A constructive view on ergodic theorems

Abstract

Let T be a positive L₁-L_∞ contraction. We prove that the following statements are equivalent in constructive mathematics.

• 1. The projection in L₂ on the space of invariant functions exists;

• 2. The sequence (Tⁿ)_{n ∈ N} Cesáro-converges in the L₂ norm;

• 3. The sequence (Tⁿ)_{n ∈ N} Cesáro-converges almost everywhere.

Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.

As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations. This answers a question posed by Bishop.