| Lec. | Date | Topic | Section |
|---|---|---|---|
| 1 | 8/24 | Introduction, examples and questions. | |
| 2 | 8/26 | Contractions. | 1.1 |
| 3 | 8/28 | Weak contractions. Increasing interval maps. | -, 1.1 |
| 4 | 8/31 | Attracting fixed points. Circle rotations. | -, 1.3 |
| 5 | 9/2 | Circle rotations. Topological transitivity and minimality. | 1.3 |
| 6 | 9/4 | Circle rotations: equidistribution. First digits of powers. | |
| 7 | 9/7 | Translations and linear flows on the torus. | 1.4, 1.5 |
| 8 | 9/9 | Criteria for topological transitivity. | 1.4 |
| 9 | 9/11 | Transitivity of translations and linear flows on the torus. | 1.4, 1.5 |
| 10 | 9/14 | Times-m map of the circle. | 1.7 |
| 11 | 9/16 | Times-m map of the circle. Shifts on sequences and semiconjugacy. | 1.7 |
| 12 | 9/18 | Topological conjugacy and semiconjugacy. | |
| 13 | 9/21 | Full shift on m symbols. | 1.9 |
| 14 | 9/23 | Topological mixing. Subshifts of finite type. | 1.9 |
| 15 | 9/25 | Subshifts of finite type: periodic points, transitivity and mixing. | 1.9 |
| 16 | 9/28 | Exponential growth rate of the number of periodic points. | 1.9, 3.1 |
| 17 | 9/30 | Definitions of topological entropy. | 3.1 |
| 18 | 10/2 | Topological entropy: Examples of calculation. | 3.2 |
| 19 | 10/5 | Properties of topological entropy | 3.1 |
| 20 | 10/7 | Linear maps of the plane. | 1.2 |
| 21 | 10/9 | Hyperbolic automorphisms of T2 : basics and an example. | 1.8 |
| 22 | 10/12 | Hyperbolic automorphisms of T2 : mixing and periodic points. | 1.8 |
| 23 | 10/14 | Calculation of topological entropy. Hyperbolic automorphisms of TN. | |
| 24 | 10/16 | Structural stability: definition and examples, Em. | 2.3, 2.4(c) |
| 25 | 10/19 | Structural stability: Em and hyperbolic automorphisms of T2. | 2.4(c), 2.6 |
| 26 | 10/21 | Coding: general, expanding circle maps. | 2.5(a), 2.4(b) |
| 27 | 10/23 | Conjugacy to Em via coding. Smale's horseshoe. | 2.4(b), 2.5(c) |
| 28 | 10/26 | Coding of a hyperbolic automorphism of T2. | 2.5(d) |
| 29 | 10/28 | Hyperbolic dynamical systems. | |
| 30 | 10/30 | Workshop in Dynamical Systems and Related topics | |
| 31 | 11/2 | Recurrence properties: ω- and α-limit sets, recurrent points. | 3.3 |
| 32 | 11/4 | Recurrence properties: nonwandering points. | 3.3 |
| 33 | 11/6 | More on recurrence properties. Poincare Recurrence Theorem. | 4.1(f) |
| 34 | 11/9 | Rotation number. | 11.1 |
| 35 | 11/11 | Rotation number: invariance under top. conjugacy, the rational case. | 11.1 |
| 36 | 11/13 | Irrational rotation number: conjugacy and semiconjugacy. | 11.2, 12.1,2 |
| 37 | 11/16 | Continuous-time dynamical systems (Flows) | |
| 38 | 11/18 | Flows: topological mixing, recurrence, topological entropy. | |
| 39 | 11/20 | Is f the time-one map of a flow? Suspension flow. | |
| Thanksgiving Break | |||
| 40 | 11/30 | Properties of suspension flows. Equivalence for flows. | 2.2 |
| 41 | 12/2 | Geodesics and geodesic flow. | 5.4 |
| 42 | 12/4 | Project presentations. | |
| 43 | 12/7 | Project presentations. | |
| 44 | 12/9 | Project presentations. | |
| 45 | 12/11 | Project presentations. | |