ONE-DIMENSIONAL DYNAMICS
ONE-DIMENSIONAL DYNAMICS
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DOI：
10.1007/3-540-26844-8_2
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发表时间：
2013
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通讯作者：
Michał Misiurewicz
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文献类型：
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作者：
Michał Misiurewicz
一维变换的研究是一个经典课题，可以追溯到庞加莱。在这里，这些系统主要作为更高维度的动力学行为的模型而引起我们的兴趣。我们专注于连续光滑的地图，许多关键（多模态地图）表现出复杂的行为。自70年代后期以来，动力学的这一领域一直在取得显著进展。Graczyk、Swiatek [199]和Lyubich [275]撰写了最近的调查报告，而de梅洛、货车Strien [147]的书仍然是关于这个问题的基本文献。双曲性是一维映射最简单的行为形式。我们回顾一下2.1节中的定义和特征。这些映射的一个特点是，远离临界点的动态总是非常双曲的，正如我们在2.2节中所评论的那样。在这些映射中，双曲动力学是Cr稠密的，这一点已经被证明了很长时间，直到最近才被确定。这个在高维中没有对应物的显著事实将在2.3节中讨论。另一方面，临界点的存在也可能导致参数空间中持续的非一致双曲性。这是第2.4节的主题。最近在这一领域的大部分进展都是由理解重整化算子的努力所推动的，并解释了Coullet，Tresser [138]和Feigenbaum [181]在分叉级联中发现的普适性现象。参见第2.5节。在最近的主要成就，让我们提到的证据，几乎所有的映射内的典型家庭的单峰映射是双曲（定期）或混沌（随机），并在任何情况下，stochastical稳定。这在第2.6节中讨论。
The study of one-dimensional transformations is a classical subject going back to Poincare. Here these systems interest us primarily as models for dynamical behavior in higher dimensions. We focus on continuous smooth maps with finitely many critical (multimodal maps) exhibiting complicated behavior. This area of Dynamics has been experiencing remarkable progress since the late seventies. Recent surveys have been written by Graczyk, Swiatek [199] and Lyubich [275], and the book of de Melo, van Strien [147] remains the fundamental expository text on the subject. Hyperbolicity is the simplest form of behavior for one-dimensional maps. We recall the definition and characterization in Section 2.1. A special feature of these maps is that the dynamics is always very much hyperbolic far from critical points, as we comment upon in Section 2.2. It had been conjectured for a long time, and was only recently established, that hyperbolic dynamics is Cr-dense among these maps. This remarkable fact, which has no counterpart in higher dimensions, is discussed in Section 2.3. On the other hand, the presence of critical points may also give rise to non-uniform hyperbolicity persistently in parameter space. This is the theme of Section 2.4. Most of the recent progress in this area has been driven by the effort to understand the renormalization operator, and explain the universality phenomena discovered by Coullet, Tresser [138] and Feigenbaum [181] in cascades of bifurcations. See Section 2.5. Among the main recent achievements, let us mention the proof that almost all maps inside typical families of unimodal maps are either hyperbolic (regular) or chaotic (stochastic) and, in either case, stochastically stable. This is discussed in Section 2.6.