Averaging, recurrence, and decomposition

Core idea

The averaging theorems identify limits; recurrence forces returns; decomposition resolves a system into ergodic components.

Prerequisite doorway

• Lebesgue integral
• L^p space
• Conditional expectation
• Orthogonal projection
• Unique ergodicity

Reading path

• Von Neumann mean ergodic theorem
• Maximal ergodic theorem
• Birkhoff pointwise ergodic theorem
• Poincare recurrence theorem
• Probability kernel
• Disintegration of a probability measure
• Ergodic decomposition theorem
• Uniform averages characterize unique ergodicity
• Mean ergodic theorem along Følner sets
• Krylov–Bogolyubov existence theorem

Return to Ergodic theory.