Schedule
| Format | time and place | Topics to be covered | Suggested reading | Suggested exercises | Extras |
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Lecture 1
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Fri 20/1 13-15 E2 | 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 |
| Exercise | Tue 24/1 8-10 | ||||
| Lecture 2 | Fri 27.1. 13-15 E2 | 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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| Lecture 3 | Tue 31/1 8-10 E2 | 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. |
| Exercise | Fri 3/2 10-12 in Lindstedtsvägen 25, room 3418 |
Notes:ChaoticDynamicalSystemsNotes (1).pdf Download ChaoticDynamicalSystemsNotes (1).pdf |
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| Lecture 4 | Fri 10/2 13-15 | 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 | |
| HW 1 is due on 10/2 | |||||
| Exercise | Tue 14/2 | discussion of HW1 | |||
| Lecture 5 | Fri 17/2 13-15 | 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 |
| Exercise | Tue 21/2 | Entropy | EntropyAutomorphisms.pdf Download EntropyAutomorphisms.pdf | ||
| Lecture 6 | Tue 28/2 | 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 | Fri 3/3 | Homeomorphisms of the circle | |||
| HW2 is due 20/3 | |||||
| Exercise | Tue 21/3 | Discussion of HW2 | |||
| Lecture 7 | Fri 24/3 |
Maps of the interval: Sharkovsky theorem quadratic family |
Sec. 4.3 in the course book [BV] Devaney's book [D] Links to an external site.Sec. 1.4, 1.5 (pp. 24-39) and Sec. 1.17 (pp. 130-139)
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Book [BV]: ... Book [D]: Ex. 1,2 p.38. |
Lecture notes Download Lecture notes |
| Lecture 8 | Tue 28/3 |
Hyperbolic dynamics I The Grobman-Hartman and the Hadamard-Perron theorems. Structural stablity. |
Book [BV]: Chap. 6; Book [D]: Sec. 2.6 (pp.214-232) Book [D]: Sec. 1.9 (pp. 53-59) |
Book [BV]: ... Book [D]: Ex. 1 p. 229. Book [D]: Ex. 5 p. 59. |
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| Lecture 9 | Fri 31/3 | Hyperbolic dynamics II: Hyperbolic sets, Smale horseshoe and homoclinic tangle |
Book [BV]: Chap. 5
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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 |
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| Exercise | Tue 4/4 |
Maps of the interval Hyperbolic dynamics |
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| Lecture 10 | Tue 18/4 | Hyperbolic dynamics III | |||
| Lecture 11 | Fri 21/4 | 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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| Exercise | Tue 25/4 | Hyperbolic dynamics (cont.) | |||
| HW3 is due 28/4 kl 8 | ChaoticHW3.pdf Download ChaoticHW3.pdf | ||||
| Exercise | Fri 28/4 | Discussion of HW3 | |||
| Lecture 12 | Ti 2/5 | Small denominators | |||
| Exam | May 23, 13:00 |