本文關(guān)鍵詞： 指數(shù)映射族 不著陸 逃逸射線 聚點(diǎn)集 不可分割的連續(xù)統(tǒng) 奇異擾動 逃逸 擬圓周 Cantor圓周 Sierpinski地毯 無窮連通Fatou分支 Herman環(huán)　出處：《南京大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：這篇博士論文主要包含以下兩部分：第一部分是關(guān)于指數(shù)映射族逃逸射線聚點(diǎn)集的研究.作為超越整函數(shù)動力系統(tǒng)的典型研究對象,指數(shù)映射族的動力系統(tǒng)一直備受關(guān)注.其中一個重要的研究課題就是對其不著陸逃逸射線的研究.在這項(xiàng)研究之前,人們所發(fā)現(xiàn)的指數(shù)映射族的不著陸的逃逸射線都聚屬于聚點(diǎn)集無界的類型,更精確的說,它們的聚點(diǎn)集都是復(fù)平面中無界的不可分割的連續(xù)統(tǒng),并且必須包含某條逃逸射線的全部作為其聚點(diǎn)集的一部分.在本文中,作者對指數(shù)映射族構(gòu)造出了這樣的逃逸射線：它們的聚點(diǎn)集是復(fù)平面中的緊集.更進(jìn)一步,作者通過引入折疊模型,構(gòu)造出了三種新類型的逃逸射線.對每一條這樣的逃逸射線,作者定義了與之相關(guān)的一個返回序列.依據(jù)這個返回序列的組合特征,作者對聚點(diǎn)集的拓?fù)渥隽巳缦氯N分類：(1)包含部分逃逸射線的不可分割的連續(xù)統(tǒng)；(2)與逃逸射線互不相交的不可分割的連續(xù)統(tǒng)；(3)Jordan弧.第二部分是關(guān)于一族奇異擾動有理映射上的動力系統(tǒng).當(dāng)作為Pn(z)=zn的擾動時,我們構(gòu)造的函數(shù)族所擾動出來的Julia集是Cantor圓周,但是此Cantor圓周上的動力系統(tǒng)與傳統(tǒng)的McMullen映射族所得到的Cantor圓周上的動力系統(tǒng)卻不是拓?fù)涔曹椀?一方面,作者研究了此函數(shù)族在自由臨界點(diǎn)逃逸到0或者∞超吸引域的情形下(雙曲情形),按其逃逸到0或者∞超吸引域時的迭代次數(shù),對其Julia集所有可能的情形進(jìn)行了分類.這里得到的Julia集可以分為擬圓周,Cantor圓周,Sierpinski地毯和退化的Sierpinski地毯共四種情形.我們可以看出它此時具有非常豐富的動力學(xué)行為.并且,在每種情況下,作者還給出了具體的參數(shù)來說明相應(yīng)的情況的確會發(fā)生.特別地,作者給出了此情形下0和∞超吸引域邊界的正則性,證明了在這種情況下∞的直接超吸引域的邊界一定是一個擬圓周.對于Julia集是擬圓周的情形,作者給出了當(dāng)參數(shù)是實(shí)數(shù)時的精確范圍.對于Cantor圓周情形,作者給出了Cantor圓周存在性關(guān)于映射度的一個充要條件.另一方面,作者還研究了此函數(shù)族在所有情形下Julia集的連通性.通過討論其自由臨界軌道是否逃逸到0或者∞的超吸引域中,作者給出了其Julia集不連通的充要條件：其Julia集不連通當(dāng)且僅當(dāng)它是Cantor圓周.這等價于這個函數(shù)族有一個臨界值包含在0或者∞的超吸引域中,而其相應(yīng)的臨界點(diǎn)卻不在其中.這個結(jié)果可以看作是經(jīng)典二次多項(xiàng)式的Julia集連通性相關(guān)結(jié)論的一種類比.
[Abstract]:This dissertation mainly consists of the following two parts: the first part is about the study of exponential mapping family escape ray accumulation point set, as a typical research object of transcendental whole function dynamic system. The dynamical system of exponential mapping family has been paid much attention. One of the important research topics is the study of its non-landing escape ray. It is found that the untouched escape rays of exponential mapping family belong to the unbounded type of accumulation point set, and more precisely, their accumulation point sets are unbounded and indivisible continuum in complex plane. And must contain all of the escape rays as part of its set of accumulation points. The authors construct an escape ray for exponential mapping families: their set of accumulation points is a compact set in the complex plane. Furthermore, the author introduces a folding model. Three new types of escape rays are constructed. For each of these escape rays, the author defines a return sequence associated with it, according to the combined characteristics of the return sequence. The topology of the set of accumulation points is classified as follows: 1) an indivisible continuum containing partial escape rays; (2) an inseparable continuum that does not intersect with escape rays; The second part is about the dynamical system on a family of singular perturbation rational maps. The Julia set perturbed by the family of functions we constructed is the Cantor circle. But the dynamical system on the Cantor circle is not topological conjugate with the dynamic system on the Cantor circle obtained by the traditional McMullen mapping family. The authors study the iterations of this family of functions in the case of free critical point escaping to 0 or 鈭，

本文編號：1451291