發(fā)布時(shí)間：2018-10-23 18:12

【摘要】：動(dòng)力系統(tǒng)和遍歷理論是20世紀(jì)富有成就的數(shù)學(xué)分支之一,在數(shù)學(xué)的其他分支也有著十分廣泛的應(yīng)用,如函數(shù)論、組合數(shù)學(xué)以及計(jì)算數(shù)學(xué)等領(lǐng)域。從最基本的素?cái)?shù)定理到流形上的周期軌道漸近估計(jì)引起了眾多學(xué)者的關(guān)注并取得了很多重要結(jié)論。雙曲流的周期軌道分布,尤其是雙曲流的周期軌道在各種限制條件下的分布的研究是近來較為活躍的研究方向之一。本文以雙曲流、以及關(guān)于雙曲流周期軌道在同調(diào)類上分布的漸近理論為基礎(chǔ),討論了關(guān)于同調(diào)類的差分固定的雙曲流的素軌道的漸近估計(jì)。對于流形上周期軌道漸近估計(jì)問題,我們通常通過建立ξ函數(shù),研究其解析性,以及對極點(diǎn)的研究,從而得出周期軌道的漸近公式。本文改變了以往的研究方法,而是通過建立Selberg跡公式與流形上同調(diào)類上閉測地線之間的關(guān)系式,應(yīng)用Selberg跡公式和多元部分求和公式得到流形上關(guān)于同調(diào)類的漸近公式的主要項(xiàng)和誤差項(xiàng)。本文在已有的研究理論基礎(chǔ)之上,部分推廣和改進(jìn)了雙曲流上關(guān)于同調(diào)類的周期軌道漸近分布的基本理論。
[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.
【學(xué)位授予單位】：南京理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O19

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