Mathematics / advanced
Ergodic theory and invariant dynamics
Connect long-time averages to invariant measures, recurrence, mixing, entropy, and measurable dynamical systems.
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Inside this module
Module 01 advanced
Ergodic theory and invariant dynamics
Connect long-time averages to invariant measures, recurrence, mixing, entropy, and measurable dynamical systems.
- 01
Measure-preserving transformations and invariance
Specify the measure as part of a dynamical system and distinguish invariance, invertibility, and long-time averaging.
31 min - 02
Poincaré recurrence theorem
Prove almost-everywhere repeated returns from finite invariant measure and separate recurrence from return-time estimates.
32 min - 03
Birkhoff’s pointwise ergodic theorem
Identify almost-everywhere time averages with conditional expectations on invariant information, rather than assuming a global constant.
34 min - 04
Mean ergodic theorem and unitary operators
Prove norm convergence through invariant vectors and telescoping coboundaries, while keeping unitary and merely isometric dynamics distinct.
34 min - 05
Ergodicity, weak mixing, and mixing
Distinguish invariant events, Cesàro decorrelation, and pointwise decorrelation using exact stationary correlations.
35 min - 06
Rotations, Bernoulli shifts, and examples
Separate periodicity, irrational exploration, and independent-coordinate mixing through fully specified probability systems.
34 min - 07
Invariant sigma-algebras and ergodic decomposition
Represent invariant probability as a measurable mixture of ergodic probabilities and identify component expectations correctly.
37 min - 08
Measure-theoretic entropy and generating partitions
Compute information growth from ordered orbit names and distinguish process entropy from one-time uncertainty.
38 min - 09
The subadditive ergodic theorem
Identify almost-sure growth rates when composition gives an inequality instead of an additive sum.
39 min - 10
Multiplicative ergodic theorem and Lyapunov spectra
State the invertible Oseledets theorem precisely and compute growth along invariant directions rather than coordinate axes.
42 min