2005/07/31 by Ben Hambly, Terry Lyons · 158 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Operator Algebra Research #Bounded function #Congruence relation #Equivalence relation #Iterated function #Lebesgue integration #Signature (topology) #Triviality #Uniqueness #math.CA
paper · pdf · doi:10.4007/annals.2010.171.109
published in Annals of Mathematics 171(1), 109-167 (Princeton University) · 52 pages - considerably extended and revised version of the previous version of the paper
arxiv created 2006/12/19 · openalex publication_date 2010/03/17 · arxiv updated 2013/05/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced. The set of these paths is the Reduced Path Group. It is a continuous analogue of the group of reduced words. The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parametrisation in the general setting. It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.