It is an old Weis-Adler result that the Boole mapping R∋x→x-1/x∈R is a Lebesgue measure preserving and ergodic. What one can state about the mappings R²∋(x,y)→(x-1/y,y+1/x)∈R² and about R²∋(x,y)→(y-1/x,x+1/y)∈R² ?
The Conjecture was recently claimed in the note attached.
Could you please precise in what sense do you use the words ``measure preserving''? If in the usual sense that for every set the measure of preimage equals the measure of the original set, then the mapping f: x→1-1/x does not look measure preserving: f-1((0,1)) = (1, +\infty). Maybe there is a misprint in the statement?
Dear Wally,
You are right! - there is a misprint, - should be x-1/x - owing to the classical Weis-Adler result the Boole mapping is measure preserving (yet in the infinitesimal sense as you mentioned) and is ergodic. Its extended development one can find in a recently enough published book written by Jon Aaronson (An introduction to infinite ergodic theory, AMS Publisher).
Sincerely,
cordial regards.
Well, now I see where the misprint is: the function is f(x) = x - 1/x, but you have written 1 - 1/x.
All the best, Vladimir
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