Dynamical Systems - Spring Semester 2021
Update: Due to the global state of emergency caused by COVID 19 , the IMM has suspended on-site lecturer visits until further notice and will perform the lessons of the Fall semester, 2020 through distance learning. This includes the Dynamical Systems course, which is taught through a combination of pre-recorded video lectures, Google Classroom assignments and interactive teacher-to-students Q&A/exercise sessions through the Zoom video conferencing tool. These Zoom interactive sessions are also recorded and made available to the students, along with all other course materials.
Dates: February-May 2021
Faculty:
Stefano Luzzatto
(Trieste, Italy)
Jeroen Lamb
(London, UK)
Liviana Palmisano
(Durham, UK)
Douglas Coates
(Tutor, Trieste, Italy)
Stefano Luzzatto
Jeroen Lamb
Liviana Palmisano
Douglas Coates
Syllabus:
Fundamental existence and uniqueness results for solutions of ODEs. Poincare Maps. Definition of Discrete and Continuous Time Dynamical Systems. Fixed points, periodic orbits, alpha and omega limits sets. Topological Conjugacy and Structural Stability. Structural Stability of Interval Diffeomorphisms. Structural Stability of Linear Maps. Structural Stability of Hyperbolic Fixed Points. Symbolic Dynamics. Structural Stability of Full Branch Piecewise Expanding Interval Maps.
Basic concepts of Ergodic Theory. Invariant Measures. Ergodic Measures. Physical Measures. Poincarè Recurrence, Birkhoff’s Ergodic Theorem (without proof). Ergodicity of Lebesgue Measure for Full Branch Piecewise Expanding Interval Maps. Distortion estimates. Existence of absolutely continuous ergodic invariant probability measures for Full Branch Piecewise Expanding Interval Maps. Induced maps.
Textbooks:
Oliveira & Viana, Foundations of Ergodic Theory, CUP https://www.cambridge.org/core/books/foundations-of-ergodic-theory/F5AE11B50B16D9FF32909EA4BBA1E7CE
An introduction to Dynamical Systems, C. Robinson, AMS https://www.amazon.com/gp/product/0821891359/ref=dbs_a_def_rwt_hsch_vapi_taft_p1_i1
Course Schedule (Pakistan time):
Week 1 (15/02-19/02):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 1
Class notes Lecture 2
Exercise Sheet 1
Week 2 (22/02-26/02):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 3
Class notes Lecture 4
Exercise Sheet 2
Week 3 (01/03-05/03):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 5
Class notes Lecture 6
Exercise Sheet 3
Week 4 (08/03-12/03):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 7
Class notes Lecture 8
Exercise Sheet 4
Week 5 (15/03-19/03):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 9
Week 6 (22/03-26/03):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 10
Class notes Lecture 11
Class notes Lecture 12
Week 7 (29/03-02/04):
Midterm break and revision
Exercise Sheet 5
Exercise Sheet 6
Week 8 (05/04-09/04):
Midterm exam
Week 9 (12/04-16/04), beginning of part 2:
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lecture 13
Class notes Lecture 14
Week 10 (19/04-23/04):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Exercise Sheet 7
Week 11 (26/04-30/04):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Class notes Lectures 17-18
Week 12 (03/05-07/05):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 13(10/05-14/05):
Holiday
Problems class on Friday 18:30-20:00h
Week 14 (17/05-21/05):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 15 (24/05-28/05):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 16 (31/05-04/06):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 17 (07/06-11/06):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 18 (14/06-18/06):
Live lecture on Monday and Thursday 14:30h-16:00h
Q&A session on Wednesday 17:00-18:30h
Problems class on Friday 14:30h-16:00h and 18:30-20:00h
Week 19 (21/06-25/06):
Break and revision
Week 20 (28/06-02/07):
Final exam
Related Materials
ICTP Summer School in Dynamics - Rotations of the circle and renormalization (video playlist, 2018)
ICTP Summer School in Dynamics - Piecewise expanding maps of the circle (video playlist, 2018)
ICTP Summer School in Dynamics - Introduction to ergodic theory (video playlist, 2018)
ICTP Summer School in Dynamics - Ergodicity in smooth dynamics (video playlist, 2018)
ICTP Diploma Programme - Differential Equations and Dynamical Systems (video playlist, 2014) by Prof. Stefano Luzzatto