本文選題：Julia集　切入點(diǎn)：轉(zhuǎn)移矩陣　出處：《清華大學(xué)》2016年博士論文

【摘要】：在這篇論文中,我們主要研究復(fù)動(dòng)力系統(tǒng)和離散薛定諤算子中的兩個(gè)問題.在第一部分中,我們根據(jù)Ahlfors覆蓋曲面的想法,建立了Julia集的幾何刻畫準(zhǔn)則,該刻畫準(zhǔn)則可適用于有理函數(shù),整函數(shù),亞純函數(shù).證明的主要是通過結(jié)合Ahlfors-Shimizu特征,Eremenko逃逸點(diǎn)的想法實(shí)現(xiàn)的.在第二部分中,我們主要是研究薛定諤算子的譜問題.我們?cè)赥hue-Morse薛定諤算子的譜集中構(gòu)造了ΣⅠ,ΣⅡ和ΣⅢ三個(gè)集合.其中ΣⅠ中的點(diǎn)是extended態(tài);ΣⅡ中的能量點(diǎn)是雙邊偽localized態(tài);ΣⅢ是單邊偽localized態(tài).首次在薛定諤算子中構(gòu)造出無界的跡軌道,并且細(xì)致地估計(jì)了對(duì)應(yīng)能量的轉(zhuǎn)移矩陣和特征方程的廣義解范數(shù)增長速度.特別地對(duì)于任意的E∈ΣⅡ∪ΣⅢ,轉(zhuǎn)移矩陣有如下的范數(shù)增長速度:另外我們也研究了這些能量點(diǎn)在譜測(cè)度下的局部維數(shù).ΣⅡ中的能量點(diǎn)對(duì)應(yīng)的局部維數(shù)是0;而ΣI∪ΣⅢ中的能量點(diǎn)對(duì)應(yīng)的局部維數(shù)至少是1.這些現(xiàn)象表明Thue-Morse薛定諤算子模型是一個(gè)具有高度復(fù)雜現(xiàn)象的混合模型.
[Abstract]:In this paper, we mainly study two problems in complex dynamical system and discrete Schrodinger operator. In the first part, we establish the geometric characterization criterion of Julia set according to the idea of Ahlfors covering surface. This characterization criterion can be applied to rational function, whole function, meromorphic function. The proof is mainly realized by combining the idea of Eremenko escape point with Ahlfors-Shimizu characteristic. In the second part, We mainly study the spectral problem of Schrodinger operator. In the spectral set of Thue-Morse Schrodinger operator, we construct three sets of 危 鈪，

本文編號(hào)：1683176