单调回复关系中的正拓扑熵-应用数学专业论文.docxVIP

单调回复关系中的正拓扑恼 英文摘要 Positive Topological Entropy for Monotone Recurrence Relations Abstract We associate the topological entropy of monotone recurrence relations with the Aubry-Mather theory. If there exists an interval [ρ0, ρ1], such that for each ω ∈ (ρ0, ρ1), all Birkhoff minimizers with rotation number ω do not form a foliation, then the d- iffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy. Therefore, we need to proof the following results: Firstly, for monotone recurrence relation, each minimizer with bounded action is Birkhoff. Secondly, we construct in configuration space with bounded action two so- lutions exchanging rotation numbers and hence arrive at the conclusion by Angenent’s criterion in [1]. Keywords: Monotone Recurrence Relation, Topological Entropy, Aubry-Mather The- ory, Birkhoff Minimizer, Gradient Flow. Written by Li Guo Supervised by Wen-Xin Qin II 目 录 §1 引言及主要结论 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2 准备知识 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 §3 具有有界作用的最小能量构型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 §4 由 Birkhoff 最小能量构型形成的十状结构 21 §5 主要定理的证明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 硕士期间撰写的论文 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 参 考 文 献 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 致 谢 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 单 单调回复关系中的正拓扑恼