The signature of a bounded variation path characterizes this path up to tree like equivalence

B. Hambly, T. Lyons: [Uniqueness for the signature of a path of bounded variation and the reduced path group][1]

and one can [reconstruct the path from its signature][2], see also this [link](https://arxiv.org/abs/2304.01862 "Fermanian, Chang, Lyons, and Biau - The insertion method to invert the signature of a path").


  [1]: https://annals.math.princeton.edu/2010/171-1/p02
  [2]: https://researchers.ms.unimelb.edu.au/~xgge@unimelb/Files/Papers/Reconstruction%20for%20the%20Signature%20of%20a%20Rough%20Path.pdf "Geng - Reconstruction from the signature of a rough path"
  [3]: https://arxiv.org/pdf/2304.01862