ergodic. We also show that this result does not hold for actions of polycyclic groups.
Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups
本文研究了局部紧群自同构在闭子群空间上的远距作用，证明了幂零群和可解李群格上的远距性等价条件，并刻画了紧生成幂零群的结构。
We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.
Mapping classes of Real Rational Surface Automorphisms
本文研究实有理曲面上微分同胚族 $F_n$ 的映射类，通过切开不变曲线证明其限制 $\hat F_n$ 为非 Penner 型伪 Anosov 映射，并在 $n=8$ 时给出 Lehmer 数的新几何实现。
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