We construct new families of examples of (real) Anosov Lie algebras starting with algebraic units. We also give examples of indecomposable Anosov Lie algebras (not a direct sum of proper Lie ideals) of dimension 13 and 16, and we conclude that for every $n \geq 6$ with $n \neq 7$ there exists an indecomposable Anosov Lie algebra of dimension $n$.
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…
In this article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation $(X,v)$, based on the study of rational factors of $(X,v)$ and of algebraic foliations on $X$, invariant under the Lie-derivative of the vector field $v$. We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disint…
A few years ago, the first example of a closed manifold admitting an Anosov diffeomorphism but no expanding map was given. Unfortunately, this example is not explicit and is high-dimensional, although its exact dimension is unknown due to the type of construction. In this paper, we present a family of concrete 12-dimensional nilmanifolds with an Anosov diffeomorphism but no expanding map, where nilmanifolds are defined as the quotient of a 1-connected nilpotent Lie group by a cocompact lattice. We show that this family has the smallest possible dimension in the class of infra-nilmanifolds, whi…
In the late seventies, Sullivan showed that for a convex cocompact subgroup $Γ$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $δ>0$, any $Γ$-conformal measure on $\partial \mathbb{H}^n$ of dimension $δ$ is necessarily supported on the limit set $Λ$ and that the conformal measure of dimension $δ$ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$ of rank at most $3$. We also obtain the local mixing for generalized BMS measures on $Γ\backslash G$ including Haar measures.
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…
On topologically transitive subsets of flag manifolds
Tata Institute of Fundamental Research（塔塔基础研究所）；Mathematisches Institut der Universität Bonn（波恩大学数学研究所）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进会数学研究所）
