The periodic points of renormalization
The periodic points of renormalization
复制标题
重正化的周期点
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
M. Martens
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文献类型：
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作者：
M. Martens
证明了作用于临界指数为a>；1的光滑单峰映射空间上的重整化算子具有任意组合类型的周期点。动力系统理论中的一个中心问题是，动力系统的小尺度几何性质是否由系统的组合性质决定。事实上，库莱-特雷瑟和费根鲍姆就发现了这种小尺度几何的普遍性。他们研究了倍周期类型的无限可重整化单峰映射，并观察到当尺度越来越小时，这类映射的不变Cantor集的几何收敛。此外，他们观察到极限几何是通用的，从这个意义上说，这些康托集的小尺度几何只取决于临界点附近映射的局部行为。这种局部行为由临界指数指定。为了解释几何的普适性，他们在一个合适的单峰映射空间上定义了倍周期重整化算子。这个操作符的作用就像一个显微镜：重整化操作符下的图像是一个在较小尺度上描述几何和动力学的单峰映射。通过猜想重整化算子有唯一的双曲不动点来理解几何的普适性。特别地，无限可重整化的单峰映射构成了重整化算子不动点的稳定流形。证明这些猜想的第一步是证明重整化不动点的存在性。
It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent a > 1, has periodic points of any combinatorial type. A central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system. Indeed, such universality of small scale geometry was discovered by Coullet-Tresser and Feigenbaum. They studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant Cantor set of such maps converges when looking at smaller and smaller scales. Furthermore they observed that the limiting geometry was universal, in the sense that the small scale geometry of these Cantor sets depends only on the local behavior of the map around the critical point. This local behavior is specified by the critical exponent. To explain the universality of geometry, they defined the period doubling renormalization operator on a suitable space of unimodal maps. This operator acts like a microscope: the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale. The universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point. In particular, the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator. The first step in proving these conjectures is showing the existence of a renormalization fixed point.