Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Critical recurrence in real quadratic and rational dynamics

Bylund, Mats LU (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:
author
supervisor
opponent
  • Prof. Graczyk, Jacek, Universite Paris-Sud XI, France.
organization
publishing date
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}},
}