周期点与不动点课程详细信息 课程号 英文名称 先修课程 00102509 Periodic and Fixed points 无 课程主要介绍如下内容： 周期点个数的同余。Bernoulli移位和Markov链。整数矩阵的Fermat同余： Tr A^p≡Tr A mod p。重合点（根和不动点）。不动点指数。根的多重性。三个问题的等价性：根，重合点，不动点。 映射度和根的多重性的计算。 一维情形。Bezout定理。 Lefshetz数和Lefshetz-Hopf定理。一维情况下的不动点指数和吸引点指数。多维全纯映射。多重性Palamodov定理。Newton多面体域上的全纯函数芽。Minkowski和。 Bernshtein-Kushnirenko-Khovanski定理。 一些例子的计算。Jenning群。 中文简介 Jenning型的Galois群。Steinlein，Zabreiko和Krasnosel'skii关于连续映射不动点迭代指标的Dold定理。Babenko-Bogatyy，Franks-Fried实现定理。关于光滑映射的不动点迭代指标的Shub-Sullivan定理。 二维光滑情形。全纯映射的Poincare正规型。 二维复全纯情形。自由映射和自由开覆盖。Stone-Cech紧致化和有限自由覆盖。 Schauder-Tikhonov不动点定理。 不变测度的Bogolubov定理。 整数集上的有限可加平移不变测度。 Amenable群和Banah-Tarski悖论。 Borsuk-ulam定理。Lusternik-Schnirel’man定理。Steinlein定理。 有限群的自由作用：不同特征的比较。周期点的Sharkovskii共存性。不同型的回归性。概周期点 的Xiangdang Ye共存性。Markov链概周期点的Bogatyy-Redkozubov共存性。The course introduces the following topics: Congruences for the number of periodic points. Bernoulli's shift and Markov's chain. Fermat's congruence for the integer matrix: TrAp TrA mod p. Coincidence point (root and xed point). Index of the xed point. Multiplicity of the root. Equivalence of three problems: root, coincidence and xed point. Degree of the mapping and calculation of multiplicity of the root. One-dimensional case. Bezout's theorem.. Lefshetz number and LefshetzHopf theorem. Index of the xed point in one-dimensional case and for attracting point. Multi-dimensional holomorphic map. 英文简介 Palamodov's theorem about multiplicity. Newton polyhedron of the holomorphic germ. Minkowski sum BernshteinKushnirenkoKhovanski theorem. Calculations of some examples. Jenning's group. Jenning's group as Galois group. Steinlein, ZabreikoKrasnosel'skii, Dold theorem about indexes of iterates of xed point for continuous map. BabenkoBogatyy, FranksFried realization theorem. ShubSullivan theorem about indexes of iterates of xed point for smooth map. Two-dimensional smooth case. Poincare normal form of the holomorphic map. Complex two-dimensional holomorphic case. Free map and free open covering. StoneCech compactication and free nite 学分 3
coverings. SchauderTikhonov's xed point theorem. Bogolubov's theorem about invariant measure. Finitly additive shift-invariant measure on integer numbers Amenable groups and BanachTarski paradox. BorsukUlam theorem. LusternikSchnirel'man theorem. Steinlein theorem. Free action of a nite group: comparison of dierent characteristics. Sharkovskii coexistence of periodic points. Dierent types of recurrency. Xiangdong Ye coexistence of almost periodic points. BogatyyRedkozubov coexistence of almost periodic points for Markov chain. 开课院系 通选课领域 是否属于艺术与美育 平台课性质 平台课类型 授课语言 教材 参考书 数学科学学院 否 英文 无； 周期性是随机过程和遍历论的重要概念，不动点是距离空间的重要工具，周期性和不动点联系起来是怎样的工作呢 1. Congruences for the number of periodic points. 2. Bernoulli`s shift and Markov`s chain. 3. Fermat`s congruence for the integer matrix: TrAp TrA mod p. 4. Coincidence point (root and xed point). 5. Index of the xed point. 6. Multiplicity of the root. 7. Equivalence of three problems: root, coincidence and xed point. 8. Degree of the mapping and calculation of multiplicity of the root. 9. One-dimensional case. Bezout`s theorem. 教学大纲 10. Lefshetz number and LefshetzHopf theorem. 11. Index of the xed point in one-dimensional case and for attracting point. 12. Multi-dimensional holomorphic map. Palamodov`s theorem about multiplicity. 13. Newton polyhedron of the holomorphic germ. 14. Minkowski sum. 15. BernshteinKushnirenkoKhovanski theorem. 16. Calculations of some examples. 17. Jenning`s group. 18. Jenning`s group as Galois group. 19. Steinlein, ZabreikoKrasnosel`skii, Dold theorem about indexes of
iterates of xed point for continuous map. 20. BabenkoBogatyy, FranksFried realization theorem. 21. ShubSullivan theorem about indexes of iterates of xed point for smooth map. 22. Two-dimensional smooth case. 23. Poincare normal form of the holomorphic map. 24. Complex two-dimensional holomorphic case. 25. Free map and free open covering. 26. StoneCech compactication and free nite coverings. 27. SchauderTikhonov`s xed point theorem. 28. Bogolubov`s theorem about invariant measure. 29. Finitly additive shift-invariant measure on integer numbers 30. Amenable groups and BanachTarski paradox. 31. BorsukUlam theorem. 32. LusternikSchnirel`man theorem. 33. Steinlein theorem. 34. Free action of a nite group: comparison of dierent characteristics. 35. Sharkovskii coexistence of periodic points. 36. Dierent types of recurrency. 37. Xiangdong Ye coexistence of almost periodic points. 38. BogatyyRedkozubov coexistence of almost periodic points for Markov chain. 课堂讲授 按考试或课堂表现 教学评估
周期点与不动点
