(retyped from a hand-written letter submitted to the editor-in-chief by one of the greatest living experts in dynamical systems; only small part of the comments is included; the rest sent to the author via email.)
The physical content of the "attractors" should not be mixed with their mathematical axiomatic definition, which loses and important part of the physical notion of "limiting behavior". This difference produced a lot of physically wrong statements on the growth of the dimensions of the attractors (for the Navie-Stokes, etc.) while the Reynolds number is growing: the attracting set of a large dimension does not imply the higher-dimensional limiting behavior!
I was excited after reading the paper. It is very reach in contents, concentrated and clear. A minor remark below does not change this impression.
Remark. I disagree with the following statement at the end.
As an example, for the complex exponential map, most orbits spend most of the time very far away from zero: Almost all orbits converge statistically towards the point at infinity. Thus the statistical global attracting set consists of the single point \(\infty\).
Indeed, under the exponential map a "left half" of a small neighborhhod of infinity is mapped close to zero under one iterate; "three quarters" are mapped close to 0 and 1 after two iterates, and so on. Hence, I am sure that for the exponential map the statistical attractor coincides with the measure one shown in the last display of p 6.
Reply: I believe that my statement was correct, but have to agree that it wasn't properly justified. Hence I have added several sentences to the text.
Reply: Thanks; I completely agree with the explanation.