Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title THE MAGIC OF COMPLEX NUMBERS: COMPLEX DYNAMICS, CHAOS AND THE MANDELBROT SET
Code MATH345
Coordinator Professor L Rempe
Mathematical Sciences
L.Rempe@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2021-22 Level 6 FHEQ Second Semester 15

Aims

1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects;

2. To introduce students to some topics of current and recent research in the field;

3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting;

4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics;

5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family.

6. To deepen students' appreciations for formal reasoning and proof. After completing the module, students should be able to:
1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.
2. us e Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.
3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.
4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.
5. apply advanced results from complex analysis in the setting of complex dynamics.
6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.


Learning Outcomes

(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

(LO5) To know how to apply advanced results from complex analysis in a dynamical setting.

(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills


Syllabus

 

Syllabus:
1. Introduction; review of complex functions and related topics; iteration.
2. The Riemann sphere: spherical distance, spherical derivative, Möbius transformations.
3. The Julia and Fatou sets; (in)stability of periodic cycles; filled Julia set of a polynomial; simple examples.
4. Normal families: Weierstraß theorem, Hurwitz theorem, Marty's theorem, Montel's theorem, Picard's theorem
5. Properties of the Julia set, dense orbits and sensitive dependence on initial conditions.
6. Zalcman's lemma and chaos of the Julia set.
7. The Mandelbrot set: Definition, characterisation, connection with the logistic family.


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

MATH243 COMPLEX FUNCTIONS 

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment open book and remote Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Examination period (second semester)  1 hour time on task    50       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
short quiz 2 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Semester 2  around 60-90 minutes    10       
short quiz 1 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Semester 2  around 60-90 minutes    10       
short quiz 4 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Semester 2  around 60-90 minutes    10       
short quiz 3 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Semester 2  around 60-90 minutes    10       
short quiz 5 Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule (When) :Semester 2  around 60-90 minutes    10