An Elementary Proof of the Hambly-Lyons Uniqueness Theorem
Abstract
We give a self-contained proof, in the bounded variation setting, of the Hambly--Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree τ:[0,1]→ T is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree T through which the original path factors by virtue of the sub-interval lemma.
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BibTeX
@misc{schachermayer2026elementary,
title = {{An Elementary Proof of the Hambly-Lyons Uniqueness Theorem}},
author = {Walter Schachermayer and Josef Teichmann and Valentin Tissot-Daguette},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12283},
eprint = {2609.12283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12283},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Elementary Proof of the Hambly-Lyons Uniqueness Theorem
AU - Walter Schachermayer
AU - Josef Teichmann
AU - Valentin Tissot-Daguette
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.12283
UR - https://arxiv.org/abs/2609.12283
AB - We give a self-contained proof, in the bounded variation setting, of the Hambly--Lyons uniqueness theorem, which states that (total) signature identifies the path up to tree-like equivalences. The argument is organized around two key geometric observations. First, tree-like paths have trivial signature because factorization over a loop in a tree τ:[0,1]→ T is preserved under signature lifts, which follows from an elementary property of planar curves. Second, a path with trivial total signature contains a nontrivial subpath with trivial total signature (the sub-interval lemma). This is proven by applying a winding-number argument to a two-dimensional projection of the signature lift. Collapsing all trivial-signature sub-intervals then defines a compact metric tree T through which the original path factors by virtue of the sub-interval lemma.
ER -