We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its epsilon-connected components. By extrapolating the limiting behavior of these quantities as epsilon goes to zero we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Henon attractor, and cantori from 4-d symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of epsilon-components as epsilon goes to zero is equivalent to the box-counting dimension.
PACS: 07.05.Kf, 02.40.Pc, 47.53.+n
Keywords: Computational topology, fractal geometry, minimal spanning tree.
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