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.