Combinatorics, geometry and attractors of quasi-quadratic maps
Combinatorics, geometry and attractors of quasi-quadratic maps
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DOI：
10.2307/2118604
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发表时间：
1992-12
期刊：
影响因子：
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通讯作者：
M. Lyubich
中科院分区：
文献类型：
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作者：
M. Lyubich
求解了具有非退化临界点c的s -单峰映射的一维吸引子上的Milnor问题，使我们对几乎每一点的Lebesgue可能的极限行为有了一个完整的认识。这个定理来源于对“不可重整”映射的临界集$\ (c)$的几何研究。证明了表征该集合几何形状的标度因子至少指数地趋于0。解决了小尺度下的非线性控制问题。这些证明强烈地涉及重整化理论和全纯动力学的思想。
The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.