双曲流上关于同调类的周期轨道的渐近估计
发布时间：2018-10-23 18:12
【摘要】：动力系统和遍历理论是20世纪富有成就的数学分支之一,在数学的其他分支也有着十分广泛的应用,如函数论、组合数学以及计算数学等领域。从最基本的素数定理到流形上的周期轨道渐近估计引起了众多学者的关注并取得了很多重要结论。双曲流的周期轨道分布,尤其是双曲流的周期轨道在各种限制条件下的分布的研究是近来较为活跃的研究方向之一。本文以双曲流、以及关于双曲流周期轨道在同调类上分布的渐近理论为基础,讨论了关于同调类的差分固定的双曲流的素轨道的渐近估计。对于流形上周期轨道渐近估计问题,我们通常通过建立ξ函数,研究其解析性,以及对极点的研究,从而得出周期轨道的渐近公式。本文改变了以往的研究方法,而是通过建立Selberg迹公式与流形上同调类上闭测地线之间的关系式,应用Selberg迹公式和多元部分求和公式得到流形上关于同调类的渐近公式的主要项和误差项。本文在已有的研究理论基础之上,部分推广和改进了双曲流上关于同调类的周期轨道渐近分布的基本理论。
[Abstract]:Dynamical systems and ergodic theory are one of the most successful branches of mathematics in the 20th century. They are also widely used in other branches of mathematics, such as function theory, combinatorial mathematics and computational mathematics. From the most basic prime theorem to the asymptotic estimation of periodic orbits on manifolds, many scholars have paid close attention to it and obtained many important conclusions. The research on the distribution of periodic orbits of hyperbolic flows, especially the periodic orbits of hyperbolic flows under various restricted conditions, is one of the more active research directions recently. In this paper, based on the asymptotic theory of hyperbolic flow and the distribution of periodic orbits on homology classes, the asymptotic estimates of prime orbits for homology difference fixed hyperbolic flows are discussed. For the asymptotic estimation of periodic orbits on manifolds, we usually obtain the asymptotic formula of periodic orbits by establishing 尉 function, studying its analytic property and studying the poles. In this paper, the previous research methods are changed, and the relationship between the Selberg trace formula and the upper closed geodesic of the cohomology of manifolds is established. By using the Selberg trace formula and the multivariate partial summation formula, the main terms and the error terms of the asymptotic formula for homology on manifold are obtained. In this paper, based on the existing theories, we extend and improve the basic theory of the asymptotic distribution of periodic orbits on hyperbolic flows.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19
本文编号：2290074
[Abstract]:Dynamical systems and ergodic theory are one of the most successful branches of mathematics in the 20th century. They are also widely used in other branches of mathematics, such as function theory, combinatorial mathematics and computational mathematics. From the most basic prime theorem to the asymptotic estimation of periodic orbits on manifolds, many scholars have paid close attention to it and obtained many important conclusions. The research on the distribution of periodic orbits of hyperbolic flows, especially the periodic orbits of hyperbolic flows under various restricted conditions, is one of the more active research directions recently. In this paper, based on the asymptotic theory of hyperbolic flow and the distribution of periodic orbits on homology classes, the asymptotic estimates of prime orbits for homology difference fixed hyperbolic flows are discussed. For the asymptotic estimation of periodic orbits on manifolds, we usually obtain the asymptotic formula of periodic orbits by establishing 尉 function, studying its analytic property and studying the poles. In this paper, the previous research methods are changed, and the relationship between the Selberg trace formula and the upper closed geodesic of the cohomology of manifolds is established. By using the Selberg trace formula and the multivariate partial summation formula, the main terms and the error terms of the asymptotic formula for homology on manifold are obtained. In this paper, based on the existing theories, we extend and improve the basic theory of the asymptotic distribution of periodic orbits on hyperbolic flows.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19
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