Ergodic Theory - Convergence of The Ergodic Means in The Lp Norms

Convergence of The Ergodic Means in The Lp Norms

Let (X, Σ, μ) be as above a probability space with a measure preserving transformation T, and let 1 ≤ p ≤ ∞. The conditional expectation with respect to the sub-σ-algebra Σ_T of the T-invariant sets is a linear projector E_T of norm 1 of the Banach space Lp(X, Σ, μ) onto its closed subspace Lp(X, Σ_T, μ) The latter may also be characterized as the space of all T-invariant Lp-functions on X. The ergodic means, as linear operators on Lp(X, Σ, μ) also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projector E_T in the strong operator topology of Lp if 1 ≤ p ≤ ∞, and in the weak operator topology if p = ∞. More is true if 1 < p ≤ ∞ then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of ƒ ∈ Lp are dominated in Lp; however, if ƒ ∈ L1, the ergodic means may fail to be equidominated in Lp. Finally, if ƒ is assumed to be in the Zygmund class, that is |ƒ| log+(|ƒ|) is integrable, then the ergodic means are even dominated in L1.