Smooth Dynamical Systems

我们提出研究低维流形光滑微分同态的动力学。这里的一个重要主题是同斜切线的丰度及其影响的研究。同斜切线产生有趣的现象已经有一段时间了。它们对结构稳定性构成阻碍，并对分岔现象产生深远影响。虽然我们所拥有的大多数关于同斜切线的信息在我们理解潜在动力学的能力方面可能被认为是负面的，但最近我们获得了一个积极的结果：一般来说，它们会导致最大Hausdorff维的拓扑传递集。当前建议的一部分涉及理解这些极大维集和勒贝格渐近测度之间的关系。这些测度是通过取相对于勒贝格测度绝对连续的测度的迭代平均值的弱极限而得到的。最近我们证明了在许多耗散情况下，表面上的SRB测度只存在于一致双曲吸引子上。我们希望将此推广到保面积的情况，并在高度光滑系统中，曲面的非anosov保面积微分同态一般具有度量熵为零的猜想上取得进展。我们将考虑的另一个问题是Henon族是否没有残差参数集的SRB度量。这是Benedicks-Carleson和Benedicks-Young的一个著名的结果，SRB测量存在于一个正的勒贝格测量参数集。除此之外，我们还将研究光滑系统中符号扩展存在性的各种问题。我们提出研究非线性光滑动力系统的轨道结构的各个方面。我们将特别强调其轨道无限频繁地返回嵌入横向表面的三维系统。这些系统出现在从生物学到物理学甚至经济学的许多科学领域的模型中。例如，它们包括一般强迫振荡问题和三个物体在一个平面上的牛顿运动。考虑到所谓的到嵌入曲面的第一次返回映射，我们被引导研究曲面到自身的光滑变换（映射）的迭代。有一种特殊的运动叫做同斜向运动（最早是由庞加莱在三体问题中发现并命名的），它可以产生丰富、有趣和复杂的轨道结构。特别是，典型的同斜运动意味着存在无限多的不稳定周期轨道和其他以不稳定和不可预测的方式运行的轨道。现在我们知道，在许多情况下，这样的系统可以通过将它们与统计对象（如随机抛一枚加权硬币）进行比较来研究。这就产生了所谓的“符号模型”，其轨道结构是可以理解的。我们的研究涉及各种光滑系统建模中可能出现的符号系统类型。包括对某些被称为“熵”和“维”的数值量的估计，这些数值量可以用来量化不同层次的复杂运动。

We propose to study the dynamics of smooth diffeomorphisms of low dimensional manifolds. A significant theme here is the study of the abundance of homoclinic tangencies and its effects. It has been known for some time that homoclinic tangencies produce interesting phenomena. They form an obstruction to structural stability, and have a profound influence on bifurcations phenomena. While most of the information we have about homoclinic tangencies may be considered negative in regard to our ability to understand the underlying dynamics, recently we obtained a positive result: generically they lead to topologically transitive sets of maximal Hausdorff dimension. One part of the current proposal involves understanding the relation between these maximal dimension sets and Lebesgue asymptotic measures. These are measures obtained by taking weak limits of the averages of the iterates of measures which are absolutely continuous with respect to Lebesgue measure. Recently we have shown that in many dissipative cases SRB measures on surfaces only exist on uniformly hyperbolic attractors. We wish to extend this to area preserving cases and to make progress on the conjecture that generically in highly smooth systems a non-Anosov area preserving diffeomorphism of a surface has metric entropy zero. Another question we will consider is whether the Henon family has no SRB measure for a residual set of parameters. It is a celebrated result of Benedicks-Carleson and Benedicks-Young that SRB measures exist for a positive Lebesgue measure set of parameters. In addition to the above, we will study various questions on the existence of symbolic extensions in smooth systems. We propose to study aspects of the orbit structure of non-linear smooth dynamical systems. Our particular emphasis will be on three dimensional systems whose orbits return infinitely often to an embedded transverse surface. These systems occur in models which arise in many fields of science from Biology to Physics and even Economics. For instance, they include the problems of general forced oscillations and the Newtoninan motion of three bodies in a plane. Considering the so-called first return map to the embedded surface we are led to study the iterations of smooth transformations (mappings) of a surface to itself. There are special motions called homoclinic motions (first discovered and named by Poincare in the three body problem) which are known to produce a rich, interesting, and complicated orbit structure. In particular, typical homoclinic motions imply the existence of infinitely many unstable periodic orbits and other orbits which behave in an erratic and unpredictable way. It is now known that in many cases such systems can be studied by comparing them to statistical objects such as the random flipping of a weighted coin. This gives rise to so-called "symbolic models" whose orbit structure can be understood. Our research concerns the types of symbolic systems which can occur in modeling various smooth systems. including the estimation of certain numerical quantities called "entropy" and "dimension" which can be used to quantify different levels of complicated motion.