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Jan 8, 2010 at 22:17 comment added Steve Huntsman After thinking about this for a while I realized that there are topological obstructions to these sorts of approximations. For example even with the quadratic Hamiltonian corresponding to the cat map the set $[0, .5)$ is a repeller. This is probably due to the same sort of thing you're mentioning.
Jan 8, 2010 at 18:05 history answered Richard Montgomery CC BY-SA 2.5