Definition talk:Gauss Map

Is it not the case that the base interval can also be defined as $\openint 0 1$? Is it worth raising as an alternative definition with a trivial equivalence proof? --prime mover (talk) 22:27, 23 May 2023 (UTC)
 * Do you mean, you want to define $ T' : \openint 0 1 \setminus \Q \to \openint 0 1 \setminus \Q$ in the exactly same way, but as another mapping (because the domain looks different) and prove it is the same as $ T : \closedint 0 1 \setminus \Q \to \closedint 0 1 \setminus \Q$?
 * The domains are not different at all. Just that I can't conceive of a universe in which one would even consider $\closedint 0 1 \setminus \Q$ in the first place. --prime mover (talk) 05:23, 24 May 2023 (UTC)
 * The following is just my point of view. Since $\closedint 0 1 \cap \Q$ is a Lebesgue null set, the values of $T$ there do not contribute in measure theory (w.r.t. Lebesgue measure) at all. So it basically does not matter, whether such points belong or not. In addition, $\closedint 0 1 \cap \Q $ are the exceptional points for this $T$, in the sense that:
 * $\forall x \in \hointl 0 1 : \quad x \in \Q \iff \exists n \in \N : \map {T^n} x = 0$
 * But, surely, there are various definitions of this map in the literature. --Usagiop (talk) 17:32, 24 May 2023 (UTC)


 * But these two things are not only equivalent, but really the same thing, since $\openint 0 1 \setminus \Q = \closedint 0 1 \setminus \Q$.
 * The difference is only apparent, not essential. Usagiop (talk) 22:50, 23 May 2023 (UTC)


 * Wikipedia, incidentally, has a completely different definition from this. Hence we may (will) need to disambiguate by renaming. --prime mover (talk) 22:30, 23 May 2023 (UTC)


 * In Wikipedia, this Gauss map is just still missing, except for being mentioned on the page of Gauss–Kuzmin–Wirsing operator.
 * However, we can rename this to continued fraction map if it is better. --Usagiop (talk) 22:50, 23 May 2023 (UTC)