Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a pgeometric rough path and let $\mathcal{G}_p^d$ be the collection of all such paths. Does the Lyons signature map define a continuous bijection between $\mathcal{G}_p^d$ and $T(\mathbb{R}^d)$?
The signature is continuous on the space of $p$geometric rough paths, but it is not injective since it is parametrisationindependent and invariant under concatenation with "treelike" pieces. Boedihardjo, Geng, Lyons and Yang showed in this article 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 $T(\mathbb{R}^d)$.

1$\begingroup$ Would there be an intuition for the final topology on $T(\mathbb{R}^d)$ induced by the (quotient of) the signature map (defined by the identification you mentioned)? This would guarantee that the signature map is continuous... $\endgroup$ – AnnieLeKatsu Feb 4 at 12:44

2$\begingroup$ Unfortunately, there isn't even an alternative characterisation of the image of the signature map... $\endgroup$ – Martin Hairer Feb 4 at 14:22

1$\begingroup$ Then could we not define a new topology on $T(\mathbb{R}^d)$ by pushingforward the topology on the (quotient/identified) space of $p$geometric rough paths via the (now bijective) "signature" map? $\endgroup$ – AIM_BLB Feb 9 at 17:18