动力系统中的度量丢番图逼近

发布时间：2021-11-07 23:46

　　设（X,d）是紧致度量空间,T:X→为连续映射,则称（X,d,T）为拓扑动力系统。动力系统主要研究连续映射轨道渐近性质,通常利用拓扑熵、拓扑压、混沌和Lyapunov指数等来刻画这种轨道性质。动力系统轨道的回复性是动力系统研究的重要课题,它与数论,分形几何,微分方程等学科有着深刻的关联。我们把重点放在动力系统中度量丢番图逼近问题相关回复性质的量化研究,也就是利用拓扑熵,拓扑压,Hausdorff维数等来对动力系统中的回复行为,收缩靶问题进行量化的研究。本文主要利用动力系统中的轨道跟踪性质来构造Moran分形集来刻画动力丢番图集合占系统的比重。在重分形分析的观点来看,是研究一类具有一定的回复、收缩行为的度量丢番图逼近水平集。第一章主要是推广了 Bosher-nitzan的关于定量回复的结果到半群作用的动力系统;第二章,研究满足非一致结构的子位移系统中的回复行为,给出了关于回复的动力丢番图的水平集的Hausdorff维数的估计;第三章,研究非一致系统的的饱和集并建立了相应的条件变分原理;第四章,定义了一类新的水平集刻画拓扑混合有限型的发散点集中的回复性;第五章,对于具有specificat... 

【文章来源】：南京师范大学江苏省 211工程院校

【文章页数】：123 页

【学位级别】：博士

【文章目录】：
摘要
Abstract
Preface
    0.1 Metric Diophantine approximation
    0.2 Nondense orbit set
    0.3 Multifractal analysis
Chapter 1 Quantitative recurrence properties for free semigroup actions
    1.1 Preliminaries and main results
    1.2 Proof of Theorem 1.1.4 and 1.1.5
Chapter 2 Quantitative recurrence properties for systems with non-uniformstructure
    2.1 Preliminaries and main results
        2.1.1 Topological pressure
    2.2 Proof of Theorem 2.1.1
        2.2.1 Proof of upper bound
        2.2.2 Proof of lower bound
    2.3 Proof of Theorem 2.1.2
        2.3.1 Proof of upper bound
        2.3.2 Proof of lower bound
    2.4 Applications
Chapter 3 Topological pressure of generic points sets with non-unifromstructure
    3.1 Preliminaries and main results
    3.2 Proof of Theorem 3.1.2
        3.2.1 Choose the sequence {n_j}_(j≥1)
        3.2.2 Construction of fractal set H
        3.2.3 To estimate the lower bound
    3.3 Applications
Chapter 4 Quantitative recurrence properties in the historic set for sym-bolic systems
    4.1 Preliminaries and main results
    4.2 Some important lemmas
    4.3 Proof of Theorem 4.1.1
        4.3.1 Proof of upper bound
        4.3.2 Proof of lower bound
    4.4 Proof of Theorem 4.1.2
        4.4.1 Proof of upper bound
        4.4.2 Proof of lower bound
Chapter 5 On the topological entropy of the set with a special shadowingtime
    5.1 Preliminaries and main results
    5.2 Proof of Theorem 5.1.2
        5.2.1 Upper bound for h_(top)~B(D_f~(xo))
        5.2.2 Lower bound for h_(top)~B(D_f~(xo))
        5.2.3 Construction of the Fractal F
        5.2.4 Construction of a special sequence of measure μ_k
    5.3 Applications
Chapter 6 Non-dense orbits on topological dynamical systems
    6.1 Preliminaries and main results
    6.2 Proof of Theorem 6.1.1
        6.2.1 Construction of the Fractal F
        6.2.2 Construction of a special sequence of measures μ)k
    6.3 Applications
Bibliography
Acknowledgements
Publications and Preprints
Further researches



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