Finitely-additive measures on the asymptotic foliations of a Markov compactum

Finitely Suslinian models for planar compacta with applications to Julia sets

A compactum $X\subset \C$ is unshielded if it coincides with the boundary of the unbounded component of $\C\sm X$. Call a compactum $X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $X$ admits a topologically unique monotone map $m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $X$ onto a finitely Suslinian quotient factors through $m_X$. We call the pair $(X_{FS},m_X)$ (or, more loosely, $X_{FS}$) the finest finitely Suslinian mo…

由 $\mathbb{P}^2$ 上凸与齐次叶状结构的乘积生成的网

Webs Generated by Products of convex and homogeneous Foliations on $\mathbb{P}^2$

本文研究射影平面上的平坦网，提出通过有限多个凸约化或凸齐次叶状结构与不变直线的乘积构造此类网的两种方法，并证明在这两种情况下其对偶网均为平坦的。

Sliding invariants and classification of singular holomorphic foliations in the plane

By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.

Vector fields and foliations associated to groups of projective automorphisms

We introduce and give normal forms for (one-dimensional) Riccati foliations (vector fields) on $\ov \bc \times \bc P(2)$ and $\ov \bc \times \ov \bc^n$. These are foliations are characterized by transversality with the generic fiber of the first projection and we prove they are conjugate {\em in some invariant Zariski open subset} to the suspension of a group of automorphisms of the fiber, $\bc P(2)$ or $\ov \bc^n$, this group called {\it global holonomy}. Our main result states that given a finitely generated subgroup $G$ of $\Aut(\bc P (2))$, there is a Riccati foliation on $\ov \bc \times \…

Minimal Følner foliations are amenable

For finitely generated groups, amenability and Følner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Følner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Følner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to…

Foliated hyperbolicity and foliations with hyperbolic leaves

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-m…