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 Discrete dynamical systems
Code MATH440
Coordinator Dr TDH Hall
Mathematical Sciences
T.Hall@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2016-17 Level 7 FHEQ Second Semester 15

Aims

To give an introduction to the theory of topological dynamical systems in low dimensions, emphasizing the use of Markov partitions and symbolic techniques as tools to analyse and understand dynamical properties.

Learning Outcomes

After completing the module, students should be able to understand the iteration of functions from the dynamical systems point of view.

After completing the module, students should be able to demonstrate familiarity with a range of examples of low-dimensional dynamical systems.
After completing the module, students should understand and be able to calculate with Markov partitions and symbolic dynamics, to analyse the behaviour of dynamical systems.
After completing the module, students should understand the concept of forcing for periodic orbits of interval maps and disk homeomorphisms, and be able to compute the forcing relation in examples.
After completing the module, students should be able to use train tracks to calculate persistent dynamics in isotopy classes of surface homeomorphisms.

Syllabus
1 
  1.  Introductory examples, including the doubling map on the circle and the logistic family.

    2.       Necessary concepts from the theory of metric spaces.

    3.       General  theory of discrete dynamical systems: periodic orbits, recurrence, topological conjugacy and semi-conjugacy, top ological transitivity, symbolic dynamics, subshifts of finite type, chaos and topological entropy.

    4.       Periodic orbits in one-dimensional dynamics: the Markov graph theorem, period 3 implies chaos, Sharkovsky’s theorem, the forcing relation on patterns, unimodal patterns.

    5.       Smale’s horseshoe map: relationship to transverse homoclinic intersections and positive topological entropy.

    6.       Dynamics of homeomorphisms of the disk: braids and braid types of periodic orbits, pseudo-Anosov homeomorphisms and Thurston’s classification theorem; train tracks.

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List:

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):
MATH101; MATH102; MATH103  

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:
Programme:G101 Year 3,4 Programme:MMAS Year:1

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Unseen Written Exam  2.5 hours  Second Semester  100  Yes  Standard UoL penalty applies  Assessment 1 Notes (applying to all assessments) Candidates may attempt all questions. Best FIVE answers taken into account. The questions carry equal weight. 
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes