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Measurements of the local dynamics on the surface of a fluid undergoing complicated motion allow prediction of the measured fractal dimension of an aggregate of passive, floating tracers. This realization of a strange attractor in physical space is a rare instance where there is a firm quantitative connection between the dimension of an experimentally observed fractal spatial pattern and the process producing it.

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Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor's basin has pieces of another attractor's basin arbitrarily nearby (the basin is "riddled" with holes). Here we report quantitative theoretical results for such basins and compare with numerical experiments on a simple physical model. PACS numbers: 05.45.+b