This shadowing result is very strong, because it is special to the Gauss map. We have shown that every computed orbit is uniformly shadowed by a true orbit for all iterations, and that the distance to the true orbit is uniformly bounded by a small multiple of the machine epsilon u. (Incidentally, I now believe (1995) that we can replace the 4 in the above with 2, but it's not worth writing the changes down, really). Most general shadowing results seem to hold only for finite times, and have bounds more like . Most computational procedures for a posteriori verification of shadowing also fit this model.
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