LECTURE 7: Irrational Rotation Theorem | Every Orbit Is Dense on the Circle | Dynamical Systems

In this lesson, we study the Irrational Rotation Theorem, an important result in Dynamical Systems concerning rotations of the circle. Consider the circle rotation: fθ(x) = x + θ (mod 1) The Irrational Rotation Theorem states that: If θ ∉ ℚ, then every orbit of fθ is dense in S¹. In this lesson, we explain the meaning and significance of this theorem and examine how the behaviour of an orbit depends on whether the rotation number θ is rational or irrational. 📌 TOPICS COVERED: • Irrational Rotation Theorem • Circle rotations • Rotation numbers • Rational rotations • Irrational rotations • Periodic orbits • Dense orbits • Orbit behaviour on the circle • Rotations preserving distance • Derivative of a rotation • Long-term behaviour of dynamical systems The key dichotomy is: θ ∈ ℚ → Periodic orbits θ ∉ ℚ → Dense orbits Thus, a simple rotation of the circle can produce two fundamentally different types of dynamical behaviour depending on whether the rotation number is rational or irrational. This lesson is useful for students studying Dynamical Systems, Discrete Dynamical Systems, Real Analysis, Advanced Mathematics and Undergraduate Mathematics. 📚 Subscribe for more mathematics lessons, worked examples, revision content and university mathematics. #IrrationalRotationTheorem #DynamicalSystems #DenseOrbits #CircleRotation #Mathematics