论文标题
几乎是搅拌机和脱皮产品
Almost blenders and parablenders
论文作者
论文摘要
表面内态的搅拌器是肥大的基本集合，局部不稳定歧管的结合包含一个牢固的开放集。 Bonatti和D {í} AZ在90年代引入的搅拌机原来具有许多功能强大的应用程序。特别是，以称为脱伞类药物的喷气机的概括，允许伯格证明存在着无限无限多个下沉的通用家庭的存在。在本文中，我们在可衡量的角度引入了类似的概念。我们将几乎搅拌器定义为双曲基本集合，为此，普遍的扰动具有具有正面测量的局部不稳定集。几乎在喷气机上类似地定义了脱纸制剂。我们研究了R2的内态性家庭家庭，使双曲线基本组的延续不变。当满足涉及熵和最大收缩的某些不平等时，我们就会获得几乎搅拌机或Parabrender。这部分回答了伯格的猜想。证明基于热力学形式主义：遵循Mihailescu，Simon，Solomyak和Urba {outa}滑雪的作品，我们研究了光纤单位偏斜产物的家族，并且我们提供了这些地图在其纤维内限制阳性度量的条件。
A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and D{í}az in the 90s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper, we introduce analogous notions in a measurable point of view. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of R2 leaving invariant the continuation of a hyperbolic basic set. When some inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak and Urba{ń}ski, we study families of fiberwise unipotent skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.
