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.