Comments on the article "Invariant distributions and mixing"

On the finiteness assumption of the measure

Françoise Pène pointed out that there is no need to assume that the measure μ is finite. All the results of the article still hold if there exists a countable family of open sets of finite μ measure, that covers almost all of X.

There are three ingredients in the proofs of the theorems in the article. Neither the L² ergodic theorem, the Banach-alaoglu nor the Banach-Saks results rely on the finiteness of the measure. These are abstract results that hold on any Hilbert space.

We also need the density of the Lipschitz square integrable bounded functions in L². This is where the assumption of a countable cover with elements of finite measure is needed.

So the result can be used to prove the weak convergence of the sequence f∘Tⁿ in an infinite measure setting. Arguably, the function f is assumed to be square integrable, so this does not really help if one wants to prove ergodicity in an infinite measure setting. See my article entitled "The Hopf argument" for some extension of the results to functions that are not square integrable.

A result of Derriennic and Downarowicz

The main theorem of the article can be deduced from an elegant unpublished result communicated to us by Yves Derriennic, a joint work of Yves together with Tomasz Downarowicz.

Theorem
Let X be a metric space together with a Borel probability measure μ. Let T a Borel transformation or flow that preserves μ. We denote by E_± the set of all weak accumulation points of the sequence F∘Tᵗ, t → ±∞, for all square integrable F.

E_± = { F' ∈ L² | ∃ F ∈ L² , ∃ tᵢ → ±∞, F∘Tᵗⁱ → F' weakly }

Then E_+ = E_-.

The proof relies on the spectral theorem for unitary operators.

Yves Coudene 14/10/2011