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This page gives a rough summary of each lecture, and contains my lecture notes and the problem sets for each week.
WEEK 1 - Problem Set #1 (due 16/01)
06/01 - Introduction to dynamical systems. Action of a semigroup. Discrete time example.
07/01 - Continuous time example. Question of existence and uniqueness of solutions to Initial Value Problems (IVP). Banach spaces, convergence, continuity, contractions. Banach Fixed Point Theorem.
09/01 - Closedness. Proof of the Banach Fixed Point Theorem. Weissinger's Theorem. The integral operator associated to an IVP.
WEEK 2 - Problem Set #2 (due 23/01)
13/01 - Proof of the Picard-Lindelöf Theorem. Sufficient condition for the existence and uniqueness of solution to an IVP for all times.
14/01 - Gronwall Inequality. Dependence of solutions to an IVP upon initial conditions.
16/01 - The flow generated by an autonomous IVP defines a group action. Orbits, fixed points, periodic points. Invariant sets and their properties.
WEEK 3 - Problem Set #3 (due 30/01)
20/01 - 𝜔-limit sets and their properties (compactedness, connectedness).
21/01 - Trapping regions, attracting sets, domains of attraction, attrasctors. Lyapunov stability and asymptotic stability of fixed points.
23/01 - Criterion for asymptotic stability. Jordan Canonical form. Matrix exponential and the solution to linear systems.
WEEK 4 - Problem Set #4 (due 06/02)
27/01 - Time evolution of a linear dynamical system with distinct eigenvalues. Limit Cycles. Eigenvalues at a periodic point.
28/01 - Criterion for stability of a limit cycle. Hilbert's 16th problem.
30/01 - Poincaré-Bendixon Theorem.
WEEK 5 - Problem Set #5 (due 14/02)
03/02 - Discrete time dynamics. Topological notions. Topologically transitive and minimal homeomorphisms. Irrational rotations are minimal.
04/02 - Equidistribution of orbits of irrational rotations.
06/02 - Application of equidistribution to digit distribution. Unique ergodicity for irrational rotations.
10/02 - Beyond rotations. Orientation-preserving homeomorphisms of the circle. Rotation number.
11/02 - Periodic points of an orientation-preserving homeomorphism of the circle with rational rotation number. Asymptotic behaviour of non-periodic point.
13/02 - Semi-conjugacies and conjugacies. The rotation number is a topological invariant. Poincaré Classification theorem.
OPTIONAL: 14/02 - 10:30 am in Jeffery 319 - Dynamics, Geometry, & Groups Seminar: Denjoy's non-transitive diffeomorphisms of the circle.
Reading Week 17/02 - 21/02
WEEK 7 - Problem Set #6 (due 06/03)
24/02 - Devaney's definition of chaos. Topological transitivity revisited (without proof). Topological mixing.
25/02 - Topological transitivity revisited (proof).
27/02 - Project Outline due today - Expanding maps of the circle are topologically mixing (and hence transitive). Linear expanding maps of the circle are chaotic. Chaotic maps exhibit sensitive dependence on initial conditions.
WEEK 8 - Problem Set #7 (due 13/03)
02/03 - Chaotic maps exhibit sensitive dependence on initial conditions (proof continued). Topologically mixing maps exhibit sensitive dependence on initial conditions.
03/03 - Semiconjugacy of the doubling map with the shift map on the space of binary sequences. Periodic points for this shift give periodic points for the doubling map. Construction of a dense orbit.
05/03 - Sequence spaces as metric spaces. Balls and cylinders. Topological Markov chains.
09/03 - Full shifts are topologically mixing and their periodic points are dense. Topological Markov chains corresponding to irreducible matrices.
10/03 - Topologically mixing Markov chains with dense periodic points. Coding for expanding maps of the circle.
12/03 - Coding for expanding maps of the circle. Expanding maps of the circle of degree 2 are all topologically conjugate.
*** THERE WILL BE NO CLASSES DURING WEEK 10 ***
16/03 - no class
17/03 - no class
19/03 - no class
*** WEEKS 11 AND 12 WILL BE DELIVERED REMOTELY, AS A SERIES OF VIDEOS ***
23/03 - Project Draft due today
- Introduction to topological entropy as invariant (2'42'').
- Capacity and box dimension of a compact metric space (24'03'').
- A dynamically defined sequence of metrics and the definition of the topological entropy (11'56'').
- Isometries and contractions have zero entropy. Some properties of the topological entropy (11'12'').
24/03
- Recall the definition of topological entropy (4'53'').
- Definition of local entropy (11'59'').
- The local entropy is bounded above by the topological entropy (3'04'').
- At some point, the reverse inequality holds before taking the limit r->0 (21'58'').
- Final remark: the topological entropy is the supremum of the local entropies (4'23'').
26/03
- The logistic map for parameters between 0 and 1: a unique attracting fixed point (6'34'').
- The logistic map for parameters between 1 and 3: a new attracting fixed point (13'03'').
- The logistic map for parameters between 3 and 1+√6: an attractive orbit of period 2 (13'41'').
- The logistic map for parameters between 3 and 3.56994567...: period doubling bifurcations (10'25'') and Feigenbaum universality (4'01'').
WEEK 12
30/03
31/03 - Feedback on others' drafts due today
- Recall the statement of Sharkovskii's theorem (1'39'').
- Definitions and constructions: h-intervals and A-graphs (8'05'').
- Three key lemmas: Lemma 1 (3'44''), Lemma 2 (11'08''), and Lemma 3 (2'40'').
- The strategy of the proof, divided in four cases, and the particular case of period 3 (16'34'').
- The proof of case 1 of Sharkovskii's theorem: coexistence of periodic orbit of periods powers of two (6'53'').
02/04
- The proof of case 2 of Sharkovskii's theorem: the existence of an orbit of period n=p*2^m (with odd p≥3 and m≥0) implies the existence of an orbit of period k=q*2^m (with even q≥2) (19'01'').
- The proof of case 3 of Sharkovskii's theorem: the existence of an orbit of period n=p*2^m (with odd p≥3 and m≥0) implies the existence of an orbit of period k=q*2^m (with odd q>p) (9'33'').
- The proof of case 4 of Sharkovskii's theorem: the existence of an orbit of period n=p*2^m (with odd p≥3 and m≥0) implies the existence of an orbit of period k=2^l (with 0<l<m) (8'04'').
- In case you want to see the proof of Lemma 1, Lemma2, and Lemma 3 (not covered in these lectures), you can look at my notes.
Other Deadlines:
15/04 - Final paper due today
22/04 - Feedback on others' papers due today
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