Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on some domain $D(x)$ around x. Moreover let $(e^k,g^k)$ converge to another distribution $(e,g)$


i). The ball-box theorem states that there exists constants $C\_k,c\_k$ and $r$ s.t


$exp\_x(Box(c\_kr)) \subset B^k\_{sR}(r) \subset exp\_x(Box(C\_kr))$


where $B^k\_{sR}(r)$ is the subriemannian ball of $k^{th}$ distribution. Now my first question is how does these constants depend on the distribution. From the proofs I have studies it seemes upper constant can be chosen to only depend on behaviour of $e^k,g^k$. The lower constant however depends on the map which exponentiates $e^k,g^k,[e^k,g^k]$. Thus it seems to me it would be uniform if for instance $(f^k,[e^k,g^k])$ is uniformly bigger than some constant and decreasing if this quantity goes to zero? However the magn. of $c\_k$ depends on the implicit inner workings of inverse function theorem (in the proofs I have seen).


ii). Looking back at the ball box theorem if $(f^k,[e^k,g^k])$ goes to 0 then the boxes (or parallelopipeds) $exp\_x(Box(c\_kr))$ are becoming increasingly thin in the $f^k$ direction therefore the subRiemannian balls are converging to a 2 dimensional hypersurface? I am not sure how this reasoning could be true since it suggest that also the orbit of this distribution for time less than r also converges to a plane. But even if $(f^k,[e^k,g^k])$ goes to 0, further Lie brackets may be non-zero thus contributing to the orbit. Moreover even the limit distribution e,g may be non-involutive since the Lie brackets might not converge. These are the reasons why this reasoning seems faulty to me.


I am new to sR geometry so far have looked at book/notes of ivanov, agrachev and gromov. I would also be grateful if you have seen and could direct me to any resources where sequence of distributions are handled or studied. 



Ball-Box Theorem and Sequence of Distributions
I do not understand exactly what you mean by $\exp\_x$, but I guess it is something like the diffeomorphism proposed by Sergei Ivanov. I want to stress out that the ball-box theorem holds only if the coordinates are privileged w.r.t. $(e\_k,g\_k)$, so you should check that the $exp\_x$ you are using defines privileged coordinates for any $k$, in order to apply this theorem.


Regarding the question about $[e\_k,g\_k]\rightarrow [e,g]\in\text{span}(e,g)$, it is a quite delicate matter. Indeed, this is exactly what happens when we try to get uniform estimates for the sub-Riemannian balls on a sequence of points approaching a singular point (think for example of the Martinet distribution, which is of step 1 outside the plane $\{y=0\}$, where it is step 2, and take any sequence of points $(x\_k,y\_k,z\_k)$ with $y\_k>0$ but $y\_k\rightarrow 0$). This problem has been treated in the paper 


F. Jean, "Uniform Estimation of Sub-Riemannian Balls", Journal on Dynamical and Control Systems, vol. 7 (4), 2001
(<http://www.ensta.fr/~fjean/jdcs03_07_01.pdf>)


Also, in the notes on s-R geometry by Jean (<http://arxiv.org/abs/1209.4387>) you can find a somewhat more detailed discussion of privileged coordinates and their subtleties than in the Agrachev, Barilari, Boscain ones. 


