1MAT217 - Raluca

1MAT217: Dynamiska System (Dynamical Systems)
Fall 2020
Raluca Tanase

Lectures, Notes & Homework

Textbook:

Introduction to Dynamical Systems by Michael Brin and Garrett Stuck

Description:

Dynamical systems are abundant in the real world. This course will investigate some fundamental concepts about dynamical systems, both in the real and in the complex setting. We will present various types of dynamics, and discuss, as time permits, the following topics: Topological transitivity/Mixing, Symbolic Dynamics (Shift Spaces), Smale's Horseshoe, Coding & Topological conjugacy, Translations and linear flow on n-dimensional torus, Subshifts of finite type & Perron-Frobenius Theorem, Topological Entropy & The variational principle, Poincare Recurrence Theorem and applications, Invariant Measures, Krylov-Bogolyubov Theorem, Ergodic Measures, Birkhoff Ergodic Theorem, Mixing, Weak Mixing, Ergodicity & Unique Ergodicity, Applications of Ergodicity to Number Theory. Szemeredi's Theorem, One-dimensional complex dynamics, Two-dymensional complex dynamics, Fatou-Bieberbach domains, invariant manifolds, linearisation, structural stability, hyperbolicty in dimension one and in dimension greater than one, stable and unstable manifolds, homo- and heteroclinic phenomena, chaos and sensitive dependence on initial values, strange attractors.

Grading Scheme:

The final grade will be determined by the following formula:
Homework: 50% Final Exam: 50%

Schdule:

This is our tentative weekly schedule and it will be updated as we advance in the semester, please check regularly.

Week	Lectures (Assigned Reading & Notes)	Assignments

Week 36	Examples of Dynamical Systems 1.1, 1.2, 1.3 Limit & Minimal Sets, Topological Transitivity/Mixing 2.1, 2.2 Lecture 1	Homework 1 Due Nov 4
Week 37	Symbolic Dynamics (Shift Spaces) 1.4, 3.1 Expanding Endomorphisms of the Circle 1.3 Smale's Horseshoe, Coding & Topological conjugacy 1.8 The Solenoid 1.9 Lecture 2 Lecture 3
Week 38	Translations and Linear Flow on the n-dimensional Torus 1.7 Topological Classification of Expanding Maps on S¹ 5.1 Homeomorphisms of the Circle, Rotation Number 7.1 Lecture 4 Lecture 5
Week 39	Poincaré Classification of Homeomoprhisms of the Circle 7.1 Diffeomorphisms of the Circle. Denjoy's Theorem & Example 7.2 Lecture 6 Lecture 7
Week 40	Subshifts of Finite Type / Topological Markov Chains 1.4, 3.2 Transitive Matrices & Perron-Frobenius Theorem 3.3 Mathematics of Internet Search 4.12 Lecture 8 Lecture 9 (Page Rank Algorithm & HITS Algorithm )
Week 41	Topological Entropy 2.5, 2.6 Lecture 10 Lecture 11
Week 42	Topological Entropy and Recurrent Behavior 2.6 The Variational Principle for Topological Entropy 9.5 Poincaré Recurrence Theorem and its applications 4.2 Invariant Measures, Krylov-Bogolyubov Theorem 4.1, 4.6 Lecture 12 Lecture 13
Week 43
Week 44	Ergodic Measures 4.3, 4.6 Lecture 14 Homework Discussion	Homework 2 Due Dec 20
Week 45	Ergodicity and Unique Ergodicity (Examples) 4.4 Ergodic Theorems. Birkhoff Ergodic Theorem 4.5 Lecture 15 Lecture 16
Week 46	Unique Ergodicity (revisited) Strong Mixing. Weak Mixing 4.4 Furstenberg's Multiple Reccurence Theorem 4.11 Szemeredi & Green-Tao Thm on Arithmetic Progressions 4.11 Lecture 17 Lecture 18
Week 47	Other applications of Ergodicity to Number Theory 4.7, 4.8 Dynamics of Quadratic Polynomials 8.3, 8.4, 8.5 Lecture 19 Lecture 20
Week 48	External Rays and Connectivity of the Julia Set 8.5 The Mandelbrot Set 8.6 Lecture 21
Week 49	Local Dynamics of Neutral Fixed Points Sullivan's Classification of Fatou components Hyperbolicity in 1D and Structural Stability Lecture 22 Lecture 23
Week 50	Hyperbolicity in Higher Dimensions 5.2 Invariant Families of Cones 5.4 Stability of Hyperbolic Sets 5.5 Stable and Unstable Manifolds 5.6 Lecture 24
Week 51

Jan 15	Take home Final Exam the final covers everything!