Schedule
| Format | time and place | Topics to be covered | Suggested reading | Suggested exercises | Extras |
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Lecture 1 (D)
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24/1, Q17 | Introduction, basic notions Ch 2. | Sections 2.3.1, 2.3.2 and 2.4 | Example 2.10 on page 21, Exercises 2.1, 2.8, 2.10, 2.14, 2.18 and 2.19 |
old lecture 1 zoom notes Download old lecture 1 zoom notes Intro into dynamics Download Intro into dynamics From order to chaos: the prize competition in honour of king Oscar II Download From order to chaos: the prize competition in honour of king Oscar II |
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Lecture 2 (D) |
28/1, 8-10 Q15 | Topological transitivity, mixing, topological equivalence, coding of doubling map 3.1-3.3, 3.4.2 |
3.1-3.3, 3.4.2 and Ch 7. Proof of Proposition 2.2 |
From the handout quadratic family Download quadratic family : Exercise A.5.1, def A.5.1, Exercise A.5.5, Exercise A.5.7 Logistic map animation Links to an external site. From the book: exercises 3.8, 3.10, |
old lecture 2 zoom notes Download old lecture 2 zoom notes
Fourier analysis on the torus Download Fourier analysis on the torus
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| Exercise | 29/1, 10-12, Q22 |
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Lecture 3 (D) |
4/2, 8-10 Q15 |
Sensitive dependence on initial conditions, toral automorphisms, topological mixing, more examples: baker map, more on coding, bit on fractals. (Ch 3, Ch 7) |
3.3.2, 7.1.1, 7.1.3, 7.2.1, 7.2.2 Baker's map coding Download Baker's map coding
Minimality of irrational flows on the torus Download Minimality of irrational flows on the torus
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Exercises 11, 12, 14 (take L=0, R=2), 15, 16, 17,
Cantor set + exercises.pdf Download Cantor set + exercises.pdf (from Hirsch-Smale-Devaney book) |
old lecture 3 Download old lecture 3 Box dimension of fractals Links to an external site. Fractals in nature Links to an external site. A propos Cantor set: Devil's staircase Links to an external site. For those who like complex analysis: fractals in complex dynamics Links to an external site.
Billiards Links to an external site. Motion of planets: 3 body problem Links to an external site. |
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Lecture 4 (M) |
11/2, 8-10 V01
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Topological entropy: motivation; several definitions and their equivalence; examples. Topological invariance of the topological entropy; some other properties of the topological entropy. |
Sections 3.4.1-3.4.3, 7.1.2, 7.2.1. You can have a look at Katok's survey Links to an external site. on entropy (just to get an impression) |
Old Lecture 4 Download Old Lecture 4 A note on the history by Sinai Links to an external site. As an analogy: definition of the box-counting dimension Links to an external site. For inspiration: You can look at the introduction to a recent paper by S.Crovisier & E.Pujals Links to an external site. |
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| Exercise |
12/2, 10-12 U61 |
Study Example 3.15 in the book (p.48). Do Exercises from the book: 3.11-3.15, 3.17. | |||
| HW 1 due on 17/2 | |||||
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Lecture 5 (M) |
18/2, 8-10, V01 |
Homeomorphisms of the circle I, Ch 4.1 | Section 4.1; can also look at Milnor's lecture notes Links to an external site. | Exercises from the book: 3.13-3.15, 3.17; 4.1-4.8. |
Old Lecture 5 Download Old Lecture 5 An overview of the work of J.-Ch. Yoccoz (Fields medal) on circle diffeomorphisms Links to an external site., as a motivation. |
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Lecture 6 (M) |
25/2, 8-10 Q15 |
Diffeomorphisms of the circle. Denjoy's theorem; Denjoy's counterexample. | Section 4.2, Denjoy counterexample Download Denjoy counterexample | Exercises from the book: 4.10-4.16 | Old Lecture 6 Download Old Lecture 6 |
| Exercise |
26/2, 10-12 E33 |
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| HW2 is due 17/3 | |||||
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Lecture 7 (M) |
18/3, 8-10 Q22 |
Maps of the interval: Sharkovsky theorem quadratic family |
Sec. 4.3 in the course book [BV] Devaney's book [D] Download Devaney's book [D] Sec. 1.4, 1.5 (pp. 24-39)
Please look at the following announcement Links to an external site. of the Master class in Dynamical systems! And tell me if you want to attend! |
Book [BV]: Book [D]: Ex. 1-6 pp. 29-31; Ex. 1,2 p.38. |
Old lecture notes Download Old lecture notes A book by Strogatz Links to an external site., Saddle-node bifurcation: p. 49; Logistic map: Chap. 10 (p.355-394) |
| Exercise |
18/3, 10-12 Q26 |
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Book [BV]: Ex. 4.1--4.16 |
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Lecture 8 (M) |
25/3, 8-10 Q15 |
Hyperbolic dynamics I Hyperbolic sets, Smale horseshoe and homoclinic tangle |
Book [BV]: Chap. 5; |
Book [BV]: Ex. 5.1, 5.2, 5.7--5.10, 5.13
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A nice overview of the hyperbolic dynamics Links to an external site. Smale's article Download Smale's article Quanta article on the horseshoe Links to an external site. Animated horseshoe Links to an external site. Horseshoe in Copacabana story Links to an external site. What is a Horseshoe? (by Shub) Links to an external site. From Poincare to horseshoe and back Links to an external site. When physicists discover homoclinic tangle Links to an external site. and |
| Exercise |
26/3, 10-12 D34 |
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Lecture 9 (M) |
1/4, 8-10 Q15 |
Hyperbolic dynamics II: The Grobman-Hartman and the Hadamard-Perron theorems. Structural stablity.
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Book [BV]: Chap. 6
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Exercises from the book: 6.3, 6.7, 6.8, 6.9. |
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| Exercise |
2/4, 10-12 Q17 |
ChaoticDynamicalSystemsNotes-1.pdf Download ChaoticDynamicalSystemsNotes-1.pdf |
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Lecture 10 (D) |
8/4, 8-10 Q15 |
Hyperbolic dynamics: stability |
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Lecture 11 (D) |
9/4, 10-12 E33 |
Non-hyperbolic systems, examples | |||
| Exercise |
15/4, 8-10 Q17 |
ChaoticDynamicalSystemsNotes2.pdf Download ChaoticDynamicalSystemsNotes2.pdf | |||
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Lecture 12 (D) |
29/4, 8-10 Q15 |
Geodesic flows |
Section 6.3 (If you wish to read a more detailed introduction into geodesic flow on negatively curved surfaces, Chapter 9 in this book Links to an external site. may help. Also, exercises 9.1.1, 9.1.2, 9.2.1, and exercises for section 9.3 may be useful.) My old zoom lecture notes on geodesic flow Download My old zoom lecture notes on geodesic flow
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Short film explaining basics of curved spaces and geodesics Links to an external site. Geodesic flow on curved surface Links to an external site. Geodesics and relativity Links to an external site. Animation: stable and unstable foliations of the geodesic flow Links to an external site.
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Lecture 13 (D) |
NO LECTURE | ||||
| Exercise |
7/5, 10-12 V11 |
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| HW3 is due 12/5 | |||||
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Lecture 14 (M) |
13/5, 8-10 Q15 |
Geodesic flows, cont. | |||
| Exercise |
14/5, 10-12 Q22 |
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| Exam | 26/5, 8-13 |