For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
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Don N. Page (1984) studied this question.
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