相关论文: Algebraic Anosov actions of Nilpotent Lie groups

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…

We prove that an Anosov action of $\mathbb{R}^k$ over a compact manifold $M$ transitive on regular sub-cones satisfies the dichotomy: each stable and unstable leaf is dense or the Anosov action is topologically conjugated to a suspension of…

We show that there are no Anosov actions by (n-1)-dimensional unimodular Lie groups on closed n-dimensional manifolds.

Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are…

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…

We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\R$-covered,…

An infra-nilmanifold is a manifold which is constructed as a quotient space $\Gamma\backslash G$ of a simply connected nilpotent Lie group $G$, where $\Gamma$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so…

It is conjectured that every manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov…

Anosov diffeomorphisms are an important class of dynamical systems with many peculiar properties. Ever since they were introduced in the sixties, it has been an open question which manifolds can admit such diffeomorphisms, where tori of…

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

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).…

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.

We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie…

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the…

We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of…

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic…

We study $\mathbb{R}^k \times \mathbb{Z}^\ell$ actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We…

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

With the goal to study and better understand algebraic Anosov actions of $\mathbb R^k$, we develop a higher codimensional analogue of the contact distribution on odd dimensional manifolds, call such structure a generalized $k$-contact…

We show that for a locally free action of a simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the…