arXiv subjects
Publications and source records attributed to Matilde Martínez.
We prove a Mostow rigidity theorem for foliated bundles over closed hyperbolic manifolds of dimension $n \geq 3$ endowed with a completely invariant measure of full support. These include solenoidal manifolds obtained as inverse limits of directed systems of finite coverings of closed hyperbolic manifolds. This theorem then extends to skew solenoidal manifolds for which the action of the holonomy group is twisted by means of a cocycle.
We study the dynamical properties of the laminated horocycle flow on the unit tangent bundles of 2-dimensional smooth solenoidal manifolds of finite type. These laminations are the analog of complete hyperbolic surfaces of finite area.
We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the \emph{second systole} of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.
We construct an uncountable family of transversely Cantor laminations of compact spaces defined by free minimal actions of solvable groups, which are not affable and whose orbits are not quasi-isometric to Cayley graphs.
We describe several methods to construct minimal foliations by hyperbolic surfaces on closed 3-manifolds, and discuss the properties of the examples thus obtained.
A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène's theorem to other kinds of foliations.
We prove that if $Γ$ is a countable group without a subgroup isomorphic to $\mathbb{Z}^2$ that acts faithfully and minimally by orientation preserving homeomorphisms on the circle, then it has a free orbit. We give examples showing that this does not hold for actions by homeomorphisms of the line.
We consider a minimal compact lamination by hyperbolic surfaces. We prove that if it admits a leaf whose holonomy covering is not topologically trivial, then the horocycle flow on its unitary tangent bundle is minimal.
Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behaviour of almost every $X$-orbit in every leaf, that we call u-Gibbs states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such an ergodic u-Gibbs states are negative, it is an SRB-measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by L. Garnett. If furthermore the foliation is transversally conformal and does not admit a transverse invariant measure we show that the ergodic u-Gibbs states are finitely many, supported each in one minimal set of the foliation, have negative Lyapunov exponents and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics is described by each of these ergodic u-Gibbs states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellors of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only 1 attractor, we obtain a North to South pole dynamics.
We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq Γ_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathcal{L}}$ which are right invariant under a minimal parabolic (Borel) subgroup $B < G$. In the special case when $G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $\hat{\mathcal{L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy-invariant ones.
We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.