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https://ptsldigital.ukm.my/jspui/handle/123456789/519586| Title: | On conjugations of circle homeomorphisms with break and critical points |
| Authors: | Sokhobiddin Akhatkulov (P67478) |
| Supervisor: | Mohd Salmi Md Noorani, Prof. Dr. |
| Keywords: | Homeomorphisms Conjugations Dissertations, Academic -- Malaysia Universiti Kebangsaan Malaysia -- Dissertations |
| Issue Date: | 25-Mar-2015 |
| Description: | This thesis deals with circle homeomorphisms with break, critical and infinite type of singularities. The main purpose of this thesis is to study the existence and smoothness of conjugation between linear rotation and circle homeomorphisms with such singularities. The main results are proved in Chapters II and IV-VI by using cross-ratio distortion techniques and properties of dynamical partition of the circle. The first chapter consists from brief introduction with problem statements and objectives of the thesis. The basic concepts: definitions, notations and classical results on circle homeomorphisms with break points and critical points are provided and briefly discussed in the second and third chapters, respectively. At the end of second chapter we prove our first main result on smoothness of conjugation between two circle homeomorphisms with infinite number of break points and identical irrational rotation number. That is, we provide a sufficient condition for Holder continuity of this conjugation and we also provide a necessary and sufficient condition for C1smoothness of this conjugation. This result extends Herman's result. In Chapter IV, we prove that the conjugation between linear rotation and circle homeomorphisms with infinite number of break points and one critical point is quasi-symmetric. This result extends Swiatek's result. Besides we prove in this chapter the existence of conjugation between linear rotation and circle homeomorphisms with finite number of infinite and break type of singularities. Because the invariant measure of circle homeomorphism is closely related with conjugation, we prove in Chapter V, that the invariant measure of circle homeomorphisms with finite number of break points which satisfy certain Zygmund condition except these points is singular with respect to Lebesgue measure. This result extends the result of Dzhalilov et al. Finally, we prove in Chapter VI, that two circle homeomorphisms with one break point and the same irrational rotation number with non-trivial jump ratios which satisfy certain Zygmund condition are singularly conjugated, that is the conjugation between them is singular function.,Ph.D. |
| Pages: | 109 |
| Call Number: | QA614.S677 2015 tesis |
| Publisher: | UKM, Bangi |
| Appears in Collections: | Faculty of Science and Technology / Fakulti Sains dan Teknologi |
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