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.
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Famous quotes containing the words branched and/or manifold:
“I am secretly afraid of animals.... I think it is because of the usness in their eyes, with the underlying not-usness which belies it, and is so tragic a reminder of the lost age when we human beings branched off and left them: left them to eternal inarticulateness and slavery. Why? their eyes seem to ask us.”
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“There is then creative reading as well as creative writing. When the mind is braced by labor and invention, the page of whatever book we read becomes luminous with manifold allusion. Every sentence is doubly significant, and the sense of our author is as broad as the world.”
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