Topologies on unparameterised path space

Understand

The signature of a path, introduced by K.T.

• Chen [5] in $1954$, has been extensively studied in recent years.
• The $2010$ paper [12] of Hambly and Lyons showed that the signature is injective on the space of continuous finite-variation paths up to a general notion of reparameterisation called tree-like equivalence.
• The signature has been widely used in applications, underpinned by the result [15] that guarantees uniform approximation of a continuous function on a compact set by a linear functional of the signature.