Chaotic period doubling
Chaotic period doubling
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混沌周期加倍
DOI：
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发表时间：
2007
影响因子：
0.9
通讯作者：
C. Tresser
中科院分区：
文献类型：
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作者：
V. Chandramouli;M. Martens;W. D. Melo;C. Tresser
摘要倍周期重整化算子是Feigenbaum和Coullet和Tresser在20世纪70年代提出来研究从简单动力学到混沌动力学过渡的一维系统的吸引子的渐近小尺度几何的。这种几何原来不依赖于选择的地图下，而温和的光滑条件。在一般的足够光滑映射中，存在唯一的重正化不动点且该不动点也是双曲的，这在相应的重正化理论中起着至关重要的作用。首先在全纯情形下证明了重整化不动点的唯一性和双曲性，并推广到其他重整化算子。证明了在C2+α单峰映射空间中，当α>0时，倍周期重整化不动点也是双曲的.在本文中，我们研究会发生什么事时，从最小光滑阈值的唯一性和双曲性的倍周期重整化一般不动点的方法。事实上，我们的主要结果表明，在空间中的C2单峰映射的解析不动点是不是双曲的，这同样是真实的，当添加足够的光滑度，以获得先验界。在这个更平滑的类中，称为C2+双曲型，双曲性的失败比C2中更温和。如果光滑度比C2小一点，情况就会变得更糟，因为这样就失去了唯一性，其他渐近行为也成为可能。证明了作用在C1+Lip单峰映射空间上的倍周期重整化算子具有无穷大的拓扑熵。
Abstract The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space of C2+α unimodal maps, for α>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space of C2 unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called C2+∣⋅∣, the failure of hyperbolicity is tamer than in C2. Things get much worse with just a bit less smoothness than C2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy.