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This course is part of the MAGIC core.


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.


Spring 2017 (Monday, January 23 to Friday, March 31)


  • Wed 09:05 - 09:55


Undergraduate-level courses in metric spaces and complex analysis.


  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


Toby Hall
Phone (0151) 7944065
Photo of Toby Hall


Photo of Abeer Al Balahi
Abeer Al Balahi
(East Anglia)
Photo of Marco Baffetti
Marco Baffetti
Photo of Jonathan Brooks
Jonathan Brooks
Photo of Daniel Evans
Daniel Evans
Photo of Andrea Foiniotis
Andrea Foiniotis
Photo of Joel Mitchell
Joel Mitchell
Photo of Marzia Romano
Marzia Romano
Photo of Luke Warren
Luke Warren


Please see the file "Comments on the bibliography" in the "Files" tab
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
Iteration of Rational Functions: Complex Analytic Dynamical SystemsBeardon
Dynamics in one complex variable: introductory lecturesMilnor
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
Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjectureLyubich
Dynamics of quadratic polynomials. I, II.Lyubich, Mikhail
Generic hyperbolicity in the logistic familyGraczyk and {'S}wiatek
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


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 will be by a take-home exam in April/May 2017. You will have 2 weeks to complete the exam and submit your solutions online. There will be 5 questions, and full marks can be obtained if you submit correct solutions to 3 of them.

No assignments have been set for this course.

Recorded Lectures

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