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| ``Re: alpha-limit set ???''
by Koro on 2006-06-12 08:25:21 |
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| The definition of omega-limit set is the same for non-invertible maps. The usual definition of alpha-limit set (altough not very useful) is by taking sequences of preimages, i.e. sequences {x_n} such that f(x_n)=x_{n-1} and x_0 = x, and consider the alpha-limit set of x to be the set of all acummulation points of all such sequences.
I decided not to add this to the entry for two reasons: A reader who is looking for the definition of omega-limit set will most likely not be discouraged by the fact that it is defined for homeomorphisms and assume that the definition is the same for non-homeomorphisms; on the other hand the definition of alpha-limit set is of such limited use that i didn't think it was worth mentioning. It is unlikely that anyone using alpha-limit sets of non-invertible maps will not define it first, so i don't see a good reason to obfuscate this entry by adding that definition. |
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