$\mathrm{SL}_{2}(\mathbb{R})$-developments and signature asymptotics for planar paths with bounded variation

Abstract

The signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as stochastic analysis. An elegant isometry conjecture states that the length of a bounded variation path \gamma can be recovered from the asymptotics of its normalised signature: \text{Length}(\gamma)= \lim_{n\rightarrow\infty} \Big\Vert n! \int_{0<t_{1}<\cdots<t_{n}<T} d\gamma_{t_{1}}\otimes\cdots\otimes d\gamma_{t_{n}} \Big\Vert^{1/n}. This property depends on a key topological non-degeneracy notion known as tree-reducedness (namely, with no tree-like pieces). Existing arguments have relied crucially on \gamma having a continuous derivative under the unit speed parametrisation. In this article, we prove the above isometry conjecture for planar paths by assuming only local bounds on the angle of \gamma' (which ensures the absence of tree-like pieces). Our technique is based on lifting the path onto the special linear group \mathrm{SL}_{2}(\mathbb{R}) and analysing the behaviour of the associated angle dynamics at a microscopic level.