Branched manifold
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 R. 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: These data must satisfy the following requirements: Then the space "K" is a branched "n"-manifold of class "C". The standard machinery of differential topology can be adapted to the case of branched manifolds. This leads to the definition of the tangent space "T""K" to a branched "n"-manifold "K" at a given point "p", which is an "n"-dimensional real vector space; a natural notion of a "C" differentiable map  f: "K" → "L" between branched manifolds, its differential "df": "T""K" → "T""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.