MAT 555, Introduction to Dynamical Systems

Spring 2019

Christopher Bishop

Professor, Mathematics
Stony Brook University

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.

Problem Sets

Wed Jan 30: Nothing due
Wed Feb 6: Exercises 1.2.2, 1.2.3, 1.3.5, 1.4.5
Wed Feb 13: Nothing due
Wed Feb 20: Exercises 2.1.5, 2.3.3, 2.2.6, 2.4.1
Wed Feb 27: Nothing due
Wed Mar 6: Exercises 3.2.1, 3.2.2, 3.3.1, 3.3.2, 3.7.1
Wed Mar 13: Nothing due (Spring Break)
Wed Mar 27: Nothing due
Wed April 3: Exercises 4.4.1, 4.4.2, 4.5.1, 4.5.6
Wed April 10: Nothing due
Wed April 17: Exercises 4.7.1, 4.7.6, 4.9.3, 4.10.5
Wed April 24: Nothing due
Wed May 1: Exercises 5.2.1, 5.2.5, 5.3.2, 5.3.4

MATLAB Scripts

Some MATLAB scripts illustrating examples in the text.

Tentative Lecture Schedule

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),


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:

The not too short introduction to LaTex

Link to history of mathematics

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Send me email at: bishop at

Email all MAT 555 participants (Bishop, Alland, Burkart, Chen, Drillick, Kolomatski, Mazor, Puszklewicz, Tham)

Link to history of mathematics