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Fractals in Dynamical Systems

杨路,张景中,曾振柄
Yang Lu Zhang Jingzhong Zeng Zhenbing

中国科学院成都分院数理科学研究室,中国科学院成都分院数理科学研究室,中国科学院成都分院数理科学研究室
(The Institute of Mathematical Sciences Academia Sinica, Chengdu, China

收稿日期: 1990-06-25
出版日期: 2015-05-15

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摘要  函数迭代的研究,可以追溯到一百多年以前,其中对后来影响较深者,有E.Schr(o| ¨)der、N.H.Abel和Charles Babbage,文献的附录中介绍了他们的工作.但是由于迭代运算与代数运算的迥然不同,研究工作艰难曲折.直到1974年Li and Yorke发表了Periodthree implies chaos和Sharkovski工作的重新发现,才带来迭代论研究的一系列突破性进
Abstract:This survey gives a comprehensive report of the recent progress in the studies of fractal geometry, especially its relation to dynamical systems. The paper contains six parts.I.Introduction, fractals, self-similarity, a brief description about geometry of fractal sets, principal properties of fractals and its relation to fractional dimensional sets, general methods in constructing of fractals, Weierstrass fractal curves.II. Fractal geometry of strange attractors. what is an attractor? fractal, static aspect of strange 1. attractors, 2. basin boundaries, 3. classification of fractal basin boundaries. Ⅲ, Fractals in analytic dynamical systems, 1. Julia sets as fractals, 2. iteration of entire transcendental functions, 3. Mandelbrot sets, 4. iterated function systems.Ⅳ.Quantitative analysis of fractals in analytic dynamical systems, definitions and properties of dimensions,some relations between different scales, computation and estimate methods.Ⅴ.Some problems, reconstruct of fractals, Julia exploision, algebraic points in fractals.Ⅵ.An appendix of references.
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