Critical recurrence in real quadratic and rational dynamics
(2022)- Abstract
- In this thesis we study the dynamics of real quadratic functions on the interval, and rational functions on the Riemann sphere. The problems we are considering are concerned with the recurrent nature of the critical orbit(s). In Paper I we investigate the real quadratic family and prove a theorem regarding the rate of recurrence of the critical point to itself, extending a previous result by Avila and Moreira. In Paper II and Paper III we consider rational functions. Here we do not study the rate of recurrence, rather we assume that the critical points approach each other at a slow rate, and investigate some of the consequences. Assuming this slow recurrence condition, we prove in Paper II that certain Collet--Eckmann rational functions... (More)
- In this thesis we study the dynamics of real quadratic functions on the interval, and rational functions on the Riemann sphere. The problems we are considering are concerned with the recurrent nature of the critical orbit(s). In Paper I we investigate the real quadratic family and prove a theorem regarding the rate of recurrence of the critical point to itself, extending a previous result by Avila and Moreira. In Paper II and Paper III we consider rational functions. Here we do not study the rate of recurrence, rather we assume that the critical points approach each other at a slow rate, and investigate some of the consequences. Assuming this slow recurrence condition, we prove in Paper II that certain Collet--Eckmann rational functions can in a strong sense be approximated by hyperbolic ones. In Paper III we observe that within the family of slowly recurrent rational maps, the well-known Collet--Eckmann conditions are all equivalent. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/537bfca2-a0b9-404f-8317-7718819dbd3d
- author
- Bylund, Mats LU
- supervisor
- opponent
-
- Prof. Graczyk, Jacek, Universite Paris-Sud XI, France.
- organization
- publishing date
- 2022-09-30
- type
- Thesis
- publication status
- published
- subject
- pages
- 136 pages
- publisher
- Lund University / Centre for Mathematical Sciences /LTH
- defense location
- Lecture hall MH:Riesz, Centre of Mathematical Sciences, Sölvegatan 18, Faculty of Engineering LTH, Lund University, Lund.
- defense date
- 2022-10-26 10:00:00
- ISBN
- 978-91-8039-342-3
- 978-91-8039-341-6
- language
- English
- LU publication?
- yes
- id
- 537bfca2-a0b9-404f-8317-7718819dbd3d
- date added to LUP
- 2022-09-29 15:04:46
- date last changed
- 2022-10-03 08:35:35
@phdthesis{537bfca2-a0b9-404f-8317-7718819dbd3d, abstract = {{In this thesis we study the dynamics of real quadratic functions on the interval, and rational functions on the Riemann sphere. The problems we are considering are concerned with the recurrent nature of the critical orbit(s). In Paper I we investigate the real quadratic family and prove a theorem regarding the rate of recurrence of the critical point to itself, extending a previous result by Avila and Moreira. In Paper II and Paper III we consider rational functions. Here we do not study the rate of recurrence, rather we assume that the critical points approach each other at a slow rate, and investigate some of the consequences. Assuming this slow recurrence condition, we prove in Paper II that certain Collet--Eckmann rational functions can in a strong sense be approximated by hyperbolic ones. In Paper III we observe that within the family of slowly recurrent rational maps, the well-known Collet--Eckmann conditions are all equivalent.}}, author = {{Bylund, Mats}}, isbn = {{978-91-8039-342-3}}, language = {{eng}}, month = {{09}}, publisher = {{Lund University / Centre for Mathematical Sciences /LTH}}, school = {{Lund University}}, title = {{Critical recurrence in real quadratic and rational dynamics}}, url = {{https://lup.lub.lu.se/search/files/124752976/Avhandling_Mats_Bylund_WEB.pdf}}, year = {{2022}}, }