Privileged coordinates for Carnot-Carathéodory spaces of lower smoothness (Q2202759)

Let \(\mathbb{M}\) be a sub-Riemannian manifold (called in this paper a Carnot-Carathéodory space) with horizontal distribution \(H \subset T\mathbb{M}\) and sub-Riemannian metric \(\langle \cdot, \cdot \rangle\) on \(H\). We suppose that \(H\) is totally non-holonomic (i.e., bracket generating); then the horizontal distribution \(H\) generates a filtration of sub-bundles \(H = H_1 \subset H_2 \subset \dots \subset H_m = T\mathbb{M}\), where \(H_{k+1} = H_k + [H_k, H]\). We further suppose that the fibers \(H_k(x)\), \(x \in \mathbb{M}\), are of constant dimension with respect to \(x\), which is to say that the sub-Riemannian manifold is equiregular. In a neighborhood of a point \(p \in \mathbb{M}\), given a frame of vector fields \(X_1, \dots, X_N\) adapted to the filtration, one may form the canonical coordinates of the first kind \(\theta_p(x_1, \dots, x_N) = \exp(x_1 X_1 + \dots + x_N X_n)(p)\). Let \(\sigma_j = \min\{k : X_j \in H_k\}\) be the weight of the vector field \(X_j\) and \(\delta_\epsilon(x_1, \dots, x_N) = (\epsilon^{\sigma_1} x_1, \dots, \epsilon^{\sigma_N} x_N)\) the corresponding dilation on \(\mathbb{R}^N\), which under \(\theta_p\) induces a dilation \(\Delta_\epsilon^p\) on a neighborhood of \(p\). When \(H\) is \(C^\infty\)-smooth, several results about \(\theta_p\) and \(\Delta_\epsilon^p\) are classical: \begin{itemize} \item[(1)] (Rothschild-Stein local approximation theorem) Under rescaling by the dilation \(\Delta_\epsilon^p\), the vector fields \(X_i\) converge, and the limit is a system of homogeneous vector fields \(\widehat{X}_k^p\) which generate the Lie algebra of some Carnot group \(\mathbb{G}^p\); \item[(2)] (Ball-box theorem) The ``boxes'' centered at \(p\) with respect to the coordinates \(\theta_p\) are comparable to the balls of the Carnot-Carathéodory distance \(d_{cc}\); \item[(3)] (Gromov local approximation theorem) Under rescaling by the dilation \(\Delta_\epsilon^p\), the metric \(d_{cc}\) converges, and the limit is the Carnot-Carathéodory metric of \(\mathbb{G}^p\). \end{itemize} In this paper, the author considers whether statements similar to (1) and (3) still hold when the distribution \(H\) is allowed to be less smooth, say \(C^1\) or \(C^m\), and the canonical coordinates \(\theta_p\) are replaced by some other coordinates \(\phi_p\), which also are not necessarily smooth. In this setting, one may seek conditions under which the vector fields \(X_i\) converge when rescaled by dilation \(\delta_\epsilon\) with respect to the new coordinates \(\phi_p\), and ask what can be said about the limits. Likewise, under what conditions does \(d_{cc}\), or some other metric \(d\), converge under such rescaling, and is the limit \(\hat{d}\) at least a quasimetric? Such properties of \(\phi_p\) are related to the notion of privileged coordinates discussed in [\textit{W. Choi} and \textit{R. Ponge}, J. Dyn. Control Syst. 25, No. 1, 109--157 (2019; Zbl 1410.53033); ibid. 25, No. 4, 631--670 (2019; Zbl 1432.53047)] and other works of the same authors. The author proves several results giving necessary and sufficient conditions for such statements as well as some counterexamples. The conditions are somewhat technical and I will not attempt to state the results precisely here, but they generally relate to the infinitesimal behavior of the transition function \(\phi_p^{-1} \circ \theta_p\) on \(\mathbb{R}^N\) with respect to the dilation \(\delta_\epsilon\). Under other conditions, it turns out that, if some version of the ball-box theorem (2) holds, then versions of (1) and (3) hold as well. A particular case considered in the paper is where \(\phi_p\) is the canonical coordinates of the second kind, \(\phi_p(x_1, \dots, X_N) = \exp(x_1 X_1) \circ \dots \circ \exp(x_1 X_N)\).