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General


This course is part of the MAGIC core.

Description

The course provides an introduction to the topological dynamics of iterated maps: circle maps, unimodal maps, Smale's horseshoe, Hyperbolic toral automorphisms, and polynomials CC. There is an emphasis on symbolic techniques (kneading theory, Markov partitions, etc.)
The aim is to provide an introduction to a broad range of topics, rather than to discuss any of the topics in great depth.
Semester

Spring 2012 (Monday, January 16 to Friday, March 23)

Timetable
  • Fri 09:05 - 09:55

Lecturer


Toby Hall
Email tobyhall@liverpool.ac.uk
Phone (0151) 7944065
vcard
Photo of Toby Hall


Students


Photo of Abeer AL-NAHDI
Abeer AL-NAHDI
(Leeds)
Photo of Nada Alhabib
Nada Alhabib
(Liverpool)
Photo of Michael Bennett
Michael Bennett
(Cardiff)
Photo of Nasser Bin Turki
Nasser Bin Turki
(Liverpool)
Photo of Yumi Boote
Yumi Boote
(Manchester)
Photo of Anthony Chiu
Anthony Chiu
(Manchester)
Photo of Mariano Galvagno
Mariano Galvagno
(Loughborough)
Photo of Alena Jassova
Alena Jassova
(Liverpool)
Photo of Youssef Lazar
Youssef Lazar
(East Anglia)
Photo of Poj Lertchoosakul
Poj Lertchoosakul
(Liverpool)
Photo of Liangang Ma
Liangang Ma
(Liverpool)
Photo of Adam  Newman
Adam Newman
(Loughborough)
Photo of Robert Pattinson
Robert Pattinson
(Newcastle)
Photo of Jeevan Rai
Jeevan Rai
(Surrey)
Photo of Heather Riley
Heather Riley
(Liverpool)
Photo of Robert Royals
Robert Royals
(East Anglia)
Photo of James Wright
James Wright
(Surrey)
Photo of Stefanie Zegowitz
Stefanie Zegowitz
(East Anglia)


Prerequisites


Undergraduate-level courses in metric spaces and complex analysis.

Syllabus


  1. One-dimensional maps, circle maps, logistic map, period doubling cascade
  2. Chaos, symbolic coding of trajectories
  3. Two or higher dimensional maps, examples: cat map
  4. Hyperbolicity, homoclinic intersections, Smale horseshoe
  5. Quadratic polynomials, Julia and Mandelbrot sets

Bibliography


Please see the file "Comments on the bibliography" in the "Files" tab
GENERAL TEXTS
An introduction to chaotic dynamical systemsDevaney
A first course in dynamics: with a panorama of recent developmentsHasselblatt and Katok
Introduction to the modern theory of dynamical systemsKatok and Hasselblatt
One-dimensional dynamicsde Melo and van Strien
Dynamics in one complex variable: introductory lecturesMilnor
TEXTS DEALING WITH SPECIFIC TOPICS
Quantitative universality for a class of nonlinear transformationsFeigenbaum
A computer-assisted proof of the Feigenbaum conjecturesLanford
A shorter proof of the existence of the Feigenbaum fixed pointLanford
Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functionsBaldwin
Topological entropyAdler, Konheim and McAndrew
Entropy for group endomorphisms and homogeneous spacesBowen
Horseshoes for mappings of the intervalMisiurewicz
Topological entropy of Devaney chaotic mapsBalibrea and Snoha
On iterated maps of the intervalMilnor and Thurston
Kneading theoryScholarpedia
Entropy of piecewise monotone mappingsMisiurewicz and Szlenk
Bifurcations in one dimension. I. The nonwandering setJonker and Rand
Differentiable dynamical systemsSmale
Lyapunov exponents, entropy and periodic orbits for diffeomorphismsKatok
A two-dimensional mapping with a strange attractorH{\'e}non
The dynamics of the Hénon mapBenedicks and Carleson
Antimonotonicity: concurrent creation and annihilation of periodic orbitsKan, Ko{\c{c}}ak and Yorke


Note:

Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)

Assessment


Assessment will be by a take home exam at the end of the course. Further details will be provided in due course.

Assignments


MAGIC060 Exam

Files:Exam paper
Deadline: Friday 27 April 2012 (907.2 days ago)
Instructions:The questions are a combination of
  • bookwork;
  • examples similar to those presented in lectures or in the problem sheets; and
  • exercises which go slightly further than that.
The markscheme is such that you are able to pass by submitting good answers to the parts of questions of the first two types.

Note that you may find the exercises and their solutions helpful, as well as your lecture notes.

Good luck!