Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a [p-geometric rough path](https://en.wikipedia.org/wiki/Rough_path#Definition_of_a_rough_path) and let $\mathcal{G}\_p^d$ be the collection of all such paths. Does the [Lyons signature map](https://en.wikipedia.org/wiki/Rough_path#Signature) define a continuous bijection between
$\mathcal{G}\_p^d$ and $T(\mathbb{R}^d)$?



Signature Map From $p$-Geometric Rough Paths to $T(\mathbb{R})$
The signature is continuous on the space of $p$-geometric rough paths, but it is not injective since it is parametrisation-independent and invariant under concatenation with "tree-like" pieces. Boedihardjo, Geng, Lyons and Yang showed in [this article](https://arxiv.org/abs/1406.7871/) that these are the only constraints, so that the signature can be inverted if we consider reduced paths modulo reparametrisation. However, even if we identify paths in this way, one does not expect the inverse map to be continuous in the topology of $T(\mathbb{R}^d)$.


