In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly-Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.where x : [0, T ] → R d is a path with bounded variation and X 1 , . . . , X d are formal non-commutative indeterminates. After proving a homomorphism property of the map S ([8], see (2.1) below), he gave an argument [10] that the map S restricted to appropriate classes of paths is, up to translation and reparametrisation, injective. Hambly and Lyons [16], motivated by the application of the map S in rough path theory, posed the following problem:How to characterise the kernel of the map S?Hambly and Lyons [16] proved that for a bounded variation path x, S (x) = 1 if and only if x is tree-like. They conjectured that the result extends to weakly geometric rough paths, a fundamental class of control paths for which controlled differential equations can be defined. Their result directly implies that the space *
We embed the rough integration in a larger geometrical/algebraic framework of integrating oneforms against group-valued paths, and reduce the rough integral to an inhomogeneous analogue of the classical Young integral. We define dominated paths as integrals of one-forms, and demonstrate that they are stable under basic operations.
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