Calculus without limits

The Dyadic integers $X=\mathbb{Z}_2$ form a compact topological group.Is it natural? In other words: Is there a metric (X,d) such that the group structure $(X,+,0)$ is forced? Unfortunately, this is not yet clear. But the Dyadic world is fascinating. It is a world on which there is a smallest group translation, the adding machine $x \to x+1$, also known as the von Neumann-Kakutani system, a uniquely ergodic group translation with Haar measure as invariant measure. This system can be visualized also through the solenoid which is the dual group of the dyadic rationals. Not knowing whether the dyadic group is natural or not, I try to build some structure. Here is also an attempt to build a doubling operation $G \to \phi(G))$ on the category of groups so that the upgraded group $\phi(G)$ is natural if the original group $G$ was natural. The idea is to mimic the dihedral construction: $\mathbb{Z} \to \mathbb{D}_{\infty}$: take two copies G,G’ of G. On G’ invert every group operation. If $a: G \to G'$ is the involution switching between the two parts and g is a group element, define generators $b_g = a g$. Now, since $a g a$ is the inverse of g, we have $a^2=b_g^2=0$. The upgraded group is the presented group $\langle a,b_g, a^2=b_g^2=0, a b_g = g \rangle$. As in Spiegelungsgeometrie (a branch of transformation geometry which I was exposed to as a freshmen and at that time not appreciated, but do appreciate now), now all group elements are products of involutions. This is prototyped by the finite and infinite dihedral groups, where the later is $\langle a,b, a^2=b^2=0 \rangle$ and the finite versions are $\langle a,b, a^2=b^2=(ab)^n=0 \rangle$. In the youtube improvisation, there is a sketch of a proof that if G is natural, then $\phi(G)$ is natural (at least in the case where not all real distances are taken in G, like in the compact or discrete case) so that one can define a distance between the two worlds $G,G'$ which is not already a distance in the group. This forces that the distance between 0 and 0′ is unique so that a’=0 is an involution. The rest then should be forced as the two worlds G,G’ (Site A and Site B, if you like Jurassic Park) are natural and carry only one group operation. We just have to fix a metric so that $d(0,a g_b)$ and $d(0,g_b a)$ have different distance, then we exclude the other product group structure $G \times \mathbb{Z}/(2 \mathbb{Z})$. This is exactly as we have forced the dihedral group structure on a metric space with 2n elements and so rendered that group natural. That nature (a natural group) is strong has been known already in the Spielberg movie of 1993: life finds its way. A natural question is then whether this doubling renormalization map has fixed points, what they are and whether they are natural and what the structure of the metrics are which render the group natural. In some sense, I feel that the dyadic world will play a role in answering the question. Having the free group with infinitely many involutive generators as the group of isometries of the dyadic group of integers, it looks like a good candidate for a fixed point of the doubling map. While talking on Saturday, I was under the impression that in the compact case there should be a unique fixed point, the reason being that there is a Haar measure $m$ which renders $(X,\mathcal{A},m)$ a Lebesgue probability space (every probability space obtained by taking the normalized Haar measure on a compact topological group definitely is a Lebesgue probability space) so that the doubling map make the original group relevant on half of the space. At the moment, I’m no more convinced about the fixed point thing and start to think that the doubling operation $\phi$ can be much more complicated and that there can be many fixed points, many periodic orbits, maybe even have some chaotic components if one looks at the system in the Edwards-Gromov-Hausdorff metric. It would be nice however to have an attractor consisting of natural groups. Completely speculative is that the map $\phi$ on that attractor then is a chaotic map.