Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631
Time and place: M-W 10-11:20, Physics P-129
Text: Introduction to Dynamical Systems, Brin and Struct, cambridge University Press, 2002.
My office hours will be M-W, 11:30-1 in my office, 4-112 in the Math Building, and by appointment. Please feel free to drop by other times as well.
This is an introductory course on dynamics systems, and covers a bit of topological dynamics, symbolic dynamics, and some ergodic theory. We will asumme some familiarity with point-set topology and measure theory; the first year Fall graduate courses in analysis and topology such be sufficient. I hope to cover Chapters 1-4 more or less completely, do parts of Chapter 5 and 7, and discuss as much of Chapters 8 and 9 as time permits.
Grades will be based on problem sets, generally due in class on Wednesdays. In general, exercises will be selected from the sections covered the previous week. There is no midterm or final exam.
Some MATLAB scripts illustrating examples in the text.
Monday, Jan 28:
   
   
1.1 (notation),
   
   
1.2 (circle rotations),
   
   
1.3 (expanding endomorphisms),
Wednesday, Jan 30:
   
   
1.4 (shifts and subshifts)
   
   
1.5 (quadratic maps),
   
   
1.6 (the Gauss transformation),
Monday, Feb 4:
   
   
1.7 (hyperbolic toral automorphism),
   
   
1.8 (the horseshoe),
   
   
1.9 (the solenoid)
Wednesday, Feb 6:
   
   
1.10 (flows and differential equations),
   
   
1.11 (suspensions and cross-sections),
   
   
1.12 (attractors)
Monday, Feb 11: Chapter 2
   
   
2.1 (liit sets and reccurence)
   
   
2.2 (topological transitivity)),
   
   
2.3 (topological mixing, examples)
Wednesday, Feb 13: Chapter 2
   
   
2.4 (expansiveness),
Monday, Feb 18: Chapter 2
   
   
2.5 (topological entropy),
   
   
2.6 (topological entropy of some examples, summary)
Wednesday, Feb 20: Chapter 3
   
   
2.7 (Equicontinuity, distality, proximality; summary),
   
   
2.8 (applications to Ramsey theory),
Monday, Feb 25: Chapter 3
   
   
3.1 (Subshifts and codes),
   
   
3.2 (Subshifts of finite type),
   
   
3.3 (The Perron-Frobenius theorem),
Wednesday, Feb 27: Chapter 3
   
   
3.4 (Topological entropy and the Zeta function of an SFT),
   
   
3.5 (Strong shift equivalence and shift equivalence),
   
   
3.6 (Substitutions),
Monday, Mar 4: Chapter 4 (No class - snow day)
Wednesday, Mar 6: Chapter 4
   
4.2 (Recurrence),
   
4.3 (Ergodicity and mixing),
Monday, Mar 11: Chapter 4
   
4.4 (Examples),
   
4.5 (Ergodic theorems),
Wednesday, Mar 13: Chapter 4 (class starts 10:30)
   
4.5 (Ergodic theorems), continued
   
4.6 (Invariant measures for continuous maps),
Monday, Mar 18: NO CLASS, SPRING BREAK
Wednesday, Mar 20: NO CLASS, SPRING BREAK
Monday, Mar 25: Chapter 4
   
4.7 (Unique ergodicity and Weyl's theorem),
Wednesday, Mar 27: Chapter 4
   
4.9 (Discrete Spectrum),
Monday, Apr 1 : Chapter 4
   
4.10 (Weak mixing),
Wednesday, Apr 3: Chapter 4
   
4.11 (Applications to Number Theory),
Monday, Apr 8 : Chapter 5
   
5.1 (Expanding endomorphisms revisited),
   
5.2 (Hyperbolic sets),
   
5.3 (epsilon-orbits)
Wednesday, Apr 10 : no class
Monday, Apr 15: Chapter 5
   
5.3 (epsilon-orbits)
   
5.4 (Invariant cones),
Wednesday, Apr 17: Chapter 5
   
5.5 (Stability of hyperbolic sets),
   
5.6 (Stable and unstable manifolds),
Monday, Apr 22: Chapter 7
   
7.1 (Circle homeomorphisms),
   
7.2 (Circle diffeomorphisms),
Wednesday, Apr 24: Chapter 7
   
7.3 (The Sharkovsky theorem),
Monday, Apr 29: Chapter 7
   
7.4 (Cominatorial theory),
   
7.5 (Schwarzian derivative),
Wednesday, May 1: Chapter 7
   
7.6 (Real quadratic maps),
   
7.7 (Bifurcations of periodic points),
Monday, May 6: Chapter 7
   
7.8 (The Feigenbaum phenomenon),
Wednesday, May 8:
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