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)