In mathematics, a branched manifold is a generalization of a differentiable manifold which may have singularities of very restricted type and admits a well-defined tangent space at each point. A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable n-disk in Rn. Branched manifolds first appeared in the dynamical systems theory, in connection with one-dimensional hyperbolic attractors constructed by Smale and were formalized by R. F. Williams in a series of papers on expanding attractors. Special cases of low dimensions are known as train tracks (n = 1) and branched surfaces (n = 2) and play prominent role in the geometry of three-manifolds after Thurston. Definition Let K be a metrizable space, together with: a collection {Ui} of closed subsets of K; for each Ui, a finite collection {Dij} of closed subsets of Ui; for each i, a map πi: Ui → Din to a closed n-disk of class Ck in Rn. These data must satisfy the following requirements: ∪j Dij = Ui and ∪i Int Ui = K; the restriction of πi to Dij is a homeomorphism onto its image πi(Dij) which is a closed class Ck n-disk relative to the boundary of Din; there is a cocycle of diffeomorphisms {αlm} of class Ck (k ≥ 1) such that πl = αlm · πm when defined. The domain of αlm is πm(Ul ∩ Um). Then the space K is a branched n-manifold of class Ck. The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space TpK to a branched n-manifold K at a given point p, which is an n-dimensional real vector space; a natural notion of a Ck differentiable map f: K → L between branched manifolds, its differential df: TpK → Tf(p)L, the germ of f at p, jet spaces, and other related notions. Examples Extrinsically, branched n-manifolds are n-dimensional complexes embedded into some Euclidean space such that each point has a well-defined n-dimensional tangent space. A finite graph whose edges are smoothly embedded arcs in a surface, such that all edges incident to a given vertex v have the same tangent line at v, is a branched one-manifold, or train track (there are several variants of the notion of a train track — here no restriction is placed on