Computing connectedness: disconnectedness and discreteness

We consider finite point-set approximations of a manifold or fractal with t he goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the numbe r and size of its epsilon-connected components. By extrapolating the limiti ng behavior of these quantities as epsilon --> 0 we can say whether the und erlying set appears to be connected, totally disconnected, or perfect. Ne d emonstrate the effectiveness of our techniques for a number of examples, in cluding a family of fractals related to the Sierpinski triangle, Canter sub sets of the plane, the Henon attractor, and cantori from four-dimensional s ymplectic sawtooth maps. For zero-measure Canter sets, we conjecture that t he growth rate of the number of epsilon-components as epsilon --> 0 is equi valent to the box-counting dimension. (C) 2000 Elsevier Science B.V. All ri ghts reserved.