arXiv subjects
Publications and source records attributed to Carlos Maquera.
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 a $\mathbb{Z}^k$-Anosov action. This represents an important progress toward addressing Verjovsky's extended conjecture for Anosov actions, as developed by Barbot and Maquera.
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 group. In particular:i) if $n \geq 3,$ then the action is topologically transitive,ii) if $n=2,$ then the action is a Nil-extension over an Anosov flow.
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.
Let $Y:\R^n\to\R^n$ be a polynomial local diffeomorphism and let $S_Y$ denote the set of not proper points of $Y$. The Jelonek's real Jacobian Conjecture states that if $\codim(S_Y)\geq2$, then $Y$ is bijective. We prove a weak version of such conjecture establishing the sufficiency of a necessary condition for bijectivity. Furthermore, we generalize our result on bijectivity to semialgebraic local diffeomorphisms.
In this paper, we consider codimension one Anosov actions of IR^k, k ? 1, on closed connected orientable manifolds of dimension n+k with n? 3. We show that the fundamental group of the ambient manifold is solvable if and only if the weak foliation of codimension one is transversely affine. We also study the situation where one 1-parameter subgroup of IR^k admits a cross-section, and compare this to the case where the whole action is transverse to a fibration over a manifold of dimension n. As a byproduct, generalizing a Theorem by Ghys in the case k=1, we show that, under some assumptions about the smoothness of the sub-bundle E^ss ? E^uu, and in the case where the action preserves the volume, it is topologically equivalent to a suspension of a linear Anosov action of Z^k on T^n.
We show that if $N$, an open connected $n$-manifold with finitely generated fundamental group, is $C^{2}$ foliated by closed planes, then $π_{1}(N)$ is a free group. This implies that if $π_{1}(N)$ has an Abelian subgroup of rank greater than one, then $\mathcal{F}$ has at least a non closed leaf. Next, we show that if $N$ is three dimensional with fundamental group abelian of rank greater than one, then $N$ is homeomorphic to $\mathbb{T}^2\times \mathbb{R}.$ Furthermore, in this case we give a complete description of the foliation.
In this paper, we define codimension one Anosov actions of $\RR^k, k\geq 2,$ on a closed connected orientable manifold $M$. We prove that if the ambient manifold has dimension greater than $k+2$, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension one Anosov flows.
Let $Y=(f,g,h):\mathbb{R}^{3} \to \mathbb{R}^{3}$ be a $C^{2}$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $DY_p$, when $p$ varies in $\mathbb{R}^3$. We begin proving that if, for some $ε>0,$ $\spec(Y)\cap (-ε,ε)=\emptyset,$ then the foliation $\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces $\{k={\rm constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of $\mathbb{R}^n.$
In this paper, we define $C^1$-robust transitivity for actions of $\RR^2$ on closed connected orientable manifolds. We prove that if the ambient manifold is three dimensional and the dense orbit of a robustly transitive action is not planar, then it is "degenerate" and the action is defined by an Anosov flow.