孤立集、悬链线李雅普诺夫函数与扩张系统

Expansive Measures versus Lyapunov exponents

In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive is expanding. Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can not be accumulated by positively measure expansive diffeomorphis have a dominated sppliting.

Expansiveness, Lyapunov exponents and entropy for set valued maps

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for se…

扩张同胚的李雅普诺夫指数

Lyapunov exponents for expansive homeomorphisms

设（M，d）是紧度量空间，f：M --> M是扩张同胚.定义了f-不变测度m的李雅普诺夫指数L（f，m）_{max}和l（f，mu）_{min}.当L（f，m）_{max} &gt; 0，l（f，mu）_{min} < 0 can be interpreted as a weak form of hyperbolicity for f. We prove that if M is a Peano space then there is g>0，使得L（f，m）_{max} &gt; g，l（f，m）_{min}&lt; - g.我们还表明，假设M是一个Peano空间是必要的，以获得最大的李雅普诺夫指数积极和最小的李雅普诺夫指数负。此外，我们还定义了M的紧f-不变子集K的李雅普诺夫指数，并证明了如果K的最大李雅普诺夫指数为负，则K是吸引子。当f是紧致流形上的一个同态时，这些李雅普诺夫指数与通常的一致。

基于惠特尼尺寸函数的李雅普诺夫函数

Lyapunov functions via Whitney's size functions

在本文中，我们提出了一种技术，用于构建基于惠特尼的大小函数的李雅普诺夫函数。给出了渐近稳定平衡点、孤立集、扩张同胚和连续扩张同胚的应用。

膨胀同胚的维数与熵

Dimensions and entropies for an expansive homeomorphism

针对膨胀同胚，受Young曲面微分同胚公式启发构造具双Lyapunov指数的双曲度量，研究多种熵与维数关系，证明双曲度量膨胀拓扑系统的Eckmann-Ruelle猜想并建立熵的变分原理。

平面的扩张同胚

Expansive homeomorphisms of the plane

本文讨论了平面的可扩同胚的分类问题。给出了同胚与线性双曲自同构共轭的充要条件。这些技术涉及平面的拓扑和度量方面。使用的李雅普诺夫度量函数定义了相同的拓扑作为一个诱导的通常的度量，但在一般情况下，是不等同于它是这样的技术的一个例子。关于李雅普诺夫函数在无穷远处的行为的一个假设的发现使我们能够推广一些在紧致上下文中有效的结果。附加的局部性质允许我们得到另一个分类定理。