Ergodic Theory - Ergodic Theorems

Ergodic Theorems

Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ) and suppose ƒ is a μ-integrable function, i.e. ƒ ∈ L1(μ). Then we define the following averages:

Time Average: This is defined as the average (if it exists) over iterations of T starting from some initial point x:

Space Average: If μ(X) is finite and nonzero, we can consider the space or phase average of ƒ:

In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval. More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of ƒ exists for almost every x and that the (almost everywhere defined) limit function ƒ̂ is integrable:

Furthermore, ƒ̂ is T-invariant, that is to say

holds almost everywhere, and if μ(X) is finite, then the normalization is the same:

In particular, if T is ergodic, then ƒ̂ must be a constant (almost everywhere), and so one has that

almost everywhere. Joining the first to the last claim and assuming that μ(X) is finite and nonzero, one has that

for almost all x, i.e., for all x except for a set of measure zero. For an ergodic transformation, the time average equals the space average almost surely. As an example, assume that the measure space (X, Σ, μ) models the particles of a gas as above, and let ƒ(x) denotes the velocity of the particle at position x. Then the pointwise ergodic theorems says that the average velocity of all particles at some given time is equal to the average velocity of one particle over time.