南方科技大学知识苑(SUSTech KC): STABLE ESSENTIAL DENSITY FOR ERGODIC HOMOCLINIC CLASSES

题名	STABLE ESSENTIAL DENSITY FOR ERGODIC HOMOCLINIC CLASSES
其他题名	关于遍历同宿类的稳定必要稠密性
姓名	金蒙召
姓名拼音	JIN Mengzhao
学号	12032850
学位类型	硕士
学位专业	070101 基础数学
学科门类/专业学位类别	07 理学
导师	Maria Alejandra Rodriguez Hertz
导师单位	数学系
论文答辩日期	2022-05-05
论文提交日期	2022-06-20
学位授予单位	南方科技大学
学位授予地点	深圳
摘要	We study the stable essential density for ergodic homoclinic classes in this paper. This question comes from the research of stable ergodicity. Firstly, we give an introduction to hyperbolic dynamics. We introduce the different definitions of hyperbolic set. We prove these different definitions are equivalent. Then we introduce some property about hyperbolic set, which are local maximality, local product structure and shadowing property. Next, We introduce two important results which are 𝜆-lemma and the spectral decomposition theorem. Then we give a proof of the spectral decomposition theorem. We also introduce some contents about partially hyperbolic dynamics. Secondly, we introduce ergodic theory. We introduce Poincaré recurrence theorem to help people understand the ergodicity and stable ergodicity. Then we introduce ergodicity and mixing. At last, we introduce weak ergodicity which is possible mechanism helping study stable ergodicity. Finally, we give a summary of the proof of the following theorem. For a 𝐶^1 -generic volume preserving diffeomorphism 𝑓 in a closed Riemannian manifold 𝑀, if it has an expanding invariant foliation, then there is an ergodic homoclinic class for 𝑓 which is a hyperbolic ergodic component, and it has stable essential density. This theorem will help study the stable minimality of minimal expanding invariant foliation.
关键词	Ergodic Homoclinic Class Expanding Foliation Generic Diffeomorphism Hyperbolic Dynamics Stable Essential Density
语种	英语
培养类别	独立培养
入学年份	2020
学位授予年份	2022-07
参考文献列表	[1] NÚÑEZ G, RODRIGUEZ HERTZ J. Stable minimality of expanding foliations[J]. J Dyn Diff Equat, 2021, 33(4): 2075-2089. [2] ANOSOV D V, SINAI Y G. Some smooth ergodic systems.[J]. Russ. Math. Surv., 1967, 22(5): 103-167. [3] AVILA A, CROVISIER S, WILKINSON A. 𝐶^1 density of stable ergodicity[Z]. 2017. [4] NÚÑEZ G, RODRIGUEZ HERTZ J. Minimality and stable Bernoulliness in dimension 3[J]. Disc. Cont. Dyn. Syst., 2020, 40(3): 1879-1887. [5] MATHER J N. Characterization of Anosov diffeomorphisms[J]. Indagationes Mathematicae (Proceedings), 1968, 71: 479-483. [6] BRIN M, STUCK G. Introduction to dynamical systems[M]. Cambridge, U.K ; New York: Cambridge University Press, 2002. [7] KATOK A, HASSELBLATT B. Introduction to the modern theory of dynamical systems[M]. Cambridge, U.K ; New York: Cambridge University Press, 1997. [8] VIANA M, OLIVEIRA K. Foundations of ergodic theory[M]. Cambridge, U.K ; New York: Cambridge University Press, 2016. [9] BURNS K, DOLGOPYAT D, PESIN Y. Partial hyperbolicity, Lyapunov exponents and stable ergodicity[J]. J Stat Phys, 2002, 108: 927-942. [10] OBATA D. On the stable ergodicity of diffeomorphisms with dominated splitting[J]. Nonlinearity, 2019, 32(2): 445-463. [11] LEE J M. Introduction to Smooth Manifolds[M]. New York: Springer New York, 2012. [12] PESIN Y B. Characteristic Lyapunov exponents and smooth ergodic theory[J]. Russ. Math. Surv, 1977, 32(4): 55-114. [13] MANÉ R. Ergodic theory and differentiable dynamics[M]. New York: Springer Science & Business Media, 2012. [14] PUGH C C. The 𝐶^(1+𝛼) hypothesis in Pesin theory[J]. Publ. Math. IHES, 1984, 59: 143-161. [15] RODRIGUEZ HERTZ F, RODRIGUEZ HERTZ M A, TAHZIBI A, et al. New criteria for ergodicity and nonuniform hyperbolicity[J]. Duke Math. J., 2011, 160(3): 599-629. [16] AVILA A, CROVISIER S, WILKINSON A. Diffeomorphisms with positive metric entropy[J]. Publ.Math.IHES, 2016, 124(1): 319-347. [17] BOCHI J. Genericity of zero Lyapunov exponents[J]. Ergod. Th. Dynam. Sys., 2002, 22(6): 1667-1696. [18] RODRIGUEZ HERTZ M A. Genericity of nonuniform hyperbolicity in dimension 3[J]. J. Mod. Dyn., 2012, 6(1): 121-138. [19] MANÉ R. Oseledec’s theorem from the generic viewpoint[J]. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2(Warsaw 1983)pp 1269–1276. PWN, Warsaw, 1984. [20] MOREIRA C, SILVA W. On the geometry of horseshoes in higher dimensions[Z]. 2012. [21] RUELLE D. An inequality for the entropy of differentiable maps[J]. Bol. Soc. Bras. Mat, 1978, 9(1): 83-87. [22] TAHZIBI A. 𝐶^1-generic Pesin’s entropy formula.[J]. C. R. Acad. Sci., 2002, 335: 1057-1062. [23] SUN X, TIAN X. Dominated splitting and Pesin’s entropy formula[J]. Discrete Contin. Dyn. Syst., 2012, 32: 1421-1434. [24] ABDENUR F, CROVISIER S. Transitivity and topological mixing for 𝐶1 diffeomorphisms[Z]. 2016. [25] AVILA A, BOCHI J. Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms[J]. Trans. Amer. Math. Soc., 2012, 364(6): 2883-2907.
所在学位评定分委会	数学系
国内图书分类号	O192
来源库	人工提交
成果类型	学位论文
条目标识符	http://sustech.caswiz.com/handle/2SGJ60CL/336332
专题	理学院_数学系
推荐引用方式 GB/T 7714	Jin MZ. STABLE ESSENTIAL DENSITY FOR ERGODIC HOMOCLINIC CLASSES[D]. 深圳. 南方科技大学,2022.

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