I'm reading paper by Arnol'd and Krylov ( UNIFORM DISTRIBUTION OF POINTS ON A SPHERE AND SOME ERGODIC PROPERTIES OF SOLUTIONS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS IN A COMPLEX REGION ).
I need help in understanding the proof of the following theorem:
Theorem 1. Let A and B be two rotations of the sphere $S^2$ and let $x$ be a point of the sphere. If the sequence of points $$x, Ax, Bx, A^2x, ABx, BAx, B^2x, \ldots \ \ \ \ \ \ (1) $$ is dense on the sphere, then it is uniformly distributed.
Where uniformly distributed means that: if we consider the $2^n$ points produced with exactly $n$ iterations of the rotations,
$$ A^nx , A^{n-1}Bx , A^{n-2}BAx , \ldots , B^nx \ \ \ \ \ \ \ \ \ \ \ \ (2) $$ and we count the proportion of them that lie in a region of the sphere $\Delta$ which is bounded by a piecewise smooth curve, then these ratios converge to the measure of $\Delta$; $$ \lim_{n\to \infty} \frac{\mbox{Number of points among } (2) \mbox{ lie in } \Delta}{2^n} = \frac{mes \ \Delta}{mes \ S^2}. \ \ \ \ (3)$$
The proof of the theorem as the following:
If any one can clarify to me why $(7)$ is impossible, how by density of the points in $(1)$ we get the commutativity on $R_{l}$ ?
Thanks in advance!
If $x,Ax,Bx,\ldots$ were dense in $S^2$, then their closure gives $S^2$. So any rotation $R$ can be approximated by products $R=\lim_{n\to\infty}C_n$ where $C_n$ is a product of $A$s and $B$s.$\dagger$
$A,B$ commute on $R_l$, since any $g\in R_l$ is $Rf$ for some rotation $R$; hence using (7) $$ABg=ABRf=\lim_{n\to\infty}ABC_nf=\lim_{n\to\infty}BAC_nf=BAg$$ But then, any two rotations $R_1$, $R_2$, commute on $R_l$ since for any $g\in R_l$, $$R_1R_2g=\lim_{n\to\infty}C_nD_ng=\lim_{n\to\infty}D_nC_ng=R_2R_1g$$ This is a contradiction since, in fact, the group of rotations is not commutative on $R_l$ for $l>0$.
$\dagger$ (I don't fully understand this part yet, but the paper says it is easy, so it must follow trivially from some other result; I think one can show this by approximating $R\approx S^m$, where $S$ is a rotation by a small angle; then $Sx\approx C_nx$, and show $R\approx C_n^m$. Perhaps you can ask another question on this point.)